LMFDB - Embedded newform 4014.2.d.a.4013.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4014,2,Mod(4013,4014)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4014, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4014.4013");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4014 = 2 \cdot 3^{2} \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4014.d (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:	$32.0519513713$
Analytic rank:	$0$
Dimension:	$72$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4013.16
Character	$\chi$	$=$	4014.4013
Dual form			4014.2.d.a.4013.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} -2.37383 q^{5} +3.81286 q^{7} -1.00000i q^{8} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000 q^{4} -2.37383 q^{5} +3.81286 q^{7} -1.00000i q^{8} -2.37383i q^{10} +5.17940 q^{11} +3.48605i q^{13} +3.81286i q^{14} +1.00000 q^{16} -0.101895i q^{17} +5.26741 q^{19} +2.37383 q^{20} +5.17940i q^{22} +0.790299 q^{23} +0.635090 q^{25} -3.48605 q^{26} -3.81286 q^{28} -6.06335i q^{29} -1.76161 q^{31} +1.00000i q^{32} +0.101895 q^{34} -9.05109 q^{35} +1.07706 q^{37} +5.26741i q^{38} +2.37383i q^{40} -4.43752i q^{41} +11.8232 q^{43} -5.17940 q^{44} +0.790299i q^{46} +1.34764i q^{47} +7.53788 q^{49} +0.635090i q^{50} -3.48605i q^{52} +1.07438i q^{53} -12.2950 q^{55} -3.81286i q^{56} +6.06335 q^{58} -9.91887 q^{59} -10.4401i q^{61} -1.76161i q^{62} -1.00000 q^{64} -8.27530i q^{65} +5.01846i q^{67} +0.101895i q^{68} -9.05109i q^{70} -8.04905 q^{71} +6.98599 q^{73} +1.07706i q^{74} -5.26741 q^{76} +19.7483 q^{77} -4.21693i q^{79} -2.37383 q^{80} +4.43752 q^{82} -8.79070i q^{83} +0.241881i q^{85} +11.8232i q^{86} -5.17940i q^{88} -1.98237i q^{89} +13.2918i q^{91} -0.790299 q^{92} -1.34764 q^{94} -12.5040 q^{95} +2.45972i q^{97} +7.53788i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 72 q - 72 q^{4} + 16 q^{7}+O(q^{10}) $ 72 * q - 72 * q^4 + 16 * q^7	$ 72 q - 72 q^{4} + 16 q^{7} + 72 q^{16} - 40 q^{19} + 96 q^{25} - 16 q^{28} - 24 q^{37} - 8 q^{43} + 56 q^{49} + 40 q^{58} - 72 q^{64} - 32 q^{73} + 40 q^{76} + 16 q^{82}+O(q^{100}) $ 72 * q - 72 * q^4 + 16 * q^7 + 72 * q^16 - 40 * q^19 + 96 * q^25 - 16 * q^28 - 24 * q^37 - 8 * q^43 + 56 * q^49 + 40 * q^58 - 72 * q^64 - 32 * q^73 + 40 * q^76 + 16 * q^82

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4014\mathbb{Z}\right)^\times$.

$n$	$893$	$2233$
$\chi(n)$	$-1$	$-1$

Coefficient data

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

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

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

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$	−2.37383			−1.06161			−0.530806	−	0.847494i	$-0.678111\pi$
$5$	−2.37383			−1.06161			−0.530806	+	0.847494i	$0.678111\pi$
$6$		0			0
$7$	3.81286			1.44112			0.720562	−	0.693390i	$-0.243884\pi$
$7$	3.81286			1.44112			0.720562	+	0.693390i	$0.243884\pi$
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$		−	2.37383i		−	0.750672i
$11$	5.17940			1.56165			0.780823	−	0.624752i	$-0.214800\pi$
$11$	5.17940			1.56165			0.780823	+	0.624752i	$0.214800\pi$
$12$		0			0
$13$			3.48605i			0.966856i	0.875384	+	0.483428i	$0.160608\pi$
$13$			3.48605i			0.966856i	−0.875384	+	0.483428i	$0.839392\pi$
$14$			3.81286i			1.01903i
$15$		0			0
$16$	1.00000			0.250000
$17$		−	0.101895i		−	0.0247131i	−0.999924	−	0.0123565i	$-0.996067\pi$
$17$		−	0.101895i		−	0.0247131i	0.999924	−	0.0123565i	$-0.00393331\pi$
$18$		0			0
$19$	5.26741			1.20843			0.604214	−	0.796822i	$-0.293487\pi$
$19$	5.26741			1.20843			0.604214	+	0.796822i	$0.293487\pi$
$20$	2.37383			0.530806
$21$		0			0
$22$			5.17940i			1.10425i
$23$	0.790299			0.164789			0.0823943	−	0.996600i	$-0.473743\pi$
$23$	0.790299			0.164789			0.0823943	+	0.996600i	$0.473743\pi$
$24$		0			0
$25$	0.635090			0.127018
$26$	−3.48605			−0.683670
$27$		0			0
$28$	−3.81286			−0.720562
$29$		−	6.06335i		−	1.12594i	−0.826479	−	0.562968i	$-0.809659\pi$
$29$		−	6.06335i		−	1.12594i	0.826479	−	0.562968i	$-0.190341\pi$
$30$		0			0
$31$	−1.76161			−0.316394			−0.158197	−	0.987408i	$-0.550568\pi$
$31$	−1.76161			−0.316394			−0.158197	+	0.987408i	$0.550568\pi$
$32$			1.00000i			0.176777i
$33$		0			0
$34$	0.101895			0.0174748
$35$	−9.05109			−1.52991
$36$		0			0
$37$	1.07706			0.177068			0.0885338	−	0.996073i	$-0.471782\pi$
$37$	1.07706			0.177068			0.0885338	+	0.996073i	$0.471782\pi$
$38$			5.26741i			0.854487i
$39$		0			0
$40$			2.37383i			0.375336i
$41$		−	4.43752i		−	0.693025i	−0.938045	−	0.346512i	$-0.887366\pi$
$41$		−	4.43752i		−	0.693025i	0.938045	−	0.346512i	$-0.112634\pi$
$42$		0			0
$43$	11.8232			1.80302			0.901508	−	0.432763i	$-0.142461\pi$
$43$	11.8232			1.80302			0.901508	+	0.432763i	$0.142461\pi$
$44$	−5.17940			−0.780823
$45$		0			0
$46$			0.790299i			0.116523i
$47$			1.34764i			0.196573i	0.995158	+	0.0982866i	$0.0313362\pi$
$47$			1.34764i			0.196573i	−0.995158	+	0.0982866i	$0.968664\pi$
$48$		0			0
$49$	7.53788			1.07684
$50$			0.635090i			0.0898153i
$51$		0			0
$52$		−	3.48605i		−	0.483428i
$53$			1.07438i			0.147577i	0.997274	+	0.0737884i	$0.0235090\pi$
$53$			1.07438i			0.147577i	−0.997274	+	0.0737884i	$0.976491\pi$
$54$		0			0
$55$	−12.2950			−1.65786
$56$		−	3.81286i		−	0.509514i
$57$		0			0
$58$	6.06335			0.796157
$59$	−9.91887			−1.29133			−0.645664	−	0.763622i	$-0.723419\pi$
$59$	−9.91887			−1.29133			−0.645664	+	0.763622i	$0.723419\pi$
$60$		0			0
$61$		−	10.4401i		−	1.33671i	−0.743841	−	0.668356i	$-0.766998\pi$
$61$		−	10.4401i		−	1.33671i	0.743841	−	0.668356i	$-0.233002\pi$
$62$		−	1.76161i		−	0.223724i
$63$		0			0
$64$	−1.00000			−0.125000
$65$		−	8.27530i		−	1.02642i
$66$		0			0
$67$			5.01846i			0.613102i	0.951854	+	0.306551i	$0.0991750\pi$
$67$			5.01846i			0.613102i	−0.951854	+	0.306551i	$0.900825\pi$
$68$			0.101895i			0.0123565i
$69$		0			0
$70$		−	9.05109i		−	1.08181i
$71$	−8.04905			−0.955246			−0.477623	−	0.878565i	$-0.658502\pi$
$71$	−8.04905			−0.955246			−0.477623	+	0.878565i	$0.658502\pi$
$72$		0			0
$73$	6.98599			0.817649			0.408824	−	0.912613i	$-0.365939\pi$
$73$	6.98599			0.817649			0.408824	+	0.912613i	$0.365939\pi$
$74$			1.07706i			0.125206i
$75$		0			0
$76$	−5.26741			−0.604214
$77$	19.7483			2.25053
$78$		0			0
$79$		−	4.21693i		−	0.474442i	−0.971456	−	0.237221i	$-0.923763\pi$
$79$		−	4.21693i		−	0.474442i	0.971456	−	0.237221i	$-0.0762365\pi$
$80$	−2.37383			−0.265403
$81$		0			0
$82$	4.43752			0.490043
$83$		−	8.79070i		−	0.964905i	−0.875922	−	0.482452i	$-0.839746\pi$
$83$		−	8.79070i		−	0.964905i	0.875922	−	0.482452i	$-0.160254\pi$
$84$		0			0
$85$			0.241881i			0.0262357i
$86$			11.8232i			1.27492i
$87$		0			0
$88$		−	5.17940i		−	0.552126i
$89$		−	1.98237i		−	0.210131i	−0.994465	−	0.105065i	$-0.966495\pi$
$89$		−	1.98237i		−	0.210131i	0.994465	−	0.105065i	$-0.0335052\pi$
$90$		0			0
$91$			13.2918i			1.39336i
$92$	−0.790299			−0.0823943
$93$		0			0
$94$	−1.34764			−0.138998
$95$	−12.5040			−1.28288
$96$		0			0
$97$			2.45972i			0.249747i	0.992173	+	0.124874i	$0.0398525\pi$
$97$			2.45972i			0.249747i	−0.992173	+	0.124874i	$0.960147\pi$
$98$			7.53788i			0.761441i
$99$		0			0
$100$	−0.635090			−0.0635090
$101$			1.87370i			0.186440i	0.995646	+	0.0932202i	$0.0297161\pi$
$101$			1.87370i			0.186440i	−0.995646	+	0.0932202i	$0.970284\pi$
$102$		0			0
$103$			10.1449i			0.999610i	0.866138	+	0.499805i	$0.166595\pi$
$103$			10.1449i			0.999610i	−0.866138	+	0.499805i	$0.833405\pi$
$104$	3.48605			0.341835
$105$		0			0
$106$	−1.07438			−0.104353
$107$	−2.16115			−0.208926			−0.104463	−	0.994529i	$-0.533312\pi$
$107$	−2.16115			−0.208926			−0.104463	+	0.994529i	$0.533312\pi$
$108$		0			0
$109$	4.84570			0.464134			0.232067	−	0.972700i	$-0.425451\pi$
$109$	4.84570			0.464134			0.232067	+	0.972700i	$0.425451\pi$
$110$		−	12.2950i		−	1.17229i
$111$		0			0
$112$	3.81286			0.360281
$113$	14.3821			1.35296			0.676478	−	0.736463i	$-0.263506\pi$
$113$	14.3821			1.35296			0.676478	+	0.736463i	$0.263506\pi$
$114$		0			0
$115$	−1.87604			−0.174941
$116$			6.06335i			0.562968i
$117$		0			0
$118$		−	9.91887i		−	0.913106i
$119$		−	0.388510i		−	0.0356146i
$120$		0			0
$121$	15.8262			1.43874
$122$	10.4401			0.945199
$123$		0			0
$124$	1.76161			0.158197
$125$	10.3616			0.926767
$126$		0			0
$127$	7.88379			0.699573			0.349786	−	0.936829i	$-0.386254\pi$
$127$	7.88379			0.699573			0.349786	+	0.936829i	$0.386254\pi$
$128$		−	1.00000i		−	0.0883883i
$129$		0			0
$130$	8.27530			0.725792
$131$		−	12.5585i		−	1.09724i	−0.836072	−	0.548620i	$-0.815154\pi$
$131$		−	12.5585i		−	1.09724i	0.836072	−	0.548620i	$-0.184846\pi$
$132$		0			0
$133$	20.0839			1.74149
$134$	−5.01846			−0.433529
$135$		0			0
$136$	−0.101895			−0.00873740
$137$	3.17746			0.271469			0.135734	−	0.990745i	$-0.456661\pi$
$137$	3.17746			0.271469			0.135734	+	0.990745i	$0.456661\pi$
$138$		0			0
$139$	−0.439141			−0.0372475			−0.0186237	−	0.999827i	$-0.505928\pi$
$139$	−0.439141			−0.0372475			−0.0186237	+	0.999827i	$0.505928\pi$
$140$	9.05109			0.764957
$141$		0			0
$142$		−	8.04905i		−	0.675461i
$143$			18.0556i			1.50989i
$144$		0			0
$145$			14.3934i			1.19531i
$146$			6.98599i			0.578165i
$147$		0			0
$148$	−1.07706			−0.0885338
$149$	−0.955626			−0.0782879			−0.0391439	−	0.999234i	$-0.512463\pi$
$149$	−0.955626			−0.0782879			−0.0391439	+	0.999234i	$0.512463\pi$
$150$		0			0
$151$			19.8110i			1.61220i	0.591781	+	0.806099i	$0.298425\pi$
$151$			19.8110i			1.61220i	−0.591781	+	0.806099i	$0.701575\pi$
$152$		−	5.26741i		−	0.427244i
$153$		0			0
$154$			19.7483i			1.59136i
$155$	4.18176			0.335887
$156$		0			0
$157$			2.58499i			0.206305i	0.994666	+	0.103152i	$0.0328929\pi$
$157$			2.58499i			0.206305i	−0.994666	+	0.103152i	$0.967107\pi$
$158$	4.21693			0.335481
$159$		0			0
$160$		−	2.37383i		−	0.187668i
$161$	3.01330			0.237481
$162$		0			0
$163$			11.5166i			0.902050i	0.892511	+	0.451025i	$0.148942\pi$
$163$			11.5166i			0.902050i	−0.892511	+	0.451025i	$0.851058\pi$
$164$			4.43752i			0.346512i
$165$		0			0
$166$	8.79070			0.682291
$167$	4.77205			0.369273			0.184636	−	0.982807i	$-0.440889\pi$
$167$	4.77205			0.369273			0.184636	+	0.982807i	$0.440889\pi$
$168$		0			0
$169$	0.847470			0.0651900
$170$	−0.241881			−0.0185514
$171$		0			0
$172$	−11.8232			−0.901508
$173$	−10.8496			−0.824877			−0.412438	−	0.910986i	$-0.635323\pi$
$173$	−10.8496			−0.824877			−0.412438	+	0.910986i	$0.635323\pi$
$174$		0			0
$175$	2.42151			0.183049
$176$	5.17940			0.390412
$177$		0			0
$178$	1.98237			0.148585
$179$			0.570257i			0.0426230i	0.999773	+	0.0213115i	$0.00678418\pi$
$179$			0.570257i			0.0426230i	−0.999773	+	0.0213115i	$0.993216\pi$
$180$		0			0
$181$	−1.69469			−0.125966			−0.0629828	−	0.998015i	$-0.520061\pi$
$181$	−1.69469			−0.125966			−0.0629828	+	0.998015i	$0.520061\pi$
$182$	−13.2918			−0.985254
$183$		0			0
$184$		−	0.790299i		−	0.0582616i
$185$	−2.55676			−0.187977
$186$		0			0
$187$		−	0.527753i		−	0.0385931i
$188$		−	1.34764i		−	0.0982866i
$189$		0			0
$190$		−	12.5040i		−	0.907133i
$191$	19.4923			1.41041			0.705206	−	0.709002i	$-0.250854\pi$
$191$	19.4923			1.41041			0.705206	+	0.709002i	$0.250854\pi$
$192$		0			0
$193$		−	10.1556i		−	0.731014i	−0.930809	−	0.365507i	$-0.880896\pi$
$193$		−	10.1556i		−	0.731014i	0.930809	−	0.365507i	$-0.119104\pi$
$194$	−2.45972			−0.176598
$195$		0			0
$196$	−7.53788			−0.538420
$197$			23.3107i			1.66082i	0.557156	+	0.830408i	$0.311893\pi$
$197$			23.3107i			1.66082i	−0.557156	+	0.830408i	$0.688107\pi$
$198$		0			0
$199$	−19.9760			−1.41606			−0.708030	−	0.706182i	$-0.750416\pi$
$199$	−19.9760			−1.41606			−0.708030	+	0.706182i	$0.750416\pi$
$200$		−	0.635090i		−	0.0449076i
$201$		0			0
$202$	−1.87370			−0.131833
$203$		−	23.1187i		−	1.62261i
$204$		0			0
$205$			10.5339i			0.735723i
$206$	−10.1449			−0.706831
$207$		0			0
$208$			3.48605i			0.241714i
$209$	27.2820			1.88714
$210$		0			0
$211$	−4.74432			−0.326613			−0.163306	−	0.986575i	$-0.552216\pi$
$211$	−4.74432			−0.326613			−0.163306	+	0.986575i	$0.552216\pi$
$212$		−	1.07438i		−	0.0737884i
$213$		0			0
$214$		−	2.16115i		−	0.147733i
$215$	−28.0662			−1.91410
$216$		0			0
$217$	−6.71675			−0.455963
$218$			4.84570i			0.328192i
$219$		0			0
$220$	12.2950			0.828931
$221$	0.355210			0.0238940
$222$		0			0
$223$	13.8971	−	5.46551i	0.930616	−	0.365997i
$223$	13.8971	−	5.46551i	0.930616	−	0.365997i
$224$			3.81286i			0.254757i
$225$		0			0
$226$			14.3821i			0.956684i
$227$			2.72963i			0.181172i	0.995889	+	0.0905860i	$0.0288740\pi$
$227$			2.72963i			0.181172i	−0.995889	+	0.0905860i	$0.971126\pi$
$228$		0			0
$229$			14.7268i			0.973174i	0.873632	+	0.486587i	$0.161758\pi$
$229$			14.7268i			0.973174i	−0.873632	+	0.486587i	$0.838242\pi$
$230$		−	1.87604i		−	0.123702i
$231$		0			0
$232$	−6.06335			−0.398079
$233$	−3.89027			−0.254860			−0.127430	−	0.991848i	$-0.540673\pi$
$233$	−3.89027			−0.254860			−0.127430	+	0.991848i	$0.540673\pi$
$234$		0			0
$235$		−	3.19907i		−	0.208684i
$236$	9.91887			0.645664
$237$		0			0
$238$	0.388510			0.0251833
$239$			27.8431i			1.80102i	0.434836	+	0.900510i	$0.356806\pi$
$239$			27.8431i			1.80102i	−0.434836	+	0.900510i	$0.643194\pi$
$240$		0			0
$241$	15.7354			1.01361			0.506804	−	0.862061i	$-0.330827\pi$
$241$	15.7354			1.01361			0.506804	+	0.862061i	$0.330827\pi$
$242$			15.8262i			1.01734i
$243$		0			0
$244$			10.4401i			0.668356i
$245$	−17.8937			−1.14318
$246$		0			0
$247$			18.3625i			1.16837i
$248$			1.76161i			0.111862i
$249$		0			0
$250$			10.3616i			0.655323i
$251$			17.1836i			1.08462i	0.840178	+	0.542310i	$0.182450\pi$
$251$			17.1836i			1.08462i	−0.840178	+	0.542310i	$0.817550\pi$
$252$		0			0
$253$	4.09327			0.257342
$254$			7.88379i			0.494673i
$255$		0			0
$256$	1.00000			0.0625000
$257$			4.84249i			0.302066i	0.988529	+	0.151033i	$0.0482600\pi$
$257$			4.84249i			0.302066i	−0.988529	+	0.151033i	$0.951740\pi$
$258$		0			0
$259$	4.10668			0.255176
$260$			8.27530i			0.513212i
$261$		0			0
$262$	12.5585			0.775865
$263$	8.91243			0.549564			0.274782	−	0.961507i	$-0.411394\pi$
$263$	8.91243			0.549564			0.274782	+	0.961507i	$0.411394\pi$
$264$		0			0
$265$		−	2.55039i		−	0.156669i
$266$			20.0839i			1.23142i
$267$		0			0
$268$		−	5.01846i		−	0.306551i
$269$	10.4489			0.637078			0.318539	−	0.947910i	$-0.396808\pi$
$269$	10.4489			0.637078			0.318539	+	0.947910i	$0.396808\pi$
$270$		0			0
$271$			12.2581i			0.744625i	0.928107	+	0.372312i	$0.121435\pi$
$271$			12.2581i			0.744625i	−0.928107	+	0.372312i	$0.878565\pi$
$272$		−	0.101895i		−	0.00617827i
$273$		0			0
$274$			3.17746i			0.191957i
$275$	3.28938			0.198357
$276$		0			0
$277$		−	30.1726i		−	1.81290i	−0.422315	−	0.906449i	$-0.638783\pi$
$277$		−	30.1726i		−	1.81290i	0.422315	−	0.906449i	$-0.361217\pi$
$278$		−	0.439141i		−	0.0263379i
$279$		0			0
$280$			9.05109i			0.540906i
$281$			13.4224i			0.800712i	0.916360	+	0.400356i	$0.131114\pi$
$281$			13.4224i			0.800712i	−0.916360	+	0.400356i	$0.868886\pi$
$282$		0			0
$283$	16.0181			0.952179			0.476089	−	0.879397i	$-0.342054\pi$
$283$	16.0181			0.952179			0.476089	+	0.879397i	$0.342054\pi$
$284$	8.04905			0.477623
$285$		0			0
$286$	−18.0556			−1.06765
$287$		−	16.9196i		−	0.998735i
$288$		0			0
$289$	16.9896			0.999389
$290$	−14.3934			−0.845209
$291$		0			0
$292$	−6.98599			−0.408824
$293$	−22.4623			−1.31226			−0.656132	−	0.754646i	$-0.727809\pi$
$293$	−22.4623			−1.31226			−0.656132	+	0.754646i	$0.727809\pi$
$294$		0			0
$295$	23.5458			1.37089
$296$		−	1.07706i		−	0.0626028i
$297$		0			0
$298$		−	0.955626i		−	0.0553579i
$299$			2.75502i			0.159327i
$300$		0			0
$301$	45.0800			2.59837
$302$	−19.8110			−1.14000
$303$		0			0
$304$	5.26741			0.302107
$305$			24.7830i			1.41907i
$306$		0			0
$307$			7.83251i			0.447025i	0.974701	+	0.223513i	$0.0717524\pi$
$307$			7.83251i			0.447025i	−0.974701	+	0.223513i	$0.928248\pi$
$308$	−19.7483			−1.12526
$309$		0			0
$310$			4.18176i			0.237508i
$311$	−23.5998			−1.33822			−0.669111	−	0.743163i	$-0.733325\pi$
$311$	−23.5998			−1.33822			−0.669111	+	0.743163i	$0.733325\pi$
$312$		0			0
$313$		−	26.4657i		−	1.49593i	−0.663738	−	0.747965i	$-0.731031\pi$
$313$		−	26.4657i		−	1.49593i	0.663738	−	0.747965i	$-0.268969\pi$
$314$	−2.58499			−0.145879
$315$		0			0
$316$			4.21693i			0.237221i
$317$			6.76749i			0.380100i	0.981774	+	0.190050i	$0.0608651\pi$
$317$			6.76749i			0.380100i	−0.981774	+	0.190050i	$0.939135\pi$
$318$		0			0
$319$		−	31.4045i		−	1.75832i
$320$	2.37383			0.132701
$321$		0			0
$322$			3.01330i			0.167924i
$323$		−	0.536721i		−	0.0298640i
$324$		0			0
$325$			2.21395i			0.122808i
$326$	−11.5166			−0.637846
$327$		0			0
$328$	−4.43752			−0.245021
$329$			5.13835i			0.283286i
$330$		0			0
$331$			17.8060i			0.978704i	0.872086	+	0.489352i	$0.162767\pi$
$331$			17.8060i			0.978704i	−0.872086	+	0.489352i	$0.837233\pi$
$332$			8.79070i			0.482452i
$333$		0			0
$334$			4.77205i			0.261115i
$335$		−	11.9130i		−	0.650876i
$336$		0			0
$337$		−	3.37707i		−	0.183961i	−0.995761	−	0.0919803i	$-0.970680\pi$
$337$		−	3.37707i		−	0.183961i	0.995761	−	0.0919803i	$-0.0293197\pi$
$338$			0.847470i			0.0460963i
$339$		0			0
$340$		−	0.241881i		−	0.0131178i
$341$	−9.12406			−0.494095
$342$		0			0
$343$	2.05085			0.110736
$344$		−	11.8232i		−	0.637462i
$345$		0			0
$346$		−	10.8496i		−	0.583276i
$347$			16.5597i			0.888970i	0.895786	+	0.444485i	$0.146613\pi$
$347$			16.5597i			0.888970i	−0.895786	+	0.444485i	$0.853387\pi$
$348$		0			0
$349$	−36.1906			−1.93724			−0.968619	−	0.248550i	$-0.920046\pi$
$349$	−36.1906			−1.93724			−0.968619	+	0.248550i	$0.920046\pi$
$350$			2.42151i			0.129435i
$351$		0			0
$352$			5.17940i			0.276063i
$353$		−	8.51205i		−	0.453051i	−0.974005	−	0.226525i	$-0.927263\pi$
$353$		−	8.51205i		−	0.453051i	0.974005	−	0.226525i	$-0.0727366\pi$
$354$		0			0
$355$	19.1071			1.01410
$356$			1.98237i			0.105065i
$357$		0			0
$358$	−0.570257			−0.0301390
$359$		−	24.0471i		−	1.26916i	−0.772858	−	0.634579i	$-0.781173\pi$
$359$		−	24.0471i		−	1.26916i	0.772858	−	0.634579i	$-0.218827\pi$
$360$		0			0
$361$	8.74563			0.460297
$362$		−	1.69469i		−	0.0890711i
$363$		0			0
$364$		−	13.2918i		−	0.696680i
$365$	−16.5836			−0.868025
$366$		0			0
$367$	17.6156			0.919529			0.459765	−	0.888041i	$-0.347934\pi$
$367$	17.6156			0.919529			0.459765	+	0.888041i	$0.347934\pi$
$368$	0.790299			0.0411972
$369$		0			0
$370$		−	2.55676i		−	0.132920i
$371$			4.09644i			0.212677i
$372$		0			0
$373$		−	13.1443i		−	0.680585i	−0.940320	−	0.340293i	$-0.889474\pi$
$373$		−	13.1443i		−	0.680585i	0.940320	−	0.340293i	$-0.110526\pi$
$374$	0.527753			0.0272895
$375$		0			0
$376$	1.34764			0.0694991
$377$	21.1371			1.08862
$378$		0			0
$379$	−6.29447			−0.323325			−0.161662	−	0.986846i	$-0.551686\pi$
$379$	−6.29447			−0.323325			−0.161662	+	0.986846i	$0.551686\pi$
$380$	12.5040			0.641440
$381$		0			0
$382$			19.4923i			0.997312i
$383$	−22.5354			−1.15151			−0.575753	−	0.817624i	$-0.695291\pi$
$383$	−22.5354			−1.15151			−0.575753	+	0.817624i	$0.695291\pi$
$384$		0			0
$385$	−46.8792			−2.38918
$386$	10.1556			0.516905
$387$		0			0
$388$		−	2.45972i		−	0.124874i
$389$		−	6.85656i		−	0.347641i	−0.984777	−	0.173821i	$-0.944389\pi$
$389$		−	6.85656i		−	0.347641i	0.984777	−	0.173821i	$-0.0556113\pi$
$390$		0			0
$391$		−	0.0805272i		−	0.00407244i
$392$		−	7.53788i		−	0.380720i
$393$		0			0
$394$	−23.3107			−1.17437
$395$			10.0103i			0.503673i
$396$		0			0
$397$		−	23.9226i		−	1.20064i	−0.799759	−	0.600321i	$-0.795039\pi$
$397$		−	23.9226i		−	1.20064i	0.799759	−	0.600321i	$-0.204961\pi$
$398$		−	19.9760i		−	1.00131i
$399$		0			0
$400$	0.635090			0.0317545
$401$		−	2.07056i		−	0.103399i	−0.998663	−	0.0516994i	$-0.983536\pi$
$401$		−	2.07056i		−	0.103399i	0.998663	−	0.0516994i	$-0.0164638\pi$
$402$		0			0
$403$		−	6.14104i		−	0.305907i
$404$		−	1.87370i		−	0.0932202i
$405$		0			0
$406$	23.1187			1.14736
$407$	5.57852			0.276517
$408$		0			0
$409$		−	17.9200i		−	0.886086i	−0.896500	−	0.443043i	$-0.853899\pi$
$409$		−	17.9200i		−	0.886086i	0.896500	−	0.443043i	$-0.146101\pi$
$410$	−10.5339			−0.520235
$411$		0			0
$412$		−	10.1449i		−	0.499805i
$413$	−37.8192			−1.86096
$414$		0			0
$415$			20.8677i			1.02435i
$416$	−3.48605			−0.170918
$417$		0			0
$418$			27.2820i			1.33441i
$419$			7.84081i			0.383048i	0.981488	+	0.191524i	$0.0613431\pi$
$419$			7.84081i			0.383048i	−0.981488	+	0.191524i	$0.938657\pi$
$420$		0			0
$421$			23.9282i			1.16619i	0.812403	+	0.583096i	$0.198159\pi$
$421$			23.9282i			1.16619i	−0.812403	+	0.583096i	$0.801841\pi$
$422$		−	4.74432i		−	0.230950i
$423$		0			0
$424$	1.07438			0.0521763
$425$		−	0.0647123i		−	0.00313901i
$426$		0			0
$427$		−	39.8065i		−	1.92637i
$428$	2.16115			0.104463
$429$		0			0
$430$		−	28.0662i		−	1.35347i
$431$	18.3018			0.881565			0.440783	−	0.897614i	$-0.354701\pi$
$431$	18.3018			0.881565			0.440783	+	0.897614i	$0.354701\pi$
$432$		0			0
$433$	12.8198			0.616078			0.308039	−	0.951374i	$-0.400327\pi$
$433$	12.8198			0.616078			0.308039	+	0.951374i	$0.400327\pi$
$434$		−	6.71675i		−	0.322414i
$435$		0			0
$436$	−4.84570			−0.232067
$437$	4.16283			0.199135
$438$		0			0
$439$		−	26.4696i		−	1.26333i	−0.775243	−	0.631663i	$-0.782373\pi$
$439$		−	26.4696i		−	1.26333i	0.775243	−	0.631663i	$-0.217627\pi$
$440$			12.2950i			0.586143i
$441$		0			0
$442$			0.355210i			0.0168956i
$443$			10.4827i			0.498048i	0.968497	+	0.249024i	$0.0801098\pi$
$443$			10.4827i			0.498048i	−0.968497	+	0.249024i	$0.919890\pi$
$444$		0			0
$445$			4.70582i			0.223077i
$446$	5.46551	+	13.8971i	0.258799	+	0.658045i
$447$		0			0
$448$	−3.81286			−0.180141
$449$	−8.43736			−0.398184			−0.199092	−	0.979981i	$-0.563799\pi$
$449$	−8.43736			−0.398184			−0.199092	+	0.979981i	$0.563799\pi$
$450$		0			0
$451$		−	22.9837i		−	1.08226i
$452$	−14.3821			−0.676478
$453$		0			0
$454$	−2.72963			−0.128108
$455$		−	31.5525i		−	1.47921i
$456$		0			0
$457$		−	20.0102i		−	0.936039i	−0.883718	−	0.468019i	$-0.844968\pi$
$457$		−	20.0102i		−	0.936039i	0.883718	−	0.468019i	$-0.155032\pi$
$458$	−14.7268			−0.688138
$459$		0			0
$460$	1.87604			0.0874707
$461$		−	2.02738i		−	0.0944244i	−0.998885	−	0.0472122i	$-0.984966\pi$
$461$		−	2.02738i		−	0.0944244i	0.998885	−	0.0472122i	$-0.0150337\pi$
$462$		0			0
$463$	−35.2124			−1.63646			−0.818228	−	0.574893i	$-0.805044\pi$
$463$	−35.2124			−1.63646			−0.818228	+	0.574893i	$0.805044\pi$
$464$		−	6.06335i		−	0.281484i
$465$		0			0
$466$		−	3.89027i		−	0.180213i
$467$	15.7635			0.729448			0.364724	−	0.931116i	$-0.381163\pi$
$467$	15.7635			0.729448			0.364724	+	0.931116i	$0.381163\pi$
$468$		0			0
$469$			19.1347i			0.883556i
$470$	3.19907			0.147562
$471$		0			0
$472$			9.91887i			0.456553i
$473$	61.2368			2.81567
$474$		0			0
$475$	3.34528			0.153492
$476$			0.388510i			0.0178073i
$477$		0			0
$478$	−27.8431			−1.27351
$479$		−	26.5148i		−	1.21149i	−0.795659	−	0.605745i	$-0.792875\pi$
$479$		−	26.5148i		−	1.21149i	0.795659	−	0.605745i	$-0.207125\pi$
$480$		0			0
$481$			3.75468i			0.171199i
$482$			15.7354i			0.716729i
$483$		0			0
$484$	−15.8262			−0.719371
$485$		−	5.83898i		−	0.265134i
$486$		0			0
$487$	17.0785			0.773900			0.386950	−	0.922101i	$-0.373529\pi$
$487$	17.0785			0.773900			0.386950	+	0.922101i	$0.373529\pi$
$488$	−10.4401			−0.472599
$489$		0			0
$490$		−	17.8937i		−	0.808354i
$491$	10.1201			0.456712			0.228356	−	0.973578i	$-0.426665\pi$
$491$	10.1201			0.456712			0.228356	+	0.973578i	$0.426665\pi$
$492$		0			0
$493$	−0.617823			−0.0278254
$494$	−18.3625			−0.826166
$495$		0			0
$496$	−1.76161			−0.0790985
$497$	−30.6899			−1.37663
$498$		0			0
$499$	−16.5278			−0.739888			−0.369944	−	0.929054i	$-0.620623\pi$
$499$	−16.5278			−0.739888			−0.369944	+	0.929054i	$0.620623\pi$
$500$	−10.3616			−0.463384
$501$		0			0
$502$	−17.1836			−0.766942
$503$	37.8046			1.68562			0.842812	−	0.538208i	$-0.180899\pi$
$503$	37.8046			1.68562			0.842812	+	0.538208i	$0.180899\pi$
$504$		0			0
$505$		−	4.44786i		−	0.197927i
$506$			4.09327i			0.181968i
$507$		0			0
$508$	−7.88379			−0.349786
$509$		−	19.3227i		−	0.856464i	−0.903669	−	0.428232i	$-0.859137\pi$
$509$		−	19.3227i		−	0.856464i	0.903669	−	0.428232i	$-0.140863\pi$
$510$		0			0
$511$	26.6366			1.17833
$512$			1.00000i			0.0441942i
$513$		0			0
$514$	−4.84249			−0.213593
$515$		−	24.0824i		−	1.06120i
$516$		0			0
$517$			6.97995i			0.306978i
$518$			4.10668i			0.180437i
$519$		0			0
$520$	−8.27530			−0.362896
$521$	4.60973			0.201956			0.100978	−	0.994889i	$-0.467803\pi$
$521$	4.60973			0.201956			0.100978	+	0.994889i	$0.467803\pi$
$522$		0			0
$523$		−	20.7623i		−	0.907873i	−0.891034	−	0.453937i	$-0.850019\pi$
$523$		−	20.7623i		−	0.907873i	0.891034	−	0.453937i	$-0.149981\pi$
$524$			12.5585i			0.548620i
$525$		0			0
$526$			8.91243i			0.388601i
$527$			0.179498i			0.00781907i
$528$		0			0
$529$	−22.3754			−0.972845
$530$	2.55039			0.110782
$531$		0			0
$532$	−20.0839			−0.870747
$533$	15.4694			0.670055
$534$		0			0
$535$	5.13020			0.221798
$536$	5.01846			0.216764
$537$		0			0
$538$			10.4489i			0.450482i
$539$	39.0417			1.68164
$540$		0			0
$541$		−	20.9169i		−	0.899288i	−0.893208	−	0.449644i	$-0.851551\pi$
$541$		−	20.9169i		−	0.899288i	0.893208	−	0.449644i	$-0.148449\pi$
$542$	−12.2581			−0.526529
$543$		0			0
$544$	0.101895			0.00436870
$545$	−11.5029			−0.492729
$546$		0			0
$547$	−45.3229			−1.93787			−0.968934	−	0.247320i	$-0.920450\pi$
$547$	−45.3229			−1.93787			−0.968934	+	0.247320i	$0.920450\pi$
$548$	−3.17746			−0.135734
$549$		0			0
$550$			3.28938i			0.140260i
$551$		−	31.9382i		−	1.36061i
$552$		0			0
$553$		−	16.0786i		−	0.683730i
$554$	30.1726			1.28191
$555$		0			0
$556$	0.439141			0.0186237
$557$	20.0192			0.848242			0.424121	−	0.905606i	$-0.360583\pi$
$557$	20.0192			0.848242			0.424121	+	0.905606i	$0.360583\pi$
$558$		0			0
$559$			41.2161i			1.74326i
$560$	−9.05109			−0.382478
$561$		0			0
$562$	−13.4224			−0.566189
$563$	−14.0740			−0.593148			−0.296574	−	0.955010i	$-0.595844\pi$
$563$	−14.0740			−0.593148			−0.296574	+	0.955010i	$0.595844\pi$
$564$		0			0
$565$	−34.1408			−1.43631
$566$			16.0181i			0.673292i
$567$		0			0
$568$			8.04905i			0.337730i
$569$	31.2255			1.30904			0.654521	−	0.756044i	$-0.272870\pi$
$569$	31.2255			1.30904			0.654521	+	0.756044i	$0.272870\pi$
$570$		0			0
$571$			15.6892i			0.656574i	0.944578	+	0.328287i	$0.106471\pi$
$571$			15.6892i			0.656574i	−0.944578	+	0.328287i	$0.893529\pi$
$572$		−	18.0556i		−	0.754944i
$573$		0			0
$574$	16.9196			0.706212
$575$	0.501911			0.0209311
$576$		0			0
$577$	21.3652			0.889445			0.444722	−	0.895669i	$-0.353302\pi$
$577$	21.3652			0.889445			0.444722	+	0.895669i	$0.353302\pi$
$578$			16.9896i			0.706675i
$579$		0			0
$580$		−	14.3934i		−	0.597653i
$581$		−	33.5177i		−	1.39055i
$582$		0			0
$583$			5.56462i			0.230463i
$584$		−	6.98599i		−	0.289083i
$585$		0			0
$586$		−	22.4623i		−	0.927911i
$587$	41.4828			1.71218			0.856089	−	0.516828i	$-0.172887\pi$
$587$	41.4828			1.71218			0.856089	+	0.516828i	$0.172887\pi$
$588$		0			0
$589$	−9.27911			−0.382339
$590$			23.5458i			0.969364i
$591$		0			0
$592$	1.07706			0.0442669
$593$	−27.3734			−1.12409			−0.562046	−	0.827106i	$-0.689986\pi$
$593$	−27.3734			−1.12409			−0.562046	+	0.827106i	$0.689986\pi$
$594$		0			0
$595$			0.922258i			0.0378089i
$596$	0.955626			0.0391439
$597$		0			0
$598$	−2.75502			−0.112661
$599$		−	30.9880i		−	1.26613i	−0.774097	−	0.633067i	$-0.781796\pi$
$599$		−	30.9880i		−	1.26613i	0.774097	−	0.633067i	$-0.218204\pi$
$600$		0			0
$601$			36.3354i			1.48215i	0.671421	+	0.741076i	$0.265684\pi$
$601$			36.3354i			1.48215i	−0.671421	+	0.741076i	$0.734316\pi$
$602$			45.0800i			1.83732i
$603$		0			0
$604$		−	19.8110i		−	0.806099i
$605$	−37.5687			−1.52738
$606$		0			0
$607$			35.3941i			1.43660i	0.695732	+	0.718302i	$0.255080\pi$
$607$			35.3941i			1.43660i	−0.695732	+	0.718302i	$0.744920\pi$
$608$			5.26741i			0.213622i
$609$		0			0
$610$	−24.7830			−1.00343
$611$	−4.69793			−0.190058
$612$		0			0
$613$			2.28549i			0.0923102i	0.998934	+	0.0461551i	$0.0146968\pi$
$613$			2.28549i			0.0923102i	−0.998934	+	0.0461551i	$0.985303\pi$
$614$	−7.83251			−0.316094
$615$		0			0
$616$		−	19.7483i		−	0.795682i
$617$		−	41.8135i		−	1.68335i	−0.539984	−	0.841675i	$-0.681570\pi$
$617$		−	41.8135i		−	1.68335i	0.539984	−	0.841675i	$-0.318430\pi$
$618$		0			0
$619$		−	30.6919i		−	1.23361i	−0.787115	−	0.616806i	$-0.788426\pi$
$619$		−	30.6919i		−	1.23361i	0.787115	−	0.616806i	$-0.211574\pi$
$620$	−4.18176			−0.167944
$621$		0			0
$622$		−	23.5998i		−	0.946265i
$623$		−	7.55849i		−	0.302824i
$624$		0			0
$625$	−27.7721			−1.11088
$626$	26.4657			1.05778
$627$		0			0
$628$		−	2.58499i		−	0.103152i
$629$		−	0.109747i		−	0.00437589i
$630$		0			0
$631$			28.2034i			1.12276i	0.827559	+	0.561379i	$0.189729\pi$
$631$			28.2034i			1.12276i	−0.827559	+	0.561379i	$0.810271\pi$
$632$	−4.21693			−0.167741
$633$		0			0
$634$	−6.76749			−0.268772
$635$	−18.7148			−0.742674
$636$		0			0
$637$			26.2774i			1.04115i
$638$	31.4045			1.24332
$639$		0			0
$640$			2.37383i			0.0938340i
$641$	24.0229			0.948848			0.474424	−	0.880296i	$-0.342656\pi$
$641$	24.0229			0.948848			0.474424	+	0.880296i	$0.342656\pi$
$642$		0			0
$643$	17.2750			0.681258			0.340629	−	0.940198i	$-0.389360\pi$
$643$	17.2750			0.681258			0.340629	+	0.940198i	$0.389360\pi$
$644$	−3.01330			−0.118740
$645$		0			0
$646$	0.536721			0.0211170
$647$			48.1913i			1.89459i	0.320356	+	0.947297i	$0.396197\pi$
$647$			48.1913i			1.89459i	−0.320356	+	0.947297i	$0.603803\pi$
$648$		0			0
$649$	−51.3738			−2.01660
$650$	−2.21395			−0.0868384
$651$		0			0
$652$		−	11.5166i		−	0.451025i
$653$	−6.16949			−0.241431			−0.120715	−	0.992687i	$-0.538519\pi$
$653$	−6.16949			−0.241431			−0.120715	+	0.992687i	$0.538519\pi$
$654$		0			0
$655$			29.8117i			1.16484i
$656$		−	4.43752i		−	0.173256i
$657$		0			0
$658$	−5.13835			−0.200314
$659$			46.2607i			1.80206i	0.433758	+	0.901030i	$0.357187\pi$
$659$			46.2607i			1.80206i	−0.433758	+	0.901030i	$0.642813\pi$
$660$		0			0
$661$			31.5957i			1.22893i	0.788943	+	0.614466i	$0.210628\pi$
$661$			31.5957i			1.22893i	−0.788943	+	0.614466i	$0.789372\pi$
$662$	−17.8060			−0.692048
$663$		0			0
$664$	−8.79070			−0.341145
$665$	−47.6758			−1.84879
$666$		0			0
$667$		−	4.79186i		−	0.185542i
$668$	−4.77205			−0.184636
$669$		0			0
$670$	11.9130			0.460239
$671$		−	54.0732i		−	2.08747i
$672$		0			0
$673$	−38.0592			−1.46708			−0.733538	−	0.679648i	$-0.762132\pi$
$673$	−38.0592			−1.46708			−0.733538	+	0.679648i	$0.762132\pi$
$674$	3.37707			0.130080
$675$		0			0
$676$	−0.847470			−0.0325950
$677$			38.5718i			1.48243i	0.671265	+	0.741217i	$0.265751\pi$
$677$			38.5718i			1.48243i	−0.671265	+	0.741217i	$0.734249\pi$
$678$		0			0
$679$			9.37857i			0.359917i
$680$	0.241881			0.00927572
$681$		0			0
$682$		−	9.12406i		−	0.349378i
$683$			34.9024i			1.33550i	0.744384	+	0.667752i	$0.232743\pi$
$683$			34.9024i			1.33550i	−0.744384	+	0.667752i	$0.767257\pi$
$684$		0			0
$685$	−7.54276			−0.288194
$686$			2.05085i			0.0783019i
$687$		0			0
$688$	11.8232			0.450754
$689$	−3.74533			−0.142686
$690$		0			0
$691$			28.4702i			1.08306i	0.840682	+	0.541528i	$0.182154\pi$
$691$			28.4702i			1.08306i	−0.840682	+	0.541528i	$0.817846\pi$
$692$	10.8496			0.412438
$693$		0			0
$694$	−16.5597			−0.628597
$695$	1.04245			0.0395423
$696$		0			0
$697$	−0.452160			−0.0171268
$698$		−	36.1906i		−	1.36983i
$699$		0			0
$700$	−2.42151			−0.0915244
$701$			20.8214i			0.786412i	0.919450	+	0.393206i	$0.128634\pi$
$701$			20.8214i			0.786412i	−0.919450	+	0.393206i	$0.871366\pi$
$702$		0			0
$703$	5.67332			0.213973
$704$	−5.17940			−0.195206
$705$		0			0
$706$	8.51205			0.320355
$707$			7.14416i			0.268684i
$708$		0			0
$709$		−	5.00986i		−	0.188149i	−0.995565	−	0.0940747i	$-0.970011\pi$
$709$		−	5.00986i		−	0.188149i	0.995565	−	0.0940747i	$-0.0299893\pi$
$710$			19.1071i			0.717077i
$711$		0			0
$712$	−1.98237			−0.0742924
$713$	−1.39220			−0.0521381
$714$		0			0
$715$		−	42.8611i		−	1.60291i
$716$		−	0.570257i		−	0.0213115i
$717$		0			0
$718$	24.0471			0.897431
$719$		−	40.7138i		−	1.51837i	−0.650875	−	0.759185i	$-0.725598\pi$
$719$		−	40.7138i		−	1.51837i	0.650875	−	0.759185i	$-0.274402\pi$
$720$		0			0
$721$			38.6812i			1.44056i
$722$			8.74563i			0.325479i
$723$		0			0
$724$	1.69469			0.0629828
$725$		−	3.85077i		−	0.143014i
$726$		0			0
$727$	−5.49106			−0.203652			−0.101826	−	0.994802i	$-0.532469\pi$
$727$	−5.49106			−0.203652			−0.101826	+	0.994802i	$0.532469\pi$
$728$	13.2918			0.492627
$729$		0			0
$730$		−	16.5836i		−	0.613786i
$731$		−	1.20472i		−	0.0445581i
$732$		0			0
$733$	−33.0177			−1.21954			−0.609769	−	0.792579i	$-0.708738\pi$
$733$	−33.0177			−1.21954			−0.609769	+	0.792579i	$0.708738\pi$
$734$			17.6156i			0.650205i
$735$		0			0
$736$			0.790299i			0.0291308i
$737$			25.9926i			0.957449i
$738$		0			0
$739$		−	52.6865i		−	1.93810i	−0.246857	−	0.969052i	$-0.579398\pi$
$739$		−	52.6865i		−	1.93810i	0.246857	−	0.969052i	$-0.420602\pi$
$740$	2.55676			0.0939884
$741$		0			0
$742$	−4.09644			−0.150385
$743$		−	38.1892i		−	1.40103i	−0.713640	−	0.700513i	$-0.752955\pi$
$743$		−	38.1892i		−	1.40103i	0.713640	−	0.700513i	$-0.247045\pi$
$744$		0			0
$745$	2.26850			0.0831113
$746$	13.1443			0.481247
$747$		0			0
$748$			0.527753i			0.0192966i
$749$	−8.24014			−0.301088
$750$		0			0
$751$	9.43846			0.344414			0.172207	−	0.985061i	$-0.444910\pi$
$751$	9.43846			0.344414			0.172207	+	0.985061i	$0.444910\pi$
$752$			1.34764i			0.0491433i
$753$		0			0
$754$			21.1371i			0.769769i
$755$		−	47.0281i		−	1.71153i
$756$		0			0
$757$			6.93184i			0.251942i	0.992034	+	0.125971i	$0.0402047\pi$
$757$			6.93184i			0.251942i	−0.992034	+	0.125971i	$0.959795\pi$
$758$		−	6.29447i		−	0.228625i
$759$		0			0
$760$			12.5040i			0.453566i
$761$	16.7359			0.606677			0.303338	−	0.952883i	$-0.401899\pi$
$761$	16.7359			0.606677			0.303338	+	0.952883i	$0.401899\pi$
$762$		0			0
$763$	18.4760			0.668874
$764$	−19.4923			−0.705206
$765$		0			0
$766$		−	22.5354i		−	0.814238i
$767$		−	34.5777i		−	1.24853i
$768$		0			0
$769$	−30.7061			−1.10729			−0.553645	−	0.832753i	$-0.686764\pi$
$769$	−30.7061			−1.10729			−0.553645	+	0.832753i	$0.686764\pi$
$770$		−	46.8792i		−	1.68941i
$771$		0			0
$772$			10.1556i			0.365507i
$773$	−53.2549			−1.91545			−0.957723	−	0.287693i	$-0.907112\pi$
$773$	−53.2549			−1.91545			−0.957723	+	0.287693i	$0.907112\pi$
$774$		0			0
$775$	−1.11878			−0.0401877
$776$	2.45972			0.0882989
$777$		0			0
$778$	6.85656			0.245819
$779$		−	23.3743i		−	0.837470i
$780$		0			0
$781$	−41.6892			−1.49176
$782$	0.0805272			0.00287965
$783$		0			0
$784$	7.53788			0.269210
$785$		−	6.13634i		−	0.219015i
$786$		0			0
$787$		−	5.88885i		−	0.209915i	−0.994477	−	0.104957i	$-0.966529\pi$
$787$		−	5.88885i		−	0.209915i	0.994477	−	0.104957i	$-0.0334706\pi$
$788$		−	23.3107i		−	0.830408i
$789$		0			0
$790$	−10.0103			−0.356150
$791$	54.8369			1.94978
$792$		0			0
$793$	36.3946			1.29241
$794$	23.9226			0.848983
$795$		0			0
$796$	19.9760			0.708030
$797$			49.0796i			1.73849i	0.494384	+	0.869244i	$0.335394\pi$
$797$			49.0796i			1.73849i	−0.494384	+	0.869244i	$0.664606\pi$
$798$		0			0
$799$	0.137317			0.00485793
$800$			0.635090i			0.0224538i
$801$		0			0
$802$	2.07056			0.0731140
$803$	36.1832			1.27688
$804$		0			0
$805$	−7.15306			−0.252112
$806$	6.14104			0.216309
$807$		0			0
$808$	1.87370			0.0659166
$809$	−26.6627			−0.937411			−0.468706	−	0.883354i	$-0.655279\pi$
$809$	−26.6627			−0.937411			−0.468706	+	0.883354i	$0.655279\pi$
$810$		0			0
$811$			12.7097i			0.446297i	0.974784	+	0.223149i	$0.0716336\pi$
$811$			12.7097i			0.446297i	−0.974784	+	0.223149i	$0.928366\pi$
$812$			23.1187i			0.811307i
$813$		0			0
$814$			5.57852i			0.195527i
$815$		−	27.3385i		−	0.957626i
$816$		0			0
$817$	62.2775			2.17881
$818$	17.9200			0.626558
$819$		0			0
$820$		−	10.5339i		−	0.367861i
$821$		−	5.44720i		−	0.190109i	−0.995472	−	0.0950543i	$-0.969698\pi$
$821$		−	5.44720i		−	0.190109i	0.995472	−	0.0950543i	$-0.0303025\pi$
$822$		0			0
$823$			14.5246i			0.506295i	0.967428	+	0.253148i	$0.0814658\pi$
$823$			14.5246i			0.506295i	−0.967428	+	0.253148i	$0.918534\pi$
$824$	10.1449			0.353415
$825$		0			0
$826$		−	37.8192i		−	1.31590i
$827$	−19.7057			−0.685235			−0.342618	−	0.939475i	$-0.611314\pi$
$827$	−19.7057			−0.685235			−0.342618	+	0.939475i	$0.611314\pi$
$828$		0			0
$829$		−	52.3573i		−	1.81845i	−0.416310	−	0.909223i	$-0.636677\pi$
$829$		−	52.3573i		−	1.81845i	0.416310	−	0.909223i	$-0.363323\pi$
$830$	−20.8677			−0.724327
$831$		0			0
$832$		−	3.48605i		−	0.120857i
$833$		−	0.768070i		−	0.0266120i
$834$		0			0
$835$	−11.3281			−0.392024
$836$	−27.2820			−0.943568
$837$		0			0
$838$	−7.84081			−0.270856
$839$	−40.5060			−1.39842			−0.699212	−	0.714915i	$-0.746466\pi$
$839$	−40.5060			−1.39842			−0.699212	+	0.714915i	$0.746466\pi$
$840$		0			0
$841$	−7.76425			−0.267733
$842$	−23.9282			−0.824622
$843$		0			0
$844$	4.74432			0.163306
$845$	−2.01175			−0.0692064
$846$		0			0
$847$	60.3429			2.07340
$848$			1.07438i			0.0368942i
$849$		0			0
$850$	0.0647123			0.00221961
$851$	0.851199			0.0291787
$852$		0			0
$853$		−	22.4685i		−	0.769306i	−0.923061	−	0.384653i	$-0.874321\pi$
$853$		−	22.4685i		−	0.769306i	0.923061	−	0.384653i	$-0.125679\pi$
$854$	39.8065			1.36215
$855$		0			0
$856$			2.16115i			0.0738664i
$857$			20.1704i			0.689006i	0.938785	+	0.344503i	$0.111953\pi$
$857$			20.1704i			0.689006i	−0.938785	+	0.344503i	$0.888047\pi$
$858$		0			0
$859$		−	4.88488i		−	0.166670i	−0.996522	−	0.0833350i	$-0.973443\pi$
$859$		−	4.88488i		−	0.166670i	0.996522	−	0.0833350i	$-0.0265572\pi$
$860$	28.0662			0.957050
$861$		0			0
$862$			18.3018i			0.623361i
$863$	22.0343			0.750057			0.375028	−	0.927013i	$-0.377633\pi$
$863$	22.0343			0.750057			0.375028	+	0.927013i	$0.377633\pi$
$864$		0			0
$865$	25.7551			0.875698
$866$			12.8198i			0.435633i
$867$		0			0
$868$	6.71675			0.227981
$869$		−	21.8412i		−	0.740911i
$870$		0			0
$871$	−17.4946			−0.592781
$872$		−	4.84570i		−	0.164096i
$873$		0			0
$874$			4.16283i			0.140810i
$875$	39.5072			1.33559
$876$		0			0
$877$		−	14.5547i		−	0.491476i	−0.969336	−	0.245738i	$-0.920970\pi$
$877$		−	14.5547i		−	0.491476i	0.969336	−	0.245738i	$-0.0790304\pi$
$878$	26.4696			0.893307
$879$		0			0
$880$	−12.2950			−0.414465
$881$		−	24.7832i		−	0.834967i	−0.908685	−	0.417483i	$-0.862912\pi$
$881$		−	24.7832i		−	0.834967i	0.908685	−	0.417483i	$-0.137088\pi$
$882$		0			0
$883$			23.9696i			0.806643i	0.915058	+	0.403321i	$0.132144\pi$
$883$			23.9696i			0.806643i	−0.915058	+	0.403321i	$0.867856\pi$
$884$	−0.355210			−0.0119470
$885$		0			0
$886$	−10.4827			−0.352173
$887$		−	22.2120i		−	0.745807i	−0.927870	−	0.372904i	$-0.878362\pi$
$887$		−	22.2120i		−	0.745807i	0.927870	−	0.372904i	$-0.121638\pi$
$888$		0			0
$889$	30.0598			1.00817
$890$	−4.70582			−0.157739
$891$		0			0
$892$	−13.8971	+	5.46551i	−0.465308	+	0.182999i
$893$			7.09856i			0.237544i
$894$		0			0
$895$		−	1.35370i		−	0.0452491i
$896$		−	3.81286i		−	0.127379i
$897$		0			0
$898$		−	8.43736i		−	0.281558i
$899$			10.6812i			0.356239i
$900$		0			0
$901$	0.109473			0.00364708
$902$	22.9837			0.765273
$903$		0			0
$904$		−	14.3821i		−	0.478342i
$905$	4.02292			0.133726
$906$		0			0
$907$	18.6472			0.619169			0.309585	−	0.950872i	$-0.399810\pi$
$907$	18.6472			0.619169			0.309585	+	0.950872i	$0.399810\pi$
$908$		−	2.72963i		−	0.0905860i
$909$		0			0
$910$	31.5525			1.04596
$911$			42.4753i			1.40727i	0.710562	+	0.703634i	$0.248441\pi$
$911$			42.4753i			1.40727i	−0.710562	+	0.703634i	$0.751559\pi$
$912$		0			0
$913$		−	45.5305i		−	1.50684i
$914$	20.0102			0.661879
$915$		0			0
$916$		−	14.7268i		−	0.486587i
$917$		−	47.8837i		−	1.58126i
$918$		0			0
$919$			14.8795i			0.490828i	0.969418	+	0.245414i	$0.0789239\pi$
$919$			14.8795i			0.490828i	−0.969418	+	0.245414i	$0.921076\pi$
$920$			1.87604i			0.0618511i
$921$		0			0
$922$	2.02738			0.0667682
$923$		−	28.0594i		−	0.923585i
$924$		0			0
$925$	0.684030			0.0224908
$926$		−	35.2124i		−	1.15715i
$927$		0			0
$928$	6.06335			0.199039
$929$			29.5025i			0.967945i	0.875083	+	0.483972i	$0.160806\pi$
$929$			29.5025i			0.967945i	−0.875083	+	0.483972i	$0.839194\pi$
$930$		0			0
$931$	39.7051			1.30128
$932$	3.89027			0.127430
$933$		0			0
$934$			15.7635i			0.515797i
$935$			1.25280i			0.0409709i
$936$		0			0
$937$			23.4776i			0.766981i	0.923545	+	0.383490i	$0.125278\pi$
$937$			23.4776i			0.766981i	−0.923545	+	0.383490i	$0.874722\pi$
$938$	−19.1347			−0.624769
$939$		0			0
$940$			3.19907i			0.104342i
$941$			27.9748i			0.911952i	0.889992	+	0.455976i	$0.150710\pi$
$941$			27.9748i			0.911952i	−0.889992	+	0.455976i	$0.849290\pi$
$942$		0			0
$943$		−	3.50697i		−	0.114203i
$944$	−9.91887			−0.322832
$945$		0			0
$946$			61.2368i			1.99098i
$947$		−	1.72913i		−	0.0561890i	−0.999605	−	0.0280945i	$-0.991056\pi$
$947$		−	1.72913i		−	0.0561890i	0.999605	−	0.0280945i	$-0.00894394\pi$
$948$		0			0
$949$			24.3535i			0.790548i
$950$			3.34528i			0.108535i
$951$		0			0
$952$	−0.388510			−0.0125917
$953$	35.7987			1.15963			0.579817	−	0.814747i	$-0.303124\pi$
$953$	35.7987			1.15963			0.579817	+	0.814747i	$0.303124\pi$
$954$		0			0
$955$	−46.2715			−1.49731
$956$		−	27.8431i		−	0.900510i
$957$		0			0
$958$	26.5148			0.856653
$959$	12.1152			0.391220
$960$		0			0
$961$	−27.8967			−0.899895
$962$	−3.75468			−0.121056
$963$		0			0
$964$	−15.7354			−0.506804
$965$			24.1077i			0.776053i
$966$		0			0
$967$			20.9766i			0.674561i	0.941404	+	0.337281i	$0.109507\pi$
$967$			20.9766i			0.674561i	−0.941404	+	0.337281i	$0.890493\pi$
$968$		−	15.8262i		−	0.508672i
$969$		0			0
$970$	5.83898			0.187478
$971$	−49.0285			−1.57340			−0.786699	−	0.617337i	$-0.788212\pi$
$971$	−49.0285			−1.57340			−0.786699	+	0.617337i	$0.788212\pi$
$972$		0			0
$973$	−1.67438			−0.0536782
$974$			17.0785i			0.547230i
$975$		0			0
$976$		−	10.4401i		−	0.334178i
$977$	27.7151			0.886686			0.443343	−	0.896352i	$-0.353792\pi$
$977$	27.7151			0.886686			0.443343	+	0.896352i	$0.353792\pi$
$978$		0			0
$979$		−	10.2675i		−	0.328150i
$980$	17.8937			0.571592
$981$		0			0
$982$			10.1201i			0.322944i
$983$	−31.3762			−1.00075			−0.500373	−	0.865810i	$-0.666804\pi$
$983$	−31.3762			−1.00075			−0.500373	+	0.865810i	$0.666804\pi$
$984$		0			0
$985$		−	55.3357i		−	1.76314i
$986$		−	0.617823i		−	0.0196755i
$987$		0			0
$988$		−	18.3625i		−	0.584187i
$989$	9.34383			0.297116
$990$		0			0
$991$		−	34.7924i		−	1.10522i	−0.833441	−	0.552608i	$-0.813633\pi$
$991$		−	34.7924i		−	1.10522i	0.833441	−	0.552608i	$-0.186367\pi$
$992$		−	1.76161i		−	0.0559311i
$993$		0			0
$994$		−	30.6899i		−	0.973423i
$995$	47.4197			1.50331
$996$		0			0
$997$	−24.7127			−0.782659			−0.391330	−	0.920251i	$-0.627985\pi$
$997$	−24.7127			−0.782659			−0.391330	+	0.920251i	$0.627985\pi$
$998$		−	16.5278i		−	0.523180i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4014.2.d.a.4013.16	yes	72
3.2	odd	2	inner	4014.2.d.a.4013.39	yes	72
223.222	odd	2	inner	4014.2.d.a.4013.40	yes	72
669.668	even	2	inner	4014.2.d.a.4013.15	✓	72

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4014.2.d.a.4013.15	✓	72	669.668	even	2	inner
4014.2.d.a.4013.16	yes	72	1.1	even	1	trivial
4014.2.d.a.4013.39	yes	72	3.2	odd	2	inner
4014.2.d.a.4013.40	yes	72	223.222	odd	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:	\( N \)	\(=\)	\( 4014 = 2 \cdot 3^{2} \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4014.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} -2.37383 q^{5} +3.81286 q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} -2.37383 q^{5} +3.81286 q^{7} -1.00000i q^{8} -2.37383i q^{10} +5.17940 q^{11} +3.48605i q^{13} +3.81286i q^{14} +1.00000 q^{16} -0.101895i q^{17} +5.26741 q^{19} +2.37383 q^{20} +5.17940i q^{22} +0.790299 q^{23} +0.635090 q^{25} -3.48605 q^{26} -3.81286 q^{28} -6.06335i q^{29} -1.76161 q^{31} +1.00000i q^{32} +0.101895 q^{34} -9.05109 q^{35} +1.07706 q^{37} +5.26741i q^{38} +2.37383i q^{40} -4.43752i q^{41} +11.8232 q^{43} -5.17940 q^{44} +0.790299i q^{46} +1.34764i q^{47} +7.53788 q^{49} +0.635090i q^{50} -3.48605i q^{52} +1.07438i q^{53} -12.2950 q^{55} -3.81286i q^{56} +6.06335 q^{58} -9.91887 q^{59} -10.4401i q^{61} -1.76161i q^{62} -1.00000 q^{64} -8.27530i q^{65} +5.01846i q^{67} +0.101895i q^{68} -9.05109i q^{70} -8.04905 q^{71} +6.98599 q^{73} +1.07706i q^{74} -5.26741 q^{76} +19.7483 q^{77} -4.21693i q^{79} -2.37383 q^{80} +4.43752 q^{82} -8.79070i q^{83} +0.241881i q^{85} +11.8232i q^{86} -5.17940i q^{88} -1.98237i q^{89} +13.2918i q^{91} -0.790299 q^{92} -1.34764 q^{94} -12.5040 q^{95} +2.45972i q^{97} +7.53788i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 72 q^{4} + 16 q^{7}+O(q^{10}) \) 72 * q - 72 * q^4 + 16 * q^7	\( 72 q - 72 q^{4} + 16 q^{7} + 72 q^{16} - 40 q^{19} + 96 q^{25} - 16 q^{28} - 24 q^{37} - 8 q^{43} + 56 q^{49} + 40 q^{58} - 72 q^{64} - 32 q^{73} + 40 q^{76} + 16 q^{82}+O(q^{100}) \) 72 * q - 72 * q^4 + 16 * q^7 + 72 * q^16 - 40 * q^19 + 96 * q^25 - 16 * q^28 - 24 * q^37 - 8 * q^43 + 56 * q^49 + 40 * q^58 - 72 * q^64 - 32 * q^73 + 40 * q^76 + 16 * q^82

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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