LMFDB - Embedded newform 3344.2.o.b.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:	$64$
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.b.1519.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.58608 q^{3} +3.42833 q^{5} -4.59904i q^{7} +3.68782 q^{9} +O(q^{10})$	$q-2.58608 q^{3} +3.42833 q^{5} -4.59904i q^{7} +3.68782 q^{9} +1.00000i q^{11} -5.60992i q^{13} -8.86593 q^{15} +6.86607 q^{17} +(-3.01679 + 3.14626i) q^{19} +11.8935i q^{21} -4.41922i q^{23} +6.75343 q^{25} -1.77875 q^{27} -8.50430i q^{29} +9.23088 q^{31} -2.58608i q^{33} -15.7670i q^{35} +1.06747i q^{37} +14.5077i q^{39} -2.23290i q^{41} -5.51453i q^{43} +12.6430 q^{45} +7.03024i q^{47} -14.1511 q^{49} -17.7562 q^{51} +8.19169i q^{53} +3.42833i q^{55} +(7.80167 - 8.13649i) q^{57} +5.18092 q^{59} -10.8947 q^{61} -16.9604i q^{63} -19.2326i q^{65} +8.76156 q^{67} +11.4285i q^{69} -7.34801 q^{71} +9.44436 q^{73} -17.4649 q^{75} +4.59904 q^{77} +10.3209 q^{79} -6.46347 q^{81} -2.87340i q^{83} +23.5391 q^{85} +21.9928i q^{87} +10.3594i q^{89} -25.8002 q^{91} -23.8718 q^{93} +(-10.3426 + 10.7864i) q^{95} +5.06525i q^{97} +3.68782i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 64 q + 56 q^{9}+O(q^{10}) $ 64 * q + 56 * q^9	$ 64 q + 56 q^{9} + 16 q^{17} + 64 q^{25} + 32 q^{45} - 88 q^{49} + 32 q^{57} + 64 q^{61} + 40 q^{73} - 48 q^{81} - 24 q^{85} + 80 q^{93}+O(q^{100}) $ 64 * q + 56 * q^9 + 16 * q^17 + 64 * q^25 + 32 * q^45 - 88 * q^49 + 32 * q^57 + 64 * q^61 + 40 * q^73 - 48 * q^81 - 24 * q^85 + 80 * 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
$3$	−2.58608			−1.49307			−0.746537	−	0.665344i	$-0.768285\pi$
$3$	−2.58608			−1.49307			−0.746537	+	0.665344i	$0.768285\pi$
$4$		0			0
$5$	3.42833			1.53319			0.766597	−	0.642128i	$-0.221948\pi$
$5$	3.42833			1.53319			0.766597	+	0.642128i	$0.221948\pi$
$6$		0			0
$7$		−	4.59904i		−	1.73827i	−0.494573	−	0.869136i	$-0.664676\pi$
$7$		−	4.59904i		−	1.73827i	0.494573	−	0.869136i	$-0.335324\pi$
$8$		0			0
$9$	3.68782			1.22927
$10$		0			0
$11$			1.00000i			0.301511i
$11$			1.00000i			0.301511i
$12$		0			0
$13$		−	5.60992i		−	1.55591i	−0.628319	−	0.777956i	$-0.716257\pi$
$13$		−	5.60992i		−	1.55591i	0.628319	−	0.777956i	$-0.283743\pi$
$14$		0			0
$15$	−8.86593			−2.28917
$16$		0			0
$17$	6.86607			1.66527			0.832633	−	0.553825i	$-0.186832\pi$
$17$	6.86607			1.66527			0.832633	+	0.553825i	$0.186832\pi$
$18$		0			0
$19$	−3.01679	+	3.14626i	−0.692100	+	0.721802i
$19$	−3.01679	+	3.14626i	−0.692100	+	0.721802i
$20$		0			0
$21$			11.8935i			2.59537i
$22$		0			0
$23$		−	4.41922i		−	0.921472i	−0.887537	−	0.460736i	$-0.847586\pi$
$23$		−	4.41922i		−	0.921472i	0.887537	−	0.460736i	$-0.152414\pi$
$24$		0			0
$25$	6.75343			1.35069
$26$		0			0
$27$	−1.77875			−0.342320
$28$		0			0
$29$		−	8.50430i		−	1.57921i	−0.613617	−	0.789604i	$-0.710286\pi$
$29$		−	8.50430i		−	1.57921i	0.613617	−	0.789604i	$-0.289714\pi$
$30$		0			0
$31$	9.23088			1.65791			0.828957	−	0.559312i	$-0.188935\pi$
$31$	9.23088			1.65791			0.828957	+	0.559312i	$0.188935\pi$
$32$		0			0
$33$		−	2.58608i		−	0.450179i
$34$		0			0
$35$		−	15.7670i		−	2.66511i
$36$		0			0
$37$			1.06747i			0.175491i	0.996143	+	0.0877457i	$0.0279663\pi$
$37$			1.06747i			0.175491i	−0.996143	+	0.0877457i	$0.972034\pi$
$38$		0			0
$39$			14.5077i			2.32309i
$40$		0			0
$41$		−	2.23290i		−	0.348721i	−0.984682	−	0.174360i	$-0.944214\pi$
$41$		−	2.23290i		−	0.348721i	0.984682	−	0.174360i	$-0.0557858\pi$
$42$		0			0
$43$		−	5.51453i		−	0.840957i	−0.907302	−	0.420479i	$-0.861862\pi$
$43$		−	5.51453i		−	0.840957i	0.907302	−	0.420479i	$-0.138138\pi$
$44$		0			0
$45$	12.6430			1.88471
$46$		0			0
$47$			7.03024i			1.02547i	0.858548	+	0.512733i	$0.171367\pi$
$47$			7.03024i			1.02547i	−0.858548	+	0.512733i	$0.828633\pi$
$48$		0			0
$49$	−14.1511			−2.02159
$50$		0			0
$51$	−17.7562			−2.48637
$52$		0			0
$53$			8.19169i			1.12522i	0.826724	+	0.562608i	$0.190202\pi$
$53$			8.19169i			1.12522i	−0.826724	+	0.562608i	$0.809798\pi$
$54$		0			0
$55$			3.42833i			0.462276i
$56$		0			0
$57$	7.80167	−	8.13649i	1.03336	−	1.07770i
$58$		0			0
$59$	5.18092			0.674498			0.337249	−	0.941415i	$-0.390504\pi$
$59$	5.18092			0.674498			0.337249	+	0.941415i	$0.390504\pi$
$60$		0			0
$61$	−10.8947			−1.39493			−0.697465	−	0.716619i	$-0.745689\pi$
$61$	−10.8947			−1.39493			−0.697465	+	0.716619i	$0.745689\pi$
$62$		0			0
$63$		−	16.9604i		−	2.13681i
$64$		0			0
$65$		−	19.2326i		−	2.38552i
$66$		0			0
$67$	8.76156			1.07039			0.535197	−	0.844727i	$-0.320237\pi$
$67$	8.76156			1.07039			0.535197	+	0.844727i	$0.320237\pi$
$68$		0			0
$69$			11.4285i			1.37583i
$70$		0			0
$71$	−7.34801			−0.872048			−0.436024	−	0.899935i	$-0.643614\pi$
$71$	−7.34801			−0.872048			−0.436024	+	0.899935i	$0.643614\pi$
$72$		0			0
$73$	9.44436			1.10538			0.552690	−	0.833387i	$-0.313602\pi$
$73$	9.44436			1.10538			0.552690	+	0.833387i	$0.313602\pi$
$74$		0			0
$75$	−17.4649			−2.01668
$76$		0			0
$77$	4.59904			0.524109
$78$		0			0
$79$	10.3209			1.16119			0.580595	−	0.814193i	$-0.302820\pi$
$79$	10.3209			1.16119			0.580595	+	0.814193i	$0.302820\pi$
$80$		0			0
$81$	−6.46347			−0.718163
$82$		0			0
$83$		−	2.87340i		−	0.315396i	−0.987487	−	0.157698i	$-0.949593\pi$
$83$		−	2.87340i		−	0.315396i	0.987487	−	0.157698i	$-0.0504073\pi$
$84$		0			0
$85$	23.5391			2.55318
$86$		0			0
$87$			21.9928i			2.35788i
$88$		0			0
$89$			10.3594i			1.09809i	0.835793	+	0.549045i	$0.185009\pi$
$89$			10.3594i			1.09809i	−0.835793	+	0.549045i	$0.814991\pi$
$90$		0			0
$91$	−25.8002			−2.70460
$92$		0			0
$93$	−23.8718			−2.47539
$94$		0			0
$95$	−10.3426	+	10.7864i	−1.06112	+	1.10666i
$96$		0			0
$97$			5.06525i			0.514298i	0.966372	+	0.257149i	$0.0827832\pi$
$97$			5.06525i			0.514298i	−0.966372	+	0.257149i	$0.917217\pi$
$98$		0			0
$99$			3.68782i			0.370639i
$100$		0			0
$101$	−7.31548			−0.727918			−0.363959	−	0.931415i	$-0.618575\pi$
$101$	−7.31548			−0.727918			−0.363959	+	0.931415i	$0.618575\pi$
$102$		0			0
$103$	7.92887			0.781255			0.390628	−	0.920549i	$-0.372258\pi$
$103$	7.92887			0.781255			0.390628	+	0.920549i	$0.372258\pi$
$104$		0			0
$105$			40.7747i			3.97921i
$106$		0			0
$107$	−18.3368			−1.77268			−0.886342	−	0.463032i	$-0.846762\pi$
$107$	−18.3368			−1.77268			−0.886342	+	0.463032i	$0.846762\pi$
$108$		0			0
$109$			9.43599i			0.903804i	0.892068	+	0.451902i	$0.149254\pi$
$109$			9.43599i			0.903804i	−0.892068	+	0.451902i	$0.850746\pi$
$110$		0			0
$111$		−	2.76057i		−	0.262022i
$112$		0			0
$113$			5.08600i			0.478450i	0.970964	+	0.239225i	$0.0768934\pi$
$113$			5.08600i			0.478450i	−0.970964	+	0.239225i	$0.923107\pi$
$114$		0			0
$115$		−	15.1505i		−	1.41280i
$116$		0			0
$117$		−	20.6883i		−	1.91264i
$118$		0			0
$119$		−	31.5773i		−	2.89468i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$			5.77447i			0.520666i
$124$		0			0
$125$	6.01135			0.537671
$126$		0			0
$127$	−1.84691			−0.163887			−0.0819435	−	0.996637i	$-0.526113\pi$
$127$	−1.84691			−0.163887			−0.0819435	+	0.996637i	$0.526113\pi$
$128$		0			0
$129$			14.2610i			1.25561i
$130$		0			0
$131$		−	2.87682i		−	0.251349i	−0.992072	−	0.125675i	$-0.959890\pi$
$131$		−	2.87682i		−	0.251349i	0.992072	−	0.125675i	$-0.0401095\pi$
$132$		0			0
$133$	14.4698	+	13.8743i	1.25469	+	1.20306i
$134$		0			0
$135$	−6.09812			−0.524843
$136$		0			0
$137$	−6.85476			−0.585642			−0.292821	−	0.956167i	$-0.594594\pi$
$137$	−6.85476			−0.585642			−0.292821	+	0.956167i	$0.594594\pi$
$138$		0			0
$139$			0.344300i			0.0292031i	0.999893	+	0.0146016i	$0.00464799\pi$
$139$			0.344300i			0.0292031i	−0.999893	+	0.0146016i	$0.995352\pi$
$140$		0			0
$141$		−	18.1808i		−	1.53110i
$142$		0			0
$143$	5.60992			0.469125
$144$		0			0
$145$		−	29.1555i		−	2.42123i
$146$		0			0
$147$	36.5960			3.01838
$148$		0			0
$149$	−10.8752			−0.890934			−0.445467	−	0.895298i	$-0.646962\pi$
$149$	−10.8752			−0.890934			−0.445467	+	0.895298i	$0.646962\pi$
$150$		0			0
$151$	23.4126			1.90529			0.952644	−	0.304087i	$-0.0983515\pi$
$151$	23.4126			1.90529			0.952644	+	0.304087i	$0.0983515\pi$
$152$		0			0
$153$	25.3208			2.04706
$154$		0			0
$155$	31.6465			2.54191
$156$		0			0
$157$	12.4364			0.992529			0.496264	−	0.868171i	$-0.334705\pi$
$157$	12.4364			0.992529			0.496264	+	0.868171i	$0.334705\pi$
$158$		0			0
$159$		−	21.1844i		−	1.68003i
$160$		0			0
$161$	−20.3242			−1.60177
$162$		0			0
$163$		−	4.81988i		−	0.377522i	−0.982023	−	0.188761i	$-0.939553\pi$
$163$		−	4.81988i		−	0.377522i	0.982023	−	0.188761i	$-0.0604472\pi$
$164$		0			0
$165$		−	8.86593i		−	0.690212i
$166$		0			0
$167$	−12.8383			−0.993461			−0.496730	−	0.867905i	$-0.665466\pi$
$167$	−12.8383			−0.993461			−0.496730	+	0.867905i	$0.665466\pi$
$168$		0			0
$169$	−18.4712			−1.42086
$170$		0			0
$171$	−11.1254	+	11.6028i	−0.850778	+	0.887291i
$172$		0			0
$173$		−	5.39987i		−	0.410544i	−0.978705	−	0.205272i	$-0.934192\pi$
$173$		−	5.39987i		−	0.410544i	0.978705	−	0.205272i	$-0.0658080\pi$
$174$		0			0
$175$		−	31.0593i		−	2.34786i
$176$		0			0
$177$	−13.3983			−1.00708
$178$		0			0
$179$	−8.55865			−0.639704			−0.319852	−	0.947468i	$-0.603633\pi$
$179$	−8.55865			−0.639704			−0.319852	+	0.947468i	$0.603633\pi$
$180$		0			0
$181$		−	0.845010i		−	0.0628091i	−0.999507	−	0.0314045i	$-0.990002\pi$
$181$		−	0.845010i		−	0.0628091i	0.999507	−	0.0314045i	$-0.00999802\pi$
$182$		0			0
$183$	28.1747			2.08273
$184$		0			0
$185$			3.65964i			0.269062i
$186$		0			0
$187$			6.86607i			0.502097i
$188$		0			0
$189$			8.18051i			0.595045i
$190$		0			0
$191$		−	18.2355i		−	1.31947i	−0.751496	−	0.659737i	$-0.770668\pi$
$191$		−	18.2355i		−	1.31947i	0.751496	−	0.659737i	$-0.229332\pi$
$192$		0			0
$193$		−	15.2601i		−	1.09845i	−0.835675	−	0.549224i	$-0.814923\pi$
$193$		−	15.2601i		−	1.09845i	0.835675	−	0.549224i	$-0.185077\pi$
$194$		0			0
$195$			49.7372i			3.56175i
$196$		0			0
$197$	17.4384			1.24243			0.621217	−	0.783638i	$-0.286638\pi$
$197$	17.4384			1.24243			0.621217	+	0.783638i	$0.286638\pi$
$198$		0			0
$199$		−	6.23774i		−	0.442182i	−0.975253	−	0.221091i	$-0.929038\pi$
$199$		−	6.23774i		−	0.442182i	0.975253	−	0.221091i	$-0.0709617\pi$
$200$		0			0
$201$	−22.6581			−1.59818
$202$		0			0
$203$	−39.1116			−2.74509
$204$		0			0
$205$		−	7.65512i		−	0.534657i
$206$		0			0
$207$		−	16.2973i		−	1.13274i
$208$		0			0
$209$	−3.14626	−	3.01679i	−0.217631	−	0.208676i
$210$		0			0
$211$	−0.568833			−0.0391601			−0.0195800	−	0.999808i	$-0.506233\pi$
$211$	−0.568833			−0.0391601			−0.0195800	+	0.999808i	$0.506233\pi$
$212$		0			0
$213$	19.0026			1.30203
$214$		0			0
$215$		−	18.9056i		−	1.28935i
$216$		0			0
$217$		−	42.4531i		−	2.88191i
$218$		0			0
$219$	−24.4239			−1.65041
$220$		0			0
$221$		−	38.5181i		−	2.59101i
$222$		0			0
$223$	0.276933			0.0185448			0.00927241	−	0.999957i	$-0.497048\pi$
$223$	0.276933			0.0185448			0.00927241	+	0.999957i	$0.497048\pi$
$224$		0			0
$225$	24.9054			1.66036
$226$		0			0
$227$	−5.93585			−0.393976			−0.196988	−	0.980406i	$-0.563116\pi$
$227$	−5.93585			−0.393976			−0.196988	+	0.980406i	$0.563116\pi$
$228$		0			0
$229$	6.45100			0.426294			0.213147	−	0.977020i	$-0.431629\pi$
$229$	6.45100			0.426294			0.213147	+	0.977020i	$0.431629\pi$
$230$		0			0
$231$	−11.8935			−0.782533
$232$		0			0
$233$	−15.1654			−0.993522			−0.496761	−	0.867887i	$-0.665477\pi$
$233$	−15.1654			−0.993522			−0.496761	+	0.867887i	$0.665477\pi$
$234$		0			0
$235$			24.1020i			1.57224i
$236$		0			0
$237$	−26.6906			−1.73374
$238$		0			0
$239$		−	3.73085i		−	0.241329i	−0.992693	−	0.120664i	$-0.961498\pi$
$239$		−	3.73085i		−	0.241329i	0.992693	−	0.120664i	$-0.0385024\pi$
$240$		0			0
$241$		−	26.6667i		−	1.71775i	−0.512183	−	0.858877i	$-0.671163\pi$
$241$		−	26.6667i		−	1.71775i	0.512183	−	0.858877i	$-0.328837\pi$
$242$		0			0
$243$	22.0513			1.41459
$244$		0			0
$245$	−48.5147			−3.09949
$246$		0			0
$247$	17.6503	+	16.9240i	1.12306	+	1.07685i
$248$		0			0
$249$			7.43084i			0.470910i
$250$		0			0
$251$			0.0265278i			0.00167442i	1.00000		0.000837210i	$0.000266492\pi$
$251$			0.0265278i			0.00167442i	−1.00000		0.000837210i	$0.999734\pi$
$252$		0			0
$253$	4.41922			0.277834
$254$		0			0
$255$	−60.8741			−3.81208
$256$		0			0
$257$		−	14.6893i		−	0.916294i	−0.888877	−	0.458147i	$-0.848513\pi$
$257$		−	14.6893i		−	0.916294i	0.888877	−	0.458147i	$-0.151487\pi$
$258$		0			0
$259$	4.90934			0.305052
$260$		0			0
$261$		−	31.3623i		−	1.94128i
$262$		0			0
$263$		−	12.9602i		−	0.799159i	−0.916698	−	0.399580i	$-0.869156\pi$
$263$		−	12.9602i		−	0.799159i	0.916698	−	0.399580i	$-0.130844\pi$
$264$		0			0
$265$			28.0838i			1.72517i
$266$		0			0
$267$		−	26.7902i		−	1.63953i
$268$		0			0
$269$			15.1407i			0.923144i	0.887103	+	0.461572i	$0.152714\pi$
$269$			15.1407i			0.923144i	−0.887103	+	0.461572i	$0.847286\pi$
$270$		0			0
$271$			27.7238i			1.68410i	0.539399	+	0.842050i	$0.318651\pi$
$271$			27.7238i			1.68410i	−0.539399	+	0.842050i	$0.681349\pi$
$272$		0			0
$273$	66.7214			4.03817
$274$		0			0
$275$			6.75343i			0.407247i
$276$		0			0
$277$	−19.5680			−1.17573			−0.587863	−	0.808961i	$-0.700031\pi$
$277$	−19.5680			−1.17573			−0.587863	+	0.808961i	$0.700031\pi$
$278$		0			0
$279$	34.0418			2.03803
$280$		0			0
$281$		−	22.7616i		−	1.35784i	−0.734211	−	0.678922i	$-0.762448\pi$
$281$		−	22.7616i		−	1.35784i	0.734211	−	0.678922i	$-0.237552\pi$
$282$		0			0
$283$		−	4.39636i		−	0.261336i	−0.991426	−	0.130668i	$-0.958288\pi$
$283$		−	4.39636i		−	0.261336i	0.991426	−	0.130668i	$-0.0417123\pi$
$284$		0			0
$285$	26.7467	−	27.8946i	1.58434	−	1.65233i
$286$		0			0
$287$	−10.2692			−0.606171
$288$		0			0
$289$	30.1429			1.77311
$290$		0			0
$291$		−	13.0992i		−	0.767886i
$292$		0			0
$293$			14.2402i			0.831920i	0.909383	+	0.415960i	$0.136554\pi$
$293$			14.2402i			0.831920i	−0.909383	+	0.415960i	$0.863446\pi$
$294$		0			0
$295$	17.7619			1.03414
$296$		0			0
$297$		−	1.77875i		−	0.103213i
$298$		0			0
$299$	−24.7915			−1.43373
$300$		0			0
$301$	−25.3615			−1.46181
$302$		0			0
$303$	18.9184			1.08684
$304$		0			0
$305$	−37.3508			−2.13870
$306$		0			0
$307$	10.7282			0.612291			0.306145	−	0.951985i	$-0.400961\pi$
$307$	10.7282			0.612291			0.306145	+	0.951985i	$0.400961\pi$
$308$		0			0
$309$	−20.5047			−1.16647
$310$		0			0
$311$			7.49622i			0.425072i	0.977153	+	0.212536i	$0.0681723\pi$
$311$			7.49622i			0.425072i	−0.977153	+	0.212536i	$0.931828\pi$
$312$		0			0
$313$	−3.45589			−0.195339			−0.0976694	−	0.995219i	$-0.531139\pi$
$313$	−3.45589			−0.195339			−0.0976694	+	0.995219i	$0.531139\pi$
$314$		0			0
$315$		−	58.1458i		−	3.27614i
$316$		0			0
$317$			17.6599i			0.991879i	0.868357	+	0.495940i	$0.165176\pi$
$317$			17.6599i			0.991879i	−0.868357	+	0.495940i	$0.834824\pi$
$318$		0			0
$319$	8.50430			0.476149
$320$		0			0
$321$	47.4204			2.64675
$322$		0			0
$323$	−20.7135	+	21.6024i	−1.15253	+	1.20199i
$324$		0			0
$325$		−	37.8862i		−	2.10155i
$326$		0			0
$327$		−	24.4022i		−	1.34945i
$328$		0			0
$329$	32.3323			1.78254
$330$		0			0
$331$	−29.7578			−1.63564			−0.817819	−	0.575476i	$-0.804817\pi$
$331$	−29.7578			−1.63564			−0.817819	+	0.575476i	$0.804817\pi$
$332$		0			0
$333$			3.93664i			0.215727i
$334$		0			0
$335$	30.0375			1.64112
$336$		0			0
$337$			14.7960i			0.805989i	0.915202	+	0.402995i	$0.132031\pi$
$337$			14.7960i			0.805989i	−0.915202	+	0.402995i	$0.867969\pi$
$338$		0			0
$339$		−	13.1528i		−	0.714362i
$340$		0			0
$341$			9.23088i			0.499880i
$342$		0			0
$343$			32.8883i			1.77580i
$344$		0			0
$345$			39.1805i			2.10941i
$346$		0			0
$347$		−	0.101260i		−	0.00543590i	−0.999996	−	0.00271795i	$-0.999135\pi$
$347$		−	0.101260i		−	0.00543590i	0.999996	−	0.00271795i	$-0.000865151\pi$
$348$		0			0
$349$	−20.1533			−1.07878			−0.539390	−	0.842056i	$-0.681345\pi$
$349$	−20.1533			−1.07878			−0.539390	+	0.842056i	$0.681345\pi$
$350$		0			0
$351$			9.97862i			0.532619i
$352$		0			0
$353$	25.0446			1.33299			0.666494	−	0.745510i	$-0.267794\pi$
$353$	25.0446			1.33299			0.666494	+	0.745510i	$0.267794\pi$
$354$		0			0
$355$	−25.1914			−1.33702
$356$		0			0
$357$			81.6614i			4.32198i
$358$		0			0
$359$		−	19.7124i		−	1.04038i	−0.854050	−	0.520191i	$-0.825861\pi$
$359$		−	19.7124i		−	1.04038i	0.854050	−	0.520191i	$-0.174139\pi$
$360$		0			0
$361$	−0.797928	−	18.9832i	−0.0419962	−	0.999118i
$362$		0			0
$363$	2.58608			0.135734
$364$		0			0
$365$	32.3784			1.69476
$366$		0			0
$367$			13.0344i			0.680391i	0.940355	+	0.340196i	$0.110493\pi$
$367$			13.0344i			0.680391i	−0.940355	+	0.340196i	$0.889507\pi$
$368$		0			0
$369$		−	8.23453i		−	0.428673i
$370$		0			0
$371$	37.6739			1.95593
$372$		0			0
$373$			3.37555i			0.174779i	0.996174	+	0.0873896i	$0.0278525\pi$
$373$			3.37555i			0.174779i	−0.996174	+	0.0873896i	$0.972147\pi$
$374$		0			0
$375$	−15.5458			−0.802783
$376$		0			0
$377$	−47.7084			−2.45711
$378$		0			0
$379$	14.3941			0.739378			0.369689	−	0.929156i	$-0.379464\pi$
$379$	14.3941			0.739378			0.369689	+	0.929156i	$0.379464\pi$
$380$		0			0
$381$	4.77626			0.244695
$382$		0			0
$383$	23.8626			1.21932			0.609662	−	0.792661i	$-0.291305\pi$
$383$	23.8626			1.21932			0.609662	+	0.792661i	$0.291305\pi$
$384$		0			0
$385$	15.7670			0.803561
$386$		0			0
$387$		−	20.3365i		−	1.03376i
$388$		0			0
$389$	−17.3711			−0.880751			−0.440375	−	0.897814i	$-0.645155\pi$
$389$	−17.3711			−0.880751			−0.440375	+	0.897814i	$0.645155\pi$
$390$		0			0
$391$		−	30.3427i		−	1.53450i
$392$		0			0
$393$			7.43969i			0.375283i
$394$		0			0
$395$	35.3834			1.78033
$396$		0			0
$397$	2.37708			0.119302			0.0596511	−	0.998219i	$-0.481001\pi$
$397$	2.37708			0.119302			0.0596511	+	0.998219i	$0.481001\pi$
$398$		0			0
$399$	−37.4200	−	35.8802i	−1.87334	−	1.79625i
$400$		0			0
$401$		−	23.0128i		−	1.14921i	−0.818432	−	0.574603i	$-0.805156\pi$
$401$		−	23.0128i		−	1.14921i	0.818432	−	0.574603i	$-0.194844\pi$
$402$		0			0
$403$		−	51.7845i		−	2.57957i
$404$		0			0
$405$	−22.1589			−1.10108
$406$		0			0
$407$	−1.06747			−0.0529126
$408$		0			0
$409$		−	4.44119i		−	0.219603i	−0.993954	−	0.109801i	$-0.964979\pi$
$409$		−	4.44119i		−	0.219603i	0.993954	−	0.109801i	$-0.0350215\pi$
$410$		0			0
$411$	17.7270			0.874407
$412$		0			0
$413$		−	23.8272i		−	1.17246i
$414$		0			0
$415$		−	9.85095i		−	0.483564i
$416$		0			0
$417$		−	0.890387i		−	0.0436024i
$418$		0			0
$419$		−	15.7150i		−	0.767730i	−0.923389	−	0.383865i	$-0.874593\pi$
$419$		−	15.7150i		−	0.767730i	0.923389	−	0.383865i	$-0.125407\pi$
$420$		0			0
$421$			28.0775i			1.36842i	0.729287	+	0.684208i	$0.239852\pi$
$421$			28.0775i			1.36842i	−0.729287	+	0.684208i	$0.760148\pi$
$422$		0			0
$423$			25.9262i			1.26058i
$424$		0			0
$425$	46.3695			2.24925
$426$		0			0
$427$			50.1053i			2.42477i
$428$		0			0
$429$	−14.5077			−0.700439
$430$		0			0
$431$	−15.9226			−0.766964			−0.383482	−	0.923548i	$-0.625275\pi$
$431$	−15.9226			−0.766964			−0.383482	+	0.923548i	$0.625275\pi$
$432$		0			0
$433$		−	17.3119i		−	0.831954i	−0.909375	−	0.415977i	$-0.863440\pi$
$433$		−	17.3119i		−	0.831954i	0.909375	−	0.415977i	$-0.136560\pi$
$434$		0			0
$435$			75.3985i			3.61508i
$436$		0			0
$437$	13.9040	+	13.3319i	0.665120	+	0.637750i
$438$		0			0
$439$	−16.4325			−0.784282			−0.392141	−	0.919905i	$-0.628266\pi$
$439$	−16.4325			−0.784282			−0.392141	+	0.919905i	$0.628266\pi$
$440$		0			0
$441$	−52.1867			−2.48508
$442$		0			0
$443$			31.1867i			1.48172i	0.671658	+	0.740862i	$0.265583\pi$
$443$			31.1867i			1.48172i	−0.671658	+	0.740862i	$0.734417\pi$
$444$		0			0
$445$			35.5153i			1.68359i
$446$		0			0
$447$	28.1242			1.33023
$448$		0			0
$449$			18.7193i			0.883416i	0.897159	+	0.441708i	$0.145627\pi$
$449$			18.7193i			0.883416i	−0.897159	+	0.441708i	$0.854373\pi$
$450$		0			0
$451$	2.23290			0.105143
$452$		0			0
$453$	−60.5468			−2.84474
$454$		0			0
$455$	−88.4516			−4.14668
$456$		0			0
$457$	25.3648			1.18652			0.593258	−	0.805013i	$-0.297842\pi$
$457$	25.3648			1.18652			0.593258	+	0.805013i	$0.297842\pi$
$458$		0			0
$459$	−12.2130			−0.570053
$460$		0			0
$461$	29.4580			1.37200			0.685998	−	0.727603i	$-0.259366\pi$
$461$	29.4580			1.37200			0.685998	+	0.727603i	$0.259366\pi$
$462$		0			0
$463$		−	17.6991i		−	0.822549i	−0.911512	−	0.411274i	$-0.865084\pi$
$463$		−	17.6991i		−	0.822549i	0.911512	−	0.411274i	$-0.134916\pi$
$464$		0			0
$465$	−81.8403			−3.79525
$466$		0			0
$467$		−	4.42834i		−	0.204919i	−0.994737	−	0.102460i	$-0.967329\pi$
$467$		−	4.42834i		−	0.204919i	0.994737	−	0.102460i	$-0.0326713\pi$
$468$		0			0
$469$		−	40.2947i		−	1.86064i
$470$		0			0
$471$	−32.1614			−1.48192
$472$		0			0
$473$	5.51453			0.253558
$474$		0			0
$475$	−20.3737	+	21.2481i	−0.934810	+	0.974928i
$476$		0			0
$477$			30.2094i			1.38320i
$478$		0			0
$479$			37.1502i			1.69743i	0.528847	+	0.848717i	$0.322624\pi$
$479$			37.1502i			1.69743i	−0.528847	+	0.848717i	$0.677376\pi$
$480$		0			0
$481$	5.98843			0.273049
$482$		0			0
$483$	52.5599			2.39156
$484$		0			0
$485$			17.3653i			0.788520i
$486$		0			0
$487$	−7.07559			−0.320626			−0.160313	−	0.987066i	$-0.551250\pi$
$487$	−7.07559			−0.320626			−0.160313	+	0.987066i	$0.551250\pi$
$488$		0			0
$489$			12.4646i			0.563669i
$490$		0			0
$491$			14.7919i			0.667549i	0.942653	+	0.333775i	$0.108322\pi$
$491$			14.7919i			0.667549i	−0.942653	+	0.333775i	$0.891678\pi$
$492$		0			0
$493$		−	58.3911i		−	2.62980i
$494$		0			0
$495$			12.6430i			0.568262i
$496$		0			0
$497$			33.7938i			1.51586i
$498$		0			0
$499$			26.5174i			1.18708i	0.804804	+	0.593540i	$0.202270\pi$
$499$			26.5174i			1.18708i	−0.804804	+	0.593540i	$0.797730\pi$
$500$		0			0
$501$	33.2010			1.48331
$502$		0			0
$503$			10.4684i			0.466765i	0.972385	+	0.233382i	$0.0749793\pi$
$503$			10.4684i			0.466765i	−0.972385	+	0.233382i	$0.925021\pi$
$504$		0			0
$505$	−25.0799			−1.11604
$506$		0			0
$507$	47.7680			2.12145
$508$		0			0
$509$			4.61765i			0.204674i	0.994750	+	0.102337i	$0.0326320\pi$
$509$			4.61765i			0.204674i	−0.994750	+	0.102337i	$0.967368\pi$
$510$		0			0
$511$		−	43.4350i		−	1.92145i
$512$		0			0
$513$	5.36611	−	5.59640i	0.236919	−	0.247087i
$514$		0			0
$515$	27.1828			1.19782
$516$		0			0
$517$	−7.03024			−0.309190
$518$		0			0
$519$			13.9645i			0.612973i
$520$		0			0
$521$		−	17.8648i		−	0.782673i	−0.920248	−	0.391336i	$-0.872013\pi$
$521$		−	17.8648i		−	0.782673i	0.920248	−	0.391336i	$-0.127987\pi$
$522$		0			0
$523$	−33.7321			−1.47500			−0.737501	−	0.675346i	$-0.763994\pi$
$523$	−33.7321			−1.47500			−0.737501	+	0.675346i	$0.763994\pi$
$524$		0			0
$525$			80.3218i			3.50553i
$526$		0			0
$527$	63.3798			2.76087
$528$		0			0
$529$	3.47047			0.150890
$530$		0			0
$531$	19.1063			0.829142
$532$		0			0
$533$	−12.5264			−0.542579
$534$		0			0
$535$	−62.8645			−2.71787
$536$		0			0
$537$	22.1334			0.955126
$538$		0			0
$539$		−	14.1511i		−	0.609532i
$540$		0			0
$541$	5.34978			0.230005			0.115002	−	0.993365i	$-0.463312\pi$
$541$	5.34978			0.230005			0.115002	+	0.993365i	$0.463312\pi$
$542$		0			0
$543$			2.18526i			0.0937786i
$544$		0			0
$545$			32.3497i			1.38571i
$546$		0			0
$547$	−3.97366			−0.169901			−0.0849507	−	0.996385i	$-0.527073\pi$
$547$	−3.97366			−0.169901			−0.0849507	+	0.996385i	$0.527073\pi$
$548$		0			0
$549$	−40.1778			−1.71475
$550$		0			0
$551$	26.7567	+	25.6557i	1.13988	+	1.09297i
$552$		0			0
$553$		−	47.4661i		−	2.01846i
$554$		0			0
$555$		−	9.46414i		−	0.401730i
$556$		0			0
$557$	24.1337			1.02258			0.511289	−	0.859409i	$-0.329168\pi$
$557$	24.1337			1.02258			0.511289	+	0.859409i	$0.329168\pi$
$558$		0			0
$559$	−30.9360			−1.30846
$560$		0			0
$561$		−	17.7562i		−	0.749668i
$562$		0			0
$563$	−21.0225			−0.885994			−0.442997	−	0.896523i	$-0.646085\pi$
$563$	−21.0225			−0.885994			−0.442997	+	0.896523i	$0.646085\pi$
$564$		0			0
$565$			17.4365i			0.733558i
$566$		0			0
$567$			29.7257i			1.24836i
$568$		0			0
$569$		−	19.0052i		−	0.796740i	−0.917225	−	0.398370i	$-0.869576\pi$
$569$		−	19.0052i		−	0.796740i	0.917225	−	0.398370i	$-0.130424\pi$
$570$		0			0
$571$		−	13.9025i		−	0.581803i	−0.956753	−	0.290902i	$-0.906045\pi$
$571$		−	13.9025i		−	0.581803i	0.956753	−	0.290902i	$-0.0939553\pi$
$572$		0			0
$573$			47.1585i			1.97007i
$574$		0			0
$575$		−	29.8449i		−	1.24462i
$576$		0			0
$577$	5.33191			0.221970			0.110985	−	0.993822i	$-0.464599\pi$
$577$	5.33191			0.221970			0.110985	+	0.993822i	$0.464599\pi$
$578$		0			0
$579$			39.4639i			1.64006i
$580$		0			0
$581$	−13.2149			−0.548245
$582$		0			0
$583$	−8.19169			−0.339265
$584$		0			0
$585$		−	70.9264i		−	2.93245i
$586$		0			0
$587$		−	2.78000i		−	0.114743i	−0.998353	−	0.0573715i	$-0.981728\pi$
$587$		−	2.78000i		−	0.114743i	0.998353	−	0.0573715i	$-0.0182720\pi$
$588$		0			0
$589$	−27.8476	+	29.0428i	−1.14744	+	1.19669i
$590$		0			0
$591$	−45.0971			−1.85505
$592$		0			0
$593$	−37.5686			−1.54276			−0.771380	−	0.636375i	$-0.780433\pi$
$593$	−37.5686			−1.54276			−0.771380	+	0.636375i	$0.780433\pi$
$594$		0			0
$595$		−	108.257i		−	4.43812i
$596$		0			0
$597$			16.1313i			0.660210i
$598$		0			0
$599$	39.9866			1.63381			0.816904	−	0.576773i	$-0.195688\pi$
$599$	39.9866			1.63381			0.816904	+	0.576773i	$0.195688\pi$
$600$		0			0
$601$			24.2948i			0.991004i	0.868607	+	0.495502i	$0.165016\pi$
$601$			24.2948i			0.991004i	−0.868607	+	0.495502i	$0.834984\pi$
$602$		0			0
$603$	32.3110			1.31581
$604$		0			0
$605$	−3.42833			−0.139381
$606$		0			0
$607$	−15.4901			−0.628723			−0.314362	−	0.949303i	$-0.601790\pi$
$607$	−15.4901			−0.628723			−0.314362	+	0.949303i	$0.601790\pi$
$608$		0			0
$609$	101.146			4.09863
$610$		0			0
$611$	39.4391			1.59553
$612$		0			0
$613$	44.7184			1.80616			0.903081	−	0.429471i	$-0.141300\pi$
$613$	44.7184			1.80616			0.903081	+	0.429471i	$0.141300\pi$
$614$		0			0
$615$			19.7968i			0.798283i
$616$		0			0
$617$	23.5964			0.949956			0.474978	−	0.879998i	$-0.342456\pi$
$617$	23.5964			0.949956			0.474978	+	0.879998i	$0.342456\pi$
$618$		0			0
$619$			30.5416i			1.22757i	0.789473	+	0.613785i	$0.210354\pi$
$619$			30.5416i			1.22757i	−0.789473	+	0.613785i	$0.789646\pi$
$620$		0			0
$621$			7.86067i			0.315438i
$622$		0			0
$623$	47.6431			1.90878
$624$		0			0
$625$	−13.1583			−0.526332
$626$		0			0
$627$	8.13649	+	7.80167i	0.324940	+	0.311569i
$628$		0			0
$629$			7.32933i			0.292240i
$630$		0			0
$631$			2.67775i			0.106600i	0.998579	+	0.0532998i	$0.0169739\pi$
$631$			2.67775i			0.106600i	−0.998579	+	0.0532998i	$0.983026\pi$
$632$		0			0
$633$	1.47105			0.0584689
$634$		0			0
$635$	−6.33182			−0.251271
$636$		0			0
$637$			79.3867i			3.14541i
$638$		0			0
$639$	−27.0981			−1.07198
$640$		0			0
$641$			11.3550i			0.448496i	0.974532	+	0.224248i	$0.0719926\pi$
$641$			11.3550i			0.448496i	−0.974532	+	0.224248i	$0.928007\pi$
$642$		0			0
$643$		−	27.8287i		−	1.09746i	−0.836001	−	0.548728i	$-0.815112\pi$
$643$		−	27.8287i		−	1.09746i	0.836001	−	0.548728i	$-0.184888\pi$
$644$		0			0
$645$			48.8914i			1.92510i
$646$		0			0
$647$			44.3945i			1.74533i	0.488320	+	0.872664i	$0.337610\pi$
$647$			44.3945i			1.74533i	−0.488320	+	0.872664i	$0.662390\pi$
$648$		0			0
$649$			5.18092i			0.203369i
$650$		0			0
$651$			109.787i			4.30290i
$652$		0			0
$653$	−7.45552			−0.291757			−0.145879	−	0.989302i	$-0.546601\pi$
$653$	−7.45552			−0.291757			−0.145879	+	0.989302i	$0.546601\pi$
$654$		0			0
$655$		−	9.86269i		−	0.385367i
$656$		0			0
$657$	34.8291			1.35881
$658$		0			0
$659$	1.65692			0.0645443			0.0322721	−	0.999479i	$-0.489726\pi$
$659$	1.65692			0.0645443			0.0322721	+	0.999479i	$0.489726\pi$
$660$		0			0
$661$			39.6930i			1.54388i	0.635695	+	0.771940i	$0.280713\pi$
$661$			39.6930i			1.54388i	−0.635695	+	0.771940i	$0.719287\pi$
$662$		0			0
$663$			99.6109i			3.86857i
$664$		0			0
$665$	49.6071	+	47.5658i	1.92368	+	1.84452i
$666$		0			0
$667$	−37.5824			−1.45520
$668$		0			0
$669$	−0.716171			−0.0276888
$670$		0			0
$671$		−	10.8947i		−	0.420587i
$672$		0			0
$673$		−	19.7846i		−	0.762641i	−0.924443	−	0.381321i	$-0.875469\pi$
$673$		−	19.7846i		−	0.762641i	0.924443	−	0.381321i	$-0.124531\pi$
$674$		0			0
$675$	−12.0126			−0.462367
$676$		0			0
$677$		−	13.7145i		−	0.527092i	−0.964647	−	0.263546i	$-0.915108\pi$
$677$		−	13.7145i		−	0.527092i	0.964647	−	0.263546i	$-0.0848921\pi$
$678$		0			0
$679$	23.2953			0.893990
$680$		0			0
$681$	15.3506			0.588236
$682$		0			0
$683$	−22.8828			−0.875586			−0.437793	−	0.899076i	$-0.644240\pi$
$683$	−22.8828			−0.875586			−0.437793	+	0.899076i	$0.644240\pi$
$684$		0			0
$685$	−23.5004			−0.897903
$686$		0			0
$687$	−16.6828			−0.636489
$688$		0			0
$689$	45.9547			1.75074
$690$		0			0
$691$		−	0.562354i		−	0.0213930i	−0.999943	−	0.0106965i	$-0.996595\pi$
$691$		−	0.562354i		−	0.0213930i	0.999943	−	0.0106965i	$-0.00340486\pi$
$692$		0			0
$693$	16.9604			0.644272
$694$		0			0
$695$			1.18037i			0.0447741i
$696$		0			0
$697$		−	15.3313i		−	0.580713i
$698$		0			0
$699$	39.2191			1.48340
$700$		0			0
$701$	11.3892			0.430164			0.215082	−	0.976596i	$-0.430998\pi$
$701$	11.3892			0.430164			0.215082	+	0.976596i	$0.430998\pi$
$702$		0			0
$703$	−3.35855	−	3.22034i	−0.126670	−	0.121457i
$704$		0			0
$705$		−	62.3296i		−	2.34747i
$706$		0			0
$707$			33.6442i			1.26532i
$708$		0			0
$709$	−29.1622			−1.09521			−0.547605	−	0.836737i	$-0.684460\pi$
$709$	−29.1622			−1.09521			−0.547605	+	0.836737i	$0.684460\pi$
$710$		0			0
$711$	38.0615			1.42742
$712$		0			0
$713$		−	40.7933i		−	1.52772i
$714$		0			0
$715$	19.2326			0.719260
$716$		0			0
$717$			9.64828i			0.360321i
$718$		0			0
$719$			44.4806i			1.65885i	0.558621	+	0.829423i	$0.311331\pi$
$719$			44.4806i			1.65885i	−0.558621	+	0.829423i	$0.688669\pi$
$720$		0			0
$721$		−	36.4652i		−	1.35803i
$722$		0			0
$723$			68.9623i			2.56473i
$724$		0			0
$725$		−	57.4332i		−	2.13302i
$726$		0			0
$727$			45.8238i			1.69951i	0.527178	+	0.849755i	$0.323250\pi$
$727$			45.8238i			1.69951i	−0.527178	+	0.849755i	$0.676750\pi$
$728$		0			0
$729$	−37.6360			−1.39393
$730$		0			0
$731$		−	37.8631i		−	1.40042i
$732$		0			0
$733$	−31.9409			−1.17976			−0.589882	−	0.807490i	$-0.700826\pi$
$733$	−31.9409			−1.17976			−0.589882	+	0.807490i	$0.700826\pi$
$734$		0			0
$735$	125.463			4.62777
$736$		0			0
$737$			8.76156i			0.322736i
$738$		0			0
$739$			13.2689i			0.488104i	0.969762	+	0.244052i	$0.0784767\pi$
$739$			13.2689i			0.488104i	−0.969762	+	0.244052i	$0.921523\pi$
$740$		0			0
$741$	−45.6450	−	43.7667i	−1.67681	−	1.60781i
$742$		0			0
$743$	−11.3480			−0.416317			−0.208158	−	0.978095i	$-0.566747\pi$
$743$	−11.3480			−0.416317			−0.208158	+	0.978095i	$0.566747\pi$
$744$		0			0
$745$	−37.2839			−1.36598
$746$		0			0
$747$		−	10.5966i		−	0.387708i
$748$		0			0
$749$			84.3315i			3.08141i
$750$		0			0
$751$	29.9390			1.09249			0.546245	−	0.837626i	$-0.316057\pi$
$751$	29.9390			1.09249			0.546245	+	0.837626i	$0.316057\pi$
$752$		0			0
$753$		−	0.0686030i		−	0.00250003i
$754$		0			0
$755$	80.2660			2.92118
$756$		0			0
$757$	43.6514			1.58654			0.793268	−	0.608873i	$-0.208378\pi$
$757$	43.6514			1.58654			0.793268	+	0.608873i	$0.208378\pi$
$758$		0			0
$759$	−11.4285			−0.414827
$760$		0			0
$761$	−13.2174			−0.479130			−0.239565	−	0.970880i	$-0.577005\pi$
$761$	−13.2174			−0.479130			−0.239565	+	0.970880i	$0.577005\pi$
$762$		0			0
$763$	43.3965			1.57106
$764$		0			0
$765$	86.8080			3.13855
$766$		0			0
$767$		−	29.0645i		−	1.04946i
$768$		0			0
$769$	25.7572			0.928829			0.464415	−	0.885618i	$-0.346265\pi$
$769$	25.7572			0.928829			0.464415	+	0.885618i	$0.346265\pi$
$770$		0			0
$771$			37.9877i			1.36809i
$772$		0			0
$773$		−	20.1357i		−	0.724230i	−0.932133	−	0.362115i	$-0.882055\pi$
$773$		−	20.1357i		−	0.724230i	0.932133	−	0.362115i	$-0.117945\pi$
$774$		0			0
$775$	62.3401			2.23932
$776$		0			0
$777$	−12.6960			−0.455465
$778$		0			0
$779$	7.02530	+	6.73620i	0.251707	+	0.241349i
$780$		0			0
$781$		−	7.34801i		−	0.262932i
$782$		0			0
$783$			15.1270i			0.540594i
$784$		0			0
$785$	42.6359			1.52174
$786$		0			0
$787$	−25.3712			−0.904387			−0.452194	−	0.891920i	$-0.649358\pi$
$787$	−25.3712			−0.904387			−0.452194	+	0.891920i	$0.649358\pi$
$788$		0			0
$789$			33.5161i			1.19320i
$790$		0			0
$791$	23.3907			0.831677
$792$		0			0
$793$			61.1186i			2.17039i
$794$		0			0
$795$		−	72.6270i		−	2.57581i
$796$		0			0
$797$		−	42.2342i		−	1.49601i	−0.663693	−	0.748005i	$-0.731012\pi$
$797$		−	42.2342i		−	1.49601i	0.663693	−	0.748005i	$-0.268988\pi$
$798$		0			0
$799$			48.2701i			1.70767i
$800$		0			0
$801$			38.2034i			1.34985i
$802$		0			0
$803$			9.44436i			0.333284i
$804$		0			0
$805$	−69.6779			−2.45582
$806$		0			0
$807$		−	39.1551i		−	1.37832i
$808$		0			0
$809$	−49.5585			−1.74238			−0.871192	−	0.490943i	$-0.836652\pi$
$809$	−49.5585			−1.74238			−0.871192	+	0.490943i	$0.836652\pi$
$810$		0			0
$811$	14.1154			0.495659			0.247829	−	0.968804i	$-0.420283\pi$
$811$	14.1154			0.495659			0.247829	+	0.968804i	$0.420283\pi$
$812$		0			0
$813$		−	71.6960i		−	2.51449i
$814$		0			0
$815$		−	16.5241i		−	0.578815i
$816$		0			0
$817$	17.3501	+	16.6362i	0.607005	+	0.582026i
$818$		0			0
$819$	−95.1464			−3.32469
$820$		0			0
$821$	9.80868			0.342325			0.171163	−	0.985243i	$-0.445248\pi$
$821$	9.80868			0.342325			0.171163	+	0.985243i	$0.445248\pi$
$822$		0			0
$823$			10.2365i			0.356822i	0.983956	+	0.178411i	$0.0570956\pi$
$823$			10.2365i			0.356822i	−0.983956	+	0.178411i	$0.942904\pi$
$824$		0			0
$825$		−	17.4649i		−	0.608051i
$826$		0			0
$827$	2.20782			0.0767733			0.0383867	−	0.999263i	$-0.487778\pi$
$827$	2.20782			0.0767733			0.0383867	+	0.999263i	$0.487778\pi$
$828$		0			0
$829$		−	10.7742i		−	0.374202i	−0.982341	−	0.187101i	$-0.940091\pi$
$829$		−	10.7742i		−	0.374202i	0.982341	−	0.187101i	$-0.0599092\pi$
$830$		0			0
$831$	50.6044			1.75545
$832$		0			0
$833$	−97.1626			−3.36648
$834$		0			0
$835$	−44.0141			−1.52317
$836$		0			0
$837$	−16.4194			−0.567537
$838$		0			0
$839$	−21.2319			−0.733005			−0.366503	−	0.930417i	$-0.619445\pi$
$839$	−21.2319			−0.733005			−0.366503	+	0.930417i	$0.619445\pi$
$840$		0			0
$841$	−43.3231			−1.49390
$842$		0			0
$843$			58.8634i			2.02736i
$844$		0			0
$845$	−63.3253			−2.17846
$846$		0			0
$847$			4.59904i			0.158025i
$848$		0			0
$849$			11.3693i			0.390195i
$850$		0			0
$851$	4.71740			0.161710
$852$		0			0
$853$	−6.55636			−0.224485			−0.112243	−	0.993681i	$-0.535803\pi$
$853$	−6.55636			−0.224485			−0.112243	+	0.993681i	$0.535803\pi$
$854$		0			0
$855$	−38.1414	+	39.7783i	−1.30441	+	1.36039i
$856$		0			0
$857$			0.108265i			0.00369827i	0.999998	+	0.00184913i	$0.000588598\pi$
$857$			0.108265i			0.00369827i	−0.999998	+	0.00184913i	$0.999411\pi$
$858$		0			0
$859$		−	36.7505i		−	1.25391i	−0.779055	−	0.626956i	$-0.784301\pi$
$859$		−	36.7505i		−	1.25391i	0.779055	−	0.626956i	$-0.215699\pi$
$860$		0			0
$861$	26.5570			0.905059
$862$		0			0
$863$	4.40999			0.150118			0.0750589	−	0.997179i	$-0.476086\pi$
$863$	4.40999			0.150118			0.0750589	+	0.997179i	$0.476086\pi$
$864$		0			0
$865$		−	18.5125i		−	0.629445i
$866$		0			0
$867$	−77.9519			−2.64739
$868$		0			0
$869$			10.3209i			0.350112i
$870$		0			0
$871$		−	49.1516i		−	1.66544i
$872$		0			0
$873$			18.6797i			0.632212i
$874$		0			0
$875$		−	27.6464i		−	0.934618i
$876$		0			0
$877$			39.5801i			1.33652i	0.743926	+	0.668262i	$0.232962\pi$
$877$			39.5801i			1.33652i	−0.743926	+	0.668262i	$0.767038\pi$
$878$		0			0
$879$		−	36.8262i		−	1.24212i
$880$		0			0
$881$	−27.2196			−0.917052			−0.458526	−	0.888681i	$-0.651622\pi$
$881$	−27.2196			−0.917052			−0.458526	+	0.888681i	$0.651622\pi$
$882$		0			0
$883$		−	38.9728i		−	1.31154i	−0.754961	−	0.655770i	$-0.772344\pi$
$883$		−	38.9728i		−	1.31154i	0.754961	−	0.655770i	$-0.227656\pi$
$884$		0			0
$885$	−45.9337			−1.54404
$886$		0			0
$887$	−36.1435			−1.21358			−0.606791	−	0.794862i	$-0.707543\pi$
$887$	−36.1435			−1.21358			−0.606791	+	0.794862i	$0.707543\pi$
$888$		0			0
$889$			8.49401i			0.284880i
$890$		0			0
$891$		−	6.46347i		−	0.216534i
$892$		0			0
$893$	−22.1190	−	21.2088i	−0.740183	−	0.709724i
$894$		0			0
$895$	−29.3419			−0.980791
$896$		0			0
$897$	64.1128			2.14066
$898$		0			0
$899$		−	78.5021i		−	2.61819i
$900$		0			0
$901$			56.2447i			1.87378i
$902$		0			0
$903$	65.5869			2.18259
$904$		0			0
$905$		−	2.89697i		−	0.0962986i
$906$		0			0
$907$	36.3775			1.20789			0.603947	−	0.797025i	$-0.293594\pi$
$907$	36.3775			1.20789			0.603947	+	0.797025i	$0.293594\pi$
$908$		0			0
$909$	−26.9781			−0.894809
$910$		0			0
$911$	3.77532			0.125082			0.0625409	−	0.998042i	$-0.480080\pi$
$911$	3.77532			0.125082			0.0625409	+	0.998042i	$0.480080\pi$
$912$		0			0
$913$	2.87340			0.0950956
$914$		0			0
$915$	96.5921			3.19324
$916$		0			0
$917$	−13.2306			−0.436913
$918$		0			0
$919$			28.6687i			0.945694i	0.881144	+	0.472847i	$0.156774\pi$
$919$			28.6687i			0.945694i	−0.881144	+	0.472847i	$0.843226\pi$
$920$		0			0
$921$	−27.7440			−0.914196
$922$		0			0
$923$			41.2217i			1.35683i
$924$		0			0
$925$			7.20910i			0.237034i
$926$		0			0
$927$	29.2402			0.960375
$928$		0			0
$929$	−9.15858			−0.300483			−0.150242	−	0.988649i	$-0.548005\pi$
$929$	−9.15858			−0.300483			−0.150242	+	0.988649i	$0.548005\pi$
$930$		0			0
$931$	42.6910	−	44.5231i	1.39914	−	1.45919i
$932$		0			0
$933$		−	19.3858i		−	0.634664i
$934$		0			0
$935$			23.5391i			0.769812i
$936$		0			0
$937$	−6.47477			−0.211522			−0.105761	−	0.994392i	$-0.533728\pi$
$937$	−6.47477			−0.211522			−0.105761	+	0.994392i	$0.533728\pi$
$938$		0			0
$939$	8.93722			0.291655
$940$		0			0
$941$		−	15.8807i		−	0.517696i	−0.965918	−	0.258848i	$-0.916657\pi$
$941$		−	15.8807i		−	0.517696i	0.965918	−	0.258848i	$-0.0833429\pi$
$942$		0			0
$943$	−9.86769			−0.321336
$944$		0			0
$945$			28.0455i			0.912320i
$946$		0			0
$947$			14.0731i			0.457314i	0.973507	+	0.228657i	$0.0734334\pi$
$947$			14.0731i			0.457314i	−0.973507	+	0.228657i	$0.926567\pi$
$948$		0			0
$949$		−	52.9821i		−	1.71987i
$950$		0			0
$951$		−	45.6699i		−	1.48095i
$952$		0			0
$953$			50.4548i			1.63439i	0.576361	+	0.817195i	$0.304472\pi$
$953$			50.4548i			1.63439i	−0.576361	+	0.817195i	$0.695528\pi$
$954$		0			0
$955$		−	62.5173i		−	2.02301i
$956$		0			0
$957$	−21.9928			−0.710926
$958$		0			0
$959$			31.5253i			1.01800i
$960$		0			0
$961$	54.2091			1.74868
$962$		0			0
$963$	−67.6226			−2.17911
$964$		0			0
$965$		−	52.3167i		−	1.68413i
$966$		0			0
$967$		−	13.7979i		−	0.443712i	−0.975079	−	0.221856i	$-0.928789\pi$
$967$		−	13.7979i		−	0.443712i	0.975079	−	0.221856i	$-0.0712115\pi$
$968$		0			0
$969$	53.5668	−	55.8657i	1.72081	−	1.79466i
$970$		0			0
$971$	22.7363			0.729642			0.364821	−	0.931078i	$-0.381130\pi$
$971$	22.7363			0.729642			0.364821	+	0.931078i	$0.381130\pi$
$972$		0			0
$973$	1.58345			0.0507630
$974$		0			0
$975$			97.9768i			3.13777i
$976$		0			0
$977$		−	23.8532i		−	0.763132i	−0.924342	−	0.381566i	$-0.875385\pi$
$977$		−	23.8532i		−	0.763132i	0.924342	−	0.381566i	$-0.124615\pi$
$978$		0			0
$979$	−10.3594			−0.331087
$980$		0			0
$981$			34.7982i			1.11102i
$982$		0			0
$983$	−48.5920			−1.54984			−0.774922	−	0.632057i	$-0.782211\pi$
$983$	−48.5920			−1.54984			−0.774922	+	0.632057i	$0.782211\pi$
$984$		0			0
$985$	59.7846			1.90489
$986$		0			0
$987$	−83.6140			−2.66146
$988$		0			0
$989$	−24.3699			−0.774918
$990$		0			0
$991$	2.71808			0.0863426			0.0431713	−	0.999068i	$-0.486254\pi$
$991$	2.71808			0.0863426			0.0431713	+	0.999068i	$0.486254\pi$
$992$		0			0
$993$	76.9561			2.44213
$994$		0			0
$995$		−	21.3850i		−	0.677951i
$996$		0			0
$997$	29.2623			0.926747			0.463373	−	0.886163i	$-0.346639\pi$
$997$	29.2623			0.926747			0.463373	+	0.886163i	$0.346639\pi$
$998$		0			0
$999$		−	1.89876i		−	0.0600742i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.o.b.1519.5	✓	64
4.3	odd	2	inner	3344.2.o.b.1519.60	yes	64
19.18	odd	2	inner	3344.2.o.b.1519.59	yes	64
76.75	even	2	inner	3344.2.o.b.1519.6	yes	64

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3344.2.o.b.1519.5	✓	64	1.1	even	1	trivial
3344.2.o.b.1519.6	yes	64	76.75	even	2	inner
3344.2.o.b.1519.59	yes	64	19.18	odd	2	inner
3344.2.o.b.1519.60	yes	64	4.3	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)

\(f(q)\)	\(=\)	\(q-2.58608 q^{3} +3.42833 q^{5} -4.59904i q^{7} +3.68782 q^{9} +O(q^{10})\)	\(q-2.58608 q^{3} +3.42833 q^{5} -4.59904i q^{7} +3.68782 q^{9} +1.00000i q^{11} -5.60992i q^{13} -8.86593 q^{15} +6.86607 q^{17} +(-3.01679 + 3.14626i) q^{19} +11.8935i q^{21} -4.41922i q^{23} +6.75343 q^{25} -1.77875 q^{27} -8.50430i q^{29} +9.23088 q^{31} -2.58608i q^{33} -15.7670i q^{35} +1.06747i q^{37} +14.5077i q^{39} -2.23290i q^{41} -5.51453i q^{43} +12.6430 q^{45} +7.03024i q^{47} -14.1511 q^{49} -17.7562 q^{51} +8.19169i q^{53} +3.42833i q^{55} +(7.80167 - 8.13649i) q^{57} +5.18092 q^{59} -10.8947 q^{61} -16.9604i q^{63} -19.2326i q^{65} +8.76156 q^{67} +11.4285i q^{69} -7.34801 q^{71} +9.44436 q^{73} -17.4649 q^{75} +4.59904 q^{77} +10.3209 q^{79} -6.46347 q^{81} -2.87340i q^{83} +23.5391 q^{85} +21.9928i q^{87} +10.3594i q^{89} -25.8002 q^{91} -23.8718 q^{93} +(-10.3426 + 10.7864i) q^{95} +5.06525i q^{97} +3.68782i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q + 56 q^{9}+O(q^{10}) \) 64 * q + 56 * q^9	\( 64 q + 56 q^{9} + 16 q^{17} + 64 q^{25} + 32 q^{45} - 88 q^{49} + 32 q^{57} + 64 q^{61} + 40 q^{73} - 48 q^{81} - 24 q^{85} + 80 q^{93}+O(q^{100}) \) 64 * q + 56 * q^9 + 16 * q^17 + 64 * q^25 + 32 * q^45 - 88 * q^49 + 32 * q^57 + 64 * q^61 + 40 * q^73 - 48 * q^81 - 24 * q^85 + 80 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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