LMFDB - Embedded newform 3344.2.o.b.1519.16

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			1519.16
Character	$\chi$	$=$	3344.1519
Dual form			3344.2.o.b.1519.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.05926 q^{3} +1.15495 q^{5} -3.33583i q^{7} +1.24056 q^{9} +O(q^{10})$	$q-2.05926 q^{3} +1.15495 q^{5} -3.33583i q^{7} +1.24056 q^{9} +1.00000i q^{11} -0.438879i q^{13} -2.37834 q^{15} +0.628386 q^{17} +(2.09122 - 3.82450i) q^{19} +6.86935i q^{21} -7.88621i q^{23} -3.66609 q^{25} +3.62314 q^{27} +9.16590i q^{29} +2.45383 q^{31} -2.05926i q^{33} -3.85271i q^{35} -6.19876i q^{37} +0.903767i q^{39} +4.92095i q^{41} -3.18129i q^{43} +1.43279 q^{45} -2.72277i q^{47} -4.12776 q^{49} -1.29401 q^{51} -7.38950i q^{53} +1.15495i q^{55} +(-4.30638 + 7.87565i) q^{57} -3.45208 q^{59} +5.99179 q^{61} -4.13831i q^{63} -0.506883i q^{65} +1.93662 q^{67} +16.2398i q^{69} -15.9863 q^{71} +11.0272 q^{73} +7.54945 q^{75} +3.33583 q^{77} -9.40120 q^{79} -11.1827 q^{81} +14.0247i q^{83} +0.725754 q^{85} -18.8750i q^{87} -1.93578i q^{89} -1.46403 q^{91} -5.05307 q^{93} +(2.41526 - 4.41710i) q^{95} +1.63984i q^{97} +1.24056i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.05926			−1.18892			−0.594458	−	0.804127i	$-0.702633\pi$
$3$	−2.05926			−1.18892			−0.594458	+	0.804127i	$0.702633\pi$
$4$		0			0
$5$	1.15495			0.516509			0.258254	−	0.966077i	$-0.416853\pi$
$5$	1.15495			0.516509			0.258254	+	0.966077i	$0.416853\pi$
$6$		0			0
$7$		−	3.33583i		−	1.26083i	−0.776260	−	0.630413i	$-0.782886\pi$
$7$		−	3.33583i		−	1.26083i	0.776260	−	0.630413i	$-0.217114\pi$
$8$		0			0
$9$	1.24056			0.413521
$10$		0			0
$11$			1.00000i			0.301511i
$11$			1.00000i			0.301511i
$12$		0			0
$13$		−	0.438879i		−	0.121723i	−0.998146	−	0.0608616i	$-0.980615\pi$
$13$		−	0.438879i		−	0.121723i	0.998146	−	0.0608616i	$-0.0193848\pi$
$14$		0			0
$15$	−2.37834			−0.614086
$16$		0			0
$17$	0.628386			0.152406			0.0762030	−	0.997092i	$-0.475720\pi$
$17$	0.628386			0.152406			0.0762030	+	0.997092i	$0.475720\pi$
$18$		0			0
$19$	2.09122	−	3.82450i	0.479760	−	0.877400i
$19$	2.09122	−	3.82450i	0.479760	−	0.877400i
$20$		0			0
$21$			6.86935i			1.49902i
$22$		0			0
$23$		−	7.88621i		−	1.64439i	−0.569207	−	0.822194i	$-0.692750\pi$
$23$		−	7.88621i		−	1.64439i	0.569207	−	0.822194i	$-0.307250\pi$
$24$		0			0
$25$	−3.66609			−0.733219
$26$		0			0
$27$	3.62314			0.697274
$28$		0			0
$29$			9.16590i			1.70207i	0.525113	+	0.851033i	$0.324023\pi$
$29$			9.16590i			1.70207i	−0.525113	+	0.851033i	$0.675977\pi$
$30$		0			0
$31$	2.45383			0.440720			0.220360	−	0.975419i	$-0.429277\pi$
$31$	2.45383			0.440720			0.220360	+	0.975419i	$0.429277\pi$
$32$		0			0
$33$		−	2.05926i		−	0.358472i
$34$		0			0
$35$		−	3.85271i		−	0.651227i
$36$		0			0
$37$		−	6.19876i		−	1.01907i	−0.860450	−	0.509535i	$-0.829817\pi$
$37$		−	6.19876i		−	1.01907i	0.860450	−	0.509535i	$-0.170183\pi$
$38$		0			0
$39$			0.903767i			0.144719i
$40$		0			0
$41$			4.92095i			0.768523i	0.923224	+	0.384262i	$0.125544\pi$
$41$			4.92095i			0.768523i	−0.923224	+	0.384262i	$0.874456\pi$
$42$		0			0
$43$		−	3.18129i		−	0.485142i	−0.970134	−	0.242571i	$-0.922009\pi$
$43$		−	3.18129i		−	0.485142i	0.970134	−	0.242571i	$-0.0779908\pi$
$44$		0			0
$45$	1.43279			0.213587
$46$		0			0
$47$		−	2.72277i		−	0.397156i	−0.980085	−	0.198578i	$-0.936368\pi$
$47$		−	2.72277i		−	0.397156i	0.980085	−	0.198578i	$-0.0636324\pi$
$48$		0			0
$49$	−4.12776			−0.589680
$50$		0			0
$51$	−1.29401			−0.181198
$52$		0			0
$53$		−	7.38950i		−	1.01503i	−0.861644	−	0.507513i	$-0.830565\pi$
$53$		−	7.38950i		−	1.01503i	0.861644	−	0.507513i	$-0.169435\pi$
$54$		0			0
$55$			1.15495i			0.155733i
$56$		0			0
$57$	−4.30638	+	7.87565i	−0.570394	+	1.04315i
$58$		0			0
$59$	−3.45208			−0.449423			−0.224711	−	0.974425i	$-0.572144\pi$
$59$	−3.45208			−0.449423			−0.224711	+	0.974425i	$0.572144\pi$
$60$		0			0
$61$	5.99179			0.767170			0.383585	−	0.923506i	$-0.374689\pi$
$61$	5.99179			0.767170			0.383585	+	0.923506i	$0.374689\pi$
$62$		0			0
$63$		−	4.13831i		−	0.521378i
$64$		0			0
$65$		−	0.506883i		−	0.0628711i
$66$		0			0
$67$	1.93662			0.236595			0.118298	−	0.992978i	$-0.462256\pi$
$67$	1.93662			0.236595			0.118298	+	0.992978i	$0.462256\pi$
$68$		0			0
$69$			16.2398i			1.95504i
$70$		0			0
$71$	−15.9863			−1.89722			−0.948612	−	0.316440i	$-0.897512\pi$
$71$	−15.9863			−1.89722			−0.948612	+	0.316440i	$0.897512\pi$
$72$		0			0
$73$	11.0272			1.29064			0.645321	−	0.763912i	$-0.276724\pi$
$73$	11.0272			1.29064			0.645321	+	0.763912i	$0.276724\pi$
$74$		0			0
$75$	7.54945			0.871735
$76$		0			0
$77$	3.33583			0.380153
$78$		0			0
$79$	−9.40120			−1.05772			−0.528859	−	0.848710i	$-0.677380\pi$
$79$	−9.40120			−1.05772			−0.528859	+	0.848710i	$0.677380\pi$
$80$		0			0
$81$	−11.1827			−1.24252
$82$		0			0
$83$			14.0247i			1.53941i	0.638399	+	0.769705i	$0.279597\pi$
$83$			14.0247i			1.53941i	−0.638399	+	0.769705i	$0.720403\pi$
$84$		0			0
$85$	0.725754			0.0787191
$86$		0			0
$87$		−	18.8750i		−	2.02361i
$88$		0			0
$89$		−	1.93578i		−	0.205192i	−0.994723	−	0.102596i	$-0.967285\pi$
$89$		−	1.93578i		−	0.205192i	0.994723	−	0.102596i	$-0.0327149\pi$
$90$		0			0
$91$	−1.46403			−0.153472
$92$		0			0
$93$	−5.05307			−0.523979
$94$		0			0
$95$	2.41526	−	4.41710i	0.247800	−	0.453185i
$96$		0			0
$97$			1.63984i			0.166501i	0.996529	+	0.0832504i	$0.0265301\pi$
$97$			1.63984i			0.166501i	−0.996529	+	0.0832504i	$0.973470\pi$
$98$		0			0
$99$			1.24056i			0.124681i
$100$		0			0
$101$	−4.01480			−0.399487			−0.199744	−	0.979848i	$-0.564011\pi$
$101$	−4.01480			−0.399487			−0.199744	+	0.979848i	$0.564011\pi$
$102$		0			0
$103$	6.60977			0.651280			0.325640	−	0.945494i	$-0.394420\pi$
$103$	6.60977			0.651280			0.325640	+	0.945494i	$0.394420\pi$
$104$		0			0
$105$			7.93375i			0.774254i
$106$		0			0
$107$	10.1663			0.982817			0.491408	−	0.870929i	$-0.336482\pi$
$107$	10.1663			0.982817			0.491408	+	0.870929i	$0.336482\pi$
$108$		0			0
$109$		−	8.04352i		−	0.770429i	−0.922827	−	0.385215i	$-0.874127\pi$
$109$		−	8.04352i		−	0.770429i	0.922827	−	0.385215i	$-0.125873\pi$
$110$		0			0
$111$			12.7649i			1.21159i
$112$		0			0
$113$			8.89097i			0.836392i	0.908357	+	0.418196i	$0.137337\pi$
$113$			8.89097i			0.836392i	−0.908357	+	0.418196i	$0.862663\pi$
$114$		0			0
$115$		−	9.10817i		−	0.849341i
$116$		0			0
$117$		−	0.544457i		−	0.0503351i
$118$		0			0
$119$		−	2.09619i		−	0.192157i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$		−	10.1335i		−	0.913710i
$124$		0			0
$125$	−10.0089			−0.895223
$126$		0			0
$127$	−1.86111			−0.165147			−0.0825736	−	0.996585i	$-0.526314\pi$
$127$	−1.86111			−0.165147			−0.0825736	+	0.996585i	$0.526314\pi$
$128$		0			0
$129$			6.55112i			0.576794i
$130$		0			0
$131$		−	9.01293i		−	0.787463i	−0.919225	−	0.393732i	$-0.871184\pi$
$131$		−	9.01293i		−	0.787463i	0.919225	−	0.393732i	$-0.128816\pi$
$132$		0			0
$133$	−12.7579	−	6.97597i	−1.10625	−	0.604893i
$134$		0			0
$135$	4.18454			0.360148
$136$		0			0
$137$	−10.4605			−0.893702			−0.446851	−	0.894608i	$-0.647455\pi$
$137$	−10.4605			−0.893702			−0.446851	+	0.894608i	$0.647455\pi$
$138$		0			0
$139$		−	0.859107i		−	0.0728685i	−0.999336	−	0.0364342i	$-0.988400\pi$
$139$		−	0.859107i		−	0.0728685i	0.999336	−	0.0364342i	$-0.0115999\pi$
$140$		0			0
$141$			5.60689i			0.472185i
$142$		0			0
$143$	0.438879			0.0367009
$144$		0			0
$145$			10.5861i			0.879132i
$146$		0			0
$147$	8.50014			0.701080
$148$		0			0
$149$	−1.74967			−0.143339			−0.0716694	−	0.997428i	$-0.522833\pi$
$149$	−1.74967			−0.143339			−0.0716694	+	0.997428i	$0.522833\pi$
$150$		0			0
$151$	−17.3778			−1.41419			−0.707094	−	0.707119i	$-0.749994\pi$
$151$	−17.3778			−1.41419			−0.707094	+	0.707119i	$0.749994\pi$
$152$		0			0
$153$	0.779553			0.0630231
$154$		0			0
$155$	2.83404			0.227636
$156$		0			0
$157$	−7.30193			−0.582758			−0.291379	−	0.956608i	$-0.594114\pi$
$157$	−7.30193			−0.582758			−0.291379	+	0.956608i	$0.594114\pi$
$158$		0			0
$159$			15.2169i			1.20678i
$160$		0			0
$161$	−26.3071			−2.07329
$162$		0			0
$163$		−	17.0228i		−	1.33333i	−0.745359	−	0.666663i	$-0.767722\pi$
$163$		−	17.0228i		−	1.33333i	0.745359	−	0.666663i	$-0.232278\pi$
$164$		0			0
$165$		−	2.37834i		−	0.185154i
$166$		0			0
$167$	−19.2107			−1.48657			−0.743283	−	0.668977i	$-0.766732\pi$
$167$	−19.2107			−1.48657			−0.743283	+	0.668977i	$0.766732\pi$
$168$		0			0
$169$	12.8074			0.985183
$170$		0			0
$171$	2.59430	−	4.74453i	0.198391	−	0.362823i
$172$		0			0
$173$		−	14.6398i		−	1.11304i	−0.830834	−	0.556520i	$-0.812136\pi$
$173$		−	14.6398i		−	1.11304i	0.830834	−	0.556520i	$-0.187864\pi$
$174$		0			0
$175$			12.2295i			0.924460i
$176$		0			0
$177$	7.10874			0.534326
$178$		0			0
$179$	−0.790702			−0.0590999			−0.0295499	−	0.999563i	$-0.509407\pi$
$179$	−0.790702			−0.0590999			−0.0295499	+	0.999563i	$0.509407\pi$
$180$		0			0
$181$		−	8.73624i		−	0.649360i	−0.945824	−	0.324680i	$-0.894743\pi$
$181$		−	8.73624i		−	0.649360i	0.945824	−	0.324680i	$-0.105257\pi$
$182$		0			0
$183$	−12.3387			−0.912100
$184$		0			0
$185$		−	7.15925i		−	0.526359i
$186$		0			0
$187$			0.628386i			0.0459521i
$188$		0			0
$189$		−	12.0862i		−	0.879141i
$190$		0			0
$191$			18.8820i			1.36625i	0.730299	+	0.683127i	$0.239381\pi$
$191$			18.8820i			1.36625i	−0.730299	+	0.683127i	$0.760619\pi$
$192$		0			0
$193$		−	1.43041i		−	0.102963i	−0.998674	−	0.0514815i	$-0.983606\pi$
$193$		−	1.43041i		−	0.102963i	0.998674	−	0.0514815i	$-0.0163943\pi$
$194$		0			0
$195$			1.04381i			0.0747484i
$196$		0			0
$197$	−7.91099			−0.563635			−0.281817	−	0.959468i	$-0.590937\pi$
$197$	−7.91099			−0.563635			−0.281817	+	0.959468i	$0.590937\pi$
$198$		0			0
$199$		−	24.6789i		−	1.74944i	−0.484625	−	0.874722i	$-0.661044\pi$
$199$		−	24.6789i		−	1.74944i	0.484625	−	0.874722i	$-0.338956\pi$
$200$		0			0
$201$	−3.98800			−0.281292
$202$		0			0
$203$	30.5759			2.14601
$204$		0			0
$205$			5.68344i			0.396949i
$206$		0			0
$207$		−	9.78335i		−	0.679990i
$208$		0			0
$209$	3.82450	+	2.09122i	0.264546	+	0.144653i
$210$		0			0
$211$	−22.0527			−1.51817			−0.759085	−	0.650992i	$-0.774353\pi$
$211$	−22.0527			−1.51817			−0.759085	+	0.650992i	$0.774353\pi$
$212$		0			0
$213$	32.9200			2.25564
$214$		0			0
$215$		−	3.67423i		−	0.250580i
$216$		0			0
$217$		−	8.18554i		−	0.555671i
$218$		0			0
$219$	−22.7080			−1.53446
$220$		0			0
$221$		−	0.275786i		−	0.0185513i
$222$		0			0
$223$	−28.4233			−1.90337			−0.951684	−	0.307081i	$-0.900648\pi$
$223$	−28.4233			−1.90337			−0.951684	+	0.307081i	$0.900648\pi$
$224$		0			0
$225$	−4.54802			−0.303201
$226$		0			0
$227$	−13.1099			−0.870133			−0.435067	−	0.900398i	$-0.643275\pi$
$227$	−13.1099			−0.870133			−0.435067	+	0.900398i	$0.643275\pi$
$228$		0			0
$229$	−11.8785			−0.784956			−0.392478	−	0.919761i	$-0.628382\pi$
$229$	−11.8785			−0.784956			−0.392478	+	0.919761i	$0.628382\pi$
$230$		0			0
$231$	−6.86935			−0.451970
$232$		0			0
$233$	−19.3337			−1.26659			−0.633296	−	0.773909i	$-0.718299\pi$
$233$	−19.3337			−1.26659			−0.633296	+	0.773909i	$0.718299\pi$
$234$		0			0
$235$		−	3.14466i		−	0.205135i
$236$		0			0
$237$	19.3596			1.25754
$238$		0			0
$239$		−	2.46930i		−	0.159726i	−0.996806	−	0.0798628i	$-0.974552\pi$
$239$		−	2.46930i		−	0.159726i	0.996806	−	0.0798628i	$-0.0254482\pi$
$240$		0			0
$241$		−	11.3596i		−	0.731733i	−0.930667	−	0.365867i	$-0.880773\pi$
$241$		−	11.3596i		−	0.731733i	0.930667	−	0.365867i	$-0.119227\pi$
$242$		0			0
$243$	12.1587			0.779979
$244$		0			0
$245$	−4.76735			−0.304575
$246$		0			0
$247$	−1.67849	−	0.917795i	−0.106800	−	0.0583979i
$248$		0			0
$249$		−	28.8805i		−	1.83023i
$250$		0			0
$251$		−	10.5064i		−	0.663159i	−0.943427	−	0.331579i	$-0.892418\pi$
$251$		−	10.5064i		−	0.663159i	0.943427	−	0.331579i	$-0.107582\pi$
$252$		0			0
$253$	7.88621			0.495802
$254$		0			0
$255$	−1.49452			−0.0935903
$256$		0			0
$257$			1.71575i			0.107026i	0.998567	+	0.0535128i	$0.0170418\pi$
$257$			1.71575i			0.107026i	−0.998567	+	0.0535128i	$0.982958\pi$
$258$		0			0
$259$	−20.6780			−1.28487
$260$		0			0
$261$			11.3709i			0.703840i
$262$		0			0
$263$		−	15.4366i		−	0.951859i	−0.879484	−	0.475929i	$-0.842112\pi$
$263$		−	15.4366i		−	0.951859i	0.879484	−	0.475929i	$-0.157888\pi$
$264$		0			0
$265$		−	8.53449i		−	0.524270i
$266$		0			0
$267$			3.98628i			0.243956i
$268$		0			0
$269$			12.9724i			0.790940i	0.918479	+	0.395470i	$0.129418\pi$
$269$			12.9724i			0.790940i	−0.918479	+	0.395470i	$0.870582\pi$
$270$		0			0
$271$		−	21.3658i		−	1.29788i	−0.760839	−	0.648941i	$-0.775212\pi$
$271$		−	21.3658i		−	1.29788i	0.760839	−	0.648941i	$-0.224788\pi$
$272$		0			0
$273$	3.01481			0.182465
$274$		0			0
$275$		−	3.66609i		−	0.221074i
$276$		0			0
$277$	31.1395			1.87099			0.935495	−	0.353340i	$-0.114954\pi$
$277$	31.1395			1.87099			0.935495	+	0.353340i	$0.114954\pi$
$278$		0			0
$279$	3.04413			0.182247
$280$		0			0
$281$		−	27.2910i		−	1.62805i	−0.580833	−	0.814023i	$-0.697273\pi$
$281$		−	27.2910i		−	1.62805i	0.580833	−	0.814023i	$-0.302727\pi$
$282$		0			0
$283$			1.06577i			0.0633534i	0.999498	+	0.0316767i	$0.0100847\pi$
$283$			1.06577i			0.0633534i	−0.999498	+	0.0316767i	$0.989915\pi$
$284$		0			0
$285$	−4.97365	+	9.09597i	−0.294614	+	0.538799i
$286$		0			0
$287$	16.4154			0.968973
$288$		0			0
$289$	−16.6051			−0.976772
$290$		0			0
$291$		−	3.37687i		−	0.197955i
$292$		0			0
$293$		−	19.2574i		−	1.12503i	−0.826788	−	0.562514i	$-0.809834\pi$
$293$		−	19.2574i		−	1.12503i	0.826788	−	0.562514i	$-0.190166\pi$
$294$		0			0
$295$	−3.98698			−0.232131
$296$		0			0
$297$			3.62314i			0.210236i
$298$		0			0
$299$	−3.46109			−0.200160
$300$		0			0
$301$	−10.6122			−0.611680
$302$		0			0
$303$	8.26752			0.474957
$304$		0			0
$305$	6.92021			0.396250
$306$		0			0
$307$	6.77196			0.386496			0.193248	−	0.981150i	$-0.438098\pi$
$307$	6.77196			0.386496			0.193248	+	0.981150i	$0.438098\pi$
$308$		0			0
$309$	−13.6113			−0.774317
$310$		0			0
$311$			25.1668i			1.42708i	0.700614	+	0.713541i	$0.252910\pi$
$311$			25.1668i			1.42708i	−0.700614	+	0.713541i	$0.747090\pi$
$312$		0			0
$313$	−14.1984			−0.802543			−0.401272	−	0.915959i	$-0.631432\pi$
$313$	−14.1984			−0.802543			−0.401272	+	0.915959i	$0.631432\pi$
$314$		0			0
$315$		−	4.77953i		−	0.269296i
$316$		0			0
$317$			6.27002i			0.352160i	0.984376	+	0.176080i	$0.0563417\pi$
$317$			6.27002i			0.352160i	−0.984376	+	0.176080i	$0.943658\pi$
$318$		0			0
$319$	−9.16590			−0.513192
$320$		0			0
$321$	−20.9352			−1.16849
$322$		0			0
$323$	1.31410	−	2.40326i	0.0731183	−	0.133721i
$324$		0			0
$325$			1.60897i			0.0892497i
$326$		0			0
$327$			16.5637i			0.915976i
$328$		0			0
$329$	−9.08268			−0.500745
$330$		0			0
$331$	12.5482			0.689714			0.344857	−	0.938655i	$-0.387927\pi$
$331$	12.5482			0.689714			0.344857	+	0.938655i	$0.387927\pi$
$332$		0			0
$333$		−	7.68996i		−	0.421407i
$334$		0			0
$335$	2.23669			0.122204
$336$		0			0
$337$			25.5757i			1.39320i	0.717460	+	0.696599i	$0.245305\pi$
$337$			25.5757i			1.39320i	−0.717460	+	0.696599i	$0.754695\pi$
$338$		0			0
$339$		−	18.3088i		−	0.994400i
$340$		0			0
$341$			2.45383i			0.132882i
$342$		0			0
$343$		−	9.58131i		−	0.517342i
$344$		0			0
$345$			18.7561i			1.00980i
$346$		0			0
$347$			5.30604i			0.284843i	0.989806	+	0.142422i	$0.0454889\pi$
$347$			5.30604i			0.284843i	−0.989806	+	0.142422i	$0.954511\pi$
$348$		0			0
$349$	2.61074			0.139750			0.0698748	−	0.997556i	$-0.477740\pi$
$349$	2.61074			0.139750			0.0698748	+	0.997556i	$0.477740\pi$
$350$		0			0
$351$		−	1.59012i		−	0.0848744i
$352$		0			0
$353$	12.7259			0.677334			0.338667	−	0.940906i	$-0.390024\pi$
$353$	12.7259			0.677334			0.338667	+	0.940906i	$0.390024\pi$
$354$		0			0
$355$	−18.4634			−0.979933
$356$		0			0
$357$			4.31660i			0.228459i
$358$		0			0
$359$			8.80214i			0.464559i	0.972649	+	0.232280i	$0.0746184\pi$
$359$			8.80214i			0.464559i	−0.972649	+	0.232280i	$0.925382\pi$
$360$		0			0
$361$	−10.2536	−	15.9958i	−0.539661	−	0.841882i
$362$		0			0
$363$	2.05926			0.108083
$364$		0			0
$365$	12.7359			0.666628
$366$		0			0
$367$			28.7525i			1.50087i	0.660944	+	0.750435i	$0.270156\pi$
$367$			28.7525i			1.50087i	−0.660944	+	0.750435i	$0.729844\pi$
$368$		0			0
$369$			6.10475i			0.317801i
$370$		0			0
$371$	−24.6501			−1.27977
$372$		0			0
$373$		−	25.8700i		−	1.33950i	−0.742587	−	0.669750i	$-0.766401\pi$
$373$		−	25.8700i		−	1.33950i	0.742587	−	0.669750i	$-0.233599\pi$
$374$		0			0
$375$	20.6109			1.06434
$376$		0			0
$377$	4.02272			0.207181
$378$		0			0
$379$	35.8001			1.83893			0.919465	−	0.393172i	$-0.128622\pi$
$379$	35.8001			1.83893			0.919465	+	0.393172i	$0.128622\pi$
$380$		0			0
$381$	3.83252			0.196346
$382$		0			0
$383$	−36.3538			−1.85759			−0.928797	−	0.370588i	$-0.879156\pi$
$383$	−36.3538			−1.85759			−0.928797	+	0.370588i	$0.879156\pi$
$384$		0			0
$385$	3.85271			0.196352
$386$		0			0
$387$		−	3.94659i		−	0.200617i
$388$		0			0
$389$	6.91916			0.350815			0.175408	−	0.984496i	$-0.443876\pi$
$389$	6.91916			0.350815			0.175408	+	0.984496i	$0.443876\pi$
$390$		0			0
$391$		−	4.95559i		−	0.250615i
$392$		0			0
$393$			18.5600i			0.936227i
$394$		0			0
$395$	−10.8579			−0.546321
$396$		0			0
$397$	−0.112151			−0.00562868			−0.00281434	−	0.999996i	$-0.500896\pi$
$397$	−0.112151			−0.00562868			−0.00281434	+	0.999996i	$0.500896\pi$
$398$		0			0
$399$	26.2718	+	14.3654i	1.31524	+	0.719167i
$400$		0			0
$401$			2.40909i			0.120304i	0.998189	+	0.0601520i	$0.0191585\pi$
$401$			2.40909i			0.120304i	−0.998189	+	0.0601520i	$0.980841\pi$
$402$		0			0
$403$		−	1.07693i		−	0.0536458i
$404$		0			0
$405$	−12.9154			−0.641773
$406$		0			0
$407$	6.19876			0.307261
$408$		0			0
$409$			26.9890i			1.33452i	0.744825	+	0.667260i	$0.232533\pi$
$409$			26.9890i			1.33452i	−0.744825	+	0.667260i	$0.767467\pi$
$410$		0			0
$411$	21.5409			1.06254
$412$		0			0
$413$			11.5156i			0.566643i
$414$		0			0
$415$			16.1978i			0.795119i
$416$		0			0
$417$			1.76913i			0.0866345i
$418$		0			0
$419$			13.6549i			0.667088i	0.942735	+	0.333544i	$0.108245\pi$
$419$			13.6549i			0.667088i	−0.942735	+	0.333544i	$0.891755\pi$
$420$		0			0
$421$		−	9.09049i		−	0.443043i	−0.975155	−	0.221522i	$-0.928898\pi$
$421$		−	9.09049i		−	0.443043i	0.975155	−	0.221522i	$-0.0711024\pi$
$422$		0			0
$423$		−	3.37776i		−	0.164233i
$424$		0			0
$425$	−2.30372			−0.111747
$426$		0			0
$427$		−	19.9876i		−	0.967267i
$428$		0			0
$429$	−0.903767			−0.0436343
$430$		0			0
$431$	−15.5397			−0.748521			−0.374260	−	0.927324i	$-0.622103\pi$
$431$	−15.5397			−0.748521			−0.374260	+	0.927324i	$0.622103\pi$
$432$		0			0
$433$			15.2197i			0.731413i	0.930730	+	0.365707i	$0.119173\pi$
$433$			15.2197i			0.731413i	−0.930730	+	0.365707i	$0.880827\pi$
$434$		0			0
$435$		−	21.7997i		−	1.04521i
$436$		0			0
$437$	−30.1608	−	16.4918i	−1.44279	−	0.788912i
$438$		0			0
$439$	38.4045			1.83295			0.916474	−	0.400095i	$-0.131023\pi$
$439$	38.4045			1.83295			0.916474	+	0.400095i	$0.131023\pi$
$440$		0			0
$441$	−5.12075			−0.243845
$442$		0			0
$443$			32.4344i			1.54100i	0.637438	+	0.770502i	$0.279994\pi$
$443$			32.4344i			1.54100i	−0.637438	+	0.770502i	$0.720006\pi$
$444$		0			0
$445$		−	2.23573i		−	0.105984i
$446$		0			0
$447$	3.60304			0.170418
$448$		0			0
$449$		−	19.8995i		−	0.939117i	−0.882901	−	0.469559i	$-0.844413\pi$
$449$		−	19.8995i		−	0.939117i	0.882901	−	0.469559i	$-0.155587\pi$
$450$		0			0
$451$	−4.92095			−0.231718
$452$		0			0
$453$	35.7855			1.68135
$454$		0			0
$455$	−1.69087			−0.0792694
$456$		0			0
$457$	−2.47108			−0.115592			−0.0577961	−	0.998328i	$-0.518407\pi$
$457$	−2.47108			−0.115592			−0.0577961	+	0.998328i	$0.518407\pi$
$458$		0			0
$459$	2.27673			0.106269
$460$		0			0
$461$	11.8012			0.549635			0.274818	−	0.961496i	$-0.411383\pi$
$461$	11.8012			0.549635			0.274818	+	0.961496i	$0.411383\pi$
$462$		0			0
$463$			1.57503i			0.0731979i	0.999330	+	0.0365989i	$0.0116524\pi$
$463$			1.57503i			0.0731979i	−0.999330	+	0.0365989i	$0.988348\pi$
$464$		0			0
$465$	−5.83604			−0.270640
$466$		0			0
$467$			25.3257i			1.17193i	0.810335	+	0.585967i	$0.199285\pi$
$467$			25.3257i			1.17193i	−0.810335	+	0.585967i	$0.800715\pi$
$468$		0			0
$469$		−	6.46022i		−	0.298305i
$470$		0			0
$471$	15.0366			0.692850
$472$		0			0
$473$	3.18129			0.146276
$474$		0			0
$475$	−7.66663	+	14.0210i	−0.351769	+	0.643326i
$476$		0			0
$477$		−	9.16714i		−	0.419735i
$478$		0			0
$479$			3.13259i			0.143132i	0.997436	+	0.0715659i	$0.0227996\pi$
$479$			3.13259i			0.143132i	−0.997436	+	0.0715659i	$0.977200\pi$
$480$		0			0
$481$	−2.72051			−0.124044
$482$		0			0
$483$	54.1731			2.46496
$484$		0			0
$485$			1.89393i			0.0859991i
$486$		0			0
$487$	−8.45280			−0.383033			−0.191516	−	0.981489i	$-0.561341\pi$
$487$	−8.45280			−0.383033			−0.191516	+	0.981489i	$0.561341\pi$
$488$		0			0
$489$			35.0544i			1.58521i
$490$		0			0
$491$		−	31.6638i		−	1.42897i	−0.699652	−	0.714484i	$-0.746662\pi$
$491$		−	31.6638i		−	1.42897i	0.699652	−	0.714484i	$-0.253338\pi$
$492$		0			0
$493$			5.75973i			0.259405i
$494$		0			0
$495$			1.43279i			0.0643990i
$496$		0			0
$497$			53.3276i			2.39207i
$498$		0			0
$499$			10.7339i			0.480517i	0.970709	+	0.240259i	$0.0772323\pi$
$499$			10.7339i			0.480517i	−0.970709	+	0.240259i	$0.922768\pi$
$500$		0			0
$501$	39.5598			1.76740
$502$		0			0
$503$		−	9.70735i		−	0.432829i	−0.976302	−	0.216415i	$-0.930564\pi$
$503$		−	9.70735i		−	0.432829i	0.976302	−	0.216415i	$-0.0694363\pi$
$504$		0			0
$505$	−4.63688			−0.206339
$506$		0			0
$507$	−26.3738			−1.17130
$508$		0			0
$509$			19.3526i			0.857790i	0.903354	+	0.428895i	$0.141097\pi$
$509$			19.3526i			0.857790i	−0.903354	+	0.428895i	$0.858903\pi$
$510$		0			0
$511$		−	36.7850i		−	1.62727i
$512$		0			0
$513$	7.57680	−	13.8567i	0.334524	−	0.611788i
$514$		0			0
$515$	7.63394			0.336392
$516$		0			0
$517$	2.72277			0.119747
$518$		0			0
$519$			30.1471i			1.32331i
$520$		0			0
$521$		−	32.1279i		−	1.40755i	−0.710424	−	0.703774i	$-0.751497\pi$
$521$		−	32.1279i		−	1.40755i	0.710424	−	0.703774i	$-0.248503\pi$
$522$		0			0
$523$	−45.2757			−1.97977			−0.989885	−	0.141873i	$-0.954688\pi$
$523$	−45.2757			−1.97977			−0.989885	+	0.141873i	$0.954688\pi$
$524$		0			0
$525$		−	25.1837i		−	1.09911i
$526$		0			0
$527$	1.54195			0.0671684
$528$		0			0
$529$	−39.1923			−1.70401
$530$		0			0
$531$	−4.28253			−0.185846
$532$		0			0
$533$	2.15970			0.0935471
$534$		0			0
$535$	11.7416			0.507634
$536$		0			0
$537$	1.62826			0.0702648
$538$		0			0
$539$		−	4.12776i		−	0.177795i
$540$		0			0
$541$	12.7367			0.547596			0.273798	−	0.961787i	$-0.411720\pi$
$541$	12.7367			0.547596			0.273798	+	0.961787i	$0.411720\pi$
$542$		0			0
$543$			17.9902i			0.772034i
$544$		0			0
$545$		−	9.28985i		−	0.397933i
$546$		0			0
$547$	3.74591			0.160164			0.0800818	−	0.996788i	$-0.474482\pi$
$547$	3.74591			0.160164			0.0800818	+	0.996788i	$0.474482\pi$
$548$		0			0
$549$	7.43319			0.317241
$550$		0			0
$551$	35.0550	+	19.1680i	1.49339	+	0.816583i
$552$		0			0
$553$			31.3608i			1.33360i
$554$		0			0
$555$			14.7428i			0.625796i
$556$		0			0
$557$	10.8698			0.460568			0.230284	−	0.973123i	$-0.426034\pi$
$557$	10.8698			0.460568			0.230284	+	0.973123i	$0.426034\pi$
$558$		0			0
$559$	−1.39620			−0.0590531
$560$		0			0
$561$		−	1.29401i		−	0.0546332i
$562$		0			0
$563$	−23.4331			−0.987588			−0.493794	−	0.869579i	$-0.664390\pi$
$563$	−23.4331			−0.987588			−0.493794	+	0.869579i	$0.664390\pi$
$564$		0			0
$565$			10.2686i			0.432004i
$566$		0			0
$567$			37.3036i			1.56660i
$568$		0			0
$569$			35.4075i			1.48436i	0.670200	+	0.742181i	$0.266208\pi$
$569$			35.4075i			1.48436i	−0.670200	+	0.742181i	$0.733792\pi$
$570$		0			0
$571$			3.51720i			0.147190i	0.997288	+	0.0735951i	$0.0234472\pi$
$571$			3.51720i			0.147190i	−0.997288	+	0.0735951i	$0.976553\pi$
$572$		0			0
$573$		−	38.8830i		−	1.62436i
$574$		0			0
$575$			28.9116i			1.20570i
$576$		0			0
$577$	−18.2987			−0.761783			−0.380892	−	0.924620i	$-0.624383\pi$
$577$	−18.2987			−0.761783			−0.380892	+	0.924620i	$0.624383\pi$
$578$		0			0
$579$			2.94559i			0.122414i
$580$		0			0
$581$	46.7840			1.94093
$582$		0			0
$583$	7.38950			0.306042
$584$		0			0
$585$		−	0.628820i		−	0.0259985i
$586$		0			0
$587$		−	12.8373i		−	0.529851i	−0.964269	−	0.264926i	$-0.914653\pi$
$587$		−	12.8373i		−	0.529851i	0.964269	−	0.264926i	$-0.0853474\pi$
$588$		0			0
$589$	5.13150	−	9.38465i	0.211440	−	0.386688i
$590$		0			0
$591$	16.2908			0.670114
$592$		0			0
$593$	29.4733			1.21032			0.605161	−	0.796103i	$-0.293109\pi$
$593$	29.4733			1.21032			0.605161	+	0.796103i	$0.293109\pi$
$594$		0			0
$595$		−	2.42099i		−	0.0992510i
$596$		0			0
$597$			50.8204i			2.07994i
$598$		0			0
$599$	31.1060			1.27096			0.635478	−	0.772119i	$-0.280803\pi$
$599$	31.1060			1.27096			0.635478	+	0.772119i	$0.280803\pi$
$600$		0			0
$601$			17.9615i			0.732663i	0.930484	+	0.366332i	$0.119386\pi$
$601$			17.9615i			0.732663i	−0.930484	+	0.366332i	$0.880614\pi$
$602$		0			0
$603$	2.40250			0.0978372
$604$		0			0
$605$	−1.15495			−0.0469553
$606$		0			0
$607$	26.4120			1.07203			0.536014	−	0.844209i	$-0.319929\pi$
$607$	26.4120			1.07203			0.536014	+	0.844209i	$0.319929\pi$
$608$		0			0
$609$	−62.9638			−2.55142
$610$		0			0
$611$	−1.19497			−0.0483431
$612$		0			0
$613$	−21.2184			−0.857001			−0.428501	−	0.903541i	$-0.640958\pi$
$613$	−21.2184			−0.857001			−0.428501	+	0.903541i	$0.640958\pi$
$614$		0			0
$615$		−	11.7037i		−	0.471939i
$616$		0			0
$617$	24.0779			0.969340			0.484670	−	0.874697i	$-0.338940\pi$
$617$	24.0779			0.969340			0.484670	+	0.874697i	$0.338940\pi$
$618$		0			0
$619$		−	22.0401i		−	0.885867i	−0.896554	−	0.442934i	$-0.853938\pi$
$619$		−	22.0401i		−	0.885867i	0.896554	−	0.442934i	$-0.146062\pi$
$620$		0			0
$621$		−	28.5729i		−	1.14659i
$622$		0			0
$623$	−6.45743			−0.258711
$624$		0			0
$625$	6.77071			0.270828
$626$		0			0
$627$	−7.87565	−	4.30638i	−0.314523	−	0.171980i
$628$		0			0
$629$		−	3.89522i		−	0.155312i
$630$		0			0
$631$			7.19738i			0.286523i	0.989685	+	0.143262i	$0.0457590\pi$
$631$			7.19738i			0.286523i	−0.989685	+	0.143262i	$0.954241\pi$
$632$		0			0
$633$	45.4123			1.80498
$634$		0			0
$635$	−2.14949			−0.0852999
$636$		0			0
$637$			1.81159i			0.0717777i
$638$		0			0
$639$	−19.8320			−0.784543
$640$		0			0
$641$		−	3.29440i		−	0.130121i	−0.997881	−	0.0650605i	$-0.979276\pi$
$641$		−	3.29440i		−	0.130121i	0.997881	−	0.0650605i	$-0.0207240\pi$
$642$		0			0
$643$			29.7401i			1.17284i	0.810009	+	0.586418i	$0.199462\pi$
$643$			29.7401i			1.17284i	−0.810009	+	0.586418i	$0.800538\pi$
$644$		0			0
$645$			7.56620i			0.297919i
$646$		0			0
$647$		−	16.4430i		−	0.646441i	−0.946324	−	0.323220i	$-0.895234\pi$
$647$		−	16.4430i		−	0.646441i	0.946324	−	0.323220i	$-0.104766\pi$
$648$		0			0
$649$		−	3.45208i		−	0.135506i
$650$		0			0
$651$			16.8562i			0.660646i
$652$		0			0
$653$	45.6444			1.78620			0.893101	−	0.449857i	$-0.148525\pi$
$653$	45.6444			1.78620			0.893101	+	0.449857i	$0.148525\pi$
$654$		0			0
$655$		−	10.4095i		−	0.406732i
$656$		0			0
$657$	13.6800			0.533708
$658$		0			0
$659$	25.0545			0.975986			0.487993	−	0.872847i	$-0.337729\pi$
$659$	25.0545			0.975986			0.487993	+	0.872847i	$0.337729\pi$
$660$		0			0
$661$			10.1509i			0.394823i	0.980321	+	0.197412i	$0.0632535\pi$
$661$			10.1509i			0.394823i	−0.980321	+	0.197412i	$0.936746\pi$
$662$		0			0
$663$			0.567915i			0.0220560i
$664$		0			0
$665$	−14.7347	−	8.05689i	−0.571387	−	0.312433i
$666$		0			0
$667$	72.2842			2.79886
$668$		0			0
$669$	58.5311			2.26294
$670$		0			0
$671$			5.99179i			0.231310i
$672$		0			0
$673$		−	17.3138i		−	0.667399i	−0.942680	−	0.333699i	$-0.891703\pi$
$673$		−	17.3138i		−	0.667399i	0.942680	−	0.333699i	$-0.108297\pi$
$674$		0			0
$675$	−13.2828			−0.511254
$676$		0			0
$677$		−	14.5078i		−	0.557578i	−0.960352	−	0.278789i	$-0.910067\pi$
$677$		−	14.5078i		−	0.557578i	0.960352	−	0.278789i	$-0.0899331\pi$
$678$		0			0
$679$	5.47023			0.209928
$680$		0			0
$681$	26.9967			1.03452
$682$		0			0
$683$	50.1834			1.92021			0.960107	−	0.279634i	$-0.0902133\pi$
$683$	50.1834			1.92021			0.960107	+	0.279634i	$0.0902133\pi$
$684$		0			0
$685$	−12.0814			−0.461605
$686$		0			0
$687$	24.4610			0.933247
$688$		0			0
$689$	−3.24310			−0.123552
$690$		0			0
$691$			22.7846i			0.866769i	0.901209	+	0.433384i	$0.142681\pi$
$691$			22.7846i			0.866769i	−0.901209	+	0.433384i	$0.857319\pi$
$692$		0			0
$693$	4.13831			0.157201
$694$		0			0
$695$		−	0.992224i		−	0.0376372i
$696$		0			0
$697$			3.09226i			0.117128i
$698$		0			0
$699$	39.8132			1.50587
$700$		0			0
$701$	−5.96283			−0.225213			−0.112607	−	0.993640i	$-0.535920\pi$
$701$	−5.96283			−0.225213			−0.112607	+	0.993640i	$0.535920\pi$
$702$		0			0
$703$	−23.7071	−	12.9630i	−0.894132	−	0.488909i
$704$		0			0
$705$			6.47567i			0.243888i
$706$		0			0
$707$			13.3927i			0.503683i
$708$		0			0
$709$	17.1943			0.645747			0.322874	−	0.946442i	$-0.395351\pi$
$709$	17.1943			0.645747			0.322874	+	0.946442i	$0.395351\pi$
$710$		0			0
$711$	−11.6628			−0.437389
$712$		0			0
$713$		−	19.3514i		−	0.724715i
$714$		0			0
$715$	0.506883			0.0189563
$716$		0			0
$717$			5.08493i			0.189900i
$718$		0			0
$719$		−	21.0903i		−	0.786536i	−0.919424	−	0.393268i	$-0.871345\pi$
$719$		−	21.0903i		−	0.786536i	0.919424	−	0.393268i	$-0.128655\pi$
$720$		0			0
$721$		−	22.0491i		−	0.821150i
$722$		0			0
$723$			23.3923i			0.869969i
$724$		0			0
$725$		−	33.6031i		−	1.24799i
$726$		0			0
$727$		−	36.4200i		−	1.35074i	−0.737478	−	0.675371i	$-0.763984\pi$
$727$		−	36.4200i		−	1.35074i	0.737478	−	0.675371i	$-0.236016\pi$
$728$		0			0
$729$	8.51017			0.315191
$730$		0			0
$731$		−	1.99908i		−	0.0739386i
$732$		0			0
$733$	33.5457			1.23904			0.619519	−	0.784982i	$-0.287328\pi$
$733$	33.5457			1.23904			0.619519	+	0.784982i	$0.287328\pi$
$734$		0			0
$735$	9.81723			0.362114
$736$		0			0
$737$			1.93662i			0.0713362i
$738$		0			0
$739$			15.4951i			0.569996i	0.958528	+	0.284998i	$0.0919930\pi$
$739$			15.4951i			0.569996i	−0.958528	+	0.284998i	$0.908007\pi$
$740$		0			0
$741$	3.45646	+	1.88998i	0.126976	+	0.0694302i
$742$		0			0
$743$	18.2392			0.669131			0.334566	−	0.942372i	$-0.391410\pi$
$743$	18.2392			0.669131			0.334566	+	0.942372i	$0.391410\pi$
$744$		0			0
$745$	−2.02078			−0.0740357
$746$		0			0
$747$			17.3985i			0.636579i
$748$		0			0
$749$		−	33.9132i		−	1.23916i
$750$		0			0
$751$	−28.0458			−1.02341			−0.511703	−	0.859162i	$-0.670985\pi$
$751$	−28.0458			−1.02341			−0.511703	+	0.859162i	$0.670985\pi$
$752$		0			0
$753$			21.6355i			0.788440i
$754$		0			0
$755$	−20.0705			−0.730441
$756$		0			0
$757$	38.2523			1.39030			0.695151	−	0.718864i	$-0.255338\pi$
$757$	38.2523			1.39030			0.695151	+	0.718864i	$0.255338\pi$
$758$		0			0
$759$	−16.2398			−0.589467
$760$		0			0
$761$	34.4814			1.24995			0.624974	−	0.780645i	$-0.285109\pi$
$761$	34.4814			1.24995			0.624974	+	0.780645i	$0.285109\pi$
$762$		0			0
$763$	−26.8318			−0.971376
$764$		0			0
$765$	0.900344			0.0325520
$766$		0			0
$767$			1.51505i			0.0547052i
$768$		0			0
$769$	−42.6049			−1.53637			−0.768187	−	0.640226i	$-0.778841\pi$
$769$	−42.6049			−1.53637			−0.768187	+	0.640226i	$0.778841\pi$
$770$		0			0
$771$		−	3.53318i		−	0.127244i
$772$		0			0
$773$			25.1025i			0.902873i	0.892303	+	0.451437i	$0.149088\pi$
$773$			25.1025i			0.902873i	−0.892303	+	0.451437i	$0.850912\pi$
$774$		0			0
$775$	−8.99595			−0.323144
$776$		0			0
$777$	42.5815			1.52760
$778$		0			0
$779$	18.8202	+	10.2908i	0.674302	+	0.368707i
$780$		0			0
$781$		−	15.9863i		−	0.572035i
$782$		0			0
$783$			33.2094i			1.18681i
$784$		0			0
$785$	−8.43336			−0.300999
$786$		0			0
$787$	−17.4695			−0.622722			−0.311361	−	0.950292i	$-0.600785\pi$
$787$	−17.4695			−0.622722			−0.311361	+	0.950292i	$0.600785\pi$
$788$		0			0
$789$			31.7879i			1.13168i
$790$		0			0
$791$	29.6588			1.05454
$792$		0			0
$793$		−	2.62967i		−	0.0933823i
$794$		0			0
$795$			17.5748i			0.623313i
$796$		0			0
$797$		−	41.2762i		−	1.46208i	−0.682337	−	0.731038i	$-0.739036\pi$
$797$		−	41.2762i		−	1.46208i	0.682337	−	0.731038i	$-0.260964\pi$
$798$		0			0
$799$		−	1.71095i		−	0.0605290i
$800$		0			0
$801$		−	2.40146i		−	0.0848513i
$802$		0			0
$803$			11.0272i			0.389143i
$804$		0			0
$805$	−30.3833			−1.07087
$806$		0			0
$807$		−	26.7135i		−	0.940361i
$808$		0			0
$809$	24.0018			0.843858			0.421929	−	0.906629i	$-0.361353\pi$
$809$	24.0018			0.843858			0.421929	+	0.906629i	$0.361353\pi$
$810$		0			0
$811$	17.1892			0.603596			0.301798	−	0.953372i	$-0.402413\pi$
$811$	17.1892			0.603596			0.301798	+	0.953372i	$0.402413\pi$
$812$		0			0
$813$			43.9979i			1.54307i
$814$		0			0
$815$		−	19.6604i		−	0.688675i
$816$		0			0
$817$	−12.1668	−	6.65280i	−0.425664	−	0.232752i
$818$		0			0
$819$	−1.81622			−0.0634638
$820$		0			0
$821$	12.0198			0.419495			0.209747	−	0.977756i	$-0.432736\pi$
$821$	12.0198			0.419495			0.209747	+	0.977756i	$0.432736\pi$
$822$		0			0
$823$			23.6635i			0.824857i	0.910990	+	0.412429i	$0.135319\pi$
$823$			23.6635i			0.824857i	−0.910990	+	0.412429i	$0.864681\pi$
$824$		0			0
$825$			7.54945i			0.262838i
$826$		0			0
$827$	−30.9553			−1.07642			−0.538211	−	0.842810i	$-0.680899\pi$
$827$	−30.9553			−1.07642			−0.538211	+	0.842810i	$0.680899\pi$
$828$		0			0
$829$		−	10.7836i		−	0.374529i	−0.982310	−	0.187264i	$-0.940038\pi$
$829$		−	10.7836i		−	0.374529i	0.982310	−	0.187264i	$-0.0599621\pi$
$830$		0			0
$831$	−64.1244			−2.22445
$832$		0			0
$833$	−2.59383			−0.0898708
$834$		0			0
$835$	−22.1873			−0.767825
$836$		0			0
$837$	8.89056			0.307303
$838$		0			0
$839$	−9.16030			−0.316249			−0.158124	−	0.987419i	$-0.550545\pi$
$839$	−9.16030			−0.316249			−0.158124	+	0.987419i	$0.550545\pi$
$840$		0			0
$841$	−55.0138			−1.89703
$842$		0			0
$843$			56.1994i			1.93561i
$844$		0			0
$845$	14.7919			0.508856
$846$		0			0
$847$			3.33583i			0.114620i
$848$		0			0
$849$		−	2.19470i		−	0.0753219i
$850$		0			0
$851$	−48.8847			−1.67575
$852$		0			0
$853$	4.11144			0.140773			0.0703864	−	0.997520i	$-0.477577\pi$
$853$	4.11144			0.140773			0.0703864	+	0.997520i	$0.477577\pi$
$854$		0			0
$855$	2.99628	−	5.47969i	0.102471	−	0.187401i
$856$		0			0
$857$		−	17.8500i		−	0.609746i	−0.952393	−	0.304873i	$-0.901386\pi$
$857$		−	17.8500i		−	0.609746i	0.952393	−	0.304873i	$-0.0986140\pi$
$858$		0			0
$859$			10.5071i			0.358497i	0.983804	+	0.179248i	$0.0573665\pi$
$859$			10.5071i			0.358497i	−0.983804	+	0.179248i	$0.942633\pi$
$860$		0			0
$861$	−33.8037			−1.15203
$862$		0			0
$863$	−0.639693			−0.0217754			−0.0108877	−	0.999941i	$-0.503466\pi$
$863$	−0.639693			−0.0217754			−0.0108877	+	0.999941i	$0.503466\pi$
$864$		0			0
$865$		−	16.9082i		−	0.574895i
$866$		0			0
$867$	34.1943			1.16130
$868$		0			0
$869$		−	9.40120i		−	0.318914i
$870$		0			0
$871$		−	0.849940i		−	0.0287991i
$872$		0			0
$873$			2.03433i			0.0688516i
$874$		0			0
$875$			33.3880i			1.12872i
$876$		0			0
$877$			9.80897i			0.331225i	0.986191	+	0.165613i	$0.0529602\pi$
$877$			9.80897i			0.331225i	−0.986191	+	0.165613i	$0.947040\pi$
$878$		0			0
$879$			39.6560i			1.33756i
$880$		0			0
$881$	2.97369			0.100186			0.0500932	−	0.998745i	$-0.484048\pi$
$881$	2.97369			0.100186			0.0500932	+	0.998745i	$0.484048\pi$
$882$		0			0
$883$			14.4137i			0.485060i	0.970144	+	0.242530i	$0.0779773\pi$
$883$			14.4137i			0.485060i	−0.970144	+	0.242530i	$0.922023\pi$
$884$		0			0
$885$	8.21023			0.275984
$886$		0			0
$887$	37.7954			1.26905			0.634523	−	0.772904i	$-0.281197\pi$
$887$	37.7954			1.26905			0.634523	+	0.772904i	$0.281197\pi$
$888$		0			0
$889$			6.20836i			0.208222i
$890$		0			0
$891$		−	11.1827i		−	0.374634i
$892$		0			0
$893$	−10.4132	−	5.69392i	−0.348465	−	0.190540i
$894$		0			0
$895$	−0.913220			−0.0305256
$896$		0			0
$897$	7.12730			0.237974
$898$		0			0
$899$			22.4915i			0.750134i
$900$		0			0
$901$		−	4.64346i		−	0.154696i
$902$		0			0
$903$	21.8534			0.727236
$904$		0			0
$905$		−	10.0899i		−	0.335400i
$906$		0			0
$907$	−57.6959			−1.91576			−0.957880	−	0.287170i	$-0.907286\pi$
$907$	−57.6959			−1.91576			−0.957880	+	0.287170i	$0.907286\pi$
$908$		0			0
$909$	−4.98061			−0.165196
$910$		0			0
$911$	43.8228			1.45191			0.725957	−	0.687741i	$-0.241397\pi$
$911$	43.8228			1.45191			0.725957	+	0.687741i	$0.241397\pi$
$912$		0			0
$913$	−14.0247			−0.464150
$914$		0			0
$915$	−14.2505			−0.471108
$916$		0			0
$917$	−30.0656			−0.992853
$918$		0			0
$919$		−	58.1902i		−	1.91952i	−0.280825	−	0.959759i	$-0.590608\pi$
$919$		−	58.1902i		−	1.91952i	0.280825	−	0.959759i	$-0.409392\pi$
$920$		0			0
$921$	−13.9452			−0.459511
$922$		0			0
$923$			7.01605i			0.230936i
$924$		0			0
$925$			22.7252i			0.747201i
$926$		0			0
$927$	8.19984			0.269318
$928$		0			0
$929$	−33.3581			−1.09445			−0.547223	−	0.836987i	$-0.684315\pi$
$929$	−33.3581			−1.09445			−0.547223	+	0.836987i	$0.684315\pi$
$930$		0			0
$931$	−8.63207	+	15.7866i	−0.282905	+	0.517385i
$932$		0			0
$933$		−	51.8252i		−	1.69668i
$934$		0			0
$935$			0.725754i			0.0237347i
$936$		0			0
$937$	−26.7392			−0.873530			−0.436765	−	0.899576i	$-0.643876\pi$
$937$	−26.7392			−0.873530			−0.436765	+	0.899576i	$0.643876\pi$
$938$		0			0
$939$	29.2383			0.954157
$940$		0			0
$941$		−	12.9752i		−	0.422979i	−0.977380	−	0.211490i	$-0.932169\pi$
$941$		−	12.9752i		−	0.422979i	0.977380	−	0.211490i	$-0.0678314\pi$
$942$		0			0
$943$	38.8077			1.26375
$944$		0			0
$945$		−	13.9589i		−	0.454084i
$946$		0			0
$947$			6.93476i			0.225349i	0.993632	+	0.112675i	$0.0359418\pi$
$947$			6.93476i			0.225349i	−0.993632	+	0.112675i	$0.964058\pi$
$948$		0			0
$949$		−	4.83963i		−	0.157101i
$950$		0			0
$951$		−	12.9116i		−	0.418688i
$952$		0			0
$953$		−	43.0851i		−	1.39566i	−0.716261	−	0.697832i	$-0.754148\pi$
$953$		−	43.0851i		−	1.39566i	0.716261	−	0.697832i	$-0.245852\pi$
$954$		0			0
$955$			21.8078i			0.705682i
$956$		0			0
$957$	18.8750			0.610142
$958$		0			0
$959$			34.8945i			1.12680i
$960$		0			0
$961$	−24.9787			−0.805766
$962$		0			0
$963$	12.6120			0.406416
$964$		0			0
$965$		−	1.65205i		−	0.0531813i
$966$		0			0
$967$		−	1.65061i		−	0.0530799i	−0.999648	−	0.0265400i	$-0.991551\pi$
$967$		−	1.65061i		−	0.0530799i	0.999648	−	0.0265400i	$-0.00844893\pi$
$968$		0			0
$969$	−2.70607	+	4.94895i	−0.0869315	+	0.158983i
$970$		0			0
$971$	15.5888			0.500269			0.250135	−	0.968211i	$-0.419525\pi$
$971$	15.5888			0.500269			0.250135	+	0.968211i	$0.419525\pi$
$972$		0			0
$973$	−2.86583			−0.0918744
$974$		0			0
$975$		−	3.31330i		−	0.106110i
$976$		0			0
$977$		−	11.6049i		−	0.371272i	−0.982619	−	0.185636i	$-0.940565\pi$
$977$		−	11.6049i		−	0.371272i	0.982619	−	0.185636i	$-0.0594346\pi$
$978$		0			0
$979$	1.93578			0.0618678
$980$		0			0
$981$		−	9.97849i		−	0.318589i
$982$		0			0
$983$	44.9817			1.43469			0.717347	−	0.696716i	$-0.245356\pi$
$983$	44.9817			1.43469			0.717347	+	0.696716i	$0.245356\pi$
$984$		0			0
$985$	−9.13679			−0.291122
$986$		0			0
$987$	18.7036			0.595343
$988$		0			0
$989$	−25.0883			−0.797763
$990$		0			0
$991$	−49.7734			−1.58110			−0.790552	−	0.612395i	$-0.790206\pi$
$991$	−49.7734			−1.58110			−0.790552	+	0.612395i	$0.790206\pi$
$992$		0			0
$993$	−25.8401			−0.820011
$994$		0			0
$995$		−	28.5029i		−	0.903603i
$996$		0			0
$997$	45.9694			1.45586			0.727932	−	0.685649i	$-0.240481\pi$
$997$	45.9694			1.45586			0.727932	+	0.685649i	$0.240481\pi$
$998$		0			0
$999$		−	22.4590i		−	0.710571i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.o.b.1519.16	yes	64
4.3	odd	2	inner	3344.2.o.b.1519.49	yes	64
19.18	odd	2	inner	3344.2.o.b.1519.50	yes	64
76.75	even	2	inner	3344.2.o.b.1519.15	✓	64

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3344.2.o.b.1519.15	✓	64	76.75	even	2	inner
3344.2.o.b.1519.16	yes	64	1.1	even	1	trivial
3344.2.o.b.1519.49	yes	64	4.3	odd	2	inner
3344.2.o.b.1519.50	yes	64	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)

\(f(q)\)	\(=\)	\(q-2.05926 q^{3} +1.15495 q^{5} -3.33583i q^{7} +1.24056 q^{9} +O(q^{10})\)	\(q-2.05926 q^{3} +1.15495 q^{5} -3.33583i q^{7} +1.24056 q^{9} +1.00000i q^{11} -0.438879i q^{13} -2.37834 q^{15} +0.628386 q^{17} +(2.09122 - 3.82450i) q^{19} +6.86935i q^{21} -7.88621i q^{23} -3.66609 q^{25} +3.62314 q^{27} +9.16590i q^{29} +2.45383 q^{31} -2.05926i q^{33} -3.85271i q^{35} -6.19876i q^{37} +0.903767i q^{39} +4.92095i q^{41} -3.18129i q^{43} +1.43279 q^{45} -2.72277i q^{47} -4.12776 q^{49} -1.29401 q^{51} -7.38950i q^{53} +1.15495i q^{55} +(-4.30638 + 7.87565i) q^{57} -3.45208 q^{59} +5.99179 q^{61} -4.13831i q^{63} -0.506883i q^{65} +1.93662 q^{67} +16.2398i q^{69} -15.9863 q^{71} +11.0272 q^{73} +7.54945 q^{75} +3.33583 q^{77} -9.40120 q^{79} -11.1827 q^{81} +14.0247i q^{83} +0.725754 q^{85} -18.8750i q^{87} -1.93578i q^{89} -1.46403 q^{91} -5.05307 q^{93} +(2.41526 - 4.41710i) q^{95} +1.63984i q^{97} +1.24056i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q + 56 q^{9}+O(q^{10}) \) 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

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