LMFDB - Embedded newform 3762.2.b.a.989.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("3762.989");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$30.0397212404$
Analytic rank:	$0$
Dimension:	$36$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			989.4
Character	$\chi$	$=$	3762.989
Dual form			3762.2.b.a.989.33

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.00000 q^{4} -3.17738i q^{5} -1.82716i q^{7} -1.00000 q^{8} +3.17738i q^{10} +(-3.31203 - 0.174537i) q^{11} -1.50882i q^{13} +1.82716i q^{14} +1.00000 q^{16} +1.79436 q^{17} -1.00000i q^{19} -3.17738i q^{20} +(3.31203 + 0.174537i) q^{22} +1.69855i q^{23} -5.09572 q^{25} +1.50882i q^{26} -1.82716i q^{28} -7.31983 q^{29} +6.41670 q^{31} -1.00000 q^{32} -1.79436 q^{34} -5.80556 q^{35} +0.490103 q^{37} +1.00000i q^{38} +3.17738i q^{40} -7.55243 q^{41} -6.64939i q^{43} +(-3.31203 - 0.174537i) q^{44} -1.69855i q^{46} -13.6683i q^{47} +3.66150 q^{49} +5.09572 q^{50} -1.50882i q^{52} +1.80541i q^{53} +(-0.554569 + 10.5236i) q^{55} +1.82716i q^{56} +7.31983 q^{58} -6.18295i q^{59} -8.74905i q^{61} -6.41670 q^{62} +1.00000 q^{64} -4.79408 q^{65} -9.32749 q^{67} +1.79436 q^{68} +5.80556 q^{70} +3.66376i q^{71} +13.3378i q^{73} -0.490103 q^{74} -1.00000i q^{76} +(-0.318906 + 6.05159i) q^{77} +11.4910i q^{79} -3.17738i q^{80} +7.55243 q^{82} -5.73214 q^{83} -5.70137i q^{85} +6.64939i q^{86} +(3.31203 + 0.174537i) q^{88} -4.98909i q^{89} -2.75685 q^{91} +1.69855i q^{92} +13.6683i q^{94} -3.17738 q^{95} -13.6131 q^{97} -3.66150 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q - 36 q^{2} + 36 q^{4} - 36 q^{8} + 4 q^{11} + 36 q^{16} - 16 q^{17} - 4 q^{22} - 28 q^{25} - 8 q^{31} - 36 q^{32} + 16 q^{34} - 16 q^{35} + 16 q^{37} - 24 q^{41} + 4 q^{44} - 68 q^{49} + 28 q^{50}+ \cdots + 68 q^{98}+O(q^{100}) $ 36 * q - 36 * q^2 + 36 * q^4 - 36 * q^8 + 4 * q^11 + 36 * q^16 - 16 * q^17 - 4 * q^22 - 28 * q^25 - 8 * q^31 - 36 * q^32 + 16 * q^34 - 16 * q^35 + 16 * q^37 - 24 * q^41 + 4 * q^44 - 68 * q^49 + 28 * q^50 + 28 * q^55 + 8 * q^62 + 36 * q^64 - 32 * q^65 + 16 * q^67 - 16 * q^68 + 16 * q^70 - 16 * q^74 + 32 * q^77 + 24 * q^82 + 56 * q^83 - 4 * q^88 - 48 * q^91 + 8 * q^97 + 68 * q^98

Character values

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

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

Coefficient data

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

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

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

$2$	−1.00000			−0.707107
$2$	−1.00000			−0.707107
$3$		0			0
$3$		0			0
$4$	1.00000			0.500000
$5$		−	3.17738i		−	1.42097i	−0.703714	−	0.710483i	$-0.748476\pi$
$5$		−	3.17738i		−	1.42097i	0.703714	−	0.710483i	$-0.251524\pi$
$6$		0			0
$7$		−	1.82716i		−	0.690600i	−0.938492	−	0.345300i	$-0.887777\pi$
$7$		−	1.82716i		−	0.690600i	0.938492	−	0.345300i	$-0.112223\pi$
$8$	−1.00000			−0.353553
$9$		0			0
$10$			3.17738i			1.00477i
$11$	−3.31203	−	0.174537i	−0.998614	−	0.0526248i
$11$	−3.31203	−	0.174537i	−0.998614	−	0.0526248i
$12$		0			0
$13$		−	1.50882i		−	0.418471i	−0.977865	−	0.209235i	$-0.932903\pi$
$13$		−	1.50882i		−	0.418471i	0.977865	−	0.209235i	$-0.0670975\pi$
$14$			1.82716i			0.488328i
$15$		0			0
$16$	1.00000			0.250000
$17$	1.79436			0.435197			0.217599	−	0.976038i	$-0.430178\pi$
$17$	1.79436			0.435197			0.217599	+	0.976038i	$0.430178\pi$
$18$		0			0
$19$		−	1.00000i		−	0.229416i
$19$		−	1.00000i		−	0.229416i
$20$		−	3.17738i		−	0.710483i
$21$		0			0
$22$	3.31203	+	0.174537i	0.706127	+	0.0372114i
$23$			1.69855i			0.354173i	0.984195	+	0.177086i	$0.0566672\pi$
$23$			1.69855i			0.354173i	−0.984195	+	0.177086i	$0.943333\pi$
$24$		0			0
$25$	−5.09572			−1.01914
$26$			1.50882i			0.295904i
$27$		0			0
$28$		−	1.82716i		−	0.345300i
$29$	−7.31983			−1.35926			−0.679629	−	0.733556i	$-0.737859\pi$
$29$	−7.31983			−1.35926			−0.679629	+	0.733556i	$0.737859\pi$
$30$		0			0
$31$	6.41670			1.15247			0.576236	−	0.817283i	$-0.304521\pi$
$31$	6.41670			1.15247			0.576236	+	0.817283i	$0.304521\pi$
$32$	−1.00000			−0.176777
$33$		0			0
$34$	−1.79436			−0.307731
$35$	−5.80556			−0.981319
$36$		0			0
$37$	0.490103			0.0805724			0.0402862	−	0.999188i	$-0.487173\pi$
$37$	0.490103			0.0805724			0.0402862	+	0.999188i	$0.487173\pi$
$38$			1.00000i			0.162221i
$39$		0			0
$40$			3.17738i			0.502387i
$41$	−7.55243			−1.17949			−0.589746	−	0.807589i	$-0.700772\pi$
$41$	−7.55243			−1.17949			−0.589746	+	0.807589i	$0.700772\pi$
$42$		0			0
$43$		−	6.64939i		−	1.01402i	−0.861940	−	0.507011i	$-0.830750\pi$
$43$		−	6.64939i		−	1.01402i	0.861940	−	0.507011i	$-0.169250\pi$
$44$	−3.31203	−	0.174537i	−0.499307	−	0.0263124i
$45$		0			0
$46$		−	1.69855i		−	0.250438i
$47$		−	13.6683i		−	1.99373i	−0.0791466	−	0.996863i	$-0.525220\pi$
$47$		−	13.6683i		−	1.99373i	0.0791466	−	0.996863i	$-0.474780\pi$
$48$		0			0
$49$	3.66150			0.523071
$50$	5.09572			0.720644
$51$		0			0
$52$		−	1.50882i		−	0.209235i
$53$			1.80541i			0.247992i	0.992283	+	0.123996i	$0.0395710\pi$
$53$			1.80541i			0.247992i	−0.992283	+	0.123996i	$0.960429\pi$
$54$		0			0
$55$	−0.554569	+	10.5236i	−0.0747781	+	1.41900i
$56$			1.82716i			0.244164i
$57$		0			0
$58$	7.31983			0.961140
$59$		−	6.18295i		−	0.804952i	−0.915431	−	0.402476i	$-0.868150\pi$
$59$		−	6.18295i		−	0.804952i	0.915431	−	0.402476i	$-0.131850\pi$
$60$		0			0
$61$		−	8.74905i		−	1.12020i	−0.828425	−	0.560100i	$-0.810763\pi$
$61$		−	8.74905i		−	1.12020i	0.828425	−	0.560100i	$-0.189237\pi$
$62$	−6.41670			−0.814921
$63$		0			0
$64$	1.00000			0.125000
$65$	−4.79408			−0.594633
$66$		0			0
$67$	−9.32749			−1.13953			−0.569767	−	0.821806i	$-0.692967\pi$
$67$	−9.32749			−1.13953			−0.569767	+	0.821806i	$0.692967\pi$
$68$	1.79436			0.217599
$69$		0			0
$70$	5.80556			0.693898
$71$			3.66376i			0.434808i	0.976082	+	0.217404i	$0.0697590\pi$
$71$			3.66376i			0.434808i	−0.976082	+	0.217404i	$0.930241\pi$
$72$		0			0
$73$			13.3378i			1.56107i	0.625112	+	0.780535i	$0.285053\pi$
$73$			13.3378i			1.56107i	−0.625112	+	0.780535i	$0.714947\pi$
$74$	−0.490103			−0.0569733
$75$		0			0
$76$		−	1.00000i		−	0.114708i
$77$	−0.318906	+	6.05159i	−0.0363427	+	0.689643i
$78$		0			0
$79$			11.4910i			1.29283i	0.762984	+	0.646417i	$0.223733\pi$
$79$			11.4910i			1.29283i	−0.762984	+	0.646417i	$0.776267\pi$
$80$		−	3.17738i		−	0.355242i
$81$		0			0
$82$	7.55243			0.834026
$83$	−5.73214			−0.629184			−0.314592	−	0.949227i	$-0.601868\pi$
$83$	−5.73214			−0.629184			−0.314592	+	0.949227i	$0.601868\pi$
$84$		0			0
$85$		−	5.70137i		−	0.618401i
$86$			6.64939i			0.717022i
$87$		0			0
$88$	3.31203	+	0.174537i	0.353063	+	0.0186057i
$89$		−	4.98909i		−	0.528843i	−0.964407	−	0.264421i	$-0.914819\pi$
$89$		−	4.98909i		−	0.528843i	0.964407	−	0.264421i	$-0.0851810\pi$
$90$		0			0
$91$	−2.75685			−0.288996
$92$			1.69855i			0.177086i
$93$		0			0
$94$			13.6683i			1.40978i
$95$	−3.17738			−0.325992
$96$		0			0
$97$	−13.6131			−1.38220			−0.691101	−	0.722758i	$-0.742874\pi$
$97$	−13.6131			−1.38220			−0.691101	+	0.722758i	$0.742874\pi$
$98$	−3.66150			−0.369867
$99$		0			0
$100$	−5.09572			−0.509572
$101$	5.97341			0.594377			0.297188	−	0.954819i	$-0.403951\pi$
$101$	5.97341			0.594377			0.297188	+	0.954819i	$0.403951\pi$
$102$		0			0
$103$	−14.8793			−1.46610			−0.733050	−	0.680175i	$-0.761904\pi$
$103$	−14.8793			−1.46610			−0.733050	+	0.680175i	$0.761904\pi$
$104$			1.50882i			0.147952i
$105$		0			0
$106$		−	1.80541i		−	0.175357i
$107$	−9.02249			−0.872237			−0.436118	−	0.899889i	$-0.643647\pi$
$107$	−9.02249			−0.872237			−0.436118	+	0.899889i	$0.643647\pi$
$108$		0			0
$109$		−	7.82316i		−	0.749323i	−0.927162	−	0.374661i	$-0.877759\pi$
$109$		−	7.82316i		−	0.749323i	0.927162	−	0.374661i	$-0.122241\pi$
$110$	0.554569	−	10.5236i	0.0528761	−	1.00338i
$111$		0			0
$112$		−	1.82716i		−	0.172650i
$113$			4.63215i			0.435756i	0.975976	+	0.217878i	$0.0699135\pi$
$113$			4.63215i			0.435756i	−0.975976	+	0.217878i	$0.930087\pi$
$114$		0			0
$115$	5.39694			0.503268
$116$	−7.31983			−0.679629
$117$		0			0
$118$			6.18295i			0.569187i
$119$		−	3.27858i		−	0.300547i
$120$		0			0
$121$	10.9391	+	1.15614i	0.994461	+	0.105104i
$122$			8.74905i			0.792102i
$123$		0			0
$124$	6.41670			0.576236
$125$			0.304150i			0.0272040i
$126$		0			0
$127$			0.889796i			0.0789567i	0.999220	+	0.0394783i	$0.0125696\pi$
$127$			0.889796i			0.0789567i	−0.999220	+	0.0394783i	$0.987430\pi$
$128$	−1.00000			−0.0883883
$129$		0			0
$130$	4.79408			0.420469
$131$	2.35311			0.205592			0.102796	−	0.994702i	$-0.467221\pi$
$131$	2.35311			0.205592			0.102796	+	0.994702i	$0.467221\pi$
$132$		0			0
$133$	−1.82716			−0.158435
$134$	9.32749			0.805773
$135$		0			0
$136$	−1.79436			−0.153865
$137$			20.2195i			1.72747i	0.503948	+	0.863734i	$0.331880\pi$
$137$			20.2195i			1.72747i	−0.503948	+	0.863734i	$0.668120\pi$
$138$		0			0
$139$			20.6561i			1.75203i	0.482288	+	0.876013i	$0.339806\pi$
$139$			20.6561i			1.75203i	−0.482288	+	0.876013i	$0.660194\pi$
$140$	−5.80556			−0.490660
$141$		0			0
$142$		−	3.66376i		−	0.307456i
$143$	−0.263344	+	4.99725i	−0.0220220	+	0.417891i
$144$		0			0
$145$			23.2578i			1.93146i
$146$		−	13.3378i		−	1.10384i
$147$		0			0
$148$	0.490103			0.0402862
$149$	−1.46265			−0.119825			−0.0599127	−	0.998204i	$-0.519082\pi$
$149$	−1.46265			−0.119825			−0.0599127	+	0.998204i	$0.519082\pi$
$150$		0			0
$151$			14.3004i			1.16375i	0.813279	+	0.581874i	$0.197680\pi$
$151$			14.3004i			1.16375i	−0.813279	+	0.581874i	$0.802320\pi$
$152$			1.00000i			0.0811107i
$153$		0			0
$154$	0.318906	−	6.05159i	0.0256982	−	0.487651i
$155$		−	20.3883i		−	1.63762i
$156$		0			0
$157$	20.4483			1.63195			0.815976	−	0.578086i	$-0.196200\pi$
$157$	20.4483			1.63195			0.815976	+	0.578086i	$0.196200\pi$
$158$		−	11.4910i		−	0.914172i
$159$		0			0
$160$			3.17738i			0.251194i
$161$	3.10352			0.244592
$162$		0			0
$163$	2.42589			0.190010			0.0950052	−	0.995477i	$-0.469713\pi$
$163$	2.42589			0.190010			0.0950052	+	0.995477i	$0.469713\pi$
$164$	−7.55243			−0.589746
$165$		0			0
$166$	5.73214			0.444900
$167$	−2.76565			−0.214013			−0.107006	−	0.994258i	$-0.534127\pi$
$167$	−2.76565			−0.214013			−0.107006	+	0.994258i	$0.534127\pi$
$168$		0			0
$169$	10.7235			0.824882
$170$			5.70137i			0.437275i
$171$		0			0
$172$		−	6.64939i		−	0.507011i
$173$	18.3305			1.39365			0.696823	−	0.717244i	$-0.254597\pi$
$173$	18.3305			1.39365			0.696823	+	0.717244i	$0.254597\pi$
$174$		0			0
$175$			9.31068i			0.703821i
$176$	−3.31203	−	0.174537i	−0.249654	−	0.0131562i
$177$		0			0
$178$			4.98909i			0.373948i
$179$			6.72559i			0.502694i	0.967897	+	0.251347i	$0.0808736\pi$
$179$			6.72559i			0.502694i	−0.967897	+	0.251347i	$0.919126\pi$
$180$		0			0
$181$	10.4185			0.774400			0.387200	−	0.921996i	$-0.373442\pi$
$181$	10.4185			0.774400			0.387200	+	0.921996i	$0.373442\pi$
$182$	2.75685			0.204351
$183$		0			0
$184$		−	1.69855i		−	0.125219i
$185$		−	1.55724i		−	0.114491i
$186$		0			0
$187$	−5.94299	−	0.313183i	−0.434594	−	0.0229022i
$188$		−	13.6683i		−	0.996863i
$189$		0			0
$190$	3.17738			0.230511
$191$			0.767620i			0.0555430i	0.999614	+	0.0277715i	$0.00884108\pi$
$191$			0.767620i			0.0555430i	−0.999614	+	0.0277715i	$0.991159\pi$
$192$		0			0
$193$		−	9.46804i		−	0.681524i	−0.940150	−	0.340762i	$-0.889315\pi$
$193$		−	9.46804i		−	0.681524i	0.940150	−	0.340762i	$-0.110685\pi$
$194$	13.6131			0.977364
$195$		0			0
$196$	3.66150			0.261536
$197$	−14.0225			−0.999058			−0.499529	−	0.866297i	$-0.666494\pi$
$197$	−14.0225			−0.999058			−0.499529	+	0.866297i	$0.666494\pi$
$198$		0			0
$199$	0.968012			0.0686206			0.0343103	−	0.999411i	$-0.489077\pi$
$199$	0.968012			0.0686206			0.0343103	+	0.999411i	$0.489077\pi$
$200$	5.09572			0.360322
$201$		0			0
$202$	−5.97341			−0.420288
$203$			13.3745i			0.938703i
$204$		0			0
$205$			23.9969i			1.67602i
$206$	14.8793			1.03669
$207$		0			0
$208$		−	1.50882i		−	0.104618i
$209$	−0.174537	+	3.31203i	−0.0120730	+	0.229098i
$210$		0			0
$211$			11.5426i			0.794628i	0.917683	+	0.397314i	$0.130058\pi$
$211$			11.5426i			0.794628i	−0.917683	+	0.397314i	$0.869942\pi$
$212$			1.80541i			0.123996i
$213$		0			0
$214$	9.02249			0.616765
$215$	−21.1276			−1.44089
$216$		0			0
$217$		−	11.7243i		−	0.795898i
$218$			7.82316i			0.529851i
$219$		0			0
$220$	−0.554569	+	10.5236i	−0.0373890	+	0.709499i
$221$		−	2.70737i		−	0.182117i
$222$		0			0
$223$	−9.33268			−0.624962			−0.312481	−	0.949924i	$-0.601160\pi$
$223$	−9.33268			−0.624962			−0.312481	+	0.949924i	$0.601160\pi$
$224$			1.82716i			0.122082i
$225$		0			0
$226$		−	4.63215i		−	0.308126i
$227$	−1.35999			−0.0902658			−0.0451329	−	0.998981i	$-0.514371\pi$
$227$	−1.35999			−0.0902658			−0.0451329	+	0.998981i	$0.514371\pi$
$228$		0			0
$229$	18.1305			1.19809			0.599047	−	0.800714i	$-0.295546\pi$
$229$	18.1305			1.19809			0.599047	+	0.800714i	$0.295546\pi$
$230$	−5.39694			−0.355864
$231$		0			0
$232$	7.31983			0.480570
$233$	2.77760			0.181967			0.0909834	−	0.995852i	$-0.470999\pi$
$233$	2.77760			0.181967			0.0909834	+	0.995852i	$0.470999\pi$
$234$		0			0
$235$	−43.4293			−2.83302
$236$		−	6.18295i		−	0.402476i
$237$		0			0
$238$			3.27858i			0.212519i
$239$	−17.7215			−1.14631			−0.573153	−	0.819448i	$-0.694280\pi$
$239$	−17.7215			−1.14631			−0.573153	+	0.819448i	$0.694280\pi$
$240$		0			0
$241$			4.99032i			0.321455i	0.986999	+	0.160728i	$0.0513840\pi$
$241$			4.99032i			0.321455i	−0.986999	+	0.160728i	$0.948616\pi$
$242$	−10.9391	−	1.15614i	−0.703190	−	0.0743196i
$243$		0			0
$244$		−	8.74905i		−	0.560100i
$245$		−	11.6340i		−	0.743267i
$246$		0			0
$247$	−1.50882			−0.0960038
$248$	−6.41670			−0.407461
$249$		0			0
$250$		−	0.304150i		−	0.0192361i
$251$			13.0051i			0.820876i	0.911889	+	0.410438i	$0.134624\pi$
$251$			13.0051i			0.820876i	−0.911889	+	0.410438i	$0.865376\pi$
$252$		0			0
$253$	0.296460	−	5.62566i	0.0186383	−	0.353682i
$254$		−	0.889796i		−	0.0558308i
$255$		0			0
$256$	1.00000			0.0625000
$257$		−	3.96286i		−	0.247197i	−0.992332	−	0.123598i	$-0.960557\pi$
$257$		−	3.96286i		−	0.247197i	0.992332	−	0.123598i	$-0.0394434\pi$
$258$		0			0
$259$		−	0.895495i		−	0.0556433i
$260$	−4.79408			−0.297316
$261$		0			0
$262$	−2.35311			−0.145375
$263$	18.6239			1.14840			0.574200	−	0.818715i	$-0.305313\pi$
$263$	18.6239			1.14840			0.574200	+	0.818715i	$0.305313\pi$
$264$		0			0
$265$	5.73647			0.352389
$266$	1.82716			0.112030
$267$		0			0
$268$	−9.32749			−0.569767
$269$			25.3246i			1.54407i	0.635582	+	0.772033i	$0.280760\pi$
$269$			25.3246i			1.54407i	−0.635582	+	0.772033i	$0.719240\pi$
$270$		0			0
$271$		−	13.5880i		−	0.825412i	−0.910864	−	0.412706i	$-0.864584\pi$
$271$		−	13.5880i		−	0.825412i	0.910864	−	0.412706i	$-0.135416\pi$
$272$	1.79436			0.108799
$273$		0			0
$274$		−	20.2195i		−	1.22150i
$275$	16.8772	+	0.889391i	1.01773	+	0.0536323i
$276$		0			0
$277$		−	2.49779i		−	0.150077i	−0.997181	−	0.0750387i	$-0.976092\pi$
$277$		−	2.49779i		−	0.150077i	0.997181	−	0.0750387i	$-0.0239080\pi$
$278$		−	20.6561i		−	1.23887i
$279$		0			0
$280$	5.80556			0.346949
$281$	−3.03707			−0.181176			−0.0905882	−	0.995888i	$-0.528875\pi$
$281$	−3.03707			−0.181176			−0.0905882	+	0.995888i	$0.528875\pi$
$282$		0			0
$283$		−	13.5569i		−	0.805872i	−0.915228	−	0.402936i	$-0.867990\pi$
$283$		−	13.5569i		−	0.805872i	0.915228	−	0.402936i	$-0.132010\pi$
$284$			3.66376i			0.217404i
$285$		0			0
$286$	0.263344	−	4.99725i	0.0155719	−	0.295494i
$287$			13.7995i			0.814557i
$288$		0			0
$289$	−13.7803			−0.810603
$290$		−	23.2578i		−	1.36575i
$291$		0			0
$292$			13.3378i			0.780535i
$293$	6.35638			0.371344			0.185672	−	0.982612i	$-0.440554\pi$
$293$	6.35638			0.371344			0.185672	+	0.982612i	$0.440554\pi$
$294$		0			0
$295$	−19.6456			−1.14381
$296$	−0.490103			−0.0284867
$297$		0			0
$298$	1.46265			0.0847293
$299$	2.56281			0.148211
$300$		0			0
$301$	−12.1495			−0.700284
$302$		−	14.3004i		−	0.822894i
$303$		0			0
$304$		−	1.00000i		−	0.0573539i
$305$	−27.7990			−1.59177
$306$		0			0
$307$			10.5247i			0.600676i	0.953833	+	0.300338i	$0.0970995\pi$
$307$			10.5247i			0.600676i	−0.953833	+	0.300338i	$0.902900\pi$
$308$	−0.318906	+	6.05159i	−0.0181714	+	0.344822i
$309$		0			0
$310$			20.3883i			1.15798i
$311$			15.9661i			0.905354i	0.891675	+	0.452677i	$0.149531\pi$
$311$			15.9661i			0.905354i	−0.891675	+	0.452677i	$0.850469\pi$
$312$		0			0
$313$	−28.5739			−1.61509			−0.807547	−	0.589804i	$-0.799205\pi$
$313$	−28.5739			−1.61509			−0.807547	+	0.589804i	$0.799205\pi$
$314$	−20.4483			−1.15396
$315$		0			0
$316$			11.4910i			0.646417i
$317$		−	5.79413i		−	0.325431i	−0.986673	−	0.162715i	$-0.947975\pi$
$317$		−	5.79413i		−	0.325431i	0.986673	−	0.162715i	$-0.0520252\pi$
$318$		0			0
$319$	24.2435	+	1.27758i	1.35737	+	0.0715307i
$320$		−	3.17738i		−	0.177621i
$321$		0			0
$322$	−3.10352			−0.172953
$323$		−	1.79436i		−	0.0998411i
$324$		0			0
$325$			7.68852i			0.426482i
$326$	−2.42589			−0.134358
$327$		0			0
$328$	7.55243			0.417013
$329$	−24.9741			−1.37687
$330$		0			0
$331$	−20.7981			−1.14317			−0.571584	−	0.820544i	$-0.693671\pi$
$331$	−20.7981			−1.14317			−0.571584	+	0.820544i	$0.693671\pi$
$332$	−5.73214			−0.314592
$333$		0			0
$334$	2.76565			0.151330
$335$			29.6370i			1.61924i
$336$		0			0
$337$			1.66130i			0.0904965i	0.998976	+	0.0452483i	$0.0144079\pi$
$337$			1.66130i			0.0904965i	−0.998976	+	0.0452483i	$0.985592\pi$
$338$	−10.7235			−0.583280
$339$		0			0
$340$		−	5.70137i		−	0.309200i
$341$	−21.2523	−	1.11995i	−1.15088	−	0.0606487i
$342$		0			0
$343$		−	19.4802i		−	1.05183i
$344$			6.64939i			0.358511i
$345$		0			0
$346$	−18.3305			−0.985456
$347$	11.1045			0.596121			0.298061	−	0.954547i	$-0.403660\pi$
$347$	11.1045			0.596121			0.298061	+	0.954547i	$0.403660\pi$
$348$		0			0
$349$		−	19.0884i		−	1.02178i	−0.859646	−	0.510890i	$-0.829316\pi$
$349$		−	19.0884i		−	1.02178i	0.859646	−	0.510890i	$-0.170684\pi$
$350$		−	9.31068i		−	0.497677i
$351$		0			0
$352$	3.31203	+	0.174537i	0.176532	+	0.00930284i
$353$		−	3.14424i		−	0.167351i	−0.996493	−	0.0836756i	$-0.973334\pi$
$353$		−	3.14424i		−	0.167351i	0.996493	−	0.0836756i	$-0.0266660\pi$
$354$		0			0
$355$	11.6411			0.617848
$356$		−	4.98909i		−	0.264421i
$357$		0			0
$358$		−	6.72559i		−	0.355459i
$359$	−31.9249			−1.68493			−0.842466	−	0.538750i	$-0.818897\pi$
$359$	−31.9249			−1.68493			−0.842466	+	0.538750i	$0.818897\pi$
$360$		0			0
$361$	−1.00000			−0.0526316
$362$	−10.4185			−0.547583
$363$		0			0
$364$	−2.75685			−0.144498
$365$	42.3792			2.21823
$366$		0			0
$367$	−33.0623			−1.72584			−0.862918	−	0.505344i	$-0.831365\pi$
$367$	−33.0623			−1.72584			−0.862918	+	0.505344i	$0.831365\pi$
$368$			1.69855i			0.0885432i
$369$		0			0
$370$			1.55724i			0.0809572i
$371$	3.29877			0.171263
$372$		0			0
$373$			6.27558i			0.324937i	0.986714	+	0.162469i	$0.0519456\pi$
$373$			6.27558i			0.324937i	−0.986714	+	0.162469i	$0.948054\pi$
$374$	5.94299	+	0.313183i	0.307305	+	0.0161943i
$375$		0			0
$376$			13.6683i			0.704889i
$377$			11.0443i			0.568810i
$378$		0			0
$379$	−35.9059			−1.84436			−0.922182	−	0.386756i	$-0.873596\pi$
$379$	−35.9059			−1.84436			−0.922182	+	0.386756i	$0.873596\pi$
$380$	−3.17738			−0.162996
$381$		0			0
$382$		−	0.767620i		−	0.0392748i
$383$			21.0829i			1.07729i	0.842533	+	0.538644i	$0.181063\pi$
$383$			21.0829i			1.07729i	−0.842533	+	0.538644i	$0.818937\pi$
$384$		0			0
$385$	19.2282	+	1.01328i	0.979960	+	0.0516418i
$386$			9.46804i			0.481910i
$387$		0			0
$388$	−13.6131			−0.691101
$389$			0.203373i			0.0103114i	0.999987	+	0.00515571i	$0.00164112\pi$
$389$			0.203373i			0.0103114i	−0.999987	+	0.00515571i	$0.998359\pi$
$390$		0			0
$391$			3.04782i			0.154135i
$392$	−3.66150			−0.184934
$393$		0			0
$394$	14.0225			0.706441
$395$	36.5111			1.83707
$396$		0			0
$397$	16.9003			0.848203			0.424102	−	0.905615i	$-0.360590\pi$
$397$	16.9003			0.848203			0.424102	+	0.905615i	$0.360590\pi$
$398$	−0.968012			−0.0485221
$399$		0			0
$400$	−5.09572			−0.254786
$401$			3.35221i			0.167401i	0.996491	+	0.0837006i	$0.0266739\pi$
$401$			3.35221i			0.167401i	−0.996491	+	0.0837006i	$0.973326\pi$
$402$		0			0
$403$		−	9.68163i		−	0.482276i
$404$	5.97341			0.297188
$405$		0			0
$406$		−	13.3745i		−	0.663763i
$407$	−1.62324	−	0.0855410i	−0.0804608	−	0.00424011i
$408$		0			0
$409$		−	33.7432i		−	1.66849i	−0.551391	−	0.834247i	$-0.685903\pi$
$409$		−	33.7432i		−	1.66849i	0.551391	−	0.834247i	$-0.314097\pi$
$410$		−	23.9969i		−	1.18512i
$411$		0			0
$412$	−14.8793			−0.733050
$413$	−11.2972			−0.555900
$414$		0			0
$415$			18.2132i			0.894049i
$416$			1.50882i			0.0739759i
$417$		0			0
$418$	0.174537	−	3.31203i	0.00853687	−	0.161997i
$419$		−	16.1750i		−	0.790201i	−0.918638	−	0.395100i	$-0.870710\pi$
$419$		−	16.1750i		−	0.790201i	0.918638	−	0.395100i	$-0.129290\pi$
$420$		0			0
$421$	−39.0009			−1.90079			−0.950393	−	0.311052i	$-0.899319\pi$
$421$	−39.0009			−1.90079			−0.950393	+	0.311052i	$0.899319\pi$
$422$		−	11.5426i		−	0.561887i
$423$		0			0
$424$		−	1.80541i		−	0.0876785i
$425$	−9.14358			−0.443529
$426$		0			0
$427$	−15.9859			−0.773611
$428$	−9.02249			−0.436118
$429$		0			0
$430$	21.1276			1.01886
$431$	−12.6119			−0.607492			−0.303746	−	0.952753i	$-0.598237\pi$
$431$	−12.6119			−0.607492			−0.303746	+	0.952753i	$0.598237\pi$
$432$		0			0
$433$	30.4096			1.46139			0.730697	−	0.682702i	$-0.239195\pi$
$433$	30.4096			1.46139			0.730697	+	0.682702i	$0.239195\pi$
$434$			11.7243i			0.562785i
$435$		0			0
$436$		−	7.82316i		−	0.374661i
$437$	1.69855			0.0812528
$438$		0			0
$439$		−	13.2488i		−	0.632332i	−0.948704	−	0.316166i	$-0.897604\pi$
$439$		−	13.2488i		−	0.632332i	0.948704	−	0.316166i	$-0.102396\pi$
$440$	0.554569	−	10.5236i	0.0264380	−	0.501691i
$441$		0			0
$442$			2.70737i			0.128776i
$443$		−	40.4929i		−	1.92388i	−0.273268	−	0.961938i	$-0.588104\pi$
$443$		−	40.4929i		−	1.92388i	0.273268	−	0.961938i	$-0.411896\pi$
$444$		0			0
$445$	−15.8522			−0.751468
$446$	9.33268			0.441915
$447$		0			0
$448$		−	1.82716i		−	0.0863250i
$449$			17.6319i			0.832101i	0.909342	+	0.416050i	$0.136586\pi$
$449$			17.6319i			0.832101i	−0.909342	+	0.416050i	$0.863414\pi$
$450$		0			0
$451$	25.0139	+	1.31818i	1.17786	+	0.0620705i
$452$			4.63215i			0.217878i
$453$		0			0
$454$	1.35999			0.0638275
$455$			8.75954i			0.410654i
$456$		0			0
$457$		−	10.6107i		−	0.496348i	−0.968716	−	0.248174i	$-0.920170\pi$
$457$		−	10.6107i		−	0.496348i	0.968716	−	0.248174i	$-0.0798304\pi$
$458$	−18.1305			−0.847181
$459$		0			0
$460$	5.39694			0.251634
$461$	−7.67734			−0.357569			−0.178785	−	0.983888i	$-0.557217\pi$
$461$	−7.67734			−0.357569			−0.178785	+	0.983888i	$0.557217\pi$
$462$		0			0
$463$	−9.58193			−0.445310			−0.222655	−	0.974897i	$-0.571472\pi$
$463$	−9.58193			−0.445310			−0.222655	+	0.974897i	$0.571472\pi$
$464$	−7.31983			−0.339814
$465$		0			0
$466$	−2.77760			−0.128670
$467$		−	23.7973i		−	1.10121i	−0.834767	−	0.550603i	$-0.814398\pi$
$467$		−	23.7973i		−	1.10121i	0.834767	−	0.550603i	$-0.185602\pi$
$468$		0			0
$469$			17.0428i			0.786963i
$470$	43.4293			2.00325
$471$		0			0
$472$			6.18295i			0.284594i
$473$	−1.16056	+	22.0230i	−0.0533627	+	1.01262i
$474$		0			0
$475$			5.09572i			0.233808i
$476$		−	3.27858i		−	0.150274i
$477$		0			0
$478$	17.7215			0.810561
$479$	24.8164			1.13389			0.566945	−	0.823755i	$-0.308125\pi$
$479$	24.8164			1.13389			0.566945	+	0.823755i	$0.308125\pi$
$480$		0			0
$481$		−	0.739476i		−	0.0337172i
$482$		−	4.99032i		−	0.227303i
$483$		0			0
$484$	10.9391	+	1.15614i	0.497231	+	0.0525519i
$485$			43.2540i			1.96406i
$486$		0			0
$487$	20.6772			0.936974			0.468487	−	0.883470i	$-0.344799\pi$
$487$	20.6772			0.936974			0.468487	+	0.883470i	$0.344799\pi$
$488$			8.74905i			0.396051i
$489$		0			0
$490$			11.6340i			0.525569i
$491$	−27.4835			−1.24031			−0.620157	−	0.784477i	$-0.712931\pi$
$491$	−27.4835			−1.24031			−0.620157	+	0.784477i	$0.712931\pi$
$492$		0			0
$493$	−13.1344			−0.591545
$494$	1.50882			0.0678849
$495$		0			0
$496$	6.41670			0.288118
$497$	6.69426			0.300279
$498$		0			0
$499$	−23.4850			−1.05133			−0.525667	−	0.850691i	$-0.676184\pi$
$499$	−23.4850			−1.05133			−0.525667	+	0.850691i	$0.676184\pi$
$500$			0.304150i			0.0136020i
$501$		0			0
$502$		−	13.0051i		−	0.580447i
$503$	39.5787			1.76473			0.882364	−	0.470567i	$-0.155950\pi$
$503$	39.5787			1.76473			0.882364	+	0.470567i	$0.155950\pi$
$504$		0			0
$505$		−	18.9798i		−	0.844589i
$506$	−0.296460	+	5.62566i	−0.0131793	+	0.250091i
$507$		0			0
$508$			0.889796i			0.0394783i
$509$		−	17.1523i		−	0.760261i	−0.924933	−	0.380130i	$-0.875879\pi$
$509$		−	17.1523i		−	0.760261i	0.924933	−	0.380130i	$-0.124121\pi$
$510$		0			0
$511$	24.3702			1.07808
$512$	−1.00000			−0.0441942
$513$		0			0
$514$			3.96286i			0.174794i
$515$			47.2771i			2.08328i
$516$		0			0
$517$	−2.38562	+	45.2698i	−0.104919	+	1.99096i
$518$			0.895495i			0.0393458i
$519$		0			0
$520$	4.79408			0.210234
$521$			16.6778i			0.730669i	0.930876	+	0.365334i	$0.119045\pi$
$521$			16.6778i			0.730669i	−0.930876	+	0.365334i	$0.880955\pi$
$522$		0			0
$523$		−	26.1007i		−	1.14131i	−0.821191	−	0.570653i	$-0.806690\pi$
$523$		−	26.1007i		−	1.14131i	0.821191	−	0.570653i	$-0.193310\pi$
$524$	2.35311			0.102796
$525$		0			0
$526$	−18.6239			−0.812041
$527$	11.5139			0.501553
$528$		0			0
$529$	20.1149			0.874562
$530$	−5.73647			−0.249176
$531$		0			0
$532$	−1.82716			−0.0792173
$533$			11.3952i			0.493583i
$534$		0			0
$535$			28.6678i			1.23942i
$536$	9.32749			0.402886
$537$		0			0
$538$		−	25.3246i		−	1.09182i
$539$	−12.1270	−	0.639067i	−0.522347	−	0.0275265i
$540$		0			0
$541$		−	28.2740i		−	1.21559i	−0.794093	−	0.607796i	$-0.792054\pi$
$541$		−	28.2740i		−	1.21559i	0.794093	−	0.607796i	$-0.207946\pi$
$542$			13.5880i			0.583655i
$543$		0			0
$544$	−1.79436			−0.0769327
$545$	−24.8571			−1.06476
$546$		0			0
$547$		−	41.8979i		−	1.79142i	−0.444634	−	0.895712i	$-0.646666\pi$
$547$		−	41.8979i		−	1.79142i	0.444634	−	0.895712i	$-0.353334\pi$
$548$			20.2195i			0.863734i
$549$		0			0
$550$	−16.8772	−	0.889391i	−0.719646	−	0.0379238i
$551$			7.31983i			0.311835i
$552$		0			0
$553$	20.9958			0.892832
$554$			2.49779i			0.106121i
$555$		0			0
$556$			20.6561i			0.876013i
$557$	19.1209			0.810178			0.405089	−	0.914277i	$-0.367241\pi$
$557$	19.1209			0.810178			0.405089	+	0.914277i	$0.367241\pi$
$558$		0			0
$559$	−10.0327			−0.424339
$560$	−5.80556			−0.245330
$561$		0			0
$562$	3.03707			0.128111
$563$	−6.76385			−0.285062			−0.142531	−	0.989790i	$-0.545524\pi$
$563$	−6.76385			−0.285062			−0.142531	+	0.989790i	$0.545524\pi$
$564$		0			0
$565$	14.7181			0.619194
$566$			13.5569i			0.569837i
$567$		0			0
$568$		−	3.66376i		−	0.153728i
$569$	−34.4928			−1.44601			−0.723006	−	0.690842i	$-0.757240\pi$
$569$	−34.4928			−1.44601			−0.723006	+	0.690842i	$0.757240\pi$
$570$		0			0
$571$			27.2434i			1.14010i	0.821609	+	0.570051i	$0.193077\pi$
$571$			27.2434i			1.14010i	−0.821609	+	0.570051i	$0.806923\pi$
$572$	−0.263344	+	4.99725i	−0.0110110	+	0.208945i
$573$		0			0
$574$		−	13.7995i		−	0.575979i
$575$		−	8.65536i		−	0.360953i
$576$		0			0
$577$	27.7152			1.15380			0.576900	−	0.816815i	$-0.304262\pi$
$577$	27.7152			1.15380			0.576900	+	0.816815i	$0.304262\pi$
$578$	13.7803			0.573183
$579$		0			0
$580$			23.2578i			0.965729i
$581$			10.4735i			0.434514i
$582$		0			0
$583$	0.315111	−	5.97957i	0.0130505	−	0.247649i
$584$		−	13.3378i		−	0.551922i
$585$		0			0
$586$	−6.35638			−0.262580
$587$		−	29.7091i		−	1.22623i	−0.789995	−	0.613114i	$-0.789917\pi$
$587$		−	29.7091i		−	1.22623i	0.789995	−	0.613114i	$-0.210083\pi$
$588$		0			0
$589$		−	6.41670i		−	0.264395i
$590$	19.6456			0.808795
$591$		0			0
$592$	0.490103			0.0201431
$593$	3.20901			0.131778			0.0658890	−	0.997827i	$-0.479012\pi$
$593$	3.20901			0.131778			0.0658890	+	0.997827i	$0.479012\pi$
$594$		0			0
$595$	−10.4173			−0.427067
$596$	−1.46265			−0.0599127
$597$		0			0
$598$	−2.56281			−0.104801
$599$			29.2201i			1.19390i	0.802278	+	0.596951i	$0.203621\pi$
$599$			29.2201i			1.19390i	−0.802278	+	0.596951i	$0.796379\pi$
$600$		0			0
$601$		−	19.4961i		−	0.795262i	−0.917545	−	0.397631i	$-0.869832\pi$
$601$		−	19.4961i		−	0.795262i	0.917545	−	0.397631i	$-0.130168\pi$
$602$	12.1495			0.495175
$603$		0			0
$604$			14.3004i			0.581874i
$605$	3.67350	−	34.7576i	0.149349	−	1.41310i
$606$		0			0
$607$		−	23.1552i		−	0.939839i	−0.882709	−	0.469919i	$-0.844283\pi$
$607$		−	23.1552i		−	0.939839i	0.882709	−	0.469919i	$-0.155717\pi$
$608$			1.00000i			0.0405554i
$609$		0			0
$610$	27.7990			1.12555
$611$	−20.6230			−0.834316
$612$		0			0
$613$			16.1215i			0.651141i	0.945518	+	0.325571i	$0.105556\pi$
$613$			16.1215i			0.651141i	−0.945518	+	0.325571i	$0.894444\pi$
$614$		−	10.5247i		−	0.424742i
$615$		0			0
$616$	0.318906	−	6.05159i	0.0128491	−	0.243826i
$617$		−	5.19928i		−	0.209315i	−0.994508	−	0.104658i	$-0.966625\pi$
$617$		−	5.19928i		−	0.209315i	0.994508	−	0.104658i	$-0.0333747\pi$
$618$		0			0
$619$	−14.1637			−0.569289			−0.284644	−	0.958633i	$-0.591876\pi$
$619$	−14.1637			−0.569289			−0.284644	+	0.958633i	$0.591876\pi$
$620$		−	20.3883i		−	0.818812i
$621$		0			0
$622$		−	15.9661i		−	0.640182i
$623$	−9.11585			−0.365219
$624$		0			0
$625$	−24.5122			−0.980489
$626$	28.5739			1.14204
$627$		0			0
$628$	20.4483			0.815976
$629$	0.879423			0.0350649
$630$		0			0
$631$	−38.3095			−1.52508			−0.762538	−	0.646944i	$-0.776047\pi$
$631$	−38.3095			−1.52508			−0.762538	+	0.646944i	$0.776047\pi$
$632$		−	11.4910i		−	0.457086i
$633$		0			0
$634$			5.79413i			0.230114i
$635$	2.82722			0.112195
$636$		0			0
$637$		−	5.52454i		−	0.218890i
$638$	−24.2435	−	1.27758i	−0.959808	−	0.0505798i
$639$		0			0
$640$			3.17738i			0.125597i
$641$		−	37.3029i		−	1.47337i	−0.676233	−	0.736687i	$-0.736389\pi$
$641$		−	37.3029i		−	1.47337i	0.676233	−	0.736687i	$-0.263611\pi$
$642$		0			0
$643$	−3.25516			−0.128371			−0.0641855	−	0.997938i	$-0.520445\pi$
$643$	−3.25516			−0.128371			−0.0641855	+	0.997938i	$0.520445\pi$
$644$	3.10352			0.122296
$645$		0			0
$646$			1.79436i			0.0705983i
$647$		−	43.4360i		−	1.70765i	−0.520563	−	0.853823i	$-0.674278\pi$
$647$		−	43.4360i		−	1.70765i	0.520563	−	0.853823i	$-0.325722\pi$
$648$		0			0
$649$	−1.07915	+	20.4781i	−0.0423605	+	0.803837i
$650$		−	7.68852i		−	0.301569i
$651$		0			0
$652$	2.42589			0.0950052
$653$			11.5798i			0.453152i	0.973994	+	0.226576i	$0.0727531\pi$
$653$			11.5798i			0.453152i	−0.973994	+	0.226576i	$0.927247\pi$
$654$		0			0
$655$		−	7.47671i		−	0.292139i
$656$	−7.55243			−0.294873
$657$		0			0
$658$	24.9741			0.973592
$659$	47.6716			1.85702			0.928511	−	0.371305i	$-0.121090\pi$
$659$	47.6716			1.85702			0.928511	+	0.371305i	$0.121090\pi$
$660$		0			0
$661$	−35.3062			−1.37325			−0.686627	−	0.727010i	$-0.740909\pi$
$661$	−35.3062			−1.37325			−0.686627	+	0.727010i	$0.740909\pi$
$662$	20.7981			0.808341
$663$		0			0
$664$	5.73214			0.222450
$665$			5.80556i			0.225130i
$666$		0			0
$667$		−	12.4331i		−	0.481412i
$668$	−2.76565			−0.107006
$669$		0			0
$670$		−	29.6370i		−	1.14498i
$671$	−1.52703	+	28.9771i	−0.0589504	+	1.11865i
$672$		0			0
$673$			16.2403i			0.626018i	0.949750	+	0.313009i	$0.101337\pi$
$673$			16.2403i			0.626018i	−0.949750	+	0.313009i	$0.898663\pi$
$674$		−	1.66130i		−	0.0639907i
$675$		0			0
$676$	10.7235			0.412441
$677$	6.67184			0.256420			0.128210	−	0.991747i	$-0.459077\pi$
$677$	6.67184			0.256420			0.128210	+	0.991747i	$0.459077\pi$
$678$		0			0
$679$			24.8733i			0.954549i
$680$			5.70137i			0.218638i
$681$		0			0
$682$	21.2523	+	1.11995i	0.813792	+	0.0428851i
$683$			15.3456i			0.587184i	0.955931	+	0.293592i	$0.0948507\pi$
$683$			15.3456i			0.587184i	−0.955931	+	0.293592i	$0.905149\pi$
$684$		0			0
$685$	64.2450			2.45467
$686$			19.4802i			0.743758i
$687$		0			0
$688$		−	6.64939i		−	0.253505i
$689$	2.72404			0.103778
$690$		0			0
$691$	−2.40671			−0.0915555			−0.0457778	−	0.998952i	$-0.514577\pi$
$691$	−2.40671			−0.0915555			−0.0457778	+	0.998952i	$0.514577\pi$
$692$	18.3305			0.696823
$693$		0			0
$694$	−11.1045			−0.421521
$695$	65.6322			2.48957
$696$		0			0
$697$	−13.5518			−0.513311
$698$			19.0884i			0.722508i
$699$		0			0
$700$			9.31068i			0.351911i
$701$	42.0831			1.58946			0.794728	−	0.606965i	$-0.207613\pi$
$701$	42.0831			1.58946			0.794728	+	0.606965i	$0.207613\pi$
$702$		0			0
$703$		−	0.490103i		−	0.0184846i
$704$	−3.31203	−	0.174537i	−0.124827	−	0.00657810i
$705$		0			0
$706$			3.14424i			0.118335i
$707$		−	10.9144i		−	0.410477i
$708$		0			0
$709$	−32.6922			−1.22778			−0.613890	−	0.789392i	$-0.710396\pi$
$709$	−32.6922			−1.22778			−0.613890	+	0.789392i	$0.710396\pi$
$710$	−11.6411			−0.436885
$711$		0			0
$712$			4.98909i			0.186974i
$713$			10.8991i			0.408174i
$714$		0			0
$715$	15.8781	+	0.836744i	0.593809	+	0.0312925i
$716$			6.72559i			0.251347i
$717$		0			0
$718$	31.9249			1.19143
$719$			26.5299i			0.989398i	0.869064	+	0.494699i	$0.164722\pi$
$719$			26.5299i			0.989398i	−0.869064	+	0.494699i	$0.835278\pi$
$720$		0			0
$721$			27.1868i			1.01249i
$722$	1.00000			0.0372161
$723$		0			0
$724$	10.4185			0.387200
$725$	37.2998			1.38528
$726$		0			0
$727$	0.487378			0.0180759			0.00903793	−	0.999959i	$-0.497123\pi$
$727$	0.487378			0.0180759			0.00903793	+	0.999959i	$0.497123\pi$
$728$	2.75685			0.102176
$729$		0			0
$730$	−42.3792			−1.56852
$731$		−	11.9314i		−	0.441300i
$732$		0			0
$733$			46.5957i			1.72105i	0.509406	+	0.860526i	$0.329865\pi$
$733$			46.5957i			1.72105i	−0.509406	+	0.860526i	$0.670135\pi$
$734$	33.0623			1.22035
$735$		0			0
$736$		−	1.69855i		−	0.0626095i
$737$	30.8929	+	1.62799i	1.13796	+	0.0599678i
$738$		0			0
$739$		−	3.29041i		−	0.121040i	−0.998167	−	0.0605198i	$-0.980724\pi$
$739$		−	3.29041i		−	0.121040i	0.998167	−	0.0605198i	$-0.0192758\pi$
$740$		−	1.55724i		−	0.0572454i
$741$		0			0
$742$	−3.29877			−0.121102
$743$	2.11739			0.0776794			0.0388397	−	0.999245i	$-0.487634\pi$
$743$	2.11739			0.0776794			0.0388397	+	0.999245i	$0.487634\pi$
$744$		0			0
$745$			4.64740i			0.170268i
$746$		−	6.27558i		−	0.229765i
$747$		0			0
$748$	−5.94299	−	0.313183i	−0.217297	−	0.0114511i
$749$			16.4855i			0.602367i
$750$		0			0
$751$	2.66024			0.0970734			0.0485367	−	0.998821i	$-0.484544\pi$
$751$	2.66024			0.0970734			0.0485367	+	0.998821i	$0.484544\pi$
$752$		−	13.6683i		−	0.498431i
$753$		0			0
$754$		−	11.0443i		−	0.402209i
$755$	45.4377			1.65365
$756$		0			0
$757$	−11.7079			−0.425532			−0.212766	−	0.977103i	$-0.568247\pi$
$757$	−11.7079			−0.425532			−0.212766	+	0.977103i	$0.568247\pi$
$758$	35.9059			1.30416
$759$		0			0
$760$	3.17738			0.115256
$761$	−2.37573			−0.0861202			−0.0430601	−	0.999072i	$-0.513711\pi$
$761$	−2.37573			−0.0861202			−0.0430601	+	0.999072i	$0.513711\pi$
$762$		0			0
$763$	−14.2941			−0.517482
$764$			0.767620i			0.0277715i
$765$		0			0
$766$		−	21.0829i		−	0.761758i
$767$	−9.32895			−0.336849
$768$		0			0
$769$			10.2270i			0.368793i	0.982852	+	0.184397i	$0.0590331\pi$
$769$			10.2270i			0.368793i	−0.982852	+	0.184397i	$0.940967\pi$
$770$	−19.2282	−	1.01328i	−0.692936	−	0.0365162i
$771$		0			0
$772$		−	9.46804i		−	0.340762i
$773$		−	34.1124i		−	1.22694i	−0.789719	−	0.613469i	$-0.789773\pi$
$773$		−	34.1124i		−	1.22694i	0.789719	−	0.613469i	$-0.210227\pi$
$774$		0			0
$775$	−32.6977			−1.17454
$776$	13.6131			0.488682
$777$		0			0
$778$		−	0.203373i		−	0.00729128i
$779$			7.55243i			0.270594i
$780$		0			0
$781$	0.639461	−	12.1345i	0.0228817	−	0.434206i
$782$		−	3.04782i		−	0.108990i
$783$		0			0
$784$	3.66150			0.130768
$785$		−	64.9719i		−	2.31895i
$786$		0			0
$787$			2.86764i			0.102220i	0.998693	+	0.0511102i	$0.0162760\pi$
$787$			2.86764i			0.102220i	−0.998693	+	0.0511102i	$0.983724\pi$
$788$	−14.0225			−0.499529
$789$		0			0
$790$	−36.5111			−1.29901
$791$	8.46366			0.300933
$792$		0			0
$793$	−13.2007			−0.468771
$794$	−16.9003			−0.599770
$795$		0			0
$796$	0.968012			0.0343103
$797$		−	30.7339i		−	1.08865i	−0.838874	−	0.544325i	$-0.816786\pi$
$797$		−	30.7339i		−	1.08865i	0.838874	−	0.544325i	$-0.183214\pi$
$798$		0			0
$799$		−	24.5259i		−	0.867664i
$800$	5.09572			0.180161
$801$		0			0
$802$		−	3.35221i		−	0.118371i
$803$	2.32794	−	44.1752i	0.0821511	−	1.55891i
$804$		0			0
$805$		−	9.86106i		−	0.347557i
$806$			9.68163i			0.341021i
$807$		0			0
$808$	−5.97341			−0.210144
$809$	11.6209			0.408568			0.204284	−	0.978912i	$-0.434513\pi$
$809$	11.6209			0.408568			0.204284	+	0.978912i	$0.434513\pi$
$810$		0			0
$811$		−	32.4565i		−	1.13970i	−0.821748	−	0.569851i	$-0.807001\pi$
$811$		−	32.4565i		−	1.13970i	0.821748	−	0.569851i	$-0.192999\pi$
$812$			13.3745i			0.469352i
$813$		0			0
$814$	1.62324	+	0.0855410i	0.0568944	+	0.00299821i
$815$		−	7.70797i		−	0.269998i
$816$		0			0
$817$	−6.64939			−0.232633
$818$			33.7432i			1.17980i
$819$		0			0
$820$			23.9969i			0.838009i
$821$	−17.9441			−0.626254			−0.313127	−	0.949711i	$-0.601377\pi$
$821$	−17.9441			−0.626254			−0.313127	+	0.949711i	$0.601377\pi$
$822$		0			0
$823$	−44.8724			−1.56416			−0.782078	−	0.623181i	$-0.785840\pi$
$823$	−44.8724			−1.56416			−0.782078	+	0.623181i	$0.785840\pi$
$824$	14.8793			0.518345
$825$		0			0
$826$	11.2972			0.393081
$827$	−7.92207			−0.275477			−0.137739	−	0.990469i	$-0.543983\pi$
$827$	−7.92207			−0.275477			−0.137739	+	0.990469i	$0.543983\pi$
$828$		0			0
$829$	35.8468			1.24501			0.622506	−	0.782615i	$-0.286115\pi$
$829$	35.8468			1.24501			0.622506	+	0.782615i	$0.286115\pi$
$830$		−	18.2132i		−	0.632188i
$831$		0			0
$832$		−	1.50882i		−	0.0523089i
$833$	6.57007			0.227639
$834$		0			0
$835$			8.78752i			0.304105i
$836$	−0.174537	+	3.31203i	−0.00603648	+	0.114549i
$837$		0			0
$838$			16.1750i			0.558756i
$839$		−	5.81287i		−	0.200683i	−0.994953	−	0.100341i	$-0.968007\pi$
$839$		−	5.81287i		−	0.200683i	0.994953	−	0.100341i	$-0.0319935\pi$
$840$		0			0
$841$	24.5798			0.847581
$842$	39.0009			1.34406
$843$		0			0
$844$			11.5426i			0.397314i
$845$		−	34.0725i		−	1.17213i
$846$		0			0
$847$	2.11245	−	19.9874i	0.0725847	−	0.686775i
$848$			1.80541i			0.0619981i
$849$		0			0
$850$	9.14358			0.313622
$851$			0.832466i			0.0285366i
$852$		0			0
$853$		−	36.5333i		−	1.25088i	−0.780274	−	0.625438i	$-0.784920\pi$
$853$		−	36.5333i		−	1.25088i	0.780274	−	0.625438i	$-0.215080\pi$
$854$	15.9859			0.547026
$855$		0			0
$856$	9.02249			0.308382
$857$	52.0819			1.77908			0.889542	−	0.456854i	$-0.151024\pi$
$857$	52.0819			1.77908			0.889542	+	0.456854i	$0.151024\pi$
$858$		0			0
$859$	−10.5108			−0.358624			−0.179312	−	0.983792i	$-0.557387\pi$
$859$	−10.5108			−0.358624			−0.179312	+	0.983792i	$0.557387\pi$
$860$	−21.1276			−0.720445
$861$		0			0
$862$	12.6119			0.429561
$863$			26.3954i			0.898510i	0.893404	+	0.449255i	$0.148311\pi$
$863$			26.3954i			0.898510i	−0.893404	+	0.449255i	$0.851689\pi$
$864$		0			0
$865$		−	58.2430i		−	1.98032i
$866$	−30.4096			−1.03336
$867$		0			0
$868$		−	11.7243i		−	0.397949i
$869$	2.00560	−	38.0584i	0.0680352	−	1.29104i
$870$		0			0
$871$			14.0735i			0.476862i
$872$			7.82316i			0.264926i
$873$		0			0
$874$	−1.69855			−0.0574544
$875$	0.555729			0.0187871
$876$		0			0
$877$			8.99322i			0.303680i	0.988405	+	0.151840i	$0.0485198\pi$
$877$			8.99322i			0.303680i	−0.988405	+	0.151840i	$0.951480\pi$
$878$			13.2488i			0.447126i
$879$		0			0
$880$	−0.554569	+	10.5236i	−0.0186945	+	0.354749i
$881$		−	49.6843i		−	1.67391i	−0.547275	−	0.836953i	$-0.684335\pi$
$881$		−	49.6843i		−	1.67391i	0.547275	−	0.836953i	$-0.315665\pi$
$882$		0			0
$883$	−14.1235			−0.475295			−0.237647	−	0.971352i	$-0.576376\pi$
$883$	−14.1235			−0.475295			−0.237647	+	0.971352i	$0.576376\pi$
$884$		−	2.70737i		−	0.0910587i
$885$		0			0
$886$			40.4929i			1.36039i
$887$	−40.1577			−1.34836			−0.674182	−	0.738565i	$-0.735504\pi$
$887$	−40.1577			−1.34836			−0.674182	+	0.738565i	$0.735504\pi$
$888$		0			0
$889$	1.62580			0.0545275
$890$	15.8522			0.531368
$891$		0			0
$892$	−9.33268			−0.312481
$893$	−13.6683			−0.457392
$894$		0			0
$895$	21.3697			0.714312
$896$			1.82716i			0.0610410i
$897$		0			0
$898$		−	17.6319i		−	0.588384i
$899$	−46.9691			−1.56651
$900$		0			0
$901$			3.23956i			0.107926i
$902$	−25.0139	−	1.31818i	−0.832871	−	0.0438905i
$903$		0			0
$904$		−	4.63215i		−	0.154063i
$905$		−	33.1034i		−	1.10040i
$906$		0			0
$907$	50.3788			1.67280			0.836401	−	0.548118i	$-0.184656\pi$
$907$	50.3788			1.67280			0.836401	+	0.548118i	$0.184656\pi$
$908$	−1.35999			−0.0451329
$909$		0			0
$910$		−	8.75954i		−	0.290376i
$911$			3.98937i			0.132174i	0.997814	+	0.0660869i	$0.0210515\pi$
$911$			3.98937i			0.132174i	−0.997814	+	0.0660869i	$0.978949\pi$
$912$		0			0
$913$	18.9850	+	1.00047i	0.628312	+	0.0331107i
$914$			10.6107i			0.350971i
$915$		0			0
$916$	18.1305			0.599047
$917$		−	4.29949i		−	0.141982i
$918$		0			0
$919$		−	8.58101i		−	0.283061i	−0.989934	−	0.141531i	$-0.954798\pi$
$919$		−	8.58101i		−	0.283061i	0.989934	−	0.141531i	$-0.0452024\pi$
$920$	−5.39694			−0.177932
$921$		0			0
$922$	7.67734			0.252840
$923$	5.52795			0.181955
$924$		0			0
$925$	−2.49743			−0.0821150
$926$	9.58193			0.314882
$927$		0			0
$928$	7.31983			0.240285
$929$		−	16.1305i		−	0.529225i	−0.964355	−	0.264612i	$-0.914756\pi$
$929$		−	16.1305i		−	0.529225i	0.964355	−	0.264612i	$-0.0852440\pi$
$930$		0			0
$931$		−	3.66150i		−	0.120001i
$932$	2.77760			0.0909834
$933$		0			0
$934$			23.7973i			0.778671i
$935$	−0.995099	+	18.8831i	−0.0325432	+	0.617544i
$936$		0			0
$937$		−	33.6080i		−	1.09793i	−0.835847	−	0.548963i	$-0.815023\pi$
$937$		−	33.6080i		−	1.09793i	0.835847	−	0.548963i	$-0.184977\pi$
$938$		−	17.0428i		−	0.556467i
$939$		0			0
$940$	−43.4293			−1.41651
$941$	45.2091			1.47377			0.736886	−	0.676017i	$-0.236295\pi$
$941$	45.2091			1.47377			0.736886	+	0.676017i	$0.236295\pi$
$942$		0			0
$943$		−	12.8282i		−	0.417744i
$944$		−	6.18295i		−	0.201238i
$945$		0			0
$946$	1.16056	−	22.0230i	0.0377331	−	0.716028i
$947$		−	9.53878i		−	0.309969i	−0.987917	−	0.154984i	$-0.950467\pi$
$947$		−	9.53878i		−	0.309969i	0.987917	−	0.154984i	$-0.0495327\pi$
$948$		0			0
$949$	20.1243			0.653262
$950$		−	5.09572i		−	0.165327i
$951$		0			0
$952$			3.27858i			0.106259i
$953$	33.4860			1.08472			0.542359	−	0.840147i	$-0.317531\pi$
$953$	33.4860			1.08472			0.542359	+	0.840147i	$0.317531\pi$
$954$		0			0
$955$	2.43902			0.0789247
$956$	−17.7215			−0.573153
$957$		0			0
$958$	−24.8164			−0.801782
$959$	36.9442			1.19299
$960$		0			0
$961$	10.1740			0.328193
$962$			0.739476i			0.0238417i
$963$		0			0
$964$			4.99032i			0.160728i
$965$	−30.0835			−0.968423
$966$		0			0
$967$		−	27.1687i		−	0.873686i	−0.899538	−	0.436843i	$-0.856097\pi$
$967$		−	27.1687i		−	0.873686i	0.899538	−	0.436843i	$-0.143903\pi$
$968$	−10.9391	−	1.15614i	−0.351595	−	0.0371598i
$969$		0			0
$970$		−	43.2540i		−	1.38880i
$971$			8.56106i			0.274738i	0.990520	+	0.137369i	$0.0438645\pi$
$971$			8.56106i			0.274738i	−0.990520	+	0.137369i	$0.956135\pi$
$972$		0			0
$973$	37.7419			1.20995
$974$	−20.6772			−0.662541
$975$		0			0
$976$		−	8.74905i		−	0.280050i
$977$		−	0.787147i		−	0.0251831i	−0.999921	−	0.0125915i	$-0.995992\pi$
$977$		−	0.787147i		−	0.0251831i	0.999921	−	0.0125915i	$-0.00400812\pi$
$978$		0			0
$979$	−0.870780	+	16.5240i	−0.0278303	+	0.528110i
$980$		−	11.6340i		−	0.371633i
$981$		0			0
$982$	27.4835			0.877035
$983$			56.2836i			1.79517i	0.440842	+	0.897585i	$0.354680\pi$
$983$			56.2836i			1.79517i	−0.440842	+	0.897585i	$0.645320\pi$
$984$		0			0
$985$			44.5546i			1.41963i
$986$	13.1344			0.418286
$987$		0			0
$988$	−1.50882			−0.0480019
$989$	11.2943			0.359139
$990$		0			0
$991$	−14.3654			−0.456332			−0.228166	−	0.973622i	$-0.573273\pi$
$991$	−14.3654			−0.456332			−0.228166	+	0.973622i	$0.573273\pi$
$992$	−6.41670			−0.203730
$993$		0			0
$994$	−6.69426			−0.212329
$995$		−	3.07574i		−	0.0975075i
$996$		0			0
$997$			9.15770i			0.290027i	0.989430	+	0.145014i	$0.0463226\pi$
$997$			9.15770i			0.290027i	−0.989430	+	0.145014i	$0.953677\pi$
$998$	23.4850			0.743405
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3762.2.b.a.989.4	✓	36
3.2	odd	2		3762.2.b.b.989.33	yes	36
11.10	odd	2		3762.2.b.b.989.4	yes	36
33.32	even	2	inner	3762.2.b.a.989.33	yes	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3762.2.b.a.989.4	✓	36	1.1	even	1	trivial
3762.2.b.a.989.33	yes	36	33.32	even	2	inner
3762.2.b.b.989.4	yes	36	11.10	odd	2
3762.2.b.b.989.33	yes	36	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -3.17738i q^{5} -1.82716i q^{7} -1.00000 q^{8} +3.17738i q^{10} +(-3.31203 - 0.174537i) q^{11} -1.50882i q^{13} +1.82716i q^{14} +1.00000 q^{16} +1.79436 q^{17} -1.00000i q^{19} -3.17738i q^{20} +(3.31203 + 0.174537i) q^{22} +1.69855i q^{23} -5.09572 q^{25} +1.50882i q^{26} -1.82716i q^{28} -7.31983 q^{29} +6.41670 q^{31} -1.00000 q^{32} -1.79436 q^{34} -5.80556 q^{35} +0.490103 q^{37} +1.00000i q^{38} +3.17738i q^{40} -7.55243 q^{41} -6.64939i q^{43} +(-3.31203 - 0.174537i) q^{44} -1.69855i q^{46} -13.6683i q^{47} +3.66150 q^{49} +5.09572 q^{50} -1.50882i q^{52} +1.80541i q^{53} +(-0.554569 + 10.5236i) q^{55} +1.82716i q^{56} +7.31983 q^{58} -6.18295i q^{59} -8.74905i q^{61} -6.41670 q^{62} +1.00000 q^{64} -4.79408 q^{65} -9.32749 q^{67} +1.79436 q^{68} +5.80556 q^{70} +3.66376i q^{71} +13.3378i q^{73} -0.490103 q^{74} -1.00000i q^{76} +(-0.318906 + 6.05159i) q^{77} +11.4910i q^{79} -3.17738i q^{80} +7.55243 q^{82} -5.73214 q^{83} -5.70137i q^{85} +6.64939i q^{86} +(3.31203 + 0.174537i) q^{88} -4.98909i q^{89} -2.75685 q^{91} +1.69855i q^{92} +13.6683i q^{94} -3.17738 q^{95} -13.6131 q^{97} -3.66150 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 36 q^{2} + 36 q^{4} - 36 q^{8} + 4 q^{11} + 36 q^{16} - 16 q^{17} - 4 q^{22} - 28 q^{25} - 8 q^{31} - 36 q^{32} + 16 q^{34} - 16 q^{35} + 16 q^{37} - 24 q^{41} + 4 q^{44} - 68 q^{49} + 28 q^{50}+ \cdots + 68 q^{98}+O(q^{100}) \) 36 * q - 36 * q^2 + 36 * q^4 - 36 * q^8 + 4 * q^11 + 36 * q^16 - 16 * q^17 - 4 * q^22 - 28 * q^25 - 8 * q^31 - 36 * q^32 + 16 * q^34 - 16 * q^35 + 16 * q^37 - 24 * q^41 + 4 * q^44 - 68 * q^49 + 28 * q^50 + 28 * q^55 + 8 * q^62 + 36 * q^64 - 32 * q^65 + 16 * q^67 - 16 * q^68 + 16 * q^70 - 16 * q^74 + 32 * q^77 + 24 * q^82 + 56 * q^83 - 4 * q^88 - 48 * q^91 + 8 * q^97 + 68 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more