LMFDB - Embedded newform 3344.2.o.a.1519.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3344,2,Mod(1519,3344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3344, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3344.1519");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3344 = 2^{4} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3344.o (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:	$26.7019744359$
Analytic rank:	$0$
Dimension:	$36$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1519.5
Character	$\chi$	$=$	3344.1519
Dual form			3344.2.o.a.1519.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.50928 q^{3} -1.52863 q^{5} +0.415002i q^{7} +3.29648 q^{9} +O(q^{10})$	$q-2.50928 q^{3} -1.52863 q^{5} +0.415002i q^{7} +3.29648 q^{9} -1.00000i q^{11} +4.53219i q^{13} +3.83577 q^{15} +0.133464 q^{17} +(4.33558 - 0.450279i) q^{19} -1.04136i q^{21} -3.38174i q^{23} -2.66328 q^{25} -0.743957 q^{27} +3.02960i q^{29} -4.01187 q^{31} +2.50928i q^{33} -0.634385i q^{35} +1.85333i q^{37} -11.3725i q^{39} -5.83030i q^{41} -7.93739i q^{43} -5.03911 q^{45} -12.1715i q^{47} +6.82777 q^{49} -0.334899 q^{51} +5.96262i q^{53} +1.52863i q^{55} +(-10.8792 + 1.12987i) q^{57} -7.95715 q^{59} -3.03402 q^{61} +1.36805i q^{63} -6.92806i q^{65} +12.6481 q^{67} +8.48574i q^{69} +5.95811 q^{71} +11.3806 q^{73} +6.68292 q^{75} +0.415002 q^{77} +0.959216 q^{79} -8.02265 q^{81} +11.0702i q^{83} -0.204018 q^{85} -7.60210i q^{87} +9.52486i q^{89} -1.88087 q^{91} +10.0669 q^{93} +(-6.62751 + 0.688311i) q^{95} -9.20133i q^{97} -3.29648i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q + 44 q^{9}+O(q^{10}) $ 36 * q + 44 * q^9	$ 36 q + 44 q^{9} - 16 q^{17} + 36 q^{25} - 32 q^{45} - 28 q^{49} + 24 q^{57} - 48 q^{61} - 24 q^{73} + 52 q^{81} + 24 q^{85} - 64 q^{93}+O(q^{100}) $ 36 * q + 44 * q^9 - 16 * q^17 + 36 * q^25 - 32 * q^45 - 28 * q^49 + 24 * q^57 - 48 * q^61 - 24 * q^73 + 52 * q^81 + 24 * q^85 - 64 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3344\mathbb{Z}\right)^\times$.

$n$	$705$	$837$	$2433$	$2927$
$\chi(n)$	$-1$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−2.50928			−1.44873			−0.724367	−	0.689415i	$-0.757868\pi$
$3$	−2.50928			−1.44873			−0.724367	+	0.689415i	$0.757868\pi$
$4$		0			0
$5$	−1.52863			−0.683625			−0.341813	−	0.939768i	$-0.611041\pi$
$5$	−1.52863			−0.683625			−0.341813	+	0.939768i	$0.611041\pi$
$6$		0			0
$7$			0.415002i			0.156856i	0.996920	+	0.0784280i	$0.0249901\pi$
$7$			0.415002i			0.156856i	−0.996920	+	0.0784280i	$0.975010\pi$
$8$		0			0
$9$	3.29648			1.09883
$10$		0			0
$11$		−	1.00000i		−	0.301511i
$11$		−	1.00000i		−	0.301511i
$12$		0			0
$13$			4.53219i			1.25700i	0.777808	+	0.628502i	$0.216332\pi$
$13$			4.53219i			1.25700i	−0.777808	+	0.628502i	$0.783668\pi$
$14$		0			0
$15$	3.83577			0.990391
$16$		0			0
$17$	0.133464			0.0323698			0.0161849	−	0.999869i	$-0.494848\pi$
$17$	0.133464			0.0323698			0.0161849	+	0.999869i	$0.494848\pi$
$18$		0			0
$19$	4.33558	−	0.450279i	0.994650	−	0.103301i
$19$	4.33558	−	0.450279i	0.994650	−	0.103301i
$20$		0			0
$21$		−	1.04136i		−	0.227242i
$22$		0			0
$23$		−	3.38174i		−	0.705142i	−0.935785	−	0.352571i	$-0.885307\pi$
$23$		−	3.38174i		−	0.705142i	0.935785	−	0.352571i	$-0.114693\pi$
$24$		0			0
$25$	−2.66328			−0.532656
$26$		0			0
$27$	−0.743957			−0.143175
$28$		0			0
$29$			3.02960i			0.562582i	0.959623	+	0.281291i	$0.0907626\pi$
$29$			3.02960i			0.562582i	−0.959623	+	0.281291i	$0.909237\pi$
$30$		0			0
$31$	−4.01187			−0.720554			−0.360277	−	0.932845i	$-0.617318\pi$
$31$	−4.01187			−0.720554			−0.360277	+	0.932845i	$0.617318\pi$
$32$		0			0
$33$			2.50928i			0.436809i
$34$		0			0
$35$		−	0.634385i		−	0.107231i
$36$		0			0
$37$			1.85333i			0.304685i	0.988328	+	0.152343i	$0.0486817\pi$
$37$			1.85333i			0.304685i	−0.988328	+	0.152343i	$0.951318\pi$
$38$		0			0
$39$		−	11.3725i		−	1.82106i
$40$		0			0
$41$		−	5.83030i		−	0.910540i	−0.890354	−	0.455270i	$-0.849543\pi$
$41$		−	5.83030i		−	0.910540i	0.890354	−	0.455270i	$-0.150457\pi$
$42$		0			0
$43$		−	7.93739i		−	1.21044i	−0.796058	−	0.605220i	$-0.793085\pi$
$43$		−	7.93739i		−	1.21044i	0.796058	−	0.605220i	$-0.206915\pi$
$44$		0			0
$45$	−5.03911			−0.751186
$46$		0			0
$47$		−	12.1715i		−	1.77540i	−0.460422	−	0.887700i	$-0.652302\pi$
$47$		−	12.1715i		−	1.77540i	0.460422	−	0.887700i	$-0.347698\pi$
$48$		0			0
$49$	6.82777			0.975396
$50$		0			0
$51$	−0.334899			−0.0468952
$52$		0			0
$53$			5.96262i			0.819028i	0.912304	+	0.409514i	$0.134302\pi$
$53$			5.96262i			0.819028i	−0.912304	+	0.409514i	$0.865698\pi$
$54$		0			0
$55$			1.52863i			0.206121i
$56$		0			0
$57$	−10.8792	+	1.12987i	−1.44098	+	0.149656i
$58$		0			0
$59$	−7.95715			−1.03593			−0.517966	−	0.855401i	$-0.673311\pi$
$59$	−7.95715			−1.03593			−0.517966	+	0.855401i	$0.673311\pi$
$60$		0			0
$61$	−3.03402			−0.388467			−0.194233	−	0.980955i	$-0.562222\pi$
$61$	−3.03402			−0.388467			−0.194233	+	0.980955i	$0.562222\pi$
$62$		0			0
$63$			1.36805i			0.172358i
$64$		0			0
$65$		−	6.92806i		−	0.859320i
$66$		0			0
$67$	12.6481			1.54522			0.772608	−	0.634884i	$-0.218952\pi$
$67$	12.6481			1.54522			0.772608	+	0.634884i	$0.218952\pi$
$68$		0			0
$69$			8.48574i			1.02156i
$70$		0			0
$71$	5.95811			0.707098			0.353549	−	0.935416i	$-0.384975\pi$
$71$	5.95811			0.707098			0.353549	+	0.935416i	$0.384975\pi$
$72$		0			0
$73$	11.3806			1.33200			0.665999	−	0.745953i	$-0.268006\pi$
$73$	11.3806			1.33200			0.665999	+	0.745953i	$0.268006\pi$
$74$		0			0
$75$	6.68292			0.771677
$76$		0			0
$77$	0.415002			0.0472938
$78$		0			0
$79$	0.959216			0.107920			0.0539601	−	0.998543i	$-0.482816\pi$
$79$	0.959216			0.107920			0.0539601	+	0.998543i	$0.482816\pi$
$80$		0			0
$81$	−8.02265			−0.891406
$82$		0			0
$83$			11.0702i			1.21511i	0.794277	+	0.607555i	$0.207850\pi$
$83$			11.0702i			1.21511i	−0.794277	+	0.607555i	$0.792150\pi$
$84$		0			0
$85$	−0.204018			−0.0221288
$86$		0			0
$87$		−	7.60210i		−	0.815031i
$88$		0			0
$89$			9.52486i			1.00963i	0.863227	+	0.504817i	$0.168440\pi$
$89$			9.52486i			1.00963i	−0.863227	+	0.504817i	$0.831560\pi$
$90$		0			0
$91$	−1.88087			−0.197169
$92$		0			0
$93$	10.0669			1.04389
$94$		0			0
$95$	−6.62751	+	0.688311i	−0.679968	+	0.0706192i
$96$		0			0
$97$		−	9.20133i		−	0.934254i	−0.884190	−	0.467127i	$-0.845289\pi$
$97$		−	9.20133i		−	0.934254i	0.884190	−	0.467127i	$-0.154711\pi$
$98$		0			0
$99$		−	3.29648i		−	0.331309i
$100$		0			0
$101$	−15.6889			−1.56110			−0.780552	−	0.625091i	$-0.785062\pi$
$101$	−15.6889			−1.56110			−0.780552	+	0.625091i	$0.785062\pi$
$102$		0			0
$103$	−15.8777			−1.56448			−0.782239	−	0.622979i	$-0.785922\pi$
$103$	−15.8777			−1.56448			−0.782239	+	0.622979i	$0.785922\pi$
$104$		0			0
$105$			1.59185i			0.155349i
$106$		0			0
$107$	−5.73531			−0.554453			−0.277227	−	0.960805i	$-0.589415\pi$
$107$	−5.73531			−0.554453			−0.277227	+	0.960805i	$0.589415\pi$
$108$		0			0
$109$			19.1730i			1.83644i	0.396069	+	0.918221i	$0.370374\pi$
$109$			19.1730i			1.83644i	−0.396069	+	0.918221i	$0.629626\pi$
$110$		0			0
$111$		−	4.65051i		−	0.441407i
$112$		0			0
$113$		−	7.09289i		−	0.667243i	−0.942707	−	0.333621i	$-0.891729\pi$
$113$		−	7.09289i		−	0.667243i	0.942707	−	0.333621i	$-0.108271\pi$
$114$		0			0
$115$			5.16944i			0.482053i
$116$		0			0
$117$			14.9403i			1.38123i
$118$		0			0
$119$			0.0553879i			0.00507740i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$			14.6298i			1.31913i
$124$		0			0
$125$	11.7143			1.04776
$126$		0			0
$127$	5.67698			0.503751			0.251875	−	0.967760i	$-0.418953\pi$
$127$	5.67698			0.503751			0.251875	+	0.967760i	$0.418953\pi$
$128$		0			0
$129$			19.9171i			1.75361i
$130$		0			0
$131$			8.10480i			0.708120i	0.935223	+	0.354060i	$0.115199\pi$
$131$			8.10480i			0.708120i	−0.935223	+	0.354060i	$0.884801\pi$
$132$		0			0
$133$	0.186866	+	1.79927i	0.0162034	+	0.156017i
$134$		0			0
$135$	1.13724			0.0978778
$136$		0			0
$137$	12.2712			1.04840			0.524200	−	0.851595i	$-0.324364\pi$
$137$	12.2712			1.04840			0.524200	+	0.851595i	$0.324364\pi$
$138$		0			0
$139$			15.4875i			1.31363i	0.754050	+	0.656817i	$0.228098\pi$
$139$			15.4875i			1.31363i	−0.754050	+	0.656817i	$0.771902\pi$
$140$		0			0
$141$			30.5418i			2.57208i
$142$		0			0
$143$	4.53219			0.379001
$144$		0			0
$145$		−	4.63114i		−	0.384595i
$146$		0			0
$147$	−17.1328			−1.41309
$148$		0			0
$149$	−14.2811			−1.16996			−0.584979	−	0.811049i	$-0.698897\pi$
$149$	−14.2811			−1.16996			−0.584979	+	0.811049i	$0.698897\pi$
$150$		0			0
$151$	3.72230			0.302916			0.151458	−	0.988464i	$-0.451603\pi$
$151$	3.72230			0.302916			0.151458	+	0.988464i	$0.451603\pi$
$152$		0			0
$153$	0.439962			0.0355688
$154$		0			0
$155$	6.13268			0.492589
$156$		0			0
$157$	7.37540			0.588621			0.294310	−	0.955710i	$-0.404910\pi$
$157$	7.37540			0.588621			0.294310	+	0.955710i	$0.404910\pi$
$158$		0			0
$159$		−	14.9619i		−	1.18655i
$160$		0			0
$161$	1.40343			0.110606
$162$		0			0
$163$		−	15.1527i		−	1.18685i	−0.804888	−	0.593426i	$-0.797775\pi$
$163$		−	15.1527i		−	1.18685i	0.804888	−	0.593426i	$-0.202225\pi$
$164$		0			0
$165$		−	3.83577i		−	0.298614i
$166$		0			0
$167$	6.67131			0.516241			0.258121	−	0.966113i	$-0.416897\pi$
$167$	6.67131			0.516241			0.258121	+	0.966113i	$0.416897\pi$
$168$		0			0
$169$	−7.54077			−0.580059
$170$		0			0
$171$	14.2922	−	1.48434i	1.09295	−	0.113510i
$172$		0			0
$173$			2.85272i			0.216888i	0.994103	+	0.108444i	$0.0345869\pi$
$173$			2.85272i			0.216888i	−0.994103	+	0.108444i	$0.965413\pi$
$174$		0			0
$175$		−	1.10527i		−	0.0835503i
$176$		0			0
$177$	19.9667			1.50079
$178$		0			0
$179$	−1.93159			−0.144374			−0.0721869	−	0.997391i	$-0.522998\pi$
$179$	−1.93159			−0.144374			−0.0721869	+	0.997391i	$0.522998\pi$
$180$		0			0
$181$			23.4911i			1.74608i	0.487646	+	0.873042i	$0.337856\pi$
$181$			23.4911i			1.74608i	−0.487646	+	0.873042i	$0.662144\pi$
$182$		0			0
$183$	7.61321			0.562784
$184$		0			0
$185$		−	2.83306i		−	0.208290i
$186$		0			0
$187$		−	0.133464i		−	0.00975986i
$188$		0			0
$189$		−	0.308744i		−	0.0224578i
$190$		0			0
$191$		−	4.48832i		−	0.324764i	−0.986728	−	0.162382i	$-0.948082\pi$
$191$		−	4.48832i		−	0.324764i	0.986728	−	0.162382i	$-0.0519176\pi$
$192$		0			0
$193$			27.6742i			1.99203i	0.0891825	+	0.996015i	$0.471575\pi$
$193$			27.6742i			1.99203i	−0.0891825	+	0.996015i	$0.528425\pi$
$194$		0			0
$195$			17.3844i			1.24492i
$196$		0			0
$197$	17.0101			1.21192			0.605959	−	0.795496i	$-0.292789\pi$
$197$	17.0101			1.21192			0.605959	+	0.795496i	$0.292789\pi$
$198$		0			0
$199$			23.8516i			1.69080i	0.534135	+	0.845399i	$0.320637\pi$
$199$			23.8516i			1.69080i	−0.534135	+	0.845399i	$0.679363\pi$
$200$		0			0
$201$	−31.7377			−2.23860
$202$		0			0
$203$	−1.25729			−0.0882443
$204$		0			0
$205$			8.91238i			0.622468i
$206$		0			0
$207$		−	11.1479i		−	0.774830i
$208$		0			0
$209$	−0.450279	−	4.33558i	−0.0311464	−	0.299898i
$210$		0			0
$211$	2.36986			0.163148			0.0815738	−	0.996667i	$-0.474005\pi$
$211$	2.36986			0.163148			0.0815738	+	0.996667i	$0.474005\pi$
$212$		0			0
$213$	−14.9506			−1.02440
$214$		0			0
$215$			12.1334i			0.827488i
$216$		0			0
$217$		−	1.66494i		−	0.113023i
$218$		0			0
$219$	−28.5571			−1.92971
$220$		0			0
$221$			0.604885i			0.0406890i
$222$		0			0
$223$	−18.3913			−1.23157			−0.615786	−	0.787914i	$-0.711161\pi$
$223$	−18.3913			−1.23157			−0.615786	+	0.787914i	$0.711161\pi$
$224$		0			0
$225$	−8.77946			−0.585297
$226$		0			0
$227$	−14.8563			−0.986050			−0.493025	−	0.870015i	$-0.664109\pi$
$227$	−14.8563			−0.986050			−0.493025	+	0.870015i	$0.664109\pi$
$228$		0			0
$229$	8.16192			0.539355			0.269678	−	0.962951i	$-0.413083\pi$
$229$	8.16192			0.539355			0.269678	+	0.962951i	$0.413083\pi$
$230$		0			0
$231$	−1.04136			−0.0685162
$232$		0			0
$233$	−18.0186			−1.18044			−0.590220	−	0.807242i	$-0.700959\pi$
$233$	−18.0186			−1.18044			−0.590220	+	0.807242i	$0.700959\pi$
$234$		0			0
$235$			18.6058i			1.21371i
$236$		0			0
$237$	−2.40694			−0.156348
$238$		0			0
$239$			5.14842i			0.333024i	0.986039	+	0.166512i	$0.0532504\pi$
$239$			5.14842i			0.333024i	−0.986039	+	0.166512i	$0.946750\pi$
$240$		0			0
$241$			24.7206i			1.59239i	0.605037	+	0.796197i	$0.293158\pi$
$241$			24.7206i			1.59239i	−0.605037	+	0.796197i	$0.706842\pi$
$242$		0			0
$243$	22.3629			1.43458
$244$		0			0
$245$	−10.4372			−0.666806
$246$		0			0
$247$	2.04075	+	19.6497i	0.129850	+	1.25028i
$248$		0			0
$249$		−	27.7782i		−	1.76037i
$250$		0			0
$251$			18.8129i			1.18746i	0.804665	+	0.593729i	$0.202345\pi$
$251$			18.8129i			1.18746i	−0.804665	+	0.593729i	$0.797655\pi$
$252$		0			0
$253$	−3.38174			−0.212608
$254$		0			0
$255$	0.511937			0.0320588
$256$		0			0
$257$			8.99648i			0.561185i	0.959827	+	0.280593i	$0.0905310\pi$
$257$			8.99648i			0.561185i	−0.959827	+	0.280593i	$0.909469\pi$
$258$		0			0
$259$	−0.769134			−0.0477917
$260$		0			0
$261$			9.98701i			0.618181i
$262$		0			0
$263$			3.49232i			0.215346i	0.994186	+	0.107673i	$0.0343399\pi$
$263$			3.49232i			0.215346i	−0.994186	+	0.107673i	$0.965660\pi$
$264$		0			0
$265$		−	9.11465i		−	0.559908i
$266$		0			0
$267$		−	23.9005i		−	1.46269i
$268$		0			0
$269$			1.24851i			0.0761228i	0.999275	+	0.0380614i	$0.0121183\pi$
$269$			1.24851i			0.0761228i	−0.999275	+	0.0380614i	$0.987882\pi$
$270$		0			0
$271$		−	4.56556i		−	0.277338i	−0.990339	−	0.138669i	$-0.955718\pi$
$271$		−	4.56556i		−	0.277338i	0.990339	−	0.138669i	$-0.0442824\pi$
$272$		0			0
$273$	4.71962			0.285645
$274$		0			0
$275$			2.66328i			0.160602i
$276$		0			0
$277$	−16.9947			−1.02111			−0.510557	−	0.859844i	$-0.670561\pi$
$277$	−16.9947			−1.02111			−0.510557	+	0.859844i	$0.670561\pi$
$278$		0			0
$279$	−13.2251			−0.791764
$280$		0			0
$281$		−	11.9487i		−	0.712800i	−0.934334	−	0.356400i	$-0.884004\pi$
$281$		−	11.9487i		−	0.712800i	0.934334	−	0.356400i	$-0.115996\pi$
$282$		0			0
$283$			17.3180i			1.02945i	0.857357	+	0.514723i	$0.172105\pi$
$283$			17.3180i			1.02945i	−0.857357	+	0.514723i	$0.827895\pi$
$284$		0			0
$285$	16.6303	−	1.72716i	0.985092	−	0.102308i
$286$		0			0
$287$	2.41958			0.142824
$288$		0			0
$289$	−16.9822			−0.998952
$290$		0			0
$291$			23.0887i			1.35348i
$292$		0			0
$293$		−	14.4740i		−	0.845578i	−0.906228	−	0.422789i	$-0.861051\pi$
$293$		−	14.4740i		−	0.845578i	0.906228	−	0.422789i	$-0.138949\pi$
$294$		0			0
$295$	12.1636			0.708190
$296$		0			0
$297$			0.743957i			0.0431688i
$298$		0			0
$299$	15.3267			0.886367
$300$		0			0
$301$	3.29403			0.189865
$302$		0			0
$303$	39.3678			2.26162
$304$		0			0
$305$	4.63790			0.265566
$306$		0			0
$307$	−3.26094			−0.186112			−0.0930558	−	0.995661i	$-0.529663\pi$
$307$	−3.26094			−0.186112			−0.0930558	+	0.995661i	$0.529663\pi$
$308$		0			0
$309$	39.8416			2.26651
$310$		0			0
$311$			23.0112i			1.30485i	0.757855	+	0.652423i	$0.226247\pi$
$311$			23.0112i			1.30485i	−0.757855	+	0.652423i	$0.773753\pi$
$312$		0			0
$313$	5.85527			0.330960			0.165480	−	0.986213i	$-0.447083\pi$
$313$	5.85527			0.330960			0.165480	+	0.986213i	$0.447083\pi$
$314$		0			0
$315$		−	2.09124i		−	0.117828i
$316$		0			0
$317$		−	20.9747i		−	1.17806i	−0.808112	−	0.589029i	$-0.799510\pi$
$317$		−	20.9747i		−	1.17806i	0.808112	−	0.589029i	$-0.200490\pi$
$318$		0			0
$319$	3.02960			0.169625
$320$		0			0
$321$	14.3915			0.803255
$322$		0			0
$323$	0.578644	−	0.0600960i	0.0321966	−	0.00334383i
$324$		0			0
$325$		−	12.0705i		−	0.669551i
$326$		0			0
$327$		−	48.1104i		−	2.66051i
$328$		0			0
$329$	5.05121			0.278482
$330$		0			0
$331$	0.370803			0.0203812			0.0101906	−	0.999948i	$-0.496756\pi$
$331$	0.370803			0.0203812			0.0101906	+	0.999948i	$0.496756\pi$
$332$		0			0
$333$			6.10946i			0.334796i
$334$		0			0
$335$	−19.3343			−1.05635
$336$		0			0
$337$			24.1742i			1.31685i	0.752646	+	0.658425i	$0.228777\pi$
$337$			24.1742i			1.31685i	−0.752646	+	0.658425i	$0.771223\pi$
$338$		0			0
$339$			17.7980i			0.966657i
$340$		0			0
$341$			4.01187i			0.217255i
$342$		0			0
$343$			5.73855i			0.309853i
$344$		0			0
$345$		−	12.9716i		−	0.698366i
$346$		0			0
$347$			7.61266i			0.408669i	0.978901	+	0.204335i	$0.0655031\pi$
$347$			7.61266i			0.408669i	−0.978901	+	0.204335i	$0.934497\pi$
$348$		0			0
$349$	−17.8541			−0.955708			−0.477854	−	0.878439i	$-0.658585\pi$
$349$	−17.8541			−0.955708			−0.477854	+	0.878439i	$0.658585\pi$
$350$		0			0
$351$		−	3.37176i		−	0.179971i
$352$		0			0
$353$	13.9928			0.744764			0.372382	−	0.928080i	$-0.378541\pi$
$353$	13.9928			0.744764			0.372382	+	0.928080i	$0.378541\pi$
$354$		0			0
$355$	−9.10776			−0.483390
$356$		0			0
$357$		−	0.138984i		−	0.00735579i
$358$		0			0
$359$			18.3628i			0.969150i	0.874750	+	0.484575i	$0.161026\pi$
$359$			18.3628i			0.969150i	−0.874750	+	0.484575i	$0.838974\pi$
$360$		0			0
$361$	18.5945	−	3.90444i	0.978658	−	0.205497i
$362$		0			0
$363$	2.50928			0.131703
$364$		0			0
$365$	−17.3967			−0.910587
$366$		0			0
$367$			14.3160i			0.747291i	0.927572	+	0.373645i	$0.121892\pi$
$367$			14.3160i			0.747291i	−0.927572	+	0.373645i	$0.878108\pi$
$368$		0			0
$369$		−	19.2195i		−	1.00053i
$370$		0			0
$371$	−2.47450			−0.128469
$372$		0			0
$373$		−	8.18308i		−	0.423704i	−0.977302	−	0.211852i	$-0.932051\pi$
$373$		−	8.18308i		−	0.423704i	0.977302	−	0.211852i	$-0.0679495\pi$
$374$		0			0
$375$	−29.3946			−1.51793
$376$		0			0
$377$	−13.7307			−0.707168
$378$		0			0
$379$	18.5333			0.951993			0.475996	−	0.879447i	$-0.342088\pi$
$379$	18.5333			0.951993			0.475996	+	0.879447i	$0.342088\pi$
$380$		0			0
$381$	−14.2451			−0.729801
$382$		0			0
$383$	−7.03621			−0.359534			−0.179767	−	0.983709i	$-0.557534\pi$
$383$	−7.03621			−0.359534			−0.179767	+	0.983709i	$0.557534\pi$
$384$		0			0
$385$	−0.634385			−0.0323313
$386$		0			0
$387$		−	26.1655i		−	1.33007i
$388$		0			0
$389$	24.2280			1.22841			0.614203	−	0.789148i	$-0.289477\pi$
$389$	24.2280			1.22841			0.614203	+	0.789148i	$0.289477\pi$
$390$		0			0
$391$		−	0.451342i		−	0.0228253i
$392$		0			0
$393$		−	20.3372i		−	1.02588i
$394$		0			0
$395$	−1.46629			−0.0737770
$396$		0			0
$397$	1.81975			0.0913305			0.0456653	−	0.998957i	$-0.485459\pi$
$397$	1.81975			0.0913305			0.0456653	+	0.998957i	$0.485459\pi$
$398$		0			0
$399$	−0.468900	−	4.51488i	−0.0234744	−	0.226027i
$400$		0			0
$401$		−	37.1855i		−	1.85695i	−0.371391	−	0.928477i	$-0.621119\pi$
$401$		−	37.1855i		−	1.85695i	0.371391	−	0.928477i	$-0.378881\pi$
$402$		0			0
$403$		−	18.1826i		−	0.905739i
$404$		0			0
$405$	12.2637			0.609388
$406$		0			0
$407$	1.85333			0.0918660
$408$		0			0
$409$			22.3614i			1.10570i	0.833281	+	0.552849i	$0.186459\pi$
$409$			22.3614i			1.10570i	−0.833281	+	0.552849i	$0.813541\pi$
$410$		0			0
$411$	−30.7919			−1.51885
$412$		0			0
$413$		−	3.30223i		−	0.162492i
$414$		0			0
$415$		−	16.9222i		−	0.830681i
$416$		0			0
$417$		−	38.8625i		−	1.90311i
$418$		0			0
$419$		−	12.9251i		−	0.631433i	−0.948854	−	0.315717i	$-0.897755\pi$
$419$		−	12.9251i		−	0.631433i	0.948854	−	0.315717i	$-0.102245\pi$
$420$		0			0
$421$			4.78793i			0.233349i	0.993170	+	0.116675i	$0.0372235\pi$
$421$			4.78793i			0.233349i	−0.993170	+	0.116675i	$0.962776\pi$
$422$		0			0
$423$		−	40.1232i		−	1.95086i
$424$		0			0
$425$	−0.355453			−0.0172420
$426$		0			0
$427$		−	1.25912i		−	0.0609333i
$428$		0			0
$429$	−11.3725			−0.549071
$430$		0			0
$431$	9.53476			0.459273			0.229637	−	0.973276i	$-0.426246\pi$
$431$	9.53476			0.459273			0.229637	+	0.973276i	$0.426246\pi$
$432$		0			0
$433$			2.79089i			0.134121i	0.997749	+	0.0670607i	$0.0213621\pi$
$433$			2.79089i			0.134121i	−0.997749	+	0.0670607i	$0.978638\pi$
$434$		0			0
$435$			11.6208i			0.557176i
$436$		0			0
$437$	−1.52273	−	14.6618i	−0.0728419	−	0.701370i
$438$		0			0
$439$	−2.02359			−0.0965805			−0.0482903	−	0.998833i	$-0.515377\pi$
$439$	−2.02359			−0.0965805			−0.0482903	+	0.998833i	$0.515377\pi$
$440$		0			0
$441$	22.5076			1.07179
$442$		0			0
$443$			32.1904i			1.52941i	0.644380	+	0.764705i	$0.277115\pi$
$443$			32.1904i			1.52941i	−0.644380	+	0.764705i	$0.722885\pi$
$444$		0			0
$445$		−	14.5600i		−	0.690211i
$446$		0			0
$447$	35.8354			1.69496
$448$		0			0
$449$		−	7.30244i		−	0.344624i	−0.985042	−	0.172312i	$-0.944876\pi$
$449$		−	7.30244i		−	0.344624i	0.985042	−	0.172312i	$-0.0551237\pi$
$450$		0			0
$451$	−5.83030			−0.274538
$452$		0			0
$453$	−9.34029			−0.438845
$454$		0			0
$455$	2.87516			0.134789
$456$		0			0
$457$	−4.04532			−0.189232			−0.0946161	−	0.995514i	$-0.530162\pi$
$457$	−4.04532			−0.189232			−0.0946161	+	0.995514i	$0.530162\pi$
$458$		0			0
$459$	−0.0992916			−0.00463453
$460$		0			0
$461$	23.8447			1.11056			0.555280	−	0.831664i	$-0.312611\pi$
$461$	23.8447			1.11056			0.555280	+	0.831664i	$0.312611\pi$
$462$		0			0
$463$		−	22.1537i		−	1.02957i	−0.857320	−	0.514784i	$-0.827872\pi$
$463$		−	22.1537i		−	1.02957i	0.857320	−	0.514784i	$-0.172128\pi$
$464$		0			0
$465$	−15.3886			−0.713630
$466$		0			0
$467$		−	27.9396i		−	1.29289i	−0.762960	−	0.646446i	$-0.776255\pi$
$467$		−	27.9396i		−	1.29289i	0.762960	−	0.646446i	$-0.223745\pi$
$468$		0			0
$469$			5.24900i			0.242376i
$470$		0			0
$471$	−18.5069			−0.852755
$472$		0			0
$473$	−7.93739			−0.364962
$474$		0			0
$475$	−11.5469	+	1.19922i	−0.529807	+	0.0550239i
$476$		0			0
$477$			19.6557i			0.899971i
$478$		0			0
$479$		−	22.9541i		−	1.04880i	−0.851472	−	0.524401i	$-0.824289\pi$
$479$		−	22.9541i		−	1.04880i	0.851472	−	0.524401i	$-0.175711\pi$
$480$		0			0
$481$	−8.39963			−0.382990
$482$		0			0
$483$	−3.52160			−0.160238
$484$		0			0
$485$			14.0655i			0.638680i
$486$		0			0
$487$	1.18991			0.0539202			0.0269601	−	0.999637i	$-0.491417\pi$
$487$	1.18991			0.0539202			0.0269601	+	0.999637i	$0.491417\pi$
$488$		0			0
$489$			38.0224i			1.71943i
$490$		0			0
$491$			6.54606i			0.295420i	0.989031	+	0.147710i	$0.0471902\pi$
$491$			6.54606i			0.295420i	−0.989031	+	0.147710i	$0.952810\pi$
$492$		0			0
$493$			0.404343i			0.0182107i
$494$		0			0
$495$			5.03911i			0.226491i
$496$		0			0
$497$			2.47263i			0.110912i
$498$		0			0
$499$		−	32.5231i		−	1.45593i	−0.685613	−	0.727966i	$-0.740466\pi$
$499$		−	32.5231i		−	1.45593i	0.685613	−	0.727966i	$-0.259534\pi$
$500$		0			0
$501$	−16.7402			−0.747896
$502$		0			0
$503$			40.2944i			1.79664i	0.439343	+	0.898319i	$0.355211\pi$
$503$			40.2944i			1.79664i	−0.439343	+	0.898319i	$0.644789\pi$
$504$		0			0
$505$	23.9826			1.06721
$506$		0			0
$507$	18.9219			0.840350
$508$		0			0
$509$			22.6070i			1.00204i	0.865437	+	0.501018i	$0.167041\pi$
$509$			22.6070i			1.00204i	−0.865437	+	0.501018i	$0.832959\pi$
$510$		0			0
$511$			4.72297i			0.208932i
$512$		0			0
$513$	−3.22549	+	0.334988i	−0.142409	+	0.0147901i
$514$		0			0
$515$	24.2712			1.06952
$516$		0			0
$517$	−12.1715			−0.535303
$518$		0			0
$519$		−	7.15827i		−	0.314213i
$520$		0			0
$521$		−	20.5795i		−	0.901605i	−0.892624	−	0.450803i	$-0.851138\pi$
$521$		−	20.5795i		−	0.901605i	0.892624	−	0.450803i	$-0.148862\pi$
$522$		0			0
$523$	1.64704			0.0720198			0.0360099	−	0.999351i	$-0.488535\pi$
$523$	1.64704			0.0720198			0.0360099	+	0.999351i	$0.488535\pi$
$524$		0			0
$525$			2.77342i			0.121042i
$526$		0			0
$527$	−0.535441			−0.0233242
$528$		0			0
$529$	11.5638			0.502774
$530$		0			0
$531$	−26.2306			−1.13831
$532$		0			0
$533$	26.4240			1.14455
$534$		0			0
$535$	8.76719			0.379038
$536$		0			0
$537$	4.84690			0.209159
$538$		0			0
$539$		−	6.82777i		−	0.294093i
$540$		0			0
$541$	−44.8608			−1.92872			−0.964359	−	0.264598i	$-0.914761\pi$
$541$	−44.8608			−1.92872			−0.964359	+	0.264598i	$0.914761\pi$
$542$		0			0
$543$		−	58.9459i		−	2.52961i
$544$		0			0
$545$		−	29.3085i		−	1.25544i
$546$		0			0
$547$	−36.1854			−1.54717			−0.773587	−	0.633690i	$-0.781540\pi$
$547$	−36.1854			−1.54717			−0.773587	+	0.633690i	$0.781540\pi$
$548$		0			0
$549$	−10.0016			−0.426858
$550$		0			0
$551$	1.36416	+	13.1351i	0.0581153	+	0.559572i
$552$		0			0
$553$			0.398076i			0.0169279i
$554$		0			0
$555$			7.10893i			0.301757i
$556$		0			0
$557$	3.19813			0.135509			0.0677546	−	0.997702i	$-0.478416\pi$
$557$	3.19813			0.135509			0.0677546	+	0.997702i	$0.478416\pi$
$558$		0			0
$559$	35.9738			1.52153
$560$		0			0
$561$			0.334899i			0.0141394i
$562$		0			0
$563$	20.9130			0.881379			0.440690	−	0.897660i	$-0.354734\pi$
$563$	20.9130			0.881379			0.440690	+	0.897660i	$0.354734\pi$
$564$		0			0
$565$			10.8424i			0.456144i
$566$		0			0
$567$		−	3.32941i		−	0.139822i
$568$		0			0
$569$		−	19.0229i		−	0.797482i	−0.917064	−	0.398741i	$-0.869447\pi$
$569$		−	19.0229i		−	0.797482i	0.917064	−	0.398741i	$-0.130553\pi$
$570$		0			0
$571$			14.5317i			0.608134i	0.952651	+	0.304067i	$0.0983447\pi$
$571$			14.5317i			0.608134i	−0.952651	+	0.304067i	$0.901655\pi$
$572$		0			0
$573$			11.2625i			0.470496i
$574$		0			0
$575$			9.00654i			0.375599i
$576$		0			0
$577$	−8.52413			−0.354864			−0.177432	−	0.984133i	$-0.556779\pi$
$577$	−8.52413			−0.354864			−0.177432	+	0.984133i	$0.556779\pi$
$578$		0			0
$579$		−	69.4422i		−	2.88592i
$580$		0			0
$581$	−4.59415			−0.190597
$582$		0			0
$583$	5.96262			0.246946
$584$		0			0
$585$		−	22.8382i		−	0.944244i
$586$		0			0
$587$		−	30.9751i		−	1.27848i	−0.769008	−	0.639239i	$-0.779249\pi$
$587$		−	30.9751i		−	1.27848i	0.769008	−	0.639239i	$-0.220751\pi$
$588$		0			0
$589$	−17.3938	+	1.80646i	−0.716699	+	0.0744339i
$590$		0			0
$591$	−42.6831			−1.75575
$592$		0			0
$593$	−15.5388			−0.638102			−0.319051	−	0.947737i	$-0.603364\pi$
$593$	−15.5388			−0.638102			−0.319051	+	0.947737i	$0.603364\pi$
$594$		0			0
$595$		−	0.0846677i		−	0.00347104i
$596$		0			0
$597$		−	59.8505i		−	2.44952i
$598$		0			0
$599$	−23.2889			−0.951560			−0.475780	−	0.879564i	$-0.657834\pi$
$599$	−23.2889			−0.951560			−0.475780	+	0.879564i	$0.657834\pi$
$600$		0			0
$601$			0.656076i			0.0267619i	0.999910	+	0.0133809i	$0.00425941\pi$
$601$			0.656076i			0.0267619i	−0.999910	+	0.0133809i	$0.995741\pi$
$602$		0			0
$603$	41.6943			1.69792
$604$		0			0
$605$	1.52863			0.0621478
$606$		0			0
$607$	−24.9837			−1.01406			−0.507030	−	0.861929i	$-0.669257\pi$
$607$	−24.9837			−1.01406			−0.507030	+	0.861929i	$0.669257\pi$
$608$		0			0
$609$	3.15489			0.127842
$610$		0			0
$611$	55.1637			2.23168
$612$		0			0
$613$	−45.0422			−1.81924			−0.909619	−	0.415444i	$-0.863626\pi$
$613$	−45.0422			−1.81924			−0.909619	+	0.415444i	$0.863626\pi$
$614$		0			0
$615$		−	22.3637i		−	0.901790i
$616$		0			0
$617$	34.0816			1.37207			0.686037	−	0.727567i	$-0.259349\pi$
$617$	34.0816			1.37207			0.686037	+	0.727567i	$0.259349\pi$
$618$		0			0
$619$		−	3.10524i		−	0.124810i	−0.998051	−	0.0624051i	$-0.980123\pi$
$619$		−	3.10524i		−	0.124810i	0.998051	−	0.0624051i	$-0.0198771\pi$
$620$		0			0
$621$			2.51587i			0.100958i
$622$		0			0
$623$	−3.95283			−0.158367
$624$		0			0
$625$	−4.59052			−0.183621
$626$		0			0
$627$	1.12987	+	10.8792i	0.0451229	+	0.434473i
$628$		0			0
$629$			0.247353i			0.00986260i
$630$		0			0
$631$			39.7357i			1.58185i	0.611912	+	0.790926i	$0.290401\pi$
$631$			39.7357i			1.58185i	−0.611912	+	0.790926i	$0.709599\pi$
$632$		0			0
$633$	−5.94663			−0.236357
$634$		0			0
$635$	−8.67803			−0.344377
$636$		0			0
$637$			30.9448i			1.22608i
$638$		0			0
$639$	19.6408			0.776978
$640$		0			0
$641$			12.8914i			0.509180i	0.967049	+	0.254590i	$0.0819405\pi$
$641$			12.8914i			0.509180i	−0.967049	+	0.254590i	$0.918060\pi$
$642$		0			0
$643$			7.01752i			0.276744i	0.990380	+	0.138372i	$0.0441870\pi$
$643$			7.01752i			0.276744i	−0.990380	+	0.138372i	$0.955813\pi$
$644$		0			0
$645$		−	30.4460i		−	1.19881i
$646$		0			0
$647$		−	0.511218i		−	0.0200980i	−0.999950	−	0.0100490i	$-0.996801\pi$
$647$		−	0.511218i		−	0.0200980i	0.999950	−	0.0100490i	$-0.00319876\pi$
$648$		0			0
$649$			7.95715i			0.312345i
$650$		0			0
$651$			4.17779i			0.163740i
$652$		0			0
$653$	13.3046			0.520648			0.260324	−	0.965521i	$-0.416171\pi$
$653$	13.3046			0.520648			0.260324	+	0.965521i	$0.416171\pi$
$654$		0			0
$655$		−	12.3893i		−	0.484089i
$656$		0			0
$657$	37.5159			1.46364
$658$		0			0
$659$	−23.8770			−0.930116			−0.465058	−	0.885280i	$-0.653967\pi$
$659$	−23.8770			−0.930116			−0.465058	+	0.885280i	$0.653967\pi$
$660$		0			0
$661$			24.2142i			0.941823i	0.882180	+	0.470912i	$0.156075\pi$
$661$			24.2142i			0.941823i	−0.882180	+	0.470912i	$0.843925\pi$
$662$		0			0
$663$		−	1.51783i		−	0.0589475i
$664$		0			0
$665$	−0.285650	−	2.75043i	−0.0110770	−	0.106657i
$666$		0			0
$667$	10.2453			0.396700
$668$		0			0
$669$	46.1489			1.78422
$670$		0			0
$671$			3.03402i			0.117127i
$672$		0			0
$673$		−	26.3324i		−	1.01504i	−0.861640	−	0.507520i	$-0.830562\pi$
$673$		−	26.3324i		−	1.01504i	0.861640	−	0.507520i	$-0.169438\pi$
$674$		0			0
$675$	1.98137			0.0762629
$676$		0			0
$677$			26.4383i			1.01611i	0.861326	+	0.508053i	$0.169635\pi$
$677$			26.4383i			1.01611i	−0.861326	+	0.508053i	$0.830365\pi$
$678$		0			0
$679$	3.81857			0.146543
$680$		0			0
$681$	37.2787			1.42852
$682$		0			0
$683$	−31.1385			−1.19148			−0.595742	−	0.803176i	$-0.703142\pi$
$683$	−31.1385			−1.19148			−0.595742	+	0.803176i	$0.703142\pi$
$684$		0			0
$685$	−18.7582			−0.716713
$686$		0			0
$687$	−20.4805			−0.781382
$688$		0			0
$689$	−27.0237			−1.02952
$690$		0			0
$691$			16.3375i			0.621509i	0.950490	+	0.310755i	$0.100582\pi$
$691$			16.3375i			0.621509i	−0.950490	+	0.310755i	$0.899418\pi$
$692$		0			0
$693$	1.36805			0.0519678
$694$		0			0
$695$		−	23.6747i		−	0.898034i
$696$		0			0
$697$		−	0.778136i		−	0.0294740i
$698$		0			0
$699$	45.2138			1.71014
$700$		0			0
$701$	−14.4417			−0.545454			−0.272727	−	0.962091i	$-0.587926\pi$
$701$	−14.4417			−0.545454			−0.272727	+	0.962091i	$0.587926\pi$
$702$		0			0
$703$	0.834513	+	8.03525i	0.0314743	+	0.303055i
$704$		0			0
$705$		−	46.6871i		−	1.75834i
$706$		0			0
$707$		−	6.51092i		−	0.244868i
$708$		0			0
$709$	18.8931			0.709545			0.354773	−	0.934953i	$-0.384558\pi$
$709$	18.8931			0.709545			0.354773	+	0.934953i	$0.384558\pi$
$710$		0			0
$711$	3.16204			0.118586
$712$		0			0
$713$			13.5671i			0.508093i
$714$		0			0
$715$	−6.92806			−0.259095
$716$		0			0
$717$		−	12.9188i		−	0.482463i
$718$		0			0
$719$			20.0206i			0.746644i	0.927702	+	0.373322i	$0.121781\pi$
$719$			20.0206i			0.746644i	−0.927702	+	0.373322i	$0.878219\pi$
$720$		0			0
$721$		−	6.58928i		−	0.245398i
$722$		0			0
$723$		−	62.0309i		−	2.30695i
$724$		0			0
$725$		−	8.06867i		−	0.299663i
$726$		0			0
$727$			31.6848i			1.17513i	0.809178	+	0.587563i	$0.199913\pi$
$727$			31.6848i			1.17513i	−0.809178	+	0.587563i	$0.800087\pi$
$728$		0			0
$729$	−32.0469			−1.18692
$730$		0			0
$731$		−	1.05936i		−	0.0391817i
$732$		0			0
$733$	30.5971			1.13013			0.565065	−	0.825047i	$-0.308851\pi$
$733$	30.5971			1.13013			0.565065	+	0.825047i	$0.308851\pi$
$734$		0			0
$735$	26.1897			0.966023
$736$		0			0
$737$		−	12.6481i		−	0.465900i
$738$		0			0
$739$		−	14.1820i		−	0.521695i	−0.965380	−	0.260847i	$-0.915998\pi$
$739$		−	14.1820i		−	0.521695i	0.965380	−	0.260847i	$-0.0840020\pi$
$740$		0			0
$741$	−5.12081	−	49.3065i	−0.188118	−	1.81132i
$742$		0			0
$743$	−12.3618			−0.453511			−0.226756	−	0.973952i	$-0.572812\pi$
$743$	−12.3618			−0.453511			−0.226756	+	0.973952i	$0.572812\pi$
$744$		0			0
$745$	21.8306			0.799812
$746$		0			0
$747$			36.4927i			1.33520i
$748$		0			0
$749$		−	2.38016i		−	0.0869693i
$750$		0			0
$751$	−7.04990			−0.257255			−0.128627	−	0.991693i	$-0.541057\pi$
$751$	−7.04990			−0.257255			−0.128627	+	0.991693i	$0.541057\pi$
$752$		0			0
$753$		−	47.2067i		−	1.72031i
$754$		0			0
$755$	−5.69003			−0.207081
$756$		0			0
$757$	36.0995			1.31206			0.656030	−	0.754735i	$-0.272235\pi$
$757$	36.0995			1.31206			0.656030	+	0.754735i	$0.272235\pi$
$758$		0			0
$759$	8.48574			0.308013
$760$		0			0
$761$	−40.0883			−1.45320			−0.726600	−	0.687061i	$-0.758901\pi$
$761$	−40.0883			−1.45320			−0.726600	+	0.687061i	$0.758901\pi$
$762$		0			0
$763$	−7.95683			−0.288057
$764$		0			0
$765$	−0.672541			−0.0243158
$766$		0			0
$767$		−	36.0633i		−	1.30217i
$768$		0			0
$769$	−10.7670			−0.388267			−0.194134	−	0.980975i	$-0.562190\pi$
$769$	−10.7670			−0.388267			−0.194134	+	0.980975i	$0.562190\pi$
$770$		0			0
$771$		−	22.5747i		−	0.813007i
$772$		0			0
$773$		−	32.7939i		−	1.17952i	−0.807580	−	0.589758i	$-0.799223\pi$
$773$		−	32.7939i		−	1.17952i	0.807580	−	0.589758i	$-0.200777\pi$
$774$		0			0
$775$	10.6848			0.383808
$776$		0			0
$777$	1.92997			0.0692374
$778$		0			0
$779$	−2.62526	−	25.2777i	−0.0940596	−	0.905668i
$780$		0			0
$781$		−	5.95811i		−	0.213198i
$782$		0			0
$783$		−	2.25389i		−	0.0805475i
$784$		0			0
$785$	−11.2743			−0.402396
$786$		0			0
$787$	30.4608			1.08581			0.542905	−	0.839794i	$-0.317324\pi$
$787$	30.4608			1.08581			0.542905	+	0.839794i	$0.317324\pi$
$788$		0			0
$789$		−	8.76322i		−	0.311979i
$790$		0			0
$791$	2.94356			0.104661
$792$		0			0
$793$		−	13.7508i		−	0.488304i
$794$		0			0
$795$			22.8712i			0.811158i
$796$		0			0
$797$			11.8517i			0.419808i	0.977722	+	0.209904i	$0.0673151\pi$
$797$			11.8517i			0.419808i	−0.977722	+	0.209904i	$0.932685\pi$
$798$		0			0
$799$		−	1.62446i		−	0.0574694i
$800$		0			0
$801$			31.3985i			1.10941i
$802$		0			0
$803$		−	11.3806i		−	0.401612i
$804$		0			0
$805$	−2.14533			−0.0756129
$806$		0			0
$807$		−	3.13285i		−	0.110282i
$808$		0			0
$809$	17.9936			0.632622			0.316311	−	0.948655i	$-0.397556\pi$
$809$	17.9936			0.632622			0.316311	+	0.948655i	$0.397556\pi$
$810$		0			0
$811$	−46.0359			−1.61654			−0.808269	−	0.588813i	$-0.799596\pi$
$811$	−46.0359			−1.61654			−0.808269	+	0.588813i	$0.799596\pi$
$812$		0			0
$813$			11.4563i			0.401789i
$814$		0			0
$815$			23.1629i			0.811363i
$816$		0			0
$817$	−3.57404	−	34.4132i	−0.125040	−	1.20396i
$818$		0			0
$819$	−6.20025			−0.216654
$820$		0			0
$821$	40.5384			1.41480			0.707401	−	0.706813i	$-0.249868\pi$
$821$	40.5384			1.41480			0.707401	+	0.706813i	$0.249868\pi$
$822$		0			0
$823$			35.5703i			1.23990i	0.784641	+	0.619951i	$0.212847\pi$
$823$			35.5703i			1.23990i	−0.784641	+	0.619951i	$0.787153\pi$
$824$		0			0
$825$		−	6.68292i		−	0.232669i
$826$		0			0
$827$	−35.8547			−1.24679			−0.623396	−	0.781907i	$-0.714247\pi$
$827$	−35.8547			−1.24679			−0.623396	+	0.781907i	$0.714247\pi$
$828$		0			0
$829$			13.8626i			0.481466i	0.970591	+	0.240733i	$0.0773879\pi$
$829$			13.8626i			0.481466i	−0.970591	+	0.240733i	$0.922612\pi$
$830$		0			0
$831$	42.6445			1.47932
$832$		0			0
$833$	0.911263			0.0315734
$834$		0			0
$835$	−10.1980			−0.352916
$836$		0			0
$837$	2.98466			0.103165
$838$		0			0
$839$	−21.0064			−0.725221			−0.362610	−	0.931941i	$-0.618114\pi$
$839$	−21.0064			−0.725221			−0.362610	+	0.931941i	$0.618114\pi$
$840$		0			0
$841$	19.8215			0.683501
$842$		0			0
$843$			29.9826i			1.03266i
$844$		0			0
$845$	11.5271			0.396543
$846$		0			0
$847$		−	0.415002i		−	0.0142596i
$848$		0			0
$849$		−	43.4556i		−	1.49139i
$850$		0			0
$851$	6.26748			0.214846
$852$		0			0
$853$	14.8177			0.507347			0.253673	−	0.967290i	$-0.418361\pi$
$853$	14.8177			0.507347			0.253673	+	0.967290i	$0.418361\pi$
$854$		0			0
$855$	−21.8475	+	2.26900i	−0.747168	+	0.0775983i
$856$		0			0
$857$		−	34.3717i		−	1.17412i	−0.809545	−	0.587058i	$-0.800286\pi$
$857$		−	34.3717i		−	1.17412i	0.809545	−	0.587058i	$-0.199714\pi$
$858$		0			0
$859$		−	40.1368i		−	1.36945i	−0.728801	−	0.684725i	$-0.759922\pi$
$859$		−	40.1368i		−	1.36945i	0.728801	−	0.684725i	$-0.240078\pi$
$860$		0			0
$861$	−6.07141			−0.206913
$862$		0			0
$863$	−20.1351			−0.685408			−0.342704	−	0.939443i	$-0.611343\pi$
$863$	−20.1351			−0.685408			−0.342704	+	0.939443i	$0.611343\pi$
$864$		0			0
$865$		−	4.36076i		−	0.148270i
$866$		0			0
$867$	42.6131			1.44722
$868$		0			0
$869$		−	0.959216i		−	0.0325392i
$870$		0			0
$871$			57.3238i			1.94234i
$872$		0			0
$873$		−	30.3320i		−	1.02658i
$874$		0			0
$875$			4.86147i			0.164348i
$876$		0			0
$877$			57.4953i			1.94148i	0.240133	+	0.970740i	$0.422809\pi$
$877$			57.4953i			1.94148i	−0.240133	+	0.970740i	$0.577191\pi$
$878$		0			0
$879$			36.3192i			1.22502i
$880$		0			0
$881$	37.3916			1.25976			0.629878	−	0.776694i	$-0.283105\pi$
$881$	37.3916			1.25976			0.629878	+	0.776694i	$0.283105\pi$
$882$		0			0
$883$		−	31.8089i		−	1.07046i	−0.844708	−	0.535228i	$-0.820226\pi$
$883$		−	31.8089i		−	1.07046i	0.844708	−	0.535228i	$-0.179774\pi$
$884$		0			0
$885$	−30.5218			−1.02598
$886$		0			0
$887$	47.0252			1.57895			0.789475	−	0.613782i	$-0.210353\pi$
$887$	47.0252			1.57895			0.789475	+	0.613782i	$0.210353\pi$
$888$		0			0
$889$			2.35596i			0.0790163i
$890$		0			0
$891$			8.02265i			0.268769i
$892$		0			0
$893$	−5.48058	−	52.7706i	−0.183401	−	1.76590i
$894$		0			0
$895$	2.95269			0.0986977
$896$		0			0
$897$	−38.4590			−1.28411
$898$		0			0
$899$		−	12.1544i		−	0.405371i
$900$		0			0
$901$			0.795795i			0.0265118i
$902$		0			0
$903$	−8.26564			−0.275063
$904$		0			0
$905$		−	35.9093i		−	1.19367i
$906$		0			0
$907$	13.2406			0.439648			0.219824	−	0.975540i	$-0.429452\pi$
$907$	13.2406			0.439648			0.219824	+	0.975540i	$0.429452\pi$
$908$		0			0
$909$	−51.7182			−1.71538
$910$		0			0
$911$	22.1712			0.734565			0.367283	−	0.930109i	$-0.380288\pi$
$911$	22.1712			0.734565			0.367283	+	0.930109i	$0.380288\pi$
$912$		0			0
$913$	11.0702			0.366370
$914$		0			0
$915$	−11.6378			−0.384734
$916$		0			0
$917$	−3.36351			−0.111073
$918$		0			0
$919$			17.5957i			0.580430i	0.956961	+	0.290215i	$0.0937268\pi$
$919$			17.5957i			0.580430i	−0.956961	+	0.290215i	$0.906273\pi$
$920$		0			0
$921$	8.18261			0.269626
$922$		0			0
$923$			27.0033i			0.888824i
$924$		0			0
$925$		−	4.93593i		−	0.162292i
$926$		0			0
$927$	−52.3406			−1.71909
$928$		0			0
$929$	25.3436			0.831495			0.415748	−	0.909480i	$-0.363520\pi$
$929$	25.3436			0.831495			0.415748	+	0.909480i	$0.363520\pi$
$930$		0			0
$931$	29.6024	−	3.07440i	0.970178	−	0.100759i
$932$		0			0
$933$		−	57.7415i		−	1.89037i
$934$		0			0
$935$			0.204018i			0.00667209i
$936$		0			0
$937$	−23.4346			−0.765576			−0.382788	−	0.923836i	$-0.625036\pi$
$937$	−23.4346			−0.765576			−0.382788	+	0.923836i	$0.625036\pi$
$938$		0			0
$939$	−14.6925			−0.479472
$940$		0			0
$941$			51.2071i			1.66930i	0.550779	+	0.834651i	$0.314331\pi$
$941$			51.2071i			1.66930i	−0.550779	+	0.834651i	$0.685669\pi$
$942$		0			0
$943$	−19.7166			−0.642060
$944$		0			0
$945$			0.471956i			0.0153527i
$946$		0			0
$947$			19.1234i			0.621428i	0.950504	+	0.310714i	$0.100568\pi$
$947$			19.1234i			0.621428i	−0.950504	+	0.310714i	$0.899432\pi$
$948$		0			0
$949$			51.5790i			1.67433i
$950$		0			0
$951$			52.6315i			1.70669i
$952$		0			0
$953$		−	1.09795i		−	0.0355660i	−0.999842	−	0.0177830i	$-0.994339\pi$
$953$		−	1.09795i		−	0.0355660i	0.999842	−	0.0177830i	$-0.00566080\pi$
$954$		0			0
$955$			6.86100i			0.222017i
$956$		0			0
$957$	−7.60210			−0.245741
$958$		0			0
$959$			5.09257i			0.164448i
$960$		0			0
$961$	−14.9049			−0.480802
$962$		0			0
$963$	−18.9064			−0.609249
$964$		0			0
$965$		−	42.3037i		−	1.36180i
$966$		0			0
$967$		−	6.53586i		−	0.210179i	−0.994463	−	0.105090i	$-0.966487\pi$
$967$		−	6.53586i		−	0.210179i	0.994463	−	0.105090i	$-0.0335129\pi$
$968$		0			0
$969$	−1.45198	+	0.150798i	−0.0466443	+	0.00484432i
$970$		0			0
$971$	−26.3985			−0.847168			−0.423584	−	0.905857i	$-0.639228\pi$
$971$	−26.3985			−0.847168			−0.423584	+	0.905857i	$0.639228\pi$
$972$		0			0
$973$	−6.42735			−0.206051
$974$		0			0
$975$			30.2883i			0.970001i
$976$		0			0
$977$		−	44.7159i		−	1.43059i	−0.698824	−	0.715293i	$-0.746293\pi$
$977$		−	44.7159i		−	1.43059i	0.698824	−	0.715293i	$-0.253707\pi$
$978$		0			0
$979$	9.52486			0.304416
$980$		0			0
$981$			63.2035i			2.01793i
$982$		0			0
$983$	14.5385			0.463707			0.231853	−	0.972751i	$-0.425521\pi$
$983$	14.5385			0.463707			0.231853	+	0.972751i	$0.425521\pi$
$984$		0			0
$985$	−26.0022			−0.828498
$986$		0			0
$987$	−12.6749			−0.403446
$988$		0			0
$989$	−26.8422			−0.853533
$990$		0			0
$991$	−29.4694			−0.936125			−0.468063	−	0.883695i	$-0.655048\pi$
$991$	−29.4694			−0.936125			−0.468063	+	0.883695i	$0.655048\pi$
$992$		0			0
$993$	−0.930449			−0.0295269
$994$		0			0
$995$		−	36.4604i		−	1.15587i
$996$		0			0
$997$	16.5810			0.525126			0.262563	−	0.964915i	$-0.415432\pi$
$997$	16.5810			0.525126			0.262563	+	0.964915i	$0.415432\pi$
$998$		0			0
$999$		−	1.37880i		−	0.0436232i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.o.a.1519.5	✓	36
4.3	odd	2	inner	3344.2.o.a.1519.31	yes	36
19.18	odd	2	inner	3344.2.o.a.1519.32	yes	36
76.75	even	2	inner	3344.2.o.a.1519.6	yes	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3344.2.o.a.1519.5	✓	36	1.1	even	1	trivial
3344.2.o.a.1519.6	yes	36	76.75	even	2	inner
3344.2.o.a.1519.31	yes	36	4.3	odd	2	inner
3344.2.o.a.1519.32	yes	36	19.18	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:	\( N \)	\(=\)	\( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3344.o (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.50928 q^{3} -1.52863 q^{5} +0.415002i q^{7} +3.29648 q^{9} +O(q^{10})\)	\(q-2.50928 q^{3} -1.52863 q^{5} +0.415002i q^{7} +3.29648 q^{9} -1.00000i q^{11} +4.53219i q^{13} +3.83577 q^{15} +0.133464 q^{17} +(4.33558 - 0.450279i) q^{19} -1.04136i q^{21} -3.38174i q^{23} -2.66328 q^{25} -0.743957 q^{27} +3.02960i q^{29} -4.01187 q^{31} +2.50928i q^{33} -0.634385i q^{35} +1.85333i q^{37} -11.3725i q^{39} -5.83030i q^{41} -7.93739i q^{43} -5.03911 q^{45} -12.1715i q^{47} +6.82777 q^{49} -0.334899 q^{51} +5.96262i q^{53} +1.52863i q^{55} +(-10.8792 + 1.12987i) q^{57} -7.95715 q^{59} -3.03402 q^{61} +1.36805i q^{63} -6.92806i q^{65} +12.6481 q^{67} +8.48574i q^{69} +5.95811 q^{71} +11.3806 q^{73} +6.68292 q^{75} +0.415002 q^{77} +0.959216 q^{79} -8.02265 q^{81} +11.0702i q^{83} -0.204018 q^{85} -7.60210i q^{87} +9.52486i q^{89} -1.88087 q^{91} +10.0669 q^{93} +(-6.62751 + 0.688311i) q^{95} -9.20133i q^{97} -3.29648i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 44 q^{9}+O(q^{10}) \) 36 * q + 44 * q^9	\( 36 q + 44 q^{9} - 16 q^{17} + 36 q^{25} - 32 q^{45} - 28 q^{49} + 24 q^{57} - 48 q^{61} - 24 q^{73} + 52 q^{81} + 24 q^{85} - 64 q^{93}+O(q^{100}) \) 36 * q + 44 * q^9 - 16 * q^17 + 36 * q^25 - 32 * q^45 - 28 * q^49 + 24 * q^57 - 48 * q^61 - 24 * q^73 + 52 * q^81 + 24 * q^85 - 64 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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