LMFDB - Embedded newform 3344.2.o.b.1519.4

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			1519.4
Character	$\chi$	$=$	3344.1519
Dual form			3344.2.o.b.1519.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.09364 q^{3} +1.45449 q^{5} -2.09473i q^{7} +6.57062 q^{9} +O(q^{10})$	$q-3.09364 q^{3} +1.45449 q^{5} -2.09473i q^{7} +6.57062 q^{9} -1.00000i q^{11} -2.25102i q^{13} -4.49967 q^{15} -0.922499 q^{17} +(-4.23510 - 1.03148i) q^{19} +6.48035i q^{21} +3.82395i q^{23} -2.88446 q^{25} -11.0462 q^{27} -2.53792i q^{29} -5.68717 q^{31} +3.09364i q^{33} -3.04676i q^{35} -6.08802i q^{37} +6.96386i q^{39} +7.15274i q^{41} -7.75908i q^{43} +9.55690 q^{45} +3.19968i q^{47} +2.61210 q^{49} +2.85388 q^{51} +2.13158i q^{53} -1.45449i q^{55} +(13.1019 + 3.19103i) q^{57} -4.62188 q^{59} +4.60632 q^{61} -13.7637i q^{63} -3.27409i q^{65} +9.64892 q^{67} -11.8299i q^{69} +3.33022 q^{71} -6.85856 q^{73} +8.92350 q^{75} -2.09473 q^{77} -1.26793 q^{79} +14.4612 q^{81} -2.23662i q^{83} -1.34176 q^{85} +7.85140i q^{87} +12.6122i q^{89} -4.71529 q^{91} +17.5941 q^{93} +(-6.15990 - 1.50028i) q^{95} -3.18221i q^{97} -6.57062i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−3.09364			−1.78612			−0.893058	−	0.449942i	$-0.851445\pi$
$3$	−3.09364			−1.78612			−0.893058	+	0.449942i	$0.851445\pi$
$4$		0			0
$5$	1.45449			0.650467			0.325234	−	0.945634i	$-0.394557\pi$
$5$	1.45449			0.650467			0.325234	+	0.945634i	$0.394557\pi$
$6$		0			0
$7$		−	2.09473i		−	0.791734i	−0.918308	−	0.395867i	$-0.870444\pi$
$7$		−	2.09473i		−	0.791734i	0.918308	−	0.395867i	$-0.129556\pi$
$8$		0			0
$9$	6.57062			2.19021
$10$		0			0
$11$		−	1.00000i		−	0.301511i
$11$		−	1.00000i		−	0.301511i
$12$		0			0
$13$		−	2.25102i		−	0.624321i	−0.950029	−	0.312161i	$-0.898947\pi$
$13$		−	2.25102i		−	0.624321i	0.950029	−	0.312161i	$-0.101053\pi$
$14$		0			0
$15$	−4.49967			−1.16181
$16$		0			0
$17$	−0.922499			−0.223739			−0.111869	−	0.993723i	$-0.535684\pi$
$17$	−0.922499			−0.223739			−0.111869	+	0.993723i	$0.535684\pi$
$18$		0			0
$19$	−4.23510	−	1.03148i	−0.971598	−	0.236638i
$19$	−4.23510	−	1.03148i	−0.971598	−	0.236638i
$20$		0			0
$21$			6.48035i			1.41413i
$22$		0			0
$23$			3.82395i			0.797349i	0.917092	+	0.398675i	$0.130530\pi$
$23$			3.82395i			0.797349i	−0.917092	+	0.398675i	$0.869470\pi$
$24$		0			0
$25$	−2.88446			−0.576893
$26$		0			0
$27$	−11.0462			−2.12585
$28$		0			0
$29$		−	2.53792i		−	0.471279i	−0.971841	−	0.235640i	$-0.924282\pi$
$29$		−	2.53792i		−	0.471279i	0.971841	−	0.235640i	$-0.0757185\pi$
$30$		0			0
$31$	−5.68717			−1.02145			−0.510723	−	0.859745i	$-0.670622\pi$
$31$	−5.68717			−1.02145			−0.510723	+	0.859745i	$0.670622\pi$
$32$		0			0
$33$			3.09364i			0.538534i
$34$		0			0
$35$		−	3.04676i		−	0.514997i
$36$		0			0
$37$		−	6.08802i		−	1.00086i	−0.865776	−	0.500432i	$-0.833174\pi$
$37$		−	6.08802i		−	1.00086i	0.865776	−	0.500432i	$-0.166826\pi$
$38$		0			0
$39$			6.96386i			1.11511i
$40$		0			0
$41$			7.15274i			1.11707i	0.829481	+	0.558535i	$0.188636\pi$
$41$			7.15274i			1.11707i	−0.829481	+	0.558535i	$0.811364\pi$
$42$		0			0
$43$		−	7.75908i		−	1.18325i	−0.806214	−	0.591624i	$-0.798487\pi$
$43$		−	7.75908i		−	1.18325i	0.806214	−	0.591624i	$-0.201513\pi$
$44$		0			0
$45$	9.55690			1.42466
$46$		0			0
$47$			3.19968i			0.466722i	0.972390	+	0.233361i	$0.0749723\pi$
$47$			3.19968i			0.466722i	−0.972390	+	0.233361i	$0.925028\pi$
$48$		0			0
$49$	2.61210			0.373157
$50$		0			0
$51$	2.85388			0.399623
$52$		0			0
$53$			2.13158i			0.292795i	0.989226	+	0.146398i	$0.0467679\pi$
$53$			2.13158i			0.292795i	−0.989226	+	0.146398i	$0.953232\pi$
$54$		0			0
$55$		−	1.45449i		−	0.196123i
$56$		0			0
$57$	13.1019	+	3.19103i	1.73539	+	0.422662i
$58$		0			0
$59$	−4.62188			−0.601718			−0.300859	−	0.953669i	$-0.597273\pi$
$59$	−4.62188			−0.601718			−0.300859	+	0.953669i	$0.597273\pi$
$60$		0			0
$61$	4.60632			0.589779			0.294889	−	0.955531i	$-0.404717\pi$
$61$	4.60632			0.589779			0.294889	+	0.955531i	$0.404717\pi$
$62$		0			0
$63$		−	13.7637i		−	1.73406i
$64$		0			0
$65$		−	3.27409i		−	0.406100i
$66$		0			0
$67$	9.64892			1.17880			0.589402	−	0.807840i	$-0.299363\pi$
$67$	9.64892			1.17880			0.589402	+	0.807840i	$0.299363\pi$
$68$		0			0
$69$		−	11.8299i		−	1.42416i
$70$		0			0
$71$	3.33022			0.395224			0.197612	−	0.980280i	$-0.436681\pi$
$71$	3.33022			0.395224			0.197612	+	0.980280i	$0.436681\pi$
$72$		0			0
$73$	−6.85856			−0.802734			−0.401367	−	0.915917i	$-0.631465\pi$
$73$	−6.85856			−0.802734			−0.401367	+	0.915917i	$0.631465\pi$
$74$		0			0
$75$	8.92350			1.03040
$76$		0			0
$77$	−2.09473			−0.238717
$78$		0			0
$79$	−1.26793			−0.142653			−0.0713264	−	0.997453i	$-0.522723\pi$
$79$	−1.26793			−0.142653			−0.0713264	+	0.997453i	$0.522723\pi$
$80$		0			0
$81$	14.4612			1.60680
$82$		0			0
$83$		−	2.23662i		−	0.245501i	−0.992438	−	0.122750i	$-0.960829\pi$
$83$		−	2.23662i		−	0.245501i	0.992438	−	0.122750i	$-0.0391714\pi$
$84$		0			0
$85$	−1.34176			−0.145535
$86$		0			0
$87$			7.85140i			0.841759i
$88$		0			0
$89$			12.6122i			1.33689i	0.743760	+	0.668446i	$0.233040\pi$
$89$			12.6122i			1.33689i	−0.743760	+	0.668446i	$0.766960\pi$
$90$		0			0
$91$	−4.71529			−0.494297
$92$		0			0
$93$	17.5941			1.82442
$94$		0			0
$95$	−6.15990	−	1.50028i	−0.631992	−	0.153925i
$96$		0			0
$97$		−	3.18221i		−	0.323105i	−0.986864	−	0.161552i	$-0.948350\pi$
$97$		−	3.18221i		−	0.323105i	0.986864	−	0.161552i	$-0.0516501\pi$
$98$		0			0
$99$		−	6.57062i		−	0.660372i
$100$		0			0
$101$	−2.77911			−0.276532			−0.138266	−	0.990395i	$-0.544153\pi$
$101$	−2.77911			−0.276532			−0.138266	+	0.990395i	$0.544153\pi$
$102$		0			0
$103$	−10.5159			−1.03617			−0.518083	−	0.855330i	$-0.673354\pi$
$103$	−10.5159			−1.03617			−0.518083	+	0.855330i	$0.673354\pi$
$104$		0			0
$105$			9.42560i			0.919844i
$106$		0			0
$107$	11.2545			1.08801			0.544007	−	0.839081i	$-0.316907\pi$
$107$	11.2545			1.08801			0.544007	+	0.839081i	$0.316907\pi$
$108$		0			0
$109$			7.43219i			0.711875i	0.934510	+	0.355937i	$0.115838\pi$
$109$			7.43219i			0.711875i	−0.934510	+	0.355937i	$0.884162\pi$
$110$		0			0
$111$			18.8342i			1.78766i
$112$		0			0
$113$		−	4.81434i		−	0.452895i	−0.974023	−	0.226447i	$-0.927289\pi$
$113$		−	4.81434i		−	0.452895i	0.974023	−	0.226447i	$-0.0727112\pi$
$114$		0			0
$115$			5.56190i			0.518650i
$116$		0			0
$117$		−	14.7906i		−	1.36739i
$118$		0			0
$119$			1.93239i			0.177142i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$		−	22.1280i		−	1.99522i
$124$		0			0
$125$	−11.4679			−1.02572
$126$		0			0
$127$	2.86373			0.254115			0.127057	−	0.991895i	$-0.459447\pi$
$127$	2.86373			0.254115			0.127057	+	0.991895i	$0.459447\pi$
$128$		0			0
$129$			24.0038i			2.11342i
$130$		0			0
$131$			5.88886i			0.514512i	0.966343	+	0.257256i	$0.0828184\pi$
$131$			5.88886i			0.514512i	−0.966343	+	0.257256i	$0.917182\pi$
$132$		0			0
$133$	−2.16068	+	8.87139i	−0.187354	+	0.769247i
$134$		0			0
$135$	−16.0666			−1.38279
$136$		0			0
$137$	−9.48779			−0.810597			−0.405298	−	0.914184i	$-0.632832\pi$
$137$	−9.48779			−0.810597			−0.405298	+	0.914184i	$0.632832\pi$
$138$		0			0
$139$		−	8.28223i		−	0.702490i	−0.936284	−	0.351245i	$-0.885758\pi$
$139$		−	8.28223i		−	0.702490i	0.936284	−	0.351245i	$-0.114242\pi$
$140$		0			0
$141$		−	9.89867i		−	0.833619i
$142$		0			0
$143$	−2.25102			−0.188240
$144$		0			0
$145$		−	3.69137i		−	0.306552i
$146$		0			0
$147$	−8.08089			−0.666501
$148$		0			0
$149$	−16.9889			−1.39179			−0.695894	−	0.718144i	$-0.744992\pi$
$149$	−16.9889			−1.39179			−0.695894	+	0.718144i	$0.744992\pi$
$150$		0			0
$151$	1.58023			0.128597			0.0642986	−	0.997931i	$-0.479519\pi$
$151$	1.58023			0.128597			0.0642986	+	0.997931i	$0.479519\pi$
$152$		0			0
$153$	−6.06139			−0.490035
$154$		0			0
$155$	−8.27192			−0.664417
$156$		0			0
$157$	−15.5363			−1.23993			−0.619964	−	0.784630i	$-0.712853\pi$
$157$	−15.5363			−1.23993			−0.619964	+	0.784630i	$0.712853\pi$
$158$		0			0
$159$		−	6.59435i		−	0.522966i
$160$		0			0
$161$	8.01016			0.631289
$162$		0			0
$163$			9.59526i			0.751559i	0.926709	+	0.375779i	$0.122625\pi$
$163$			9.59526i			0.751559i	−0.926709	+	0.375779i	$0.877375\pi$
$164$		0			0
$165$			4.49967i			0.350299i
$166$		0			0
$167$	14.3467			1.11018			0.555090	−	0.831790i	$-0.312684\pi$
$167$	14.3467			1.11018			0.555090	+	0.831790i	$0.312684\pi$
$168$		0			0
$169$	7.93290			0.610223
$170$		0			0
$171$	−27.8272	−	6.77747i	−2.12800	−	0.518286i
$172$		0			0
$173$			19.9136i			1.51400i	0.653414	+	0.757001i	$0.273336\pi$
$173$			19.9136i			1.51400i	−0.653414	+	0.757001i	$0.726664\pi$
$174$		0			0
$175$			6.04218i			0.456746i
$176$		0			0
$177$	14.2985			1.07474
$178$		0			0
$179$	−21.0746			−1.57519			−0.787596	−	0.616192i	$-0.788675\pi$
$179$	−21.0746			−1.57519			−0.787596	+	0.616192i	$0.788675\pi$
$180$		0			0
$181$		−	8.94552i		−	0.664915i	−0.943118	−	0.332458i	$-0.892122\pi$
$181$		−	8.94552i		−	0.664915i	0.943118	−	0.332458i	$-0.107878\pi$
$182$		0			0
$183$	−14.2503			−1.05341
$184$		0			0
$185$		−	8.85496i		−	0.651030i
$186$		0			0
$187$			0.922499i			0.0674598i
$188$		0			0
$189$			23.1389i			1.68311i
$190$		0			0
$191$			11.3232i			0.819317i	0.912239	+	0.409658i	$0.134352\pi$
$191$			11.3232i			0.819317i	−0.912239	+	0.409658i	$0.865648\pi$
$192$		0			0
$193$			1.48594i			0.106960i	0.998569	+	0.0534802i	$0.0170314\pi$
$193$			1.48594i			0.106960i	−0.998569	+	0.0534802i	$0.982969\pi$
$194$		0			0
$195$			10.1289i			0.725342i
$196$		0			0
$197$	14.9554			1.06553			0.532763	−	0.846264i	$-0.321154\pi$
$197$	14.9554			1.06553			0.532763	+	0.846264i	$0.321154\pi$
$198$		0			0
$199$			24.9852i			1.77115i	0.464495	+	0.885576i	$0.346236\pi$
$199$			24.9852i			1.77115i	−0.464495	+	0.885576i	$0.653764\pi$
$200$		0			0
$201$	−29.8503			−2.10548
$202$		0			0
$203$	−5.31625			−0.373128
$204$		0			0
$205$			10.4036i			0.726617i
$206$		0			0
$207$			25.1258i			1.74636i
$208$		0			0
$209$	−1.03148	+	4.23510i	−0.0713490	+	0.292948i
$210$		0			0
$211$	−10.9249			−0.752103			−0.376052	−	0.926599i	$-0.622718\pi$
$211$	−10.9249			−0.752103			−0.376052	+	0.926599i	$0.622718\pi$
$212$		0			0
$213$	−10.3025			−0.705916
$214$		0			0
$215$		−	11.2855i		−	0.769664i
$216$		0			0
$217$			11.9131i			0.808714i
$218$		0			0
$219$	21.2179			1.43378
$220$		0			0
$221$			2.07657i			0.139685i
$222$		0			0
$223$	1.76222			0.118007			0.0590036	−	0.998258i	$-0.481208\pi$
$223$	1.76222			0.118007			0.0590036	+	0.998258i	$0.481208\pi$
$224$		0			0
$225$	−18.9527			−1.26351
$226$		0			0
$227$	−3.56187			−0.236410			−0.118205	−	0.992989i	$-0.537714\pi$
$227$	−3.56187			−0.236410			−0.118205	+	0.992989i	$0.537714\pi$
$228$		0			0
$229$	−28.2755			−1.86850			−0.934248	−	0.356623i	$-0.883928\pi$
$229$	−28.2755			−1.86850			−0.934248	+	0.356623i	$0.883928\pi$
$230$		0			0
$231$	6.48035			0.426376
$232$		0			0
$233$	−10.7275			−0.702785			−0.351392	−	0.936228i	$-0.614292\pi$
$233$	−10.7275			−0.702785			−0.351392	+	0.936228i	$0.614292\pi$
$234$		0			0
$235$			4.65390i			0.303587i
$236$		0			0
$237$	3.92251			0.254794
$238$		0			0
$239$			24.5451i			1.58769i	0.608119	+	0.793846i	$0.291924\pi$
$239$			24.5451i			1.58769i	−0.608119	+	0.793846i	$0.708076\pi$
$240$		0			0
$241$			4.98704i			0.321243i	0.987016	+	0.160622i	$0.0513499\pi$
$241$			4.98704i			0.321243i	−0.987016	+	0.160622i	$0.948650\pi$
$242$		0			0
$243$	−11.5991			−0.744084
$244$		0			0
$245$	3.79927			0.242726
$246$		0			0
$247$	−2.32189	+	9.53330i	−0.147738	+	0.606589i
$248$		0			0
$249$			6.91930i			0.438493i
$250$		0			0
$251$			6.34764i			0.400660i	0.979728	+	0.200330i	$0.0642014\pi$
$251$			6.34764i			0.400660i	−0.979728	+	0.200330i	$0.935799\pi$
$252$		0			0
$253$	3.82395			0.240410
$254$		0			0
$255$	4.15094			0.259942
$256$		0			0
$257$		−	9.47092i		−	0.590780i	−0.955377	−	0.295390i	$-0.904550\pi$
$257$		−	9.47092i		−	0.590780i	0.955377	−	0.295390i	$-0.0954495\pi$
$258$		0			0
$259$	−12.7528			−0.792419
$260$		0			0
$261$		−	16.6757i		−	1.03220i
$262$		0			0
$263$			23.1261i			1.42602i	0.701156	+	0.713008i	$0.252668\pi$
$263$			23.1261i			1.42602i	−0.701156	+	0.713008i	$0.747332\pi$
$264$		0			0
$265$			3.10036i			0.190454i
$266$		0			0
$267$		−	39.0177i		−	2.38784i
$268$		0			0
$269$		−	11.6846i		−	0.712422i	−0.934406	−	0.356211i	$-0.884068\pi$
$269$		−	11.6846i		−	0.712422i	0.934406	−	0.356211i	$-0.115932\pi$
$270$		0			0
$271$			28.7467i			1.74624i	0.487505	+	0.873120i	$0.337907\pi$
$271$			28.7467i			1.74624i	−0.487505	+	0.873120i	$0.662093\pi$
$272$		0			0
$273$	14.5874			0.882871
$274$		0			0
$275$			2.88446i			0.173940i
$276$		0			0
$277$	−27.1821			−1.63321			−0.816607	−	0.577194i	$-0.804148\pi$
$277$	−27.1821			−1.63321			−0.816607	+	0.577194i	$0.804148\pi$
$278$		0			0
$279$	−37.3682			−2.23718
$280$		0			0
$281$		−	9.16558i		−	0.546772i	−0.961904	−	0.273386i	$-0.911856\pi$
$281$		−	9.16558i		−	0.546772i	0.961904	−	0.273386i	$-0.0881437\pi$
$282$		0			0
$283$			1.21750i			0.0723729i	0.999345	+	0.0361865i	$0.0115210\pi$
$283$			1.21750i			0.0723729i	−0.999345	+	0.0361865i	$0.988479\pi$
$284$		0			0
$285$	19.0565	+	4.64132i	1.12881	+	0.274928i
$286$		0			0
$287$	14.9831			0.884423
$288$		0			0
$289$	−16.1490			−0.949941
$290$		0			0
$291$			9.84462i			0.577102i
$292$		0			0
$293$			7.69706i			0.449667i	0.974397	+	0.224834i	$0.0721838\pi$
$293$			7.69706i			0.449667i	−0.974397	+	0.224834i	$0.927816\pi$
$294$		0			0
$295$	−6.72248			−0.391398
$296$		0			0
$297$			11.0462i			0.640967i
$298$		0			0
$299$	8.60780			0.497802
$300$		0			0
$301$	−16.2532			−0.936818
$302$		0			0
$303$	8.59758			0.493918
$304$		0			0
$305$	6.69984			0.383632
$306$		0			0
$307$	30.1994			1.72357			0.861787	−	0.507271i	$-0.169346\pi$
$307$	30.1994			1.72357			0.861787	+	0.507271i	$0.169346\pi$
$308$		0			0
$309$	32.5326			1.85071
$310$		0			0
$311$		−	20.1697i		−	1.14372i	−0.820353	−	0.571858i	$-0.806223\pi$
$311$		−	20.1697i		−	1.14372i	0.820353	−	0.571858i	$-0.193777\pi$
$312$		0			0
$313$	−6.26810			−0.354294			−0.177147	−	0.984184i	$-0.556687\pi$
$313$	−6.26810			−0.354294			−0.177147	+	0.984184i	$0.556687\pi$
$314$		0			0
$315$		−	20.0191i		−	1.12795i
$316$		0			0
$317$			21.6303i			1.21488i	0.794367	+	0.607439i	$0.207803\pi$
$317$			21.6303i			1.21488i	−0.794367	+	0.607439i	$0.792197\pi$
$318$		0			0
$319$	−2.53792			−0.142096
$320$		0			0
$321$	−34.8174			−1.94332
$322$		0			0
$323$	3.90687	+	0.951540i	0.217384	+	0.0529451i
$324$		0			0
$325$			6.49299i			0.360166i
$326$		0			0
$327$		−	22.9925i		−	1.27149i
$328$		0			0
$329$	6.70248			0.369520
$330$		0			0
$331$	−13.6759			−0.751693			−0.375847	−	0.926682i	$-0.622648\pi$
$331$	−13.6759			−0.751693			−0.375847	+	0.926682i	$0.622648\pi$
$332$		0			0
$333$		−	40.0021i		−	2.19210i
$334$		0			0
$335$	14.0342			0.766773
$336$		0			0
$337$		−	22.0825i		−	1.20291i	−0.798906	−	0.601456i	$-0.794588\pi$
$337$		−	22.0825i		−	1.20291i	0.798906	−	0.601456i	$-0.205412\pi$
$338$		0			0
$339$			14.8938i			0.808923i
$340$		0			0
$341$			5.68717i			0.307977i
$342$		0			0
$343$		−	20.1348i		−	1.08718i
$344$		0			0
$345$		−	17.2065i		−	0.926368i
$346$		0			0
$347$		−	30.1209i		−	1.61697i	−0.588514	−	0.808487i	$-0.700287\pi$
$347$		−	30.1209i		−	1.61697i	0.588514	−	0.808487i	$-0.299713\pi$
$348$		0			0
$349$	26.9792			1.44416			0.722081	−	0.691809i	$-0.243186\pi$
$349$	26.9792			1.44416			0.722081	+	0.691809i	$0.243186\pi$
$350$		0			0
$351$			24.8653i			1.32721i
$352$		0			0
$353$	−23.9113			−1.27267			−0.636336	−	0.771412i	$-0.719551\pi$
$353$	−23.9113			−1.27267			−0.636336	+	0.771412i	$0.719551\pi$
$354$		0			0
$355$	4.84377			0.257081
$356$		0			0
$357$		−	5.97812i		−	0.316396i
$358$		0			0
$359$		−	3.68462i		−	0.194467i	−0.995262	−	0.0972334i	$-0.969001\pi$
$359$		−	3.68462i		−	0.194467i	0.995262	−	0.0972334i	$-0.0309993\pi$
$360$		0			0
$361$	16.8721	+	8.73684i	0.888005	+	0.459834i
$362$		0			0
$363$	3.09364			0.162374
$364$		0			0
$365$	−9.97570			−0.522152
$366$		0			0
$367$			13.7497i			0.717729i	0.933390	+	0.358865i	$0.116836\pi$
$367$			13.7497i			0.717729i	−0.933390	+	0.358865i	$0.883164\pi$
$368$		0			0
$369$			46.9979i			2.44662i
$370$		0			0
$371$	4.46509			0.231816
$372$		0			0
$373$		−	26.4401i		−	1.36902i	−0.729005	−	0.684508i	$-0.760017\pi$
$373$		−	26.4401i		−	1.36902i	0.729005	−	0.684508i	$-0.239983\pi$
$374$		0			0
$375$	35.4775			1.83205
$376$		0			0
$377$	−5.71290			−0.294230
$378$		0			0
$379$	−38.0469			−1.95434			−0.977170	−	0.212461i	$-0.931852\pi$
$379$	−38.0469			−1.95434			−0.977170	+	0.212461i	$0.931852\pi$
$380$		0			0
$381$	−8.85936			−0.453879
$382$		0			0
$383$	12.2758			0.627265			0.313633	−	0.949544i	$-0.398454\pi$
$383$	12.2758			0.627265			0.313633	+	0.949544i	$0.398454\pi$
$384$		0			0
$385$	−3.04676			−0.155277
$386$		0			0
$387$		−	50.9820i		−	2.59156i
$388$		0			0
$389$	8.45566			0.428719			0.214359	−	0.976755i	$-0.431234\pi$
$389$	8.45566			0.428719			0.214359	+	0.976755i	$0.431234\pi$
$390$		0			0
$391$		−	3.52759i		−	0.178398i
$392$		0			0
$393$		−	18.2180i		−	0.918978i
$394$		0			0
$395$	−1.84418			−0.0927909
$396$		0			0
$397$	−5.54141			−0.278115			−0.139058	−	0.990284i	$-0.544407\pi$
$397$	−5.54141			−0.278115			−0.139058	+	0.990284i	$0.544407\pi$
$398$		0			0
$399$	6.68436	−	27.4449i	0.334636	−	1.37396i
$400$		0			0
$401$		−	27.3091i		−	1.36375i	−0.731468	−	0.681876i	$-0.761164\pi$
$401$		−	27.3091i		−	1.36375i	0.731468	−	0.681876i	$-0.238836\pi$
$402$		0			0
$403$			12.8019i			0.637710i
$404$		0			0
$405$	21.0337			1.04517
$406$		0			0
$407$	−6.08802			−0.301772
$408$		0			0
$409$			29.9297i			1.47993i	0.672645	+	0.739965i	$0.265158\pi$
$409$			29.9297i			1.47993i	−0.672645	+	0.739965i	$0.734842\pi$
$410$		0			0
$411$	29.3518			1.44782
$412$		0			0
$413$			9.68161i			0.476401i
$414$		0			0
$415$		−	3.25314i		−	0.159690i
$416$		0			0
$417$			25.6223i			1.25473i
$418$		0			0
$419$			31.2144i			1.52493i	0.647032	+	0.762463i	$0.276010\pi$
$419$			31.2144i			1.52493i	−0.647032	+	0.762463i	$0.723990\pi$
$420$		0			0
$421$			6.76294i			0.329606i	0.986327	+	0.164803i	$0.0526988\pi$
$421$			6.76294i			0.329606i	−0.986327	+	0.164803i	$0.947301\pi$
$422$		0			0
$423$			21.0239i			1.02222i
$424$		0			0
$425$	2.66091			0.129073
$426$		0			0
$427$		−	9.64901i		−	0.466948i
$428$		0			0
$429$	6.96386			0.336218
$430$		0			0
$431$	−22.0760			−1.06336			−0.531681	−	0.846945i	$-0.678439\pi$
$431$	−22.0760			−1.06336			−0.531681	+	0.846945i	$0.678439\pi$
$432$		0			0
$433$		−	6.52416i		−	0.313531i	−0.987636	−	0.156766i	$-0.949893\pi$
$433$		−	6.52416i		−	0.313531i	0.987636	−	0.156766i	$-0.0501067\pi$
$434$		0			0
$435$			11.4198i			0.547536i
$436$		0			0
$437$	3.94433	−	16.1948i	0.188683	−	0.774703i
$438$		0			0
$439$	−18.9282			−0.903394			−0.451697	−	0.892172i	$-0.649181\pi$
$439$	−18.9282			−0.903394			−0.451697	+	0.892172i	$0.649181\pi$
$440$		0			0
$441$	17.1631			0.817291
$442$		0			0
$443$			2.48101i			0.117876i	0.998262	+	0.0589382i	$0.0187715\pi$
$443$			2.48101i			0.117876i	−0.998262	+	0.0589382i	$0.981229\pi$
$444$		0			0
$445$			18.3443i			0.869605i
$446$		0			0
$447$	52.5577			2.48589
$448$		0			0
$449$			13.1372i			0.619985i	0.950739	+	0.309992i	$0.100327\pi$
$449$			13.1372i			0.619985i	−0.950739	+	0.309992i	$0.899673\pi$
$450$		0			0
$451$	7.15274			0.336809
$452$		0			0
$453$	−4.88866			−0.229689
$454$		0			0
$455$	−6.85833			−0.321524
$456$		0			0
$457$	15.2970			0.715562			0.357781	−	0.933806i	$-0.383533\pi$
$457$	15.2970			0.715562			0.357781	+	0.933806i	$0.383533\pi$
$458$		0			0
$459$	10.1901			0.475635
$460$		0			0
$461$	−19.7537			−0.920020			−0.460010	−	0.887914i	$-0.652154\pi$
$461$	−19.7537			−0.920020			−0.460010	+	0.887914i	$0.652154\pi$
$462$		0			0
$463$		−	12.9801i		−	0.603235i	−0.953429	−	0.301617i	$-0.902473\pi$
$463$		−	12.9801i		−	0.603235i	0.953429	−	0.301617i	$-0.0975265\pi$
$464$		0			0
$465$	25.5904			1.18672
$466$		0			0
$467$			40.0300i			1.85237i	0.377073	+	0.926184i	$0.376931\pi$
$467$			40.0300i			1.85237i	−0.377073	+	0.926184i	$0.623069\pi$
$468$		0			0
$469$		−	20.2119i		−	0.933299i
$470$		0			0
$471$	48.0636			2.21465
$472$		0			0
$473$	−7.75908			−0.356763
$474$		0			0
$475$	12.2160	+	2.97527i	0.560508	+	0.136515i
$476$		0			0
$477$			14.0058i			0.641283i
$478$		0			0
$479$		−	18.9829i		−	0.867351i	−0.901069	−	0.433675i	$-0.857216\pi$
$479$		−	18.9829i		−	0.867351i	0.901069	−	0.433675i	$-0.142784\pi$
$480$		0			0
$481$	−13.7043			−0.624861
$482$		0			0
$483$	−24.7806			−1.12755
$484$		0			0
$485$		−	4.62849i		−	0.210169i
$486$		0			0
$487$	−16.8080			−0.761644			−0.380822	−	0.924648i	$-0.624359\pi$
$487$	−16.8080			−0.761644			−0.380822	+	0.924648i	$0.624359\pi$
$488$		0			0
$489$		−	29.6843i		−	1.34237i
$490$		0			0
$491$			29.8173i			1.34564i	0.739808	+	0.672819i	$0.234917\pi$
$491$			29.8173i			1.34564i	−0.739808	+	0.672819i	$0.765083\pi$
$492$		0			0
$493$			2.34122i			0.105443i
$494$		0			0
$495$		−	9.55690i		−	0.429550i
$496$		0			0
$497$		−	6.97592i		−	0.312913i
$498$		0			0
$499$		−	19.9481i		−	0.893001i	−0.894783	−	0.446500i	$-0.852670\pi$
$499$		−	19.9481i		−	0.893001i	0.894783	−	0.446500i	$-0.147330\pi$
$500$		0			0
$501$	−44.3835			−1.98291
$502$		0			0
$503$		−	24.9296i		−	1.11155i	−0.831331	−	0.555777i	$-0.812421\pi$
$503$		−	24.9296i		−	1.11155i	0.831331	−	0.555777i	$-0.187579\pi$
$504$		0			0
$505$	−4.04219			−0.179875
$506$		0			0
$507$	−24.5415			−1.08993
$508$		0			0
$509$		−	20.2985i		−	0.899716i	−0.893100	−	0.449858i	$-0.851475\pi$
$509$		−	20.2985i		−	0.899716i	0.893100	−	0.449858i	$-0.148525\pi$
$510$		0			0
$511$			14.3669i			0.635552i
$512$		0			0
$513$	46.7818	+	11.3940i	2.06547	+	0.503056i
$514$		0			0
$515$	−15.2953			−0.673992
$516$		0			0
$517$	3.19968			0.140722
$518$		0			0
$519$		−	61.6055i		−	2.70418i
$520$		0			0
$521$			41.1016i			1.80069i	0.435172	+	0.900347i	$0.356688\pi$
$521$			41.1016i			1.80069i	−0.435172	+	0.900347i	$0.643312\pi$
$522$		0			0
$523$	9.59781			0.419683			0.209842	−	0.977735i	$-0.432705\pi$
$523$	9.59781			0.419683			0.209842	+	0.977735i	$0.432705\pi$
$524$		0			0
$525$		−	18.6923i		−	0.815800i
$526$		0			0
$527$	5.24641			0.228537
$528$		0			0
$529$	8.37738			0.364234
$530$		0			0
$531$	−30.3687			−1.31789
$532$		0			0
$533$	16.1010			0.697411
$534$		0			0
$535$	16.3695			0.707717
$536$		0			0
$537$	65.1974			2.81347
$538$		0			0
$539$		−	2.61210i		−	0.112511i
$540$		0			0
$541$	−6.19811			−0.266477			−0.133239	−	0.991084i	$-0.542538\pi$
$541$	−6.19811			−0.266477			−0.133239	+	0.991084i	$0.542538\pi$
$542$		0			0
$543$			27.6742i			1.18762i
$544$		0			0
$545$			10.8100i			0.463051i
$546$		0			0
$547$	11.4983			0.491633			0.245816	−	0.969316i	$-0.420944\pi$
$547$	11.4983			0.491633			0.245816	+	0.969316i	$0.420944\pi$
$548$		0			0
$549$	30.2664			1.29174
$550$		0			0
$551$	−2.61781	+	10.7483i	−0.111522	+	0.457894i
$552$		0			0
$553$			2.65596i			0.112943i
$554$		0			0
$555$			27.3941i			1.16281i
$556$		0			0
$557$	−7.22647			−0.306195			−0.153098	−	0.988211i	$-0.548925\pi$
$557$	−7.22647			−0.306195			−0.153098	+	0.988211i	$0.548925\pi$
$558$		0			0
$559$	−17.4659			−0.738727
$560$		0			0
$561$		−	2.85388i		−	0.120491i
$562$		0			0
$563$	−2.69757			−0.113689			−0.0568444	−	0.998383i	$-0.518104\pi$
$563$	−2.69757			−0.113689			−0.0568444	+	0.998383i	$0.518104\pi$
$564$		0			0
$565$		−	7.00240i		−	0.294593i
$566$		0			0
$567$		−	30.2924i		−	1.27216i
$568$		0			0
$569$		−	23.1671i		−	0.971217i	−0.874176	−	0.485609i	$-0.838598\pi$
$569$		−	23.1671i		−	0.971217i	0.874176	−	0.485609i	$-0.161402\pi$
$570$		0			0
$571$			18.2476i			0.763638i	0.924237	+	0.381819i	$0.124702\pi$
$571$			18.2476i			0.763638i	−0.924237	+	0.381819i	$0.875298\pi$
$572$		0			0
$573$		−	35.0299i		−	1.46339i
$574$		0			0
$575$		−	11.0301i		−	0.459985i
$576$		0			0
$577$	31.8208			1.32472			0.662358	−	0.749187i	$-0.269556\pi$
$577$	31.8208			1.32472			0.662358	+	0.749187i	$0.269556\pi$
$578$		0			0
$579$		−	4.59697i		−	0.191044i
$580$		0			0
$581$	−4.68512			−0.194371
$582$		0			0
$583$	2.13158			0.0882811
$584$		0			0
$585$		−	21.5128i		−	0.889444i
$586$		0			0
$587$		−	38.9434i		−	1.60737i	−0.595056	−	0.803684i	$-0.702870\pi$
$587$		−	38.9434i		−	1.60737i	0.595056	−	0.803684i	$-0.297130\pi$
$588$		0			0
$589$	24.0857	+	5.86620i	0.992434	+	0.241713i
$590$		0			0
$591$	−46.2666			−1.90315
$592$		0			0
$593$	16.9729			0.696995			0.348497	−	0.937310i	$-0.386692\pi$
$593$	16.9729			0.696995			0.348497	+	0.937310i	$0.386692\pi$
$594$		0			0
$595$			2.81064i			0.115225i
$596$		0			0
$597$		−	77.2952i		−	3.16348i
$598$		0			0
$599$	−8.60368			−0.351537			−0.175768	−	0.984432i	$-0.556241\pi$
$599$	−8.60368			−0.351537			−0.175768	+	0.984432i	$0.556241\pi$
$600$		0			0
$601$			46.4827i			1.89607i	0.318165	+	0.948035i	$0.396933\pi$
$601$			46.4827i			1.89607i	−0.318165	+	0.948035i	$0.603067\pi$
$602$		0			0
$603$	63.3994			2.58182
$604$		0			0
$605$	−1.45449			−0.0591334
$606$		0			0
$607$	−39.1029			−1.58714			−0.793570	−	0.608479i	$-0.791780\pi$
$607$	−39.1029			−1.58714			−0.793570	+	0.608479i	$0.791780\pi$
$608$		0			0
$609$	16.4466			0.666449
$610$		0			0
$611$	7.20256			0.291384
$612$		0			0
$613$	2.43236			0.0982419			0.0491210	−	0.998793i	$-0.484358\pi$
$613$	2.43236			0.0982419			0.0491210	+	0.998793i	$0.484358\pi$
$614$		0			0
$615$		−	32.1849i		−	1.29782i
$616$		0			0
$617$	15.5953			0.627841			0.313921	−	0.949449i	$-0.398357\pi$
$617$	15.5953			0.627841			0.313921	+	0.949449i	$0.398357\pi$
$618$		0			0
$619$		−	15.5364i		−	0.624461i	−0.950006	−	0.312231i	$-0.898924\pi$
$619$		−	15.5364i		−	0.624461i	0.950006	−	0.312231i	$-0.101076\pi$
$620$		0			0
$621$		−	42.2403i		−	1.69504i
$622$		0			0
$623$	26.4192			1.05846
$624$		0			0
$625$	−2.25756			−0.0903024
$626$		0			0
$627$	3.19103	−	13.1019i	0.127438	−	0.523239i
$628$		0			0
$629$			5.61620i			0.223932i
$630$		0			0
$631$		−	28.4729i		−	1.13349i	−0.823894	−	0.566745i	$-0.808203\pi$
$631$		−	28.4729i		−	1.13349i	0.823894	−	0.566745i	$-0.191797\pi$
$632$		0			0
$633$	33.7978			1.34334
$634$		0			0
$635$	4.16526			0.165293
$636$		0			0
$637$		−	5.87989i		−	0.232970i
$638$		0			0
$639$	21.8816			0.865624
$640$		0			0
$641$			16.2649i			0.642424i	0.947007	+	0.321212i	$0.104090\pi$
$641$			16.2649i			0.642424i	−0.947007	+	0.321212i	$0.895910\pi$
$642$		0			0
$643$			6.40132i			0.252443i	0.992002	+	0.126222i	$0.0402850\pi$
$643$			6.40132i			0.252443i	−0.992002	+	0.126222i	$0.959715\pi$
$644$		0			0
$645$			34.9133i			1.37471i
$646$		0			0
$647$		−	30.4676i		−	1.19780i	−0.800822	−	0.598902i	$-0.795604\pi$
$647$		−	30.4676i		−	1.19780i	0.800822	−	0.598902i	$-0.204396\pi$
$648$		0			0
$649$			4.62188i			0.181425i
$650$		0			0
$651$		−	36.8549i		−	1.44446i
$652$		0			0
$653$	−28.0368			−1.09717			−0.548583	−	0.836096i	$-0.684832\pi$
$653$	−28.0368			−1.09717			−0.548583	+	0.836096i	$0.684832\pi$
$654$		0			0
$655$			8.56528i			0.334673i
$656$		0			0
$657$	−45.0650			−1.75815
$658$		0			0
$659$	39.5348			1.54006			0.770029	−	0.638008i	$-0.220241\pi$
$659$	39.5348			1.54006			0.770029	+	0.638008i	$0.220241\pi$
$660$		0			0
$661$		−	32.7137i		−	1.27241i	−0.771518	−	0.636207i	$-0.780502\pi$
$661$		−	32.7137i		−	1.27241i	0.771518	−	0.636207i	$-0.219498\pi$
$662$		0			0
$663$		−	6.42415i		−	0.249493i
$664$		0			0
$665$	−3.14268	+	12.9033i	−0.121868	+	0.500370i
$666$		0			0
$667$	9.70487			0.375774
$668$		0			0
$669$	−5.45169			−0.210775
$670$		0			0
$671$		−	4.60632i		−	0.177825i
$672$		0			0
$673$			31.7058i			1.22217i	0.791565	+	0.611085i	$0.209267\pi$
$673$			31.7058i			1.22217i	−0.791565	+	0.611085i	$0.790733\pi$
$674$		0			0
$675$	31.8624			1.22639
$676$		0			0
$677$		−	15.1786i		−	0.583361i	−0.956516	−	0.291680i	$-0.905786\pi$
$677$		−	15.1786i		−	0.583361i	0.956516	−	0.291680i	$-0.0942144\pi$
$678$		0			0
$679$	−6.66588			−0.255813
$680$		0			0
$681$	11.0191			0.422255
$682$		0			0
$683$	−1.57966			−0.0604440			−0.0302220	−	0.999543i	$-0.509621\pi$
$683$	−1.57966			−0.0604440			−0.0302220	+	0.999543i	$0.509621\pi$
$684$		0			0
$685$	−13.7999			−0.527267
$686$		0			0
$687$	87.4742			3.33735
$688$		0			0
$689$	4.79824			0.182798
$690$		0			0
$691$			47.9684i			1.82480i	0.409297	+	0.912401i	$0.365774\pi$
$691$			47.9684i			1.82480i	−0.409297	+	0.912401i	$0.634226\pi$
$692$		0			0
$693$	−13.7637			−0.522839
$694$		0			0
$695$		−	12.0464i		−	0.456946i
$696$		0			0
$697$		−	6.59840i		−	0.249932i
$698$		0			0
$699$	33.1872			1.25525
$700$		0			0
$701$	−2.79007			−0.105379			−0.0526897	−	0.998611i	$-0.516779\pi$
$701$	−2.79007			−0.105379			−0.0526897	+	0.998611i	$0.516779\pi$
$702$		0			0
$703$	−6.27968	+	25.7834i	−0.236842	+	0.972438i
$704$		0			0
$705$		−	14.3975i		−	0.542241i
$706$		0			0
$707$			5.82150i			0.218940i
$708$		0			0
$709$	19.3592			0.727048			0.363524	−	0.931585i	$-0.381573\pi$
$709$	19.3592			0.727048			0.363524	+	0.931585i	$0.381573\pi$
$710$		0			0
$711$	−8.33106			−0.312439
$712$		0			0
$713$		−	21.7475i		−	0.814449i
$714$		0			0
$715$	−3.27409			−0.122444
$716$		0			0
$717$		−	75.9338i		−	2.83580i
$718$		0			0
$719$		−	43.8786i		−	1.63640i	−0.574936	−	0.818199i	$-0.694973\pi$
$719$		−	43.8786i		−	1.63640i	0.574936	−	0.818199i	$-0.305027\pi$
$720$		0			0
$721$			22.0281i			0.820369i
$722$		0			0
$723$		−	15.4281i		−	0.573778i
$724$		0			0
$725$			7.32052i			0.271877i
$726$		0			0
$727$			23.4244i			0.868763i	0.900729	+	0.434381i	$0.143033\pi$
$727$			23.4244i			0.868763i	−0.900729	+	0.434381i	$0.856967\pi$
$728$		0			0
$729$	−7.50008			−0.277781
$730$		0			0
$731$			7.15774i			0.264739i
$732$		0			0
$733$	1.63590			0.0604234			0.0302117	−	0.999544i	$-0.490382\pi$
$733$	1.63590			0.0604234			0.0302117	+	0.999544i	$0.490382\pi$
$734$		0			0
$735$	−11.7536			−0.433537
$736$		0			0
$737$		−	9.64892i		−	0.355423i
$738$		0			0
$739$		−	14.9002i		−	0.548113i	−0.961714	−	0.274056i	$-0.911635\pi$
$739$		−	14.9002i		−	0.548113i	0.961714	−	0.274056i	$-0.0883655\pi$
$740$		0			0
$741$	7.18308	−	29.4926i	0.263877	−	1.08344i
$742$		0			0
$743$	−4.84448			−0.177727			−0.0888634	−	0.996044i	$-0.528323\pi$
$743$	−4.84448			−0.177727			−0.0888634	+	0.996044i	$0.528323\pi$
$744$		0			0
$745$	−24.7102			−0.905313
$746$		0			0
$747$		−	14.6960i		−	0.537697i
$748$		0			0
$749$		−	23.5752i		−	0.861417i
$750$		0			0
$751$	13.6549			0.498273			0.249136	−	0.968468i	$-0.419853\pi$
$751$	13.6549			0.498273			0.249136	+	0.968468i	$0.419853\pi$
$752$		0			0
$753$		−	19.6373i		−	0.715625i
$754$		0			0
$755$	2.29842			0.0836482
$756$		0			0
$757$	−5.66432			−0.205873			−0.102937	−	0.994688i	$-0.532824\pi$
$757$	−5.66432			−0.205873			−0.102937	+	0.994688i	$0.532824\pi$
$758$		0			0
$759$	−11.8299			−0.429400
$760$		0			0
$761$	−49.0772			−1.77905			−0.889524	−	0.456889i	$-0.848964\pi$
$761$	−49.0772			−1.77905			−0.889524	+	0.456889i	$0.848964\pi$
$762$		0			0
$763$	15.5684			0.563616
$764$		0			0
$765$	−8.81623			−0.318751
$766$		0			0
$767$			10.4040i			0.375665i
$768$		0			0
$769$	4.18296			0.150842			0.0754208	−	0.997152i	$-0.475970\pi$
$769$	4.18296			0.150842			0.0754208	+	0.997152i	$0.475970\pi$
$770$		0			0
$771$			29.2996i			1.05520i
$772$		0			0
$773$			44.5975i			1.60406i	0.597283	+	0.802031i	$0.296247\pi$
$773$			44.5975i			1.60406i	−0.597283	+	0.802031i	$0.703753\pi$
$774$		0			0
$775$	16.4044			0.589264
$776$		0			0
$777$	39.4525			1.41535
$778$		0			0
$779$	7.37791	−	30.2925i	0.264341	−	1.08534i
$780$		0			0
$781$		−	3.33022i		−	0.119165i
$782$		0			0
$783$			28.0344i			1.00187i
$784$		0			0
$785$	−22.5973			−0.806532
$786$		0			0
$787$	−4.55979			−0.162539			−0.0812694	−	0.996692i	$-0.525897\pi$
$787$	−4.55979			−0.162539			−0.0812694	+	0.996692i	$0.525897\pi$
$788$		0			0
$789$		−	71.5438i		−	2.54703i
$790$		0			0
$791$	−10.0848			−0.358572
$792$		0			0
$793$		−	10.3689i		−	0.368212i
$794$		0			0
$795$		−	9.59141i		−	0.340172i
$796$		0			0
$797$		−	8.36105i		−	0.296163i	−0.988975	−	0.148082i	$-0.952690\pi$
$797$		−	8.36105i		−	0.296163i	0.988975	−	0.148082i	$-0.0473099\pi$
$798$		0			0
$799$		−	2.95170i		−	0.104424i
$800$		0			0
$801$			82.8701i			2.92807i
$802$		0			0
$803$			6.85856i			0.242033i
$804$		0			0
$805$	11.6507			0.410633
$806$		0			0
$807$			36.1479i			1.27247i
$808$		0			0
$809$	15.1477			0.532563			0.266282	−	0.963895i	$-0.414205\pi$
$809$	15.1477			0.532563			0.266282	+	0.963895i	$0.414205\pi$
$810$		0			0
$811$	−25.4169			−0.892507			−0.446253	−	0.894907i	$-0.647242\pi$
$811$	−25.4169			−0.892507			−0.446253	+	0.894907i	$0.647242\pi$
$812$		0			0
$813$		−	88.9321i		−	3.11899i
$814$		0			0
$815$			13.9562i			0.488864i
$816$		0			0
$817$	−8.00334	+	32.8604i	−0.280001	+	1.14964i
$818$		0			0
$819$	−30.9824			−1.08261
$820$		0			0
$821$	−29.2501			−1.02084			−0.510418	−	0.859926i	$-0.670509\pi$
$821$	−29.2501			−1.02084			−0.510418	+	0.859926i	$0.670509\pi$
$822$		0			0
$823$		−	11.7814i		−	0.410673i	−0.978691	−	0.205336i	$-0.934171\pi$
$823$		−	11.7814i		−	0.410673i	0.978691	−	0.205336i	$-0.0658289\pi$
$824$		0			0
$825$		−	8.92350i		−	0.310676i
$826$		0			0
$827$	43.2633			1.50441			0.752206	−	0.658928i	$-0.228990\pi$
$827$	43.2633			1.50441			0.752206	+	0.658928i	$0.228990\pi$
$828$		0			0
$829$			37.3542i			1.29736i	0.761060	+	0.648682i	$0.224679\pi$
$829$			37.3542i			1.29736i	−0.761060	+	0.648682i	$0.775321\pi$
$830$		0			0
$831$	84.0917			2.91711
$832$		0			0
$833$	−2.40966			−0.0834897
$834$		0			0
$835$	20.8671			0.722136
$836$		0			0
$837$	62.8218			2.17144
$838$		0			0
$839$	29.9940			1.03551			0.517754	−	0.855530i	$-0.326768\pi$
$839$	29.9940			1.03551			0.517754	+	0.855530i	$0.326768\pi$
$840$		0			0
$841$	22.5590			0.777896
$842$		0			0
$843$			28.3550i			0.976598i
$844$		0			0
$845$	11.5383			0.396930
$846$		0			0
$847$			2.09473i			0.0719758i
$848$		0			0
$849$		−	3.76651i		−	0.129266i
$850$		0			0
$851$	23.2803			0.798039
$852$		0			0
$853$	−47.2861			−1.61905			−0.809523	−	0.587088i	$-0.800274\pi$
$853$	−47.2861			−1.61905			−0.809523	+	0.587088i	$0.800274\pi$
$854$		0			0
$855$	−40.4744	−	9.85775i	−1.38419	−	0.337128i
$856$		0			0
$857$		−	10.5204i		−	0.359370i	−0.983724	−	0.179685i	$-0.942492\pi$
$857$		−	10.5204i		−	0.359370i	0.983724	−	0.179685i	$-0.0575079\pi$
$858$		0			0
$859$			37.1402i			1.26721i	0.773657	+	0.633604i	$0.218425\pi$
$859$			37.1402i			1.26721i	−0.773657	+	0.633604i	$0.781575\pi$
$860$		0			0
$861$	−46.3523			−1.57968
$862$		0			0
$863$	18.6132			0.633599			0.316800	−	0.948492i	$-0.397392\pi$
$863$	18.6132			0.633599			0.316800	+	0.948492i	$0.397392\pi$
$864$		0			0
$865$			28.9641i			0.984809i
$866$		0			0
$867$	49.9592			1.69670
$868$		0			0
$869$			1.26793i			0.0430114i
$870$		0			0
$871$		−	21.7199i		−	0.735952i
$872$		0			0
$873$		−	20.9091i		−	0.707666i
$874$		0			0
$875$			24.0221i			0.812095i
$876$		0			0
$877$		−	53.8696i		−	1.81905i	−0.415651	−	0.909524i	$-0.636446\pi$
$877$		−	53.8696i		−	1.81905i	0.415651	−	0.909524i	$-0.363554\pi$
$878$		0			0
$879$		−	23.8120i		−	0.803157i
$880$		0			0
$881$	−42.5578			−1.43381			−0.716904	−	0.697172i	$-0.754441\pi$
$881$	−42.5578			−1.43381			−0.716904	+	0.697172i	$0.754441\pi$
$882$		0			0
$883$		−	26.8077i		−	0.902150i	−0.892486	−	0.451075i	$-0.851041\pi$
$883$		−	26.8077i		−	0.902150i	0.892486	−	0.451075i	$-0.148959\pi$
$884$		0			0
$885$	20.7969			0.699082
$886$		0			0
$887$	31.5370			1.05891			0.529455	−	0.848338i	$-0.322396\pi$
$887$	31.5370			1.05891			0.529455	+	0.848338i	$0.322396\pi$
$888$		0			0
$889$		−	5.99875i		−	0.201192i
$890$		0			0
$891$		−	14.4612i		−	0.484469i
$892$		0			0
$893$	3.30041	−	13.5510i	0.110444	−	0.453466i
$894$		0			0
$895$	−30.6528			−1.02461
$896$		0			0
$897$	−26.6295			−0.889132
$898$		0			0
$899$			14.4336i			0.481386i
$900$		0			0
$901$		−	1.96638i		−	0.0655097i
$902$		0			0
$903$	50.2816			1.67327
$904$		0			0
$905$		−	13.0112i		−	0.432505i
$906$		0			0
$907$	−55.5165			−1.84339			−0.921697	−	0.387911i	$-0.873197\pi$
$907$	−55.5165			−1.84339			−0.921697	+	0.387911i	$0.873197\pi$
$908$		0			0
$909$	−18.2605			−0.605662
$910$		0			0
$911$	22.8334			0.756503			0.378252	−	0.925703i	$-0.376525\pi$
$911$	22.8334			0.756503			0.378252	+	0.925703i	$0.376525\pi$
$912$		0			0
$913$	−2.23662			−0.0740212
$914$		0			0
$915$	−20.7269			−0.685211
$916$		0			0
$917$	12.3356			0.407357
$918$		0			0
$919$		−	22.9073i		−	0.755641i	−0.925879	−	0.377820i	$-0.876674\pi$
$919$		−	22.9073i		−	0.755641i	0.925879	−	0.377820i	$-0.123326\pi$
$920$		0			0
$921$	−93.4263			−3.07850
$922$		0			0
$923$		−	7.49640i		−	0.246747i
$924$		0			0
$925$			17.5607i			0.577391i
$926$		0			0
$927$	−69.0963			−2.26942
$928$		0			0
$929$	−6.58923			−0.216186			−0.108093	−	0.994141i	$-0.534474\pi$
$929$	−6.58923			−0.216186			−0.108093	+	0.994141i	$0.534474\pi$
$930$		0			0
$931$	−11.0625	−	2.69433i	−0.362558	−	0.0883030i
$932$		0			0
$933$			62.3977i			2.04281i
$934$		0			0
$935$			1.34176i			0.0438804i
$936$		0			0
$937$	−8.48489			−0.277189			−0.138595	−	0.990349i	$-0.544259\pi$
$937$	−8.48489			−0.277189			−0.138595	+	0.990349i	$0.544259\pi$
$938$		0			0
$939$	19.3913			0.632810
$940$		0			0
$941$		−	14.5329i		−	0.473759i	−0.971539	−	0.236880i	$-0.923875\pi$
$941$		−	14.5329i		−	0.473759i	0.971539	−	0.236880i	$-0.0761247\pi$
$942$		0			0
$943$	−27.3517			−0.890695
$944$		0			0
$945$			33.6552i			1.09481i
$946$		0			0
$947$		−	12.4658i		−	0.405085i	−0.979274	−	0.202542i	$-0.935080\pi$
$947$		−	12.4658i		−	0.405085i	0.979274	−	0.202542i	$-0.0649204\pi$
$948$		0			0
$949$			15.4388i			0.501164i
$950$		0			0
$951$		−	66.9163i		−	2.16991i
$952$		0			0
$953$		−	58.2086i		−	1.88556i	−0.333415	−	0.942780i	$-0.608201\pi$
$953$		−	58.2086i		−	1.88556i	0.333415	−	0.942780i	$-0.391799\pi$
$954$		0			0
$955$			16.4694i			0.532938i
$956$		0			0
$957$	7.85140			0.253800
$958$		0			0
$959$			19.8744i			0.641777i
$960$		0			0
$961$	1.34389			0.0433512
$962$		0			0
$963$	73.9490			2.38297
$964$		0			0
$965$			2.16129i			0.0695742i
$966$		0			0
$967$		−	9.88032i		−	0.317730i	−0.987300	−	0.158865i	$-0.949217\pi$
$967$		−	9.88032i		−	0.317730i	0.987300	−	0.158865i	$-0.0507834\pi$
$968$		0			0
$969$	−12.0865	−	2.94372i	−0.388273	−	0.0945660i
$970$		0			0
$971$	−48.4671			−1.55538			−0.777691	−	0.628647i	$-0.783609\pi$
$971$	−48.4671			−1.55538			−0.777691	+	0.628647i	$0.783609\pi$
$972$		0			0
$973$	−17.3491			−0.556185
$974$		0			0
$975$		−	20.0870i		−	0.643299i
$976$		0			0
$977$			43.4975i			1.39161i	0.718232	+	0.695804i	$0.244952\pi$
$977$			43.4975i			1.39161i	−0.718232	+	0.695804i	$0.755048\pi$
$978$		0			0
$979$	12.6122			0.403088
$980$		0			0
$981$			48.8341i			1.55915i
$982$		0			0
$983$	32.2336			1.02809			0.514046	−	0.857763i	$-0.328146\pi$
$983$	32.2336			1.02809			0.514046	+	0.857763i	$0.328146\pi$
$984$		0			0
$985$	21.7524			0.693090
$986$		0			0
$987$	−20.7351			−0.660004
$988$		0			0
$989$	29.6704			0.943462
$990$		0			0
$991$	−9.21026			−0.292574			−0.146287	−	0.989242i	$-0.546732\pi$
$991$	−9.21026			−0.292574			−0.146287	+	0.989242i	$0.546732\pi$
$992$		0			0
$993$	42.3082			1.34261
$994$		0			0
$995$			36.3406i			1.15208i
$996$		0			0
$997$	−39.8658			−1.26256			−0.631281	−	0.775554i	$-0.717470\pi$
$997$	−39.8658			−1.26256			−0.631281	+	0.775554i	$0.717470\pi$
$998$		0			0
$999$			67.2497i			2.12769i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.o.b.1519.4	yes	64
4.3	odd	2	inner	3344.2.o.b.1519.61	yes	64
19.18	odd	2	inner	3344.2.o.b.1519.62	yes	64
76.75	even	2	inner	3344.2.o.b.1519.3	✓	64

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3344.2.o.b.1519.3	✓	64	76.75	even	2	inner
3344.2.o.b.1519.4	yes	64	1.1	even	1	trivial
3344.2.o.b.1519.61	yes	64	4.3	odd	2	inner
3344.2.o.b.1519.62	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)

\(f(q)\)	\(=\)	\(q-3.09364 q^{3} +1.45449 q^{5} -2.09473i q^{7} +6.57062 q^{9} +O(q^{10})\)	\(q-3.09364 q^{3} +1.45449 q^{5} -2.09473i q^{7} +6.57062 q^{9} -1.00000i q^{11} -2.25102i q^{13} -4.49967 q^{15} -0.922499 q^{17} +(-4.23510 - 1.03148i) q^{19} +6.48035i q^{21} +3.82395i q^{23} -2.88446 q^{25} -11.0462 q^{27} -2.53792i q^{29} -5.68717 q^{31} +3.09364i q^{33} -3.04676i q^{35} -6.08802i q^{37} +6.96386i q^{39} +7.15274i q^{41} -7.75908i q^{43} +9.55690 q^{45} +3.19968i q^{47} +2.61210 q^{49} +2.85388 q^{51} +2.13158i q^{53} -1.45449i q^{55} +(13.1019 + 3.19103i) q^{57} -4.62188 q^{59} +4.60632 q^{61} -13.7637i q^{63} -3.27409i q^{65} +9.64892 q^{67} -11.8299i q^{69} +3.33022 q^{71} -6.85856 q^{73} +8.92350 q^{75} -2.09473 q^{77} -1.26793 q^{79} +14.4612 q^{81} -2.23662i q^{83} -1.34176 q^{85} +7.85140i q^{87} +12.6122i q^{89} -4.71529 q^{91} +17.5941 q^{93} +(-6.15990 - 1.50028i) q^{95} -3.18221i q^{97} -6.57062i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q + 56 q^{9}+O(q^{10}) \) 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

Downloads

Learn more