LMFDB - Embedded newform 3344.2.o.b.1519.13

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			1519.13
Character	$\chi$	$=$	3344.1519
Dual form			3344.2.o.b.1519.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.07682 q^{3} -4.09903 q^{5} +1.91903i q^{7} +1.31320 q^{9} +O(q^{10})$	$q-2.07682 q^{3} -4.09903 q^{5} +1.91903i q^{7} +1.31320 q^{9} -1.00000i q^{11} +1.96768i q^{13} +8.51296 q^{15} +0.0670079 q^{17} +(-2.95770 - 3.20187i) q^{19} -3.98548i q^{21} -2.50007i q^{23} +11.8020 q^{25} +3.50319 q^{27} -6.01530i q^{29} -2.95859 q^{31} +2.07682i q^{33} -7.86614i q^{35} -4.44302i q^{37} -4.08653i q^{39} +1.27570i q^{41} +5.01550i q^{43} -5.38285 q^{45} -5.53314i q^{47} +3.31733 q^{49} -0.139164 q^{51} -10.9556i q^{53} +4.09903i q^{55} +(6.14263 + 6.64973i) q^{57} -13.6277 q^{59} +7.83256 q^{61} +2.52007i q^{63} -8.06557i q^{65} -5.22519 q^{67} +5.19221i q^{69} +5.12728 q^{71} +8.90263 q^{73} -24.5107 q^{75} +1.91903 q^{77} -5.98178 q^{79} -11.2151 q^{81} +0.943616i q^{83} -0.274667 q^{85} +12.4927i q^{87} +2.25817i q^{89} -3.77603 q^{91} +6.14448 q^{93} +(12.1237 + 13.1246i) q^{95} +13.1527i q^{97} -1.31320i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.07682			−1.19906			−0.599528	−	0.800354i	$-0.704645\pi$
$3$	−2.07682			−1.19906			−0.599528	+	0.800354i	$0.704645\pi$
$4$		0			0
$5$	−4.09903			−1.83314			−0.916570	−	0.399874i	$-0.869054\pi$
$5$	−4.09903			−1.83314			−0.916570	+	0.399874i	$0.869054\pi$
$6$		0			0
$7$			1.91903i			0.725324i	0.931921	+	0.362662i	$0.118132\pi$
$7$			1.91903i			0.725324i	−0.931921	+	0.362662i	$0.881868\pi$
$8$		0			0
$9$	1.31320			0.437734
$10$		0			0
$11$		−	1.00000i		−	0.301511i
$11$		−	1.00000i		−	0.301511i
$12$		0			0
$13$			1.96768i			0.545736i	0.962051	+	0.272868i	$0.0879723\pi$
$13$			1.96768i			0.545736i	−0.962051	+	0.272868i	$0.912028\pi$
$14$		0			0
$15$	8.51296			2.19804
$16$		0			0
$17$	0.0670079			0.0162518			0.00812590	−	0.999967i	$-0.497413\pi$
$17$	0.0670079			0.0162518			0.00812590	+	0.999967i	$0.497413\pi$
$18$		0			0
$19$	−2.95770	−	3.20187i	−0.678544	−	0.734560i
$19$	−2.95770	−	3.20187i	−0.678544	−	0.734560i
$20$		0			0
$21$		−	3.98548i		−	0.869704i
$22$		0			0
$23$		−	2.50007i		−	0.521301i	−0.965433	−	0.260650i	$-0.916063\pi$
$23$		−	2.50007i		−	0.521301i	0.965433	−	0.260650i	$-0.0839370\pi$
$24$		0			0
$25$	11.8020			2.36040
$26$		0			0
$27$	3.50319			0.674188
$28$		0			0
$29$		−	6.01530i		−	1.11701i	−0.829500	−	0.558507i	$-0.811374\pi$
$29$		−	6.01530i		−	1.11701i	0.829500	−	0.558507i	$-0.188626\pi$
$30$		0			0
$31$	−2.95859			−0.531379			−0.265690	−	0.964059i	$-0.585600\pi$
$31$	−2.95859			−0.531379			−0.265690	+	0.964059i	$0.585600\pi$
$32$		0			0
$33$			2.07682i			0.361529i
$34$		0			0
$35$		−	7.86614i		−	1.32962i
$36$		0			0
$37$		−	4.44302i		−	0.730428i	−0.930924	−	0.365214i	$-0.880996\pi$
$37$		−	4.44302i		−	0.730428i	0.930924	−	0.365214i	$-0.119004\pi$
$38$		0			0
$39$		−	4.08653i		−	0.654368i
$40$		0			0
$41$			1.27570i			0.199231i	0.995026	+	0.0996156i	$0.0317613\pi$
$41$			1.27570i			0.199231i	−0.995026	+	0.0996156i	$0.968239\pi$
$42$		0			0
$43$			5.01550i			0.764856i	0.923985	+	0.382428i	$0.124912\pi$
$43$			5.01550i			0.764856i	−0.923985	+	0.382428i	$0.875088\pi$
$44$		0			0
$45$	−5.38285			−0.802427
$46$		0			0
$47$		−	5.53314i		−	0.807091i	−0.914960	−	0.403545i	$-0.867778\pi$
$47$		−	5.53314i		−	0.807091i	0.914960	−	0.403545i	$-0.132222\pi$
$48$		0			0
$49$	3.31733			0.473905
$50$		0			0
$51$	−0.139164			−0.0194868
$52$		0			0
$53$		−	10.9556i		−	1.50487i	−0.658668	−	0.752434i	$-0.728880\pi$
$53$		−	10.9556i		−	1.50487i	0.658668	−	0.752434i	$-0.271120\pi$
$54$		0			0
$55$			4.09903i			0.552713i
$56$		0			0
$57$	6.14263	+	6.64973i	0.813612	+	0.880778i
$58$		0			0
$59$	−13.6277			−1.77418			−0.887089	−	0.461598i	$-0.847276\pi$
$59$	−13.6277			−1.77418			−0.887089	+	0.461598i	$0.847276\pi$
$60$		0			0
$61$	7.83256			1.00286			0.501428	−	0.865199i	$-0.332808\pi$
$61$	7.83256			1.00286			0.501428	+	0.865199i	$0.332808\pi$
$62$		0			0
$63$			2.52007i			0.317499i
$64$		0			0
$65$		−	8.06557i		−	1.00041i
$66$		0			0
$67$	−5.22519			−0.638359			−0.319179	−	0.947694i	$-0.603407\pi$
$67$	−5.22519			−0.638359			−0.319179	+	0.947694i	$0.603407\pi$
$68$		0			0
$69$			5.19221i			0.625068i
$70$		0			0
$71$	5.12728			0.608496			0.304248	−	0.952593i	$-0.401595\pi$
$71$	5.12728			0.608496			0.304248	+	0.952593i	$0.401595\pi$
$72$		0			0
$73$	8.90263			1.04197			0.520987	−	0.853565i	$-0.325564\pi$
$73$	8.90263			1.04197			0.520987	+	0.853565i	$0.325564\pi$
$74$		0			0
$75$	−24.5107			−2.83025
$76$		0			0
$77$	1.91903			0.218693
$78$		0			0
$79$	−5.98178			−0.673002			−0.336501	−	0.941683i	$-0.609244\pi$
$79$	−5.98178			−0.673002			−0.336501	+	0.941683i	$0.609244\pi$
$80$		0			0
$81$	−11.2151			−1.24612
$82$		0			0
$83$			0.943616i			0.103575i	0.998658	+	0.0517876i	$0.0164919\pi$
$83$			0.943616i			0.103575i	−0.998658	+	0.0517876i	$0.983508\pi$
$84$		0			0
$85$	−0.274667			−0.0297918
$86$		0			0
$87$			12.4927i			1.33936i
$88$		0			0
$89$			2.25817i			0.239366i	0.992812	+	0.119683i	$0.0381877\pi$
$89$			2.25817i			0.239366i	−0.992812	+	0.119683i	$0.961812\pi$
$90$		0			0
$91$	−3.77603			−0.395836
$92$		0			0
$93$	6.14448			0.637153
$94$		0			0
$95$	12.1237	+	13.1246i	1.24387	+	1.34655i
$96$		0			0
$97$			13.1527i			1.33545i	0.744408	+	0.667726i	$0.232732\pi$
$97$			13.1527i			1.33545i	−0.744408	+	0.667726i	$0.767268\pi$
$98$		0			0
$99$		−	1.31320i		−	0.131982i
$100$		0			0
$101$	2.51467			0.250219			0.125110	−	0.992143i	$-0.460072\pi$
$101$	2.51467			0.250219			0.125110	+	0.992143i	$0.460072\pi$
$102$		0			0
$103$	−6.02636			−0.593795			−0.296898	−	0.954909i	$-0.595952\pi$
$103$	−6.02636			−0.593795			−0.296898	+	0.954909i	$0.595952\pi$
$104$		0			0
$105$			16.3366i			1.59429i
$106$		0			0
$107$	0.184614			0.0178473			0.00892365	−	0.999960i	$-0.497159\pi$
$107$	0.184614			0.0178473			0.00892365	+	0.999960i	$0.497159\pi$
$108$		0			0
$109$		−	18.4830i		−	1.77035i	−0.465257	−	0.885176i	$-0.654038\pi$
$109$		−	18.4830i		−	1.77035i	0.465257	−	0.885176i	$-0.345962\pi$
$110$		0			0
$111$			9.22738i			0.875824i
$112$		0			0
$113$			2.79805i			0.263219i	0.991302	+	0.131609i	$0.0420144\pi$
$113$			2.79805i			0.263219i	−0.991302	+	0.131609i	$0.957986\pi$
$114$		0			0
$115$			10.2479i			0.955617i
$116$		0			0
$117$			2.58396i			0.238887i
$118$		0			0
$119$			0.128590i			0.0117878i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$		−	2.64941i		−	0.238889i
$124$		0			0
$125$	−27.8816			−2.49381
$126$		0			0
$127$	12.8223			1.13779			0.568896	−	0.822410i	$-0.307371\pi$
$127$	12.8223			1.13779			0.568896	+	0.822410i	$0.307371\pi$
$128$		0			0
$129$		−	10.4163i		−	0.917105i
$130$		0			0
$131$			5.57271i			0.486890i	0.969915	+	0.243445i	$0.0782775\pi$
$131$			5.57271i			0.486890i	−0.969915	+	0.243445i	$0.921722\pi$
$132$		0			0
$133$	6.14448	−	5.67592i	0.532794	−	0.492164i
$134$		0			0
$135$	−14.3597			−1.23588
$136$		0			0
$137$	14.8189			1.26607			0.633033	−	0.774125i	$-0.281810\pi$
$137$	14.8189			1.26607			0.633033	+	0.774125i	$0.281810\pi$
$138$		0			0
$139$			18.9572i			1.60793i	0.594678	+	0.803964i	$0.297280\pi$
$139$			18.9572i			1.60793i	−0.594678	+	0.803964i	$0.702720\pi$
$140$		0			0
$141$			11.4914i			0.967746i
$142$		0			0
$143$	1.96768			0.164546
$144$		0			0
$145$			24.6569i			2.04764i
$146$		0			0
$147$	−6.88952			−0.568238
$148$		0			0
$149$	12.3435			1.01122			0.505611	−	0.862761i	$-0.331267\pi$
$149$	12.3435			1.01122			0.505611	+	0.862761i	$0.331267\pi$
$150$		0			0
$151$	−22.9067			−1.86412			−0.932060	−	0.362305i	$-0.881990\pi$
$151$	−22.9067			−1.86412			−0.932060	+	0.362305i	$0.881990\pi$
$152$		0			0
$153$	0.0879948			0.00711396
$154$		0			0
$155$	12.1274			0.974093
$156$		0			0
$157$	−14.9703			−1.19476			−0.597379	−	0.801959i	$-0.703791\pi$
$157$	−14.9703			−1.19476			−0.597379	+	0.801959i	$0.703791\pi$
$158$		0			0
$159$			22.7529i			1.80442i
$160$		0			0
$161$	4.79770			0.378112
$162$		0			0
$163$			11.4587i			0.897511i	0.893655	+	0.448756i	$0.148133\pi$
$163$			11.4587i			0.897511i	−0.893655	+	0.448756i	$0.851867\pi$
$164$		0			0
$165$		−	8.51296i		−	0.662733i
$166$		0			0
$167$	−13.4407			−1.04007			−0.520037	−	0.854144i	$-0.674082\pi$
$167$	−13.4407			−1.04007			−0.520037	+	0.854144i	$0.674082\pi$
$168$		0			0
$169$	9.12823			0.702172
$170$		0			0
$171$	−3.88406	−	4.20470i	−0.297022	−	0.321542i
$172$		0			0
$173$			17.5440i			1.33384i	0.745128	+	0.666921i	$0.232388\pi$
$173$			17.5440i			1.33384i	−0.745128	+	0.666921i	$0.767612\pi$
$174$		0			0
$175$			22.6484i			1.71206i
$176$		0			0
$177$	28.3024			2.12734
$178$		0			0
$179$	1.34524			0.100548			0.0502739	−	0.998735i	$-0.483991\pi$
$179$	1.34524			0.100548			0.0502739	+	0.998735i	$0.483991\pi$
$180$		0			0
$181$			16.0658i			1.19416i	0.802180	+	0.597082i	$0.203673\pi$
$181$			16.0658i			1.19416i	−0.802180	+	0.597082i	$0.796327\pi$
$182$		0			0
$183$	−16.2668			−1.20248
$184$		0			0
$185$			18.2121i			1.33898i
$186$		0			0
$187$		−	0.0670079i		−	0.00490010i
$188$		0			0
$189$			6.72271i			0.489005i
$190$		0			0
$191$			2.91132i			0.210656i	0.994438	+	0.105328i	$0.0335892\pi$
$191$			2.91132i			0.210656i	−0.994438	+	0.105328i	$0.966411\pi$
$192$		0			0
$193$			3.35937i			0.241813i	0.992664	+	0.120907i	$0.0385801\pi$
$193$			3.35937i			0.241813i	−0.992664	+	0.120907i	$0.961420\pi$
$194$		0			0
$195$			16.7508i			1.19955i
$196$		0			0
$197$	−11.1193			−0.792221			−0.396110	−	0.918203i	$-0.629640\pi$
$197$	−11.1193			−0.792221			−0.396110	+	0.918203i	$0.629640\pi$
$198$		0			0
$199$		−	15.4833i		−	1.09758i	−0.835959	−	0.548792i	$-0.815088\pi$
$199$		−	15.4833i		−	1.09758i	0.835959	−	0.548792i	$-0.184912\pi$
$200$		0			0
$201$	10.8518			0.765427
$202$		0			0
$203$	11.5435			0.810197
$204$		0			0
$205$		−	5.22914i		−	0.365219i
$206$		0			0
$207$		−	3.28310i		−	0.228191i
$208$		0			0
$209$	−3.20187	+	2.95770i	−0.221478	+	0.204589i
$210$		0			0
$211$	−27.1620			−1.86991			−0.934955	−	0.354767i	$-0.884560\pi$
$211$	−27.1620			−1.86991			−0.934955	+	0.354767i	$0.884560\pi$
$212$		0			0
$213$	−10.6485			−0.729621
$214$		0			0
$215$		−	20.5587i		−	1.40209i
$216$		0			0
$217$		−	5.67762i		−	0.385422i
$218$		0			0
$219$	−18.4892			−1.24938
$220$		0			0
$221$			0.131850i			0.00886919i
$222$		0			0
$223$	5.07517			0.339859			0.169929	−	0.985456i	$-0.445646\pi$
$223$	5.07517			0.339859			0.169929	+	0.985456i	$0.445646\pi$
$224$		0			0
$225$	15.4984			1.03323
$226$		0			0
$227$	8.59486			0.570461			0.285230	−	0.958459i	$-0.407930\pi$
$227$	8.59486			0.570461			0.285230	+	0.958459i	$0.407930\pi$
$228$		0			0
$229$	6.83002			0.451341			0.225670	−	0.974204i	$-0.427543\pi$
$229$	6.83002			0.451341			0.225670	+	0.974204i	$0.427543\pi$
$230$		0			0
$231$	−3.98548			−0.262226
$232$		0			0
$233$	−12.4946			−0.818551			−0.409275	−	0.912411i	$-0.634218\pi$
$233$	−12.4946			−0.818551			−0.409275	+	0.912411i	$0.634218\pi$
$234$		0			0
$235$			22.6805i			1.47951i
$236$		0			0
$237$	12.4231			0.806967
$238$		0			0
$239$			8.38229i			0.542205i	0.962550	+	0.271102i	$0.0873882\pi$
$239$			8.38229i			0.542205i	−0.962550	+	0.271102i	$0.912612\pi$
$240$		0			0
$241$			11.7385i			0.756141i	0.925777	+	0.378070i	$0.123412\pi$
$241$			11.7385i			0.756141i	−0.925777	+	0.378070i	$0.876588\pi$
$242$		0			0
$243$	12.7823			0.819982
$244$		0			0
$245$	−13.5978			−0.868734
$246$		0			0
$247$	6.30026	−	5.81982i	0.400876	−	0.370306i
$248$		0			0
$249$		−	1.95972i		−	0.124193i
$250$		0			0
$251$			4.79726i			0.302800i	0.988473	+	0.151400i	$0.0483782\pi$
$251$			4.79726i			0.302800i	−0.988473	+	0.151400i	$0.951622\pi$
$252$		0			0
$253$	−2.50007			−0.157178
$254$		0			0
$255$	0.570435			0.0357220
$256$		0			0
$257$		−	30.9419i		−	1.93010i	−0.262065	−	0.965050i	$-0.584403\pi$
$257$		−	30.9419i		−	1.93010i	0.262065	−	0.965050i	$-0.415597\pi$
$258$		0			0
$259$	8.52628			0.529797
$260$		0			0
$261$		−	7.89930i		−	0.488955i
$262$		0			0
$263$		−	2.22960i		−	0.137483i	−0.997635	−	0.0687415i	$-0.978102\pi$
$263$		−	2.22960i		−	0.137483i	0.997635	−	0.0687415i	$-0.0218984\pi$
$264$		0			0
$265$			44.9073i			2.75863i
$266$		0			0
$267$		−	4.68982i		−	0.287013i
$268$		0			0
$269$			0.365994i			0.0223151i	0.999938	+	0.0111575i	$0.00355163\pi$
$269$			0.365994i			0.0223151i	−0.999938	+	0.0111575i	$0.996448\pi$
$270$		0			0
$271$			13.6749i			0.830690i	0.909664	+	0.415345i	$0.136339\pi$
$271$			13.6749i			0.830690i	−0.909664	+	0.415345i	$0.863661\pi$
$272$		0			0
$273$	7.84216			0.474629
$274$		0			0
$275$		−	11.8020i		−	0.711688i
$276$		0			0
$277$	−7.14662			−0.429399			−0.214699	−	0.976680i	$-0.568877\pi$
$277$	−7.14662			−0.429399			−0.214699	+	0.976680i	$0.568877\pi$
$278$		0			0
$279$	−3.88523			−0.232603
$280$		0			0
$281$			11.7703i			0.702159i	0.936346	+	0.351080i	$0.114185\pi$
$281$			11.7703i			0.702159i	−0.936346	+	0.351080i	$0.885815\pi$
$282$		0			0
$283$		−	28.5630i		−	1.69790i	−0.528476	−	0.848948i	$-0.677236\pi$
$283$		−	28.5630i		−	1.69790i	0.528476	−	0.848948i	$-0.322764\pi$
$284$		0			0
$285$	−25.1788	−	27.2574i	−1.49146	−	1.61459i
$286$		0			0
$287$	−2.44811			−0.144507
$288$		0			0
$289$	−16.9955			−0.999736
$290$		0			0
$291$		−	27.3158i		−	1.60128i
$292$		0			0
$293$			19.1937i			1.12131i	0.828050	+	0.560654i	$0.189450\pi$
$293$			19.1937i			1.12131i	−0.828050	+	0.560654i	$0.810550\pi$
$294$		0			0
$295$	55.8604			3.25232
$296$		0			0
$297$		−	3.50319i		−	0.203275i
$298$		0			0
$299$	4.91934			0.284493
$300$		0			0
$301$	−9.62488			−0.554769
$302$		0			0
$303$	−5.22253			−0.300027
$304$		0			0
$305$	−32.1059			−1.83838
$306$		0			0
$307$	3.55588			0.202945			0.101472	−	0.994838i	$-0.467645\pi$
$307$	3.55588			0.202945			0.101472	+	0.994838i	$0.467645\pi$
$308$		0			0
$309$	12.5157			0.711993
$310$		0			0
$311$		−	13.5252i		−	0.766945i	−0.923552	−	0.383472i	$-0.874728\pi$
$311$		−	13.5252i		−	0.766945i	0.923552	−	0.383472i	$-0.125272\pi$
$312$		0			0
$313$	17.6034			0.995000			0.497500	−	0.867464i	$-0.334251\pi$
$313$	17.6034			0.995000			0.497500	+	0.867464i	$0.334251\pi$
$314$		0			0
$315$		−	10.3298i		−	0.582020i
$316$		0			0
$317$		−	31.5494i		−	1.77199i	−0.463692	−	0.885996i	$-0.653476\pi$
$317$		−	31.5494i		−	1.77199i	0.463692	−	0.885996i	$-0.346524\pi$
$318$		0			0
$319$	−6.01530			−0.336792
$320$		0			0
$321$	−0.383411			−0.0213999
$322$		0			0
$323$	−0.198189	−	0.214551i	−0.0110276	−	0.0119379i
$324$		0			0
$325$			23.2226i			1.28816i
$326$		0			0
$327$			38.3860i			2.12275i
$328$		0			0
$329$	10.6182			0.585402
$330$		0			0
$331$	27.3378			1.50262			0.751310	−	0.659950i	$-0.229422\pi$
$331$	27.3378			1.50262			0.751310	+	0.659950i	$0.229422\pi$
$332$		0			0
$333$		−	5.83458i		−	0.319733i
$334$		0			0
$335$	21.4182			1.17020
$336$		0			0
$337$			21.4712i			1.16961i	0.811173	+	0.584806i	$0.198829\pi$
$337$			21.4712i			1.16961i	−0.811173	+	0.584806i	$0.801171\pi$
$338$		0			0
$339$		−	5.81107i		−	0.315614i
$340$		0			0
$341$			2.95859i			0.160217i
$342$		0			0
$343$			19.7992i			1.06906i
$344$		0			0
$345$		−	21.2830i		−	1.14584i
$346$		0			0
$347$		−	13.5708i		−	0.728516i	−0.931298	−	0.364258i	$-0.881323\pi$
$347$		−	13.5708i		−	0.728516i	0.931298	−	0.364258i	$-0.118677\pi$
$348$		0			0
$349$	−22.6231			−1.21099			−0.605493	−	0.795851i	$-0.707024\pi$
$349$	−22.6231			−1.21099			−0.605493	+	0.795851i	$0.707024\pi$
$350$		0			0
$351$			6.89315i			0.367929i
$352$		0			0
$353$	−3.31890			−0.176647			−0.0883236	−	0.996092i	$-0.528151\pi$
$353$	−3.31890			−0.176647			−0.0883236	+	0.996092i	$0.528151\pi$
$354$		0			0
$355$	−21.0169			−1.11546
$356$		0			0
$357$		−	0.267059i		−	0.0141342i
$358$		0			0
$359$		−	22.6677i		−	1.19635i	−0.801364	−	0.598177i	$-0.795892\pi$
$359$		−	22.6677i		−	1.19635i	0.801364	−	0.598177i	$-0.204108\pi$
$360$		0			0
$361$	−1.50396	+	18.9404i	−0.0791559	+	0.996862i
$362$		0			0
$363$	2.07682			0.109005
$364$		0			0
$365$	−36.4921			−1.91008
$366$		0			0
$367$		−	14.7205i		−	0.768406i	−0.923249	−	0.384203i	$-0.874476\pi$
$367$		−	14.7205i		−	0.768406i	0.923249	−	0.384203i	$-0.125524\pi$
$368$		0			0
$369$			1.67525i			0.0872102i
$370$		0			0
$371$	21.0241			1.09152
$372$		0			0
$373$			30.3395i			1.57092i	0.618913	+	0.785460i	$0.287574\pi$
$373$			30.3395i			1.57092i	−0.618913	+	0.785460i	$0.712426\pi$
$374$		0			0
$375$	57.9053			2.99022
$376$		0			0
$377$	11.8362			0.609595
$378$		0			0
$379$	−11.1766			−0.574104			−0.287052	−	0.957915i	$-0.592675\pi$
$379$	−11.1766			−0.574104			−0.287052	+	0.957915i	$0.592675\pi$
$380$		0			0
$381$	−26.6296			−1.36428
$382$		0			0
$383$	11.6431			0.594937			0.297468	−	0.954732i	$-0.403858\pi$
$383$	11.6431			0.594937			0.297468	+	0.954732i	$0.403858\pi$
$384$		0			0
$385$	−7.86614			−0.400896
$386$		0			0
$387$			6.58636i			0.334803i
$388$		0			0
$389$	−20.0400			−1.01607			−0.508034	−	0.861337i	$-0.669627\pi$
$389$	−20.0400			−1.01607			−0.508034	+	0.861337i	$0.669627\pi$
$390$		0			0
$391$		−	0.167524i		−	0.00847207i
$392$		0			0
$393$		−	11.5735i		−	0.583808i
$394$		0			0
$395$	24.5195			1.23371
$396$		0			0
$397$	38.1803			1.91622			0.958108	−	0.286406i	$-0.0924607\pi$
$397$	38.1803			1.91622			0.958108	+	0.286406i	$0.0924607\pi$
$398$		0			0
$399$	−12.7610	+	11.7879i	−0.638849	+	0.590132i
$400$		0			0
$401$			20.3804i			1.01775i	0.860840	+	0.508875i	$0.169939\pi$
$401$			20.3804i			1.01775i	−0.860840	+	0.508875i	$0.830061\pi$
$402$		0			0
$403$		−	5.82157i		−	0.289993i
$404$		0			0
$405$	45.9710			2.28432
$406$		0			0
$407$	−4.44302			−0.220232
$408$		0			0
$409$		−	14.6712i		−	0.725446i	−0.931897	−	0.362723i	$-0.881847\pi$
$409$		−	14.6712i		−	0.725446i	0.931897	−	0.362723i	$-0.118153\pi$
$410$		0			0
$411$	−30.7763			−1.51808
$412$		0			0
$413$		−	26.1520i		−	1.28685i
$414$		0			0
$415$		−	3.86791i		−	0.189868i
$416$		0			0
$417$		−	39.3708i		−	1.92800i
$418$		0			0
$419$			24.4545i			1.19468i	0.801988	+	0.597340i	$0.203776\pi$
$419$			24.4545i			1.19468i	−0.801988	+	0.597340i	$0.796224\pi$
$420$		0			0
$421$			4.49565i			0.219104i	0.993981	+	0.109552i	$0.0349417\pi$
$421$			4.49565i			0.219104i	−0.993981	+	0.109552i	$0.965058\pi$
$422$		0			0
$423$		−	7.26612i		−	0.353291i
$424$		0			0
$425$	0.790828			0.0383608
$426$		0			0
$427$			15.0309i			0.727396i
$428$		0			0
$429$	−4.08653			−0.197299
$430$		0			0
$431$	−6.80989			−0.328021			−0.164010	−	0.986459i	$-0.552443\pi$
$431$	−6.80989			−0.328021			−0.164010	+	0.986459i	$0.552443\pi$
$432$		0			0
$433$		−	20.9958i		−	1.00899i	−0.863414	−	0.504497i	$-0.831678\pi$
$433$		−	20.9958i		−	1.00899i	0.863414	−	0.504497i	$-0.168322\pi$
$434$		0			0
$435$		−	51.2080i		−	2.45524i
$436$		0			0
$437$	−8.00490	+	7.39447i	−0.382926	+	0.353725i
$438$		0			0
$439$	−40.4413			−1.93016			−0.965079	−	0.261960i	$-0.915631\pi$
$439$	−40.4413			−1.93016			−0.965079	+	0.261960i	$0.915631\pi$
$440$		0			0
$441$	4.35633			0.207444
$442$		0			0
$443$			0.176165i			0.00836986i	0.999991	+	0.00418493i	$0.00133211\pi$
$443$			0.176165i			0.00836986i	−0.999991	+	0.00418493i	$0.998668\pi$
$444$		0			0
$445$		−	9.25630i		−	0.438791i
$446$		0			0
$447$	−25.6354			−1.21251
$448$		0			0
$449$		−	31.8421i		−	1.50272i	−0.659891	−	0.751361i	$-0.729398\pi$
$449$		−	31.8421i		−	1.50272i	0.659891	−	0.751361i	$-0.270602\pi$
$450$		0			0
$451$	1.27570			0.0600705
$452$		0			0
$453$	47.5732			2.23518
$454$		0			0
$455$	15.4781			0.725622
$456$		0			0
$457$	−28.5076			−1.33353			−0.666764	−	0.745269i	$-0.732321\pi$
$457$	−28.5076			−1.33353			−0.666764	+	0.745269i	$0.732321\pi$
$458$		0			0
$459$	0.234741			0.0109568
$460$		0			0
$461$	−20.9758			−0.976941			−0.488470	−	0.872580i	$-0.662445\pi$
$461$	−20.9758			−0.976941			−0.488470	+	0.872580i	$0.662445\pi$
$462$		0			0
$463$			7.21951i			0.335519i	0.985828	+	0.167760i	$0.0536532\pi$
$463$			7.21951i			0.335519i	−0.985828	+	0.167760i	$0.946347\pi$
$464$		0			0
$465$	−25.1864			−1.16799
$466$		0			0
$467$		−	17.7986i		−	0.823621i	−0.911269	−	0.411811i	$-0.864897\pi$
$467$		−	17.7986i		−	0.823621i	0.911269	−	0.411811i	$-0.135103\pi$
$468$		0			0
$469$		−	10.0273i		−	0.463017i
$470$		0			0
$471$	31.0906			1.43258
$472$		0			0
$473$	5.01550			0.230613
$474$		0			0
$475$	−34.9069	−	37.7885i	−1.60164	−	1.73386i
$476$		0			0
$477$		−	14.3869i		−	0.658731i
$478$		0			0
$479$			26.4296i			1.20760i	0.797136	+	0.603799i	$0.206347\pi$
$479$			26.4296i			1.20760i	−0.797136	+	0.603799i	$0.793653\pi$
$480$		0			0
$481$	8.74245			0.398621
$482$		0			0
$483$	−9.96399			−0.453377
$484$		0			0
$485$		−	53.9131i		−	2.44807i
$486$		0			0
$487$	−11.3537			−0.514487			−0.257244	−	0.966347i	$-0.582814\pi$
$487$	−11.3537			−0.514487			−0.257244	+	0.966347i	$0.582814\pi$
$488$		0			0
$489$		−	23.7976i		−	1.07617i
$490$		0			0
$491$		−	31.2033i		−	1.40818i	−0.710109	−	0.704092i	$-0.751354\pi$
$491$		−	31.2033i		−	1.40818i	0.710109	−	0.704092i	$-0.248646\pi$
$492$		0			0
$493$		−	0.403073i		−	0.0181535i
$494$		0			0
$495$			5.38285i			0.241941i
$496$		0			0
$497$			9.83939i			0.441357i
$498$		0			0
$499$			17.8025i			0.796952i	0.917179	+	0.398476i	$0.130461\pi$
$499$			17.8025i			0.796952i	−0.917179	+	0.398476i	$0.869539\pi$
$500$		0			0
$501$	27.9140			1.24711
$502$		0			0
$503$		−	3.81638i		−	0.170164i	−0.996374	−	0.0850819i	$-0.972885\pi$
$503$		−	3.81638i		−	0.170164i	0.996374	−	0.0850819i	$-0.0271152\pi$
$504$		0			0
$505$	−10.3077			−0.458687
$506$		0			0
$507$	−18.9577			−0.841943
$508$		0			0
$509$			20.6488i			0.915240i	0.889148	+	0.457620i	$0.151298\pi$
$509$			20.6488i			0.915240i	−0.889148	+	0.457620i	$0.848702\pi$
$510$		0			0
$511$			17.0844i			0.755769i
$512$		0			0
$513$	−10.3614	−	11.2168i	−0.457467	−	0.495232i
$514$		0			0
$515$	24.7022			1.08851
$516$		0			0
$517$	−5.53314			−0.243347
$518$		0			0
$519$		−	36.4357i		−	1.59935i
$520$		0			0
$521$		−	7.88434i		−	0.345419i	−0.984973	−	0.172710i	$-0.944748\pi$
$521$		−	7.88434i		−	0.345419i	0.984973	−	0.172710i	$-0.0552522\pi$
$522$		0			0
$523$	12.6994			0.555306			0.277653	−	0.960681i	$-0.410443\pi$
$523$	12.6994			0.555306			0.277653	+	0.960681i	$0.410443\pi$
$524$		0			0
$525$		−	47.0367i		−	2.05285i
$526$		0			0
$527$	−0.198249			−0.00863587
$528$		0			0
$529$	16.7496			0.728246
$530$		0			0
$531$	−17.8959			−0.776618
$532$		0			0
$533$	−2.51017			−0.108728
$534$		0			0
$535$	−0.756737			−0.0327166
$536$		0			0
$537$	−2.79382			−0.120562
$538$		0			0
$539$		−	3.31733i		−	0.142888i
$540$		0			0
$541$	8.47444			0.364345			0.182172	−	0.983267i	$-0.441687\pi$
$541$	8.47444			0.364345			0.182172	+	0.983267i	$0.441687\pi$
$542$		0			0
$543$		−	33.3659i		−	1.43187i
$544$		0			0
$545$			75.7623i			3.24530i
$546$		0			0
$547$	−5.47072			−0.233911			−0.116955	−	0.993137i	$-0.537313\pi$
$547$	−5.47072			−0.233911			−0.116955	+	0.993137i	$0.537313\pi$
$548$		0			0
$549$	10.2857			0.438984
$550$		0			0
$551$	−19.2602	+	17.7915i	−0.820513	+	0.757943i
$552$		0			0
$553$		−	11.4792i		−	0.488145i
$554$		0			0
$555$		−	37.8233i		−	1.60551i
$556$		0			0
$557$	37.4034			1.58483			0.792417	−	0.609980i	$-0.208822\pi$
$557$	37.4034			1.58483			0.792417	+	0.609980i	$0.208822\pi$
$558$		0			0
$559$	−9.86890			−0.417410
$560$		0			0
$561$			0.139164i			0.00587549i
$562$		0			0
$563$	14.3660			0.605453			0.302726	−	0.953078i	$-0.402103\pi$
$563$	14.3660			0.605453			0.302726	+	0.953078i	$0.402103\pi$
$564$		0			0
$565$		−	11.4693i		−	0.482517i
$566$		0			0
$567$		−	21.5221i		−	0.903843i
$568$		0			0
$569$		−	21.5972i		−	0.905403i	−0.891662	−	0.452701i	$-0.850460\pi$
$569$		−	21.5972i		−	0.905403i	0.891662	−	0.452701i	$-0.149540\pi$
$570$		0			0
$571$			28.0763i			1.17496i	0.809240	+	0.587478i	$0.199879\pi$
$571$			28.0763i			1.17496i	−0.809240	+	0.587478i	$0.800121\pi$
$572$		0			0
$573$		−	6.04630i		−	0.252588i
$574$		0			0
$575$		−	29.5059i		−	1.23048i
$576$		0			0
$577$	3.48384			0.145034			0.0725170	−	0.997367i	$-0.476897\pi$
$577$	3.48384			0.145034			0.0725170	+	0.997367i	$0.476897\pi$
$578$		0			0
$579$		−	6.97683i		−	0.289947i
$580$		0			0
$581$	−1.81082			−0.0751257
$582$		0			0
$583$	−10.9556			−0.453735
$584$		0			0
$585$		−	10.5917i		−	0.437914i
$586$		0			0
$587$			20.3916i			0.841650i	0.907142	+	0.420825i	$0.138259\pi$
$587$			20.3916i			0.841650i	−0.907142	+	0.420825i	$0.861741\pi$
$588$		0			0
$589$	8.75065	+	9.47304i	0.360564	+	0.390330i
$590$		0			0
$591$	23.0929			0.949916
$592$		0			0
$593$	−10.5881			−0.434802			−0.217401	−	0.976082i	$-0.569758\pi$
$593$	−10.5881			−0.434802			−0.217401	+	0.976082i	$0.569758\pi$
$594$		0			0
$595$		−	0.527093i		−	0.0216087i
$596$		0			0
$597$			32.1562i			1.31606i
$598$		0			0
$599$	17.2983			0.706791			0.353396	−	0.935474i	$-0.385027\pi$
$599$	17.2983			0.706791			0.353396	+	0.935474i	$0.385027\pi$
$600$		0			0
$601$		−	17.3017i		−	0.705752i	−0.935670	−	0.352876i	$-0.885204\pi$
$601$		−	17.3017i		−	0.705752i	0.935670	−	0.352876i	$-0.114796\pi$
$602$		0			0
$603$	−6.86173			−0.279431
$604$		0			0
$605$	4.09903			0.166649
$606$		0			0
$607$	−2.11134			−0.0856968			−0.0428484	−	0.999082i	$-0.513643\pi$
$607$	−2.11134			−0.0856968			−0.0428484	+	0.999082i	$0.513643\pi$
$608$		0			0
$609$	−23.9739			−0.971471
$610$		0			0
$611$	10.8874			0.440459
$612$		0			0
$613$	22.4110			0.905170			0.452585	−	0.891721i	$-0.350502\pi$
$613$	22.4110			0.905170			0.452585	+	0.891721i	$0.350502\pi$
$614$		0			0
$615$			10.8600i			0.437917i
$616$		0			0
$617$	−27.2771			−1.09814			−0.549068	−	0.835778i	$-0.685017\pi$
$617$	−27.2771			−1.09814			−0.549068	+	0.835778i	$0.685017\pi$
$618$		0			0
$619$			37.0575i			1.48947i	0.667363	+	0.744733i	$0.267423\pi$
$619$			37.0575i			1.48947i	−0.667363	+	0.744733i	$0.732577\pi$
$620$		0			0
$621$		−	8.75821i		−	0.351455i
$622$		0			0
$623$	−4.33349			−0.173618
$624$		0			0
$625$	55.2775			2.21110
$626$		0			0
$627$	6.64973	−	6.14263i	0.265564	−	0.245313i
$628$		0			0
$629$		−	0.297717i		−	0.0118708i
$630$		0			0
$631$			39.3029i			1.56462i	0.622887	+	0.782311i	$0.285959\pi$
$631$			39.3029i			1.56462i	−0.622887	+	0.782311i	$0.714041\pi$
$632$		0			0
$633$	56.4107			2.24213
$634$		0			0
$635$	−52.5588			−2.08573
$636$		0			0
$637$			6.52745i			0.258627i
$638$		0			0
$639$	6.73315			0.266359
$640$		0			0
$641$			11.1441i			0.440167i	0.975481	+	0.220083i	$0.0706329\pi$
$641$			11.1441i			0.440167i	−0.975481	+	0.220083i	$0.929367\pi$
$642$		0			0
$643$			33.4830i			1.32044i	0.751073	+	0.660220i	$0.229537\pi$
$643$			33.4830i			1.32044i	−0.751073	+	0.660220i	$0.770463\pi$
$644$		0			0
$645$			42.6967i			1.68118i
$646$		0			0
$647$			3.60580i			0.141759i	0.997485	+	0.0708793i	$0.0225805\pi$
$647$			3.60580i			0.141759i	−0.997485	+	0.0708793i	$0.977419\pi$
$648$		0			0
$649$			13.6277i			0.534935i
$650$		0			0
$651$			11.7914i			0.462143i
$652$		0			0
$653$	−5.09373			−0.199333			−0.0996665	−	0.995021i	$-0.531778\pi$
$653$	−5.09373			−0.199333			−0.0996665	+	0.995021i	$0.531778\pi$
$654$		0			0
$655$		−	22.8427i		−	0.892537i
$656$		0			0
$657$	11.6909			0.456107
$658$		0			0
$659$	18.6935			0.728196			0.364098	−	0.931361i	$-0.381377\pi$
$659$	18.6935			0.728196			0.364098	+	0.931361i	$0.381377\pi$
$660$		0			0
$661$			33.4068i			1.29938i	0.760201	+	0.649688i	$0.225100\pi$
$661$			33.4068i			1.29938i	−0.760201	+	0.649688i	$0.774900\pi$
$662$		0			0
$663$		−	0.273829i		−	0.0106347i
$664$		0			0
$665$	−25.1864	+	23.2657i	−0.976686	+	0.902206i
$666$		0			0
$667$	−15.0387			−0.582300
$668$		0			0
$669$	−10.5402			−0.407509
$670$		0			0
$671$		−	7.83256i		−	0.302372i
$672$		0			0
$673$			19.9369i			0.768513i	0.923226	+	0.384256i	$0.125542\pi$
$673$			19.9369i			0.768513i	−0.923226	+	0.384256i	$0.874458\pi$
$674$		0			0
$675$	41.3447			1.59136
$676$		0			0
$677$			44.7286i			1.71906i	0.511085	+	0.859530i	$0.329244\pi$
$677$			44.7286i			1.71906i	−0.511085	+	0.859530i	$0.670756\pi$
$678$		0			0
$679$	−25.2403			−0.968635
$680$		0			0
$681$	−17.8500			−0.684014
$682$		0			0
$683$	16.9538			0.648721			0.324360	−	0.945934i	$-0.394851\pi$
$683$	16.9538			0.648721			0.324360	+	0.945934i	$0.394851\pi$
$684$		0			0
$685$	−60.7432			−2.32088
$686$		0			0
$687$	−14.1848			−0.541182
$688$		0			0
$689$	21.5571			0.821261
$690$		0			0
$691$			42.3855i			1.61242i	0.591629	+	0.806210i	$0.298485\pi$
$691$			42.3855i			1.61242i	−0.591629	+	0.806210i	$0.701515\pi$
$692$		0			0
$693$	2.52007			0.0957295
$694$		0			0
$695$		−	77.7061i		−	2.94756i
$696$		0			0
$697$			0.0854821i			0.00323786i
$698$		0			0
$699$	25.9492			0.981488
$700$		0			0
$701$	−26.2716			−0.992265			−0.496132	−	0.868247i	$-0.665247\pi$
$701$	−26.2716			−0.992265			−0.496132	+	0.868247i	$0.665247\pi$
$702$		0			0
$703$	−14.2260	+	13.1411i	−0.536543	+	0.495628i
$704$		0			0
$705$		−	47.1034i		−	1.77402i
$706$		0			0
$707$			4.82572i			0.181490i
$708$		0			0
$709$	2.29932			0.0863526			0.0431763	−	0.999067i	$-0.486252\pi$
$709$	2.29932			0.0863526			0.0431763	+	0.999067i	$0.486252\pi$
$710$		0			0
$711$	−7.85527			−0.294596
$712$		0			0
$713$			7.39669i			0.277008i
$714$		0			0
$715$	−8.06557			−0.301635
$716$		0			0
$717$		−	17.4085i		−	0.650134i
$718$		0			0
$719$			16.9910i			0.633659i	0.948482	+	0.316830i	$0.102618\pi$
$719$			16.9910i			0.633659i	−0.948482	+	0.316830i	$0.897382\pi$
$720$		0			0
$721$		−	11.5648i		−	0.430694i
$722$		0			0
$723$		−	24.3787i		−	0.906655i
$724$		0			0
$725$		−	70.9927i		−	2.63660i
$726$		0			0
$727$		−	25.4093i		−	0.942379i	−0.882032	−	0.471190i	$-0.843825\pi$
$727$		−	25.4093i		−	0.942379i	0.882032	−	0.471190i	$-0.156175\pi$
$728$		0			0
$729$	7.09882			0.262919
$730$		0			0
$731$			0.336078i			0.0124303i
$732$		0			0
$733$	−0.863375			−0.0318895			−0.0159447	−	0.999873i	$-0.505076\pi$
$733$	−0.863375			−0.0318895			−0.0159447	+	0.999873i	$0.505076\pi$
$734$		0			0
$735$	28.2403			1.04166
$736$		0			0
$737$			5.22519i			0.192472i
$738$		0			0
$739$			10.5243i			0.387143i	0.981086	+	0.193572i	$0.0620072\pi$
$739$			10.5243i			0.387143i	−0.981086	+	0.193572i	$0.937993\pi$
$740$		0			0
$741$	−13.0845	+	12.0867i	−0.480672	+	0.444018i
$742$		0			0
$743$	−3.63282			−0.133275			−0.0666376	−	0.997777i	$-0.521227\pi$
$743$	−3.63282			−0.133275			−0.0666376	+	0.997777i	$0.521227\pi$
$744$		0			0
$745$	−50.5965			−1.85371
$746$		0			0
$747$			1.23916i			0.0453384i
$748$		0			0
$749$			0.354279i			0.0129451i
$750$		0			0
$751$	4.31852			0.157585			0.0787926	−	0.996891i	$-0.474894\pi$
$751$	4.31852			0.157585			0.0787926	+	0.996891i	$0.474894\pi$
$752$		0			0
$753$		−	9.96306i		−	0.363074i
$754$		0			0
$755$	93.8951			3.41719
$756$		0			0
$757$	32.1061			1.16692			0.583459	−	0.812143i	$-0.301699\pi$
$757$	32.1061			1.16692			0.583459	+	0.812143i	$0.301699\pi$
$758$		0			0
$759$	5.19221			0.188465
$760$		0			0
$761$	44.6755			1.61949			0.809743	−	0.586785i	$-0.199607\pi$
$761$	44.6755			1.61949			0.809743	+	0.586785i	$0.199607\pi$
$762$		0			0
$763$	35.4694			1.28408
$764$		0			0
$765$	−0.360693			−0.0130409
$766$		0			0
$767$		−	26.8150i		−	0.968234i
$768$		0			0
$769$	3.55099			0.128052			0.0640260	−	0.997948i	$-0.479606\pi$
$769$	3.55099			0.128052			0.0640260	+	0.997948i	$0.479606\pi$
$770$		0			0
$771$			64.2608i			2.31430i
$772$		0			0
$773$		−	34.3455i		−	1.23532i	−0.786444	−	0.617661i	$-0.788080\pi$
$773$		−	34.3455i		−	1.23532i	0.786444	−	0.617661i	$-0.211920\pi$
$774$		0			0
$775$	−34.9174			−1.25427
$776$		0			0
$777$	−17.7076			−0.635256
$778$		0			0
$779$	4.08463	−	3.77315i	0.146347	−	0.135187i
$780$		0			0
$781$		−	5.12728i		−	0.183469i
$782$		0			0
$783$		−	21.0727i		−	0.753078i
$784$		0			0
$785$	61.3636			2.19016
$786$		0			0
$787$	−34.5276			−1.23078			−0.615388	−	0.788225i	$-0.711001\pi$
$787$	−34.5276			−1.23078			−0.615388	+	0.788225i	$0.711001\pi$
$788$		0			0
$789$			4.63048i			0.164850i
$790$		0			0
$791$	−5.36954			−0.190919
$792$		0			0
$793$			15.4120i			0.547295i
$794$		0			0
$795$		−	93.2646i		−	3.30775i
$796$		0			0
$797$		−	50.9506i		−	1.80476i	−0.430940	−	0.902381i	$-0.641818\pi$
$797$		−	50.9506i		−	1.80476i	0.430940	−	0.902381i	$-0.358182\pi$
$798$		0			0
$799$		−	0.370764i		−	0.0131167i
$800$		0			0
$801$			2.96543i			0.104778i
$802$		0			0
$803$		−	8.90263i		−	0.314167i
$804$		0			0
$805$	−19.6659			−0.693132
$806$		0			0
$807$		−	0.760106i		−	0.0267570i
$808$		0			0
$809$	20.7365			0.729058			0.364529	−	0.931192i	$-0.381230\pi$
$809$	20.7365			0.729058			0.364529	+	0.931192i	$0.381230\pi$
$810$		0			0
$811$	10.1857			0.357667			0.178833	−	0.983879i	$-0.442768\pi$
$811$	10.1857			0.357667			0.178833	+	0.983879i	$0.442768\pi$
$812$		0			0
$813$		−	28.4003i		−	0.996043i
$814$		0			0
$815$		−	46.9693i		−	1.64526i
$816$		0			0
$817$	16.0590	−	14.8344i	0.561833	−	0.518989i
$818$		0			0
$819$	−4.95869			−0.173271
$820$		0			0
$821$	9.14624			0.319206			0.159603	−	0.987181i	$-0.448979\pi$
$821$	9.14624			0.319206			0.159603	+	0.987181i	$0.448979\pi$
$822$		0			0
$823$		−	39.7117i		−	1.38426i	−0.721772	−	0.692131i	$-0.756672\pi$
$823$		−	39.7117i		−	1.38426i	0.721772	−	0.692131i	$-0.243328\pi$
$824$		0			0
$825$			24.5107i			0.853354i
$826$		0			0
$827$	−9.31510			−0.323918			−0.161959	−	0.986797i	$-0.551781\pi$
$827$	−9.31510			−0.323918			−0.161959	+	0.986797i	$0.551781\pi$
$828$		0			0
$829$		−	9.41593i		−	0.327029i	−0.986541	−	0.163514i	$-0.947717\pi$
$829$		−	9.41593i		−	0.327029i	0.986541	−	0.163514i	$-0.0522830\pi$
$830$		0			0
$831$	14.8423			0.514873
$832$		0			0
$833$	0.222287			0.00770180
$834$		0			0
$835$	55.0938			1.90660
$836$		0			0
$837$	−10.3645			−0.358250
$838$		0			0
$839$	24.0459			0.830157			0.415078	−	0.909786i	$-0.363754\pi$
$839$	24.0459			0.830157			0.415078	+	0.909786i	$0.363754\pi$
$840$		0			0
$841$	−7.18388			−0.247720
$842$		0			0
$843$		−	24.4449i		−	0.841928i
$844$		0			0
$845$	−37.4169			−1.28718
$846$		0			0
$847$		−	1.91903i		−	0.0659386i
$848$		0			0
$849$			59.3204i			2.03587i
$850$		0			0
$851$	−11.1079			−0.380773
$852$		0			0
$853$	12.1284			0.415270			0.207635	−	0.978206i	$-0.433423\pi$
$853$	12.1284			0.415270			0.207635	+	0.978206i	$0.433423\pi$
$854$		0			0
$855$	15.9209	+	17.2352i	0.544482	+	0.589431i
$856$		0			0
$857$			21.8937i			0.747876i	0.927454	+	0.373938i	$0.121993\pi$
$857$			21.8937i			0.747876i	−0.927454	+	0.373938i	$0.878007\pi$
$858$		0			0
$859$			29.6474i			1.01155i	0.862664	+	0.505777i	$0.168794\pi$
$859$			29.6474i			1.01155i	−0.862664	+	0.505777i	$0.831206\pi$
$860$		0			0
$861$	5.08429			0.173272
$862$		0			0
$863$	−30.1991			−1.02799			−0.513995	−	0.857793i	$-0.671835\pi$
$863$	−30.1991			−1.02799			−0.513995	+	0.857793i	$0.671835\pi$
$864$		0			0
$865$		−	71.9132i		−	2.44512i
$866$		0			0
$867$	35.2967			1.19874
$868$		0			0
$869$			5.98178i			0.202918i
$870$		0			0
$871$		−	10.2815i		−	0.348375i
$872$		0			0
$873$			17.2721i			0.584572i
$874$		0			0
$875$		−	53.5056i		−	1.80882i
$876$		0			0
$877$		−	48.7210i		−	1.64519i	−0.568627	−	0.822595i	$-0.692525\pi$
$877$		−	48.7210i		−	1.64519i	0.568627	−	0.822595i	$-0.307475\pi$
$878$		0			0
$879$		−	39.8619i		−	1.34451i
$880$		0			0
$881$	−32.1276			−1.08241			−0.541204	−	0.840892i	$-0.682031\pi$
$881$	−32.1276			−1.08241			−0.541204	+	0.840892i	$0.682031\pi$
$882$		0			0
$883$		−	13.7182i		−	0.461656i	−0.972995	−	0.230828i	$-0.925857\pi$
$883$		−	13.7182i		−	0.461656i	0.972995	−	0.230828i	$-0.0741434\pi$
$884$		0			0
$885$	−116.012			−3.89971
$886$		0			0
$887$	−47.2113			−1.58520			−0.792601	−	0.609741i	$-0.791274\pi$
$887$	−47.2113			−1.58520			−0.792601	+	0.609741i	$0.791274\pi$
$888$		0			0
$889$			24.6063i			0.825268i
$890$		0			0
$891$			11.2151i			0.375720i
$892$		0			0
$893$	−17.7164	+	16.3654i	−0.592856	+	0.547647i
$894$		0			0
$895$	−5.51417			−0.184318
$896$		0			0
$897$	−10.2166			−0.341122
$898$		0			0
$899$			17.7968i			0.593558i
$900$		0			0
$901$		−	0.734111i		−	0.0244568i
$902$		0			0
$903$	19.9892			0.665199
$904$		0			0
$905$		−	65.8543i		−	2.18907i
$906$		0			0
$907$	31.8476			1.05748			0.528741	−	0.848783i	$-0.322664\pi$
$907$	31.8476			1.05748			0.528741	+	0.848783i	$0.322664\pi$
$908$		0			0
$909$	3.30227			0.109529
$910$		0			0
$911$	3.65403			0.121063			0.0605317	−	0.998166i	$-0.480720\pi$
$911$	3.65403			0.121063			0.0605317	+	0.998166i	$0.480720\pi$
$912$		0			0
$913$	0.943616			0.0312291
$914$		0			0
$915$	66.6782			2.20431
$916$		0			0
$917$	−10.6942			−0.353153
$918$		0			0
$919$		−	52.4006i		−	1.72854i	−0.503030	−	0.864269i	$-0.667781\pi$
$919$		−	52.4006i		−	1.72854i	0.503030	−	0.864269i	$-0.332219\pi$
$920$		0			0
$921$	−7.38494			−0.243342
$922$		0			0
$923$			10.0889i			0.332079i
$924$		0			0
$925$		−	52.4366i		−	1.72411i
$926$		0			0
$927$	−7.91383			−0.259924
$928$		0			0
$929$	31.2435			1.02507			0.512533	−	0.858667i	$-0.328707\pi$
$929$	31.2435			1.02507			0.512533	+	0.858667i	$0.328707\pi$
$930$		0			0
$931$	−9.81170	−	10.6217i	−0.321565	−	0.348111i
$932$		0			0
$933$			28.0895i			0.919609i
$934$		0			0
$935$			0.274667i			0.00898257i
$936$		0			0
$937$	−31.0721			−1.01508			−0.507541	−	0.861627i	$-0.669446\pi$
$937$	−31.0721			−1.01508			−0.507541	+	0.861627i	$0.669446\pi$
$938$		0			0
$939$	−36.5591			−1.19306
$940$		0			0
$941$			56.1288i			1.82974i	0.403743	+	0.914872i	$0.367709\pi$
$941$			56.1288i			1.82974i	−0.403743	+	0.914872i	$0.632291\pi$
$942$		0			0
$943$	3.18934			0.103859
$944$		0			0
$945$		−	27.5566i		−	0.896415i
$946$		0			0
$947$			7.79720i			0.253375i	0.991943	+	0.126687i	$0.0404345\pi$
$947$			7.79720i			0.253375i	−0.991943	+	0.126687i	$0.959565\pi$
$948$		0			0
$949$			17.5175i			0.568643i
$950$		0			0
$951$			65.5227i			2.12472i
$952$		0			0
$953$			28.5946i			0.926269i	0.886288	+	0.463134i	$0.153275\pi$
$953$			28.5946i			0.926269i	−0.886288	+	0.463134i	$0.846725\pi$
$954$		0			0
$955$		−	11.9336i		−	0.386162i
$956$		0			0
$957$	12.4927			0.403833
$958$		0			0
$959$			28.4379i			0.918309i
$960$		0			0
$961$	−22.2467			−0.717636
$962$		0			0
$963$	0.242435			0.00781237
$964$		0			0
$965$		−	13.7702i		−	0.443277i
$966$		0			0
$967$			31.4165i			1.01029i	0.863036	+	0.505143i	$0.168560\pi$
$967$			31.4165i			1.01029i	−0.863036	+	0.505143i	$0.831440\pi$
$968$		0			0
$969$	0.411605	+	0.445584i	0.0132227	+	0.0143142i
$970$		0			0
$971$	29.3880			0.943106			0.471553	−	0.881838i	$-0.343694\pi$
$971$	29.3880			0.943106			0.471553	+	0.881838i	$0.343694\pi$
$972$		0			0
$973$	−36.3794			−1.16627
$974$		0			0
$975$		−	48.2293i		−	1.54457i
$976$		0			0
$977$			3.55709i			0.113801i	0.998380	+	0.0569007i	$0.0181218\pi$
$977$			3.55709i			0.113801i	−0.998380	+	0.0569007i	$0.981878\pi$
$978$		0			0
$979$	2.25817			0.0721714
$980$		0			0
$981$		−	24.2719i		−	0.774942i
$982$		0			0
$983$	−30.8589			−0.984246			−0.492123	−	0.870526i	$-0.663779\pi$
$983$	−30.8589			−0.984246			−0.492123	+	0.870526i	$0.663779\pi$
$984$		0			0
$985$	45.5785			1.45225
$986$		0			0
$987$	−22.0522			−0.701930
$988$		0			0
$989$	12.5391			0.398720
$990$		0			0
$991$	38.8062			1.23272			0.616360	−	0.787465i	$-0.288607\pi$
$991$	38.8062			1.23272			0.616360	+	0.787465i	$0.288607\pi$
$992$		0			0
$993$	−56.7758			−1.80172
$994$		0			0
$995$			63.4666i			2.01203i
$996$		0			0
$997$	36.1708			1.14554			0.572770	−	0.819716i	$-0.305869\pi$
$997$	36.1708			1.14554			0.572770	+	0.819716i	$0.305869\pi$
$998$		0			0
$999$		−	15.5647i		−	0.492446i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.o.b.1519.13	✓	64
4.3	odd	2	inner	3344.2.o.b.1519.51	yes	64
19.18	odd	2	inner	3344.2.o.b.1519.52	yes	64
76.75	even	2	inner	3344.2.o.b.1519.14	yes	64

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3344.2.o.b.1519.13	✓	64	1.1	even	1	trivial
3344.2.o.b.1519.14	yes	64	76.75	even	2	inner
3344.2.o.b.1519.51	yes	64	4.3	odd	2	inner
3344.2.o.b.1519.52	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)

\(f(q)\)	\(=\)	\(q-2.07682 q^{3} -4.09903 q^{5} +1.91903i q^{7} +1.31320 q^{9} +O(q^{10})\)	\(q-2.07682 q^{3} -4.09903 q^{5} +1.91903i q^{7} +1.31320 q^{9} -1.00000i q^{11} +1.96768i q^{13} +8.51296 q^{15} +0.0670079 q^{17} +(-2.95770 - 3.20187i) q^{19} -3.98548i q^{21} -2.50007i q^{23} +11.8020 q^{25} +3.50319 q^{27} -6.01530i q^{29} -2.95859 q^{31} +2.07682i q^{33} -7.86614i q^{35} -4.44302i q^{37} -4.08653i q^{39} +1.27570i q^{41} +5.01550i q^{43} -5.38285 q^{45} -5.53314i q^{47} +3.31733 q^{49} -0.139164 q^{51} -10.9556i q^{53} +4.09903i q^{55} +(6.14263 + 6.64973i) q^{57} -13.6277 q^{59} +7.83256 q^{61} +2.52007i q^{63} -8.06557i q^{65} -5.22519 q^{67} +5.19221i q^{69} +5.12728 q^{71} +8.90263 q^{73} -24.5107 q^{75} +1.91903 q^{77} -5.98178 q^{79} -11.2151 q^{81} +0.943616i q^{83} -0.274667 q^{85} +12.4927i q^{87} +2.25817i q^{89} -3.77603 q^{91} +6.14448 q^{93} +(12.1237 + 13.1246i) q^{95} +13.1527i q^{97} -1.31320i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q + 56 q^{9}+O(q^{10}) \) 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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