LMFDB - Embedded newform 3240.2.f.k.649.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3240,2,Mod(649,3240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3240, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3240.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3240 = 2^{3} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3240.f (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:	$25.8715302549$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} - x^{14} + 26 x^{13} - 44 x^{12} + 6 x^{11} + 225 x^{10} - 174 x^{9} + 102 x^{8} + \cdots + 390625 $ x^16 - 2x^15 - x^14 + 26x^13 - 44x^12 + 6x^11 + 225x^10 - 174x^9 + 102x^8 - 870x^7 + 5625x^6 + 750x^5 - 27500x^4 + 81250x^3 - 15625x^2 - 156250x + 390625
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 360)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.12
Root			$1.51358 + 1.64593i$ of defining polynomial
Character	$\chi$	$=$	3240.649
Dual form			3240.2.f.k.649.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.51358 + 1.64593i) q^{5} -4.50582i q^{7} +O(q^{10})$	$q+(1.51358 + 1.64593i) q^{5} -4.50582i q^{7} -1.26360 q^{11} +4.97755i q^{13} -4.59329i q^{17} -6.38235 q^{19} -6.16729i q^{23} +(-0.418174 + 4.98248i) q^{25} -3.90760 q^{29} -2.17120 q^{31} +(7.41626 - 6.81990i) q^{35} +0.516341i q^{37} -2.23765 q^{41} -5.88515i q^{43} -6.87443i q^{47} -13.3024 q^{49} -6.92003i q^{53} +(-1.91255 - 2.07980i) q^{55} -7.83554 q^{59} +13.2438 q^{61} +(-8.19270 + 7.53390i) q^{65} +3.04258i q^{67} -9.06407 q^{71} +6.90760i q^{73} +5.69355i q^{77} -8.18651 q^{79} +6.59742i q^{83} +(7.56023 - 6.95229i) q^{85} +0.969826 q^{89} +22.4279 q^{91} +(-9.66018 - 10.5049i) q^{95} -12.4909i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 2 q^{5}+O(q^{10}) $ 16 * q + 2 * q^5	$ 16 q + 2 q^{5} - 16 q^{11} + 4 q^{19} + 6 q^{25} - 20 q^{29} + 12 q^{31} + 2 q^{35} + 8 q^{41} - 36 q^{49} + 10 q^{55} + 20 q^{61} - 10 q^{65} + 8 q^{71} - 4 q^{79} - 36 q^{85} - 48 q^{89} - 4 q^{91} + 32 q^{95}+O(q^{100}) $ 16 * q + 2 * q^5 - 16 * q^11 + 4 * q^19 + 6 * q^25 - 20 * q^29 + 12 * q^31 + 2 * q^35 + 8 * q^41 - 36 * q^49 + 10 * q^55 + 20 * q^61 - 10 * q^65 + 8 * q^71 - 4 * q^79 - 36 * q^85 - 48 * q^89 - 4 * q^91 + 32 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3240\mathbb{Z}\right)^\times$.

$n$	$1297$	$1621$	$2431$	$3161$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	1.51358	+	1.64593i	0.676892	+	0.736082i
$5$	1.51358	+	1.64593i	0.676892	+	0.736082i
$6$		0			0
$7$		−	4.50582i		−	1.70304i	−0.524323	−	0.851519i	$-0.675682\pi$
$7$		−	4.50582i		−	1.70304i	0.524323	−	0.851519i	$-0.324318\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.26360			−0.380990			−0.190495	−	0.981688i	$-0.561009\pi$
$11$	−1.26360			−0.380990			−0.190495	+	0.981688i	$0.561009\pi$
$12$		0			0
$13$			4.97755i			1.38052i	0.723560	+	0.690262i	$0.242505\pi$
$13$			4.97755i			1.38052i	−0.723560	+	0.690262i	$0.757495\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	4.59329i		−	1.11404i	−0.830501	−	0.557018i	$-0.811946\pi$
$17$		−	4.59329i		−	1.11404i	0.830501	−	0.557018i	$-0.188054\pi$
$18$		0			0
$19$	−6.38235			−1.46421			−0.732106	−	0.681191i	$-0.761462\pi$
$19$	−6.38235			−1.46421			−0.732106	+	0.681191i	$0.761462\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	6.16729i		−	1.28597i	−0.765880	−	0.642984i	$-0.777696\pi$
$23$		−	6.16729i		−	1.28597i	0.765880	−	0.642984i	$-0.222304\pi$
$24$		0			0
$25$	−0.418174	+	4.98248i	−0.0836348	+	0.996496i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.90760			−0.725622			−0.362811	−	0.931863i	$-0.618183\pi$
$29$	−3.90760			−0.725622			−0.362811	+	0.931863i	$0.618183\pi$
$30$		0			0
$31$	−2.17120			−0.389958			−0.194979	−	0.980807i	$-0.562464\pi$
$31$	−2.17120			−0.389958			−0.194979	+	0.980807i	$0.562464\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	7.41626	−	6.81990i	1.25358	−	1.15277i
$36$		0			0
$37$			0.516341i			0.0848860i	0.999099	+	0.0424430i	$0.0135141\pi$
$37$			0.516341i			0.0848860i	−0.999099	+	0.0424430i	$0.986486\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−2.23765			−0.349462			−0.174731	−	0.984616i	$-0.555906\pi$
$41$	−2.23765			−0.349462			−0.174731	+	0.984616i	$0.555906\pi$
$42$		0			0
$43$		−	5.88515i		−	0.897476i	−0.893663	−	0.448738i	$-0.851874\pi$
$43$		−	5.88515i		−	0.897476i	0.893663	−	0.448738i	$-0.148126\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	6.87443i		−	1.00274i	−0.865233	−	0.501370i	$-0.832830\pi$
$47$		−	6.87443i		−	1.00274i	0.865233	−	0.501370i	$-0.167170\pi$
$48$		0			0
$49$	−13.3024			−1.90034
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	6.92003i		−	0.950540i	−0.879840	−	0.475270i	$-0.842350\pi$
$53$		−	6.92003i		−	0.950540i	0.879840	−	0.475270i	$-0.157650\pi$
$54$		0			0
$55$	−1.91255	−	2.07980i	−0.257889	−	0.280440i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−7.83554			−1.02010			−0.510050	−	0.860144i	$-0.670373\pi$
$59$	−7.83554			−1.02010			−0.510050	+	0.860144i	$0.670373\pi$
$60$		0			0
$61$	13.2438			1.69570			0.847850	−	0.530235i	$-0.177896\pi$
$61$	13.2438			1.69570			0.847850	+	0.530235i	$0.177896\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−8.19270	+	7.53390i	−1.01618	+	0.934465i
$66$		0			0
$67$			3.04258i			0.371710i	0.982577	+	0.185855i	$0.0595055\pi$
$67$			3.04258i			0.371710i	−0.982577	+	0.185855i	$0.940494\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−9.06407			−1.07571			−0.537853	−	0.843038i	$-0.680765\pi$
$71$	−9.06407			−1.07571			−0.537853	+	0.843038i	$0.680765\pi$
$72$		0			0
$73$			6.90760i			0.808473i	0.914654	+	0.404237i	$0.132463\pi$
$73$			6.90760i			0.808473i	−0.914654	+	0.404237i	$0.867537\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			5.69355i			0.648840i
$78$		0			0
$79$	−8.18651			−0.921055			−0.460527	−	0.887646i	$-0.652340\pi$
$79$	−8.18651			−0.921055			−0.460527	+	0.887646i	$0.652340\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			6.59742i			0.724160i	0.932147	+	0.362080i	$0.117933\pi$
$83$			6.59742i			0.724160i	−0.932147	+	0.362080i	$0.882067\pi$
$84$		0			0
$85$	7.56023	−	6.95229i	0.820022	−	0.754081i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	0.969826			0.102801			0.0514007	−	0.998678i	$-0.483631\pi$
$89$	0.969826			0.102801			0.0514007	+	0.998678i	$0.483631\pi$
$90$		0			0
$91$	22.4279			2.35109
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−9.66018	−	10.5049i	−0.991113	−	1.07778i
$96$		0			0
$97$		−	12.4909i		−	1.26826i	−0.773228	−	0.634128i	$-0.781359\pi$
$97$		−	12.4909i		−	1.26826i	0.773228	−	0.634128i	$-0.218641\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.95939			0.194967			0.0974834	−	0.995237i	$-0.468921\pi$
$101$	1.95939			0.194967			0.0974834	+	0.995237i	$0.468921\pi$
$102$		0			0
$103$			1.02245i			0.100745i	0.998730	+	0.0503725i	$0.0160409\pi$
$103$			1.02245i			0.100745i	−0.998730	+	0.0503725i	$0.983959\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			8.51447i			0.823125i	0.911382	+	0.411562i	$0.135017\pi$
$107$			8.51447i			0.823125i	−0.911382	+	0.411562i	$0.864983\pi$
$108$		0			0
$109$	−10.7130			−1.02612			−0.513058	−	0.858354i	$-0.671487\pi$
$109$	−10.7130			−1.02612			−0.513058	+	0.858354i	$0.671487\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	3.62817i		−	0.341310i	−0.985331	−	0.170655i	$-0.945412\pi$
$113$		−	3.62817i		−	0.341310i	0.985331	−	0.170655i	$-0.0545883\pi$
$114$		0			0
$115$	10.1509	−	9.33466i	0.946579	−	0.870461i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−20.6965			−1.89725
$120$		0			0
$121$	−9.40332			−0.854847
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−8.83376	+	6.85308i	−0.790115	+	0.612958i
$126$		0			0
$127$		−	0.378129i		−	0.0335535i	−0.999859	−	0.0167768i	$-0.994660\pi$
$127$		−	0.378129i		−	0.0335535i	0.999859	−	0.0167768i	$-0.00534046\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−8.08791			−0.706644			−0.353322	−	0.935502i	$-0.614948\pi$
$131$	−8.08791			−0.706644			−0.353322	+	0.935502i	$0.614948\pi$
$132$		0			0
$133$			28.7577i			2.49361i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	6.67568i		−	0.570342i	−0.958477	−	0.285171i	$-0.907950\pi$
$137$		−	6.67568i		−	0.570342i	0.958477	−	0.285171i	$-0.0920503\pi$
$138$		0			0
$139$	18.4714			1.56672			0.783360	−	0.621568i	$-0.213504\pi$
$139$	18.4714			1.56672			0.783360	+	0.621568i	$0.213504\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	6.28963i		−	0.525965i
$144$		0			0
$145$	−5.91445	−	6.43163i	−0.491168	−	0.534118i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−6.74020			−0.552179			−0.276090	−	0.961132i	$-0.589039\pi$
$149$	−6.74020			−0.552179			−0.276090	+	0.961132i	$0.589039\pi$
$150$		0			0
$151$	2.61033			0.212426			0.106213	−	0.994343i	$-0.466127\pi$
$151$	2.61033			0.212426			0.106213	+	0.994343i	$0.466127\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−3.28627	−	3.57364i	−0.263960	−	0.287042i
$156$		0			0
$157$			9.29226i			0.741603i	0.928712	+	0.370801i	$0.120917\pi$
$157$			9.29226i			0.741603i	−0.928712	+	0.370801i	$0.879083\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−27.7887			−2.19005
$162$		0			0
$163$		−	4.95234i		−	0.387897i	−0.981012	−	0.193949i	$-0.937871\pi$
$163$		−	4.95234i		−	0.387897i	0.981012	−	0.193949i	$-0.0621295\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	10.7359i		−	0.830771i	−0.909645	−	0.415386i	$-0.863647\pi$
$167$		−	10.7359i		−	0.830771i	0.909645	−	0.415386i	$-0.136353\pi$
$168$		0			0
$169$	−11.7760			−0.905846
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	24.3017i		−	1.84762i	−0.382849	−	0.923811i	$-0.625057\pi$
$173$		−	24.3017i		−	1.84762i	0.382849	−	0.923811i	$-0.374943\pi$
$174$		0			0
$175$	22.4502	+	1.88421i	1.69707	+	0.142433i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−0.821382			−0.0613930			−0.0306965	−	0.999529i	$-0.509773\pi$
$179$	−0.821382			−0.0613930			−0.0306965	+	0.999529i	$0.509773\pi$
$180$		0			0
$181$	3.35574			0.249430			0.124715	−	0.992193i	$-0.460198\pi$
$181$	3.35574			0.249430			0.124715	+	0.992193i	$0.460198\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−0.849862	+	0.781522i	−0.0624831	+	0.0574586i
$186$		0			0
$187$			5.80408i			0.424436i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	19.1459			1.38535			0.692676	−	0.721249i	$-0.256432\pi$
$191$	19.1459			1.38535			0.692676	+	0.721249i	$0.256432\pi$
$192$		0			0
$193$		−	9.92963i		−	0.714750i	−0.933961	−	0.357375i	$-0.883672\pi$
$193$		−	9.92963i		−	0.714750i	0.933961	−	0.357375i	$-0.116328\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	11.3402i		−	0.807953i	−0.914769	−	0.403977i	$-0.867628\pi$
$197$		−	11.3402i		−	0.807953i	0.914769	−	0.403977i	$-0.132372\pi$
$198$		0			0
$199$	3.24887			0.230306			0.115153	−	0.993348i	$-0.463264\pi$
$199$	3.24887			0.230306			0.115153	+	0.993348i	$0.463264\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			17.6069i			1.23576i
$204$		0			0
$205$	−3.38685	−	3.68301i	−0.236548	−	0.257233i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	8.06474			0.557850
$210$		0			0
$211$	−5.93771			−0.408769			−0.204384	−	0.978891i	$-0.565519\pi$
$211$	−5.93771			−0.408769			−0.204384	+	0.978891i	$0.565519\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	9.68654	−	8.90762i	0.660617	−	0.607494i
$216$		0			0
$217$			9.78301i			0.664114i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	22.8633			1.53795
$222$		0			0
$223$			11.5727i			0.774966i	0.921877	+	0.387483i	$0.126655\pi$
$223$			11.5727i			0.774966i	−0.921877	+	0.387483i	$0.873345\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	15.2102i		−	1.00954i	−0.863255	−	0.504769i	$-0.831578\pi$
$227$		−	15.2102i		−	1.00954i	0.863255	−	0.504769i	$-0.168422\pi$
$228$		0			0
$229$	3.47768			0.229812			0.114906	−	0.993376i	$-0.463343\pi$
$229$	3.47768			0.229812			0.114906	+	0.993376i	$0.463343\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	9.49649i		−	0.622136i	−0.950388	−	0.311068i	$-0.899313\pi$
$233$		−	9.49649i		−	0.622136i	0.950388	−	0.311068i	$-0.100687\pi$
$234$		0			0
$235$	11.3148	−	10.4050i	0.738099	−	0.678746i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	23.9378			1.54840			0.774202	−	0.632938i	$-0.218151\pi$
$239$	23.9378			1.54840			0.774202	+	0.632938i	$0.218151\pi$
$240$		0			0
$241$	−30.5838			−1.97007			−0.985037	−	0.172345i	$-0.944866\pi$
$241$	−30.5838			−1.97007			−0.985037	+	0.172345i	$0.944866\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−20.1342	−	21.8948i	−1.28633	−	1.39881i
$246$		0			0
$247$		−	31.7685i		−	2.02138i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−21.3298			−1.34633			−0.673163	−	0.739494i	$-0.735065\pi$
$251$	−21.3298			−1.34633			−0.673163	+	0.739494i	$0.735065\pi$
$252$		0			0
$253$			7.79298i			0.489941i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	21.2542i		−	1.32580i	−0.748708	−	0.662899i	$-0.769326\pi$
$257$		−	21.2542i		−	1.32580i	0.748708	−	0.662899i	$-0.230674\pi$
$258$		0			0
$259$	2.32654			0.144564
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	6.34893i		−	0.391492i	−0.980655	−	0.195746i	$-0.937287\pi$
$263$		−	6.34893i		−	0.391492i	0.980655	−	0.195746i	$-0.0627128\pi$
$264$		0			0
$265$	11.3899	−	10.4740i	0.699676	−	0.643413i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−0.830098			−0.0506120			−0.0253060	−	0.999680i	$-0.508056\pi$
$269$	−0.830098			−0.0506120			−0.0253060	+	0.999680i	$0.508056\pi$
$270$		0			0
$271$	−8.84383			−0.537225			−0.268612	−	0.963248i	$-0.586565\pi$
$271$	−8.84383			−0.537225			−0.268612	+	0.963248i	$0.586565\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.528404	−	6.29586i	0.0318640	−	0.379655i
$276$		0			0
$277$			16.3042i			0.979622i	0.871829	+	0.489811i	$0.162934\pi$
$277$			16.3042i			0.979622i	−0.871829	+	0.489811i	$0.837066\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	25.0632			1.49514			0.747572	−	0.664181i	$-0.231219\pi$
$281$	25.0632			1.49514			0.747572	+	0.664181i	$0.231219\pi$
$282$		0			0
$283$			11.1107i			0.660461i	0.943900	+	0.330230i	$0.107126\pi$
$283$			11.1107i			0.660461i	−0.943900	+	0.330230i	$0.892874\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			10.0824i			0.595147i
$288$		0			0
$289$	−4.09827			−0.241075
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	26.0531i		−	1.52204i	−0.648729	−	0.761019i	$-0.724699\pi$
$293$		−	26.0531i		−	1.52204i	0.648729	−	0.761019i	$-0.275301\pi$
$294$		0			0
$295$	−11.8597	−	12.8968i	−0.690498	−	0.750878i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	30.6980			1.77531
$300$		0			0
$301$	−26.5174			−1.52844
$302$		0			0
$303$		0			0
$304$		0			0
$305$	20.0456	+	21.7985i	1.14781	+	1.24818i
$306$		0			0
$307$		−	22.7847i		−	1.30039i	−0.759766	−	0.650197i	$-0.774686\pi$
$307$		−	22.7847i		−	1.30039i	0.759766	−	0.650197i	$-0.225314\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−24.7902			−1.40572			−0.702861	−	0.711327i	$-0.748094\pi$
$311$	−24.7902			−1.40572			−0.702861	+	0.711327i	$0.748094\pi$
$312$		0			0
$313$		−	6.50973i		−	0.367952i	−0.982931	−	0.183976i	$-0.941103\pi$
$313$		−	6.50973i		−	0.367952i	0.982931	−	0.183976i	$-0.0588968\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			25.2501i			1.41819i	0.705113	+	0.709095i	$0.250896\pi$
$317$			25.2501i			1.41819i	−0.705113	+	0.709095i	$0.749104\pi$
$318$		0			0
$319$	4.93764			0.276455
$320$		0			0
$321$		0			0
$322$		0			0
$323$			29.3160i			1.63118i
$324$		0			0
$325$	−24.8006	−	2.08148i	−1.37569	−	0.115460i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−30.9749			−1.70770
$330$		0			0
$331$	11.0285			0.606182			0.303091	−	0.952962i	$-0.401981\pi$
$331$	11.0285			0.606182			0.303091	+	0.952962i	$0.401981\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−5.00788	+	4.60518i	−0.273609	+	0.251608i
$336$		0			0
$337$			16.3723i			0.891855i	0.895069	+	0.445927i	$0.147126\pi$
$337$			16.3723i			0.891855i	−0.895069	+	0.445927i	$0.852874\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	2.74352			0.148570
$342$		0			0
$343$			28.3974i			1.53332i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	3.25412i		−	0.174691i	−0.996178	−	0.0873453i	$-0.972162\pi$
$347$		−	3.25412i		−	0.174691i	0.996178	−	0.0873453i	$-0.0278383\pi$
$348$		0			0
$349$	17.3843			0.930561			0.465280	−	0.885163i	$-0.345954\pi$
$349$	17.3843			0.930561			0.465280	+	0.885163i	$0.345954\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			6.65331i			0.354120i	0.984200	+	0.177060i	$0.0566586\pi$
$353$			6.65331i			0.354120i	−0.984200	+	0.177060i	$0.943341\pi$
$354$		0			0
$355$	−13.7192	−	14.9188i	−0.728137	−	0.791809i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	0.786305			0.0414996			0.0207498	−	0.999785i	$-0.493395\pi$
$359$	0.786305			0.0414996			0.0207498	+	0.999785i	$0.493395\pi$
$360$		0			0
$361$	21.7344			1.14392
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−11.3694	+	10.4552i	−0.595103	+	0.547249i
$366$		0			0
$367$			21.7361i			1.13462i	0.823505	+	0.567308i	$0.192015\pi$
$367$			21.7361i			1.13462i	−0.823505	+	0.567308i	$0.807985\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−31.1804			−1.61881
$372$		0			0
$373$			32.2545i			1.67007i	0.550195	+	0.835036i	$0.314553\pi$
$373$			32.2545i			1.67007i	−0.550195	+	0.835036i	$0.685447\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	19.4503i		−	1.00174i
$378$		0			0
$379$	−3.30273			−0.169650			−0.0848249	−	0.996396i	$-0.527033\pi$
$379$	−3.30273			−0.169650			−0.0848249	+	0.996396i	$0.527033\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			4.02986i			0.205916i	0.994686	+	0.102958i	$0.0328308\pi$
$383$			4.02986i			0.205916i	−0.994686	+	0.102958i	$0.967169\pi$
$384$		0			0
$385$	−9.37119	+	8.61762i	−0.477600	+	0.439195i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−35.5357			−1.80173			−0.900865	−	0.434100i	$-0.857066\pi$
$389$	−35.5357			−1.80173			−0.900865	+	0.434100i	$0.857066\pi$
$390$		0			0
$391$	−28.3281			−1.43261
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−12.3909	−	13.4744i	−0.623454	−	0.677972i
$396$		0			0
$397$		−	15.3547i		−	0.770629i	−0.922785	−	0.385315i	$-0.874093\pi$
$397$		−	15.3547i		−	0.770629i	0.922785	−	0.385315i	$-0.125907\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−11.6507			−0.581808			−0.290904	−	0.956752i	$-0.593956\pi$
$401$	−11.6507			−0.581808			−0.290904	+	0.956752i	$0.593956\pi$
$402$		0			0
$403$		−	10.8072i		−	0.538347i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	0.652449i		−	0.0323407i
$408$		0			0
$409$	12.3921			0.612752			0.306376	−	0.951911i	$-0.400884\pi$
$409$	12.3921			0.612752			0.306376	+	0.951911i	$0.400884\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			35.3055i			1.73727i
$414$		0			0
$415$	−10.8589	+	9.98569i	−0.533042	+	0.490178i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	6.26662			0.306144			0.153072	−	0.988215i	$-0.451083\pi$
$419$	6.26662			0.306144			0.153072	+	0.988215i	$0.451083\pi$
$420$		0			0
$421$	−22.2489			−1.08435			−0.542173	−	0.840267i	$-0.682398\pi$
$421$	−22.2489			−1.08435			−0.542173	+	0.840267i	$0.682398\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	22.8860	+	1.92079i	1.11013	+	0.0931721i
$426$		0			0
$427$		−	59.6744i		−	2.88784i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−24.7545			−1.19238			−0.596191	−	0.802842i	$-0.703320\pi$
$431$	−24.7545			−1.19238			−0.596191	+	0.802842i	$0.703320\pi$
$432$		0			0
$433$			26.5725i			1.27700i	0.769624	+	0.638498i	$0.220444\pi$
$433$			26.5725i			1.27700i	−0.769624	+	0.638498i	$0.779556\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			39.3618i			1.88293i
$438$		0			0
$439$	19.6813			0.939335			0.469668	−	0.882843i	$-0.344374\pi$
$439$	19.6813			0.939335			0.469668	+	0.882843i	$0.344374\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			33.2621i			1.58033i	0.612894	+	0.790165i	$0.290005\pi$
$443$			33.2621i			1.58033i	−0.612894	+	0.790165i	$0.709995\pi$
$444$		0			0
$445$	1.46791	+	1.59627i	0.0695854	+	0.0756703i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	10.8999			0.514396			0.257198	−	0.966359i	$-0.417201\pi$
$449$	10.8999			0.514396			0.257198	+	0.966359i	$0.417201\pi$
$450$		0			0
$451$	2.82749			0.133141
$452$		0			0
$453$		0			0
$454$		0			0
$455$	33.9464	+	36.9148i	1.59143	+	1.73059i
$456$		0			0
$457$		−	26.1241i		−	1.22203i	−0.791617	−	0.611017i	$-0.790761\pi$
$457$		−	26.1241i		−	1.22203i	0.791617	−	0.611017i	$-0.209239\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	9.39635			0.437631			0.218816	−	0.975766i	$-0.429781\pi$
$461$	9.39635			0.437631			0.218816	+	0.975766i	$0.429781\pi$
$462$		0			0
$463$			40.5925i			1.88649i	0.332092	+	0.943247i	$0.392246\pi$
$463$			40.5925i			1.88649i	−0.332092	+	0.943247i	$0.607754\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	9.24300i		−	0.427715i	−0.976865	−	0.213858i	$-0.931397\pi$
$467$		−	9.24300i		−	0.427715i	0.976865	−	0.213858i	$-0.0686028\pi$
$468$		0			0
$469$	13.7093			0.633037
$470$		0			0
$471$		0			0
$472$		0			0
$473$			7.43647i			0.341929i
$474$		0			0
$475$	2.66893	−	31.8000i	0.122459	−	1.45908i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−17.8359			−0.814943			−0.407471	−	0.913218i	$-0.633589\pi$
$479$	−17.8359			−0.814943			−0.407471	+	0.913218i	$0.633589\pi$
$480$		0			0
$481$	−2.57011			−0.117187
$482$		0			0
$483$		0			0
$484$		0			0
$485$	20.5591	−	18.9059i	0.933541	−	0.858472i
$486$		0			0
$487$		−	35.8484i		−	1.62445i	−0.583347	−	0.812223i	$-0.698257\pi$
$487$		−	35.8484i		−	1.62445i	0.583347	−	0.812223i	$-0.301743\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	34.8855			1.57436			0.787179	−	0.616724i	$-0.211541\pi$
$491$	34.8855			1.57436			0.787179	+	0.616724i	$0.211541\pi$
$492$		0			0
$493$			17.9487i			0.808369i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			40.8410i			1.83197i
$498$		0			0
$499$	20.8706			0.934298			0.467149	−	0.884179i	$-0.345281\pi$
$499$	20.8706			0.934298			0.467149	+	0.884179i	$0.345281\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	30.0613i		−	1.34037i	−0.742196	−	0.670183i	$-0.766216\pi$
$503$		−	30.0613i		−	1.34037i	0.742196	−	0.670183i	$-0.233784\pi$
$504$		0			0
$505$	2.96569	+	3.22502i	0.131971	+	0.143512i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	13.8119			0.612203			0.306101	−	0.951999i	$-0.400975\pi$
$509$	13.8119			0.612203			0.306101	+	0.951999i	$0.400975\pi$
$510$		0			0
$511$	31.1244			1.37686
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−1.68288	+	1.54756i	−0.0741566	+	0.0681935i
$516$		0			0
$517$			8.68653i			0.382033i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	31.6849			1.38814			0.694070	−	0.719908i	$-0.255816\pi$
$521$	31.6849			1.38814			0.694070	+	0.719908i	$0.255816\pi$
$522$		0			0
$523$			9.20648i			0.402571i	0.979533	+	0.201286i	$0.0645120\pi$
$523$			9.20648i			0.402571i	−0.979533	+	0.201286i	$0.935488\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			9.97293i			0.434427i
$528$		0			0
$529$	−15.0354			−0.653714
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	11.1380i		−	0.482440i
$534$		0			0
$535$	−14.0142	+	12.8873i	−0.605888	+	0.557167i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	16.8089			0.724010
$540$		0			0
$541$	−25.7154			−1.10559			−0.552796	−	0.833317i	$-0.686439\pi$
$541$	−25.7154			−1.10559			−0.552796	+	0.833317i	$0.686439\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−16.2149	−	17.6328i	−0.694569	−	0.755305i
$546$		0			0
$547$		−	5.04603i		−	0.215752i	−0.994164	−	0.107876i	$-0.965595\pi$
$547$		−	5.04603i		−	0.215752i	0.994164	−	0.107876i	$-0.0344050\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	24.9397			1.06247
$552$		0			0
$553$			36.8869i			1.56859i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			11.0624i			0.468731i	0.972149	+	0.234365i	$0.0753012\pi$
$557$			11.0624i			0.468731i	−0.972149	+	0.234365i	$0.924699\pi$
$558$		0			0
$559$	29.2936			1.23899
$560$		0			0
$561$		0			0
$562$		0			0
$563$			13.7795i			0.580738i	0.956915	+	0.290369i	$0.0937780\pi$
$563$			13.7795i			0.580738i	−0.956915	+	0.290369i	$0.906222\pi$
$564$		0			0
$565$	5.97172	−	5.49152i	0.251232	−	0.231030i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	0.592556			0.0248412			0.0124206	−	0.999923i	$-0.496046\pi$
$569$	0.592556			0.0248412			0.0124206	+	0.999923i	$0.496046\pi$
$570$		0			0
$571$	18.0686			0.756148			0.378074	−	0.925775i	$-0.376586\pi$
$571$	18.0686			0.756148			0.378074	+	0.925775i	$0.376586\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	30.7284	+	2.57900i	1.28146	+	0.107552i
$576$		0			0
$577$			20.3195i			0.845910i	0.906151	+	0.422955i	$0.139007\pi$
$577$			20.3195i			0.845910i	−0.906151	+	0.422955i	$0.860993\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	29.7267			1.23327
$582$		0			0
$583$			8.74416i			0.362146i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			3.54730i			0.146413i	0.997317	+	0.0732064i	$0.0233232\pi$
$587$			3.54730i			0.146413i	−0.997317	+	0.0732064i	$0.976677\pi$
$588$		0			0
$589$	13.8573			0.570982
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	10.1123i		−	0.415261i	−0.978207	−	0.207630i	$-0.933425\pi$
$593$		−	10.1123i		−	0.415261i	0.978207	−	0.207630i	$-0.0665751\pi$
$594$		0			0
$595$	−31.3257	−	34.0650i	−1.28423	−	1.39653i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−16.0683			−0.656531			−0.328266	−	0.944585i	$-0.606464\pi$
$599$	−16.0683			−0.656531			−0.328266	+	0.944585i	$0.606464\pi$
$600$		0			0
$601$	47.3923			1.93317			0.966585	−	0.256345i	$-0.0825183\pi$
$601$	47.3923			1.93317			0.966585	+	0.256345i	$0.0825183\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−14.2326	−	15.4772i	−0.578639	−	0.629238i
$606$		0			0
$607$			31.7535i			1.28883i	0.764674	+	0.644417i	$0.222900\pi$
$607$			31.7535i			1.28883i	−0.764674	+	0.644417i	$0.777100\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	34.2178			1.38431
$612$		0			0
$613$		−	9.56267i		−	0.386233i	−0.981176	−	0.193116i	$-0.938141\pi$
$613$		−	9.56267i		−	0.386233i	0.981176	−	0.193116i	$-0.0618595\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	31.5072i		−	1.26843i	−0.773155	−	0.634217i	$-0.781323\pi$
$617$		−	31.5072i		−	1.26843i	0.773155	−	0.634217i	$-0.218677\pi$
$618$		0			0
$619$	45.0081			1.80903			0.904513	−	0.426445i	$-0.140234\pi$
$619$	45.0081			1.80903			0.904513	+	0.426445i	$0.140234\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	4.36986i		−	0.175075i
$624$		0			0
$625$	−24.6503	−	4.16709i	−0.986010	−	0.166684i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	2.37170			0.0945660
$630$		0			0
$631$	4.68299			0.186427			0.0932135	−	0.995646i	$-0.470286\pi$
$631$	4.68299			0.186427			0.0932135	+	0.995646i	$0.470286\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	0.622374	−	0.572327i	0.0246982	−	0.0227121i
$636$		0			0
$637$		−	66.2133i		−	2.62347i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	8.10633			0.320181			0.160090	−	0.987102i	$-0.448821\pi$
$641$	8.10633			0.320181			0.160090	+	0.987102i	$0.448821\pi$
$642$		0			0
$643$		−	30.8126i		−	1.21513i	−0.794271	−	0.607564i	$-0.792147\pi$
$643$		−	30.8126i		−	1.21513i	0.794271	−	0.607564i	$-0.207853\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			10.2266i			0.402051i	0.979586	+	0.201025i	$0.0644274\pi$
$647$			10.2266i			0.402051i	−0.979586	+	0.201025i	$0.935573\pi$
$648$		0			0
$649$	9.90099			0.388648
$650$		0			0
$651$		0			0
$652$		0			0
$653$			18.5535i			0.726054i	0.931779	+	0.363027i	$0.118257\pi$
$653$			18.5535i			0.726054i	−0.931779	+	0.363027i	$0.881743\pi$
$654$		0			0
$655$	−12.2417	−	13.3121i	−0.478321	−	0.520148i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	4.49065			0.174931			0.0874654	−	0.996168i	$-0.472123\pi$
$659$	4.49065			0.174931			0.0874654	+	0.996168i	$0.472123\pi$
$660$		0			0
$661$	32.3543			1.25844			0.629219	−	0.777228i	$-0.283375\pi$
$661$	32.3543			1.25844			0.629219	+	0.777228i	$0.283375\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−47.3332	+	43.5270i	−1.83550	+	1.68790i
$666$		0			0
$667$			24.0993i			0.933127i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−16.7349			−0.646045
$672$		0			0
$673$			3.74568i			0.144385i	0.997391	+	0.0721927i	$0.0229997\pi$
$673$			3.74568i			0.144385i	−0.997391	+	0.0721927i	$0.977000\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			36.4837i			1.40218i	0.713071	+	0.701091i	$0.247304\pi$
$677$			36.4837i			1.40218i	−0.713071	+	0.701091i	$0.752696\pi$
$678$		0			0
$679$	−56.2816			−2.15989
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	40.6438i		−	1.55519i	−0.628765	−	0.777595i	$-0.716439\pi$
$683$		−	40.6438i		−	1.55519i	0.628765	−	0.777595i	$-0.283561\pi$
$684$		0			0
$685$	10.9877	−	10.1041i	0.419818	−	0.386060i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	34.4448			1.31224
$690$		0			0
$691$	−36.4744			−1.38755			−0.693776	−	0.720191i	$-0.744054\pi$
$691$	−36.4744			−1.38755			−0.693776	+	0.720191i	$0.744054\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	27.9578	+	30.4026i	1.06050	+	1.15323i
$696$		0			0
$697$			10.2782i			0.389313i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	49.7764			1.88003			0.940014	−	0.341136i	$-0.110812\pi$
$701$	49.7764			1.88003			0.940014	+	0.341136i	$0.110812\pi$
$702$		0			0
$703$		−	3.29547i		−	0.124291i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	8.82866i		−	0.332036i
$708$		0			0
$709$	2.71689			0.102035			0.0510174	−	0.998698i	$-0.483754\pi$
$709$	2.71689			0.102035			0.0510174	+	0.998698i	$0.483754\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			13.3904i			0.501474i
$714$		0			0
$715$	10.3523	−	9.51984i	0.387154	−	0.356022i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	25.2525			0.941759			0.470880	−	0.882197i	$-0.343937\pi$
$719$	25.2525			0.941759			0.470880	+	0.882197i	$0.343937\pi$
$720$		0			0
$721$	4.60697			0.171573
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1.63405	−	19.4695i	0.0606873	−	0.723080i
$726$		0			0
$727$			10.1590i			0.376778i	0.982094	+	0.188389i	$0.0603266\pi$
$727$			10.1590i			0.376778i	−0.982094	+	0.188389i	$0.939673\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−27.0322			−0.999820
$732$		0			0
$733$		−	23.9866i		−	0.885965i	−0.896530	−	0.442983i	$-0.853920\pi$
$733$		−	23.9866i		−	0.885965i	0.896530	−	0.442983i	$-0.146080\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	3.84460i		−	0.141618i
$738$		0			0
$739$	16.5460			0.608656			0.304328	−	0.952567i	$-0.401568\pi$
$739$	16.5460			0.608656			0.304328	+	0.952567i	$0.401568\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	6.37510i		−	0.233880i	−0.993139	−	0.116940i	$-0.962692\pi$
$743$		−	6.37510i		−	0.233880i	0.993139	−	0.116940i	$-0.0373085\pi$
$744$		0			0
$745$	−10.2018	−	11.0939i	−0.373765	−	0.406449i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	38.3646			1.40181
$750$		0			0
$751$	2.06748			0.0754434			0.0377217	−	0.999288i	$-0.487990\pi$
$751$	2.06748			0.0754434			0.0377217	+	0.999288i	$0.487990\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	3.95094	+	4.29643i	0.143789	+	0.156363i
$756$		0			0
$757$		−	3.09824i		−	0.112607i	−0.998414	−	0.0563037i	$-0.982068\pi$
$757$		−	3.09824i		−	0.112607i	0.998414	−	0.0563037i	$-0.0179315\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	7.26245			0.263264			0.131632	−	0.991299i	$-0.457978\pi$
$761$	7.26245			0.263264			0.131632	+	0.991299i	$0.457978\pi$
$762$		0			0
$763$			48.2706i			1.74751i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	39.0018i		−	1.40827i
$768$		0			0
$769$	31.9660			1.15272			0.576362	−	0.817194i	$-0.304472\pi$
$769$	31.9660			1.15272			0.576362	+	0.817194i	$0.304472\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			16.5784i			0.596283i	0.954522	+	0.298142i	$0.0963668\pi$
$773$			16.5784i			0.596283i	−0.954522	+	0.298142i	$0.903633\pi$
$774$		0			0
$775$	0.907938	−	10.8179i	0.0326141	−	0.388592i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	14.2815			0.511686
$780$		0			0
$781$	11.4534			0.409833
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−15.2944	+	14.0645i	−0.545881	+	0.501985i
$786$		0			0
$787$			17.7770i			0.633681i	0.948479	+	0.316840i	$0.102622\pi$
$787$			17.7770i			0.633681i	−0.948479	+	0.316840i	$0.897378\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−16.3479			−0.581264
$792$		0			0
$793$			65.9219i			2.34096i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			32.9692i			1.16783i	0.811815	+	0.583915i	$0.198480\pi$
$797$			32.9692i			1.16783i	−0.811815	+	0.583915i	$0.801520\pi$
$798$		0			0
$799$	−31.5762			−1.11709
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	8.72844i		−	0.308020i
$804$		0			0
$805$	−42.0603	−	45.7382i	−1.48243	−	1.61206i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−16.1251			−0.566929			−0.283464	−	0.958983i	$-0.591484\pi$
$809$	−16.1251			−0.566929			−0.283464	+	0.958983i	$0.591484\pi$
$810$		0			0
$811$	23.5077			0.825467			0.412733	−	0.910852i	$-0.364574\pi$
$811$	23.5077			0.825467			0.412733	+	0.910852i	$0.364574\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	8.15120	−	7.49574i	0.285524	−	0.262564i
$816$		0			0
$817$			37.5611i			1.31410i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	37.4755			1.30790			0.653952	−	0.756536i	$-0.273110\pi$
$821$	37.4755			1.30790			0.653952	+	0.756536i	$0.273110\pi$
$822$		0			0
$823$			12.6253i			0.440090i	0.975490	+	0.220045i	$0.0706205\pi$
$823$			12.6253i			0.440090i	−0.975490	+	0.220045i	$0.929380\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			28.7120i			0.998415i	0.866482	+	0.499208i	$0.166376\pi$
$827$			28.7120i			0.998415i	−0.866482	+	0.499208i	$0.833624\pi$
$828$		0			0
$829$	−36.7202			−1.27534			−0.637672	−	0.770308i	$-0.720103\pi$
$829$	−36.7202			−1.27534			−0.637672	+	0.770308i	$0.720103\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			61.1017i			2.11705i
$834$		0			0
$835$	17.6706	−	16.2496i	0.611516	−	0.562342i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	39.1743			1.35245			0.676223	−	0.736697i	$-0.263616\pi$
$839$	39.1743			1.35245			0.676223	+	0.736697i	$0.263616\pi$
$840$		0			0
$841$	−13.7307			−0.473472
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−17.8239	−	19.3825i	−0.613160	−	0.666777i
$846$		0			0
$847$			42.3696i			1.45584i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	3.18443			0.109161
$852$		0			0
$853$			53.7308i			1.83971i	0.392262	+	0.919853i	$0.371692\pi$
$853$			53.7308i			1.83971i	−0.392262	+	0.919853i	$0.628308\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	45.2265i		−	1.54491i	−0.635071	−	0.772454i	$-0.719029\pi$
$857$		−	45.2265i		−	1.54491i	0.635071	−	0.772454i	$-0.280971\pi$
$858$		0			0
$859$	−24.5178			−0.836538			−0.418269	−	0.908323i	$-0.637363\pi$
$859$	−24.5178			−0.836538			−0.418269	+	0.908323i	$0.637363\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			18.1003i			0.616141i	0.951363	+	0.308071i	$0.0996833\pi$
$863$			18.1003i			0.616141i	−0.951363	+	0.308071i	$0.900317\pi$
$864$		0			0
$865$	39.9989	−	36.7824i	1.36000	−	1.25064i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	10.3445			0.350912
$870$		0			0
$871$	−15.1446			−0.513155
$872$		0			0
$873$		0			0
$874$		0			0
$875$	30.8787	+	39.8033i	1.04389	+	1.34560i
$876$		0			0
$877$		−	23.5866i		−	0.796462i	−0.917285	−	0.398231i	$-0.869624\pi$
$877$		−	23.5866i		−	0.796462i	0.917285	−	0.398231i	$-0.130376\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	53.3253			1.79658			0.898288	−	0.439407i	$-0.144811\pi$
$881$	53.3253			1.79658			0.898288	+	0.439407i	$0.144811\pi$
$882$		0			0
$883$			3.16348i			0.106460i	0.998582	+	0.0532298i	$0.0169516\pi$
$883$			3.16348i			0.106460i	−0.998582	+	0.0532298i	$0.983048\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			17.9481i			0.602639i	0.953523	+	0.301319i	$0.0974270\pi$
$887$			17.9481i			0.602639i	−0.953523	+	0.301319i	$0.902573\pi$
$888$		0			0
$889$	−1.70378			−0.0571430
$890$		0			0
$891$		0			0
$892$		0			0
$893$			43.8750i			1.46822i
$894$		0			0
$895$	−1.24322	−	1.35194i	−0.0415564	−	0.0451903i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	8.48416			0.282963
$900$		0			0
$901$	−31.7857			−1.05894
$902$		0			0
$903$		0			0
$904$		0			0
$905$	5.07917	+	5.52331i	0.168837	+	0.183601i
$906$		0			0
$907$		−	2.45734i		−	0.0815948i	−0.999167	−	0.0407974i	$-0.987010\pi$
$907$		−	2.45734i		−	0.0815948i	0.999167	−	0.0407974i	$-0.0129898\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−53.5023			−1.77261			−0.886306	−	0.463100i	$-0.846737\pi$
$911$	−53.5023			−1.77261			−0.886306	+	0.463100i	$0.846737\pi$
$912$		0			0
$913$		−	8.33649i		−	0.275898i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			36.4426i			1.20344i
$918$		0			0
$919$	−46.9236			−1.54787			−0.773933	−	0.633267i	$-0.781713\pi$
$919$	−46.9236			−1.54787			−0.773933	+	0.633267i	$0.781713\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	45.1168i		−	1.48504i
$924$		0			0
$925$	−2.57266	−	0.215920i	−0.0845886	−	0.00709942i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	8.95345			0.293753			0.146877	−	0.989155i	$-0.453078\pi$
$929$	8.95345			0.293753			0.146877	+	0.989155i	$0.453078\pi$
$930$		0			0
$931$	84.9005			2.78250
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−9.55310	+	8.78491i	−0.312420	+	0.287297i
$936$		0			0
$937$		−	57.7860i		−	1.88779i	−0.330251	−	0.943893i	$-0.607134\pi$
$937$		−	57.7860i		−	1.88779i	0.330251	−	0.943893i	$-0.392866\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−47.7883			−1.55785			−0.778927	−	0.627114i	$-0.784236\pi$
$941$	−47.7883			−1.55785			−0.778927	+	0.627114i	$0.784236\pi$
$942$		0			0
$943$			13.8002i			0.449397i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	18.5038i		−	0.601292i	−0.953736	−	0.300646i	$-0.902798\pi$
$947$		−	18.5038i		−	0.601292i	0.953736	−	0.300646i	$-0.0972023\pi$
$948$		0			0
$949$	−34.3829			−1.11612
$950$		0			0
$951$		0			0
$952$		0			0
$953$			27.7246i			0.898088i	0.893510	+	0.449044i	$0.148235\pi$
$953$			27.7246i			0.898088i	−0.893510	+	0.449044i	$0.851765\pi$
$954$		0			0
$955$	28.9788	+	31.5129i	0.937733	+	1.01973i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−30.0794			−0.971314
$960$		0			0
$961$	−26.2859			−0.847932
$962$		0			0
$963$		0			0
$964$		0			0
$965$	16.3435	−	15.0292i	0.526115	−	0.483809i
$966$		0			0
$967$		−	12.1300i		−	0.390073i	−0.980796	−	0.195037i	$-0.937517\pi$
$967$		−	12.1300i		−	0.390073i	0.980796	−	0.195037i	$-0.0624826\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−40.5273			−1.30058			−0.650291	−	0.759685i	$-0.725353\pi$
$971$	−40.5273			−1.30058			−0.650291	+	0.759685i	$0.725353\pi$
$972$		0			0
$973$		−	83.2285i		−	2.66818i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			20.9384i			0.669878i	0.942240	+	0.334939i	$0.108716\pi$
$977$			20.9384i			0.669878i	−0.942240	+	0.334939i	$0.891284\pi$
$978$		0			0
$979$	−1.22547			−0.0391663
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	1.43321i		−	0.0457124i	−0.999739	−	0.0228562i	$-0.992724\pi$
$983$		−	1.43321i		−	0.0457124i	0.999739	−	0.0228562i	$-0.00727599\pi$
$984$		0			0
$985$	18.6651	−	17.1642i	0.594720	−	0.546897i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−36.2954			−1.15413
$990$		0			0
$991$	−55.4821			−1.76245			−0.881223	−	0.472700i	$-0.843279\pi$
$991$	−55.4821			−1.76245			−0.881223	+	0.472700i	$0.843279\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	4.91742	+	5.34742i	0.155893	+	0.169525i
$996$		0			0
$997$		−	43.6768i		−	1.38326i	−0.722253	−	0.691629i	$-0.756893\pi$
$997$		−	43.6768i		−	1.38326i	0.722253	−	0.691629i	$-0.243107\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3240.2.f.k.649.12		16
3.2	odd	2		3240.2.f.i.649.5		16
5.4	even	2	inner	3240.2.f.k.649.11		16
9.2	odd	6		1080.2.bi.b.1009.16		32
9.4	even	3		360.2.bi.b.169.7	yes	32
9.5	odd	6		1080.2.bi.b.289.7		32
9.7	even	3		360.2.bi.b.49.10	yes	32
15.14	odd	2		3240.2.f.i.649.6		16
36.7	odd	6		720.2.by.f.49.7		32
36.11	even	6		2160.2.by.f.1009.16		32
36.23	even	6		2160.2.by.f.289.7		32
36.31	odd	6		720.2.by.f.529.10		32
45.4	even	6		360.2.bi.b.169.10	yes	32
45.14	odd	6		1080.2.bi.b.289.16		32
45.29	odd	6		1080.2.bi.b.1009.7		32
45.34	even	6		360.2.bi.b.49.7	✓	32
180.59	even	6		2160.2.by.f.289.16		32
180.79	odd	6		720.2.by.f.49.10		32
180.119	even	6		2160.2.by.f.1009.7		32
180.139	odd	6		720.2.by.f.529.7		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.bi.b.49.7	✓	32	45.34	even	6
360.2.bi.b.49.10	yes	32	9.7	even	3
360.2.bi.b.169.7	yes	32	9.4	even	3
360.2.bi.b.169.10	yes	32	45.4	even	6
720.2.by.f.49.7		32	36.7	odd	6
720.2.by.f.49.10		32	180.79	odd	6
720.2.by.f.529.7		32	180.139	odd	6
720.2.by.f.529.10		32	36.31	odd	6
1080.2.bi.b.289.7		32	9.5	odd	6
1080.2.bi.b.289.16		32	45.14	odd	6
1080.2.bi.b.1009.7		32	45.29	odd	6
1080.2.bi.b.1009.16		32	9.2	odd	6
2160.2.by.f.289.7		32	36.23	even	6
2160.2.by.f.289.16		32	180.59	even	6
2160.2.by.f.1009.7		32	180.119	even	6
2160.2.by.f.1009.16		32	36.11	even	6
3240.2.f.i.649.5		16	3.2	odd	2
3240.2.f.i.649.6		16	15.14	odd	2
3240.2.f.k.649.11		16	5.4	even	2	inner
3240.2.f.k.649.12		16	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 3240 = 2^{3} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3240.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.51358 + 1.64593i) q^{5} -4.50582i q^{7} +O(q^{10})\)	\(q+(1.51358 + 1.64593i) q^{5} -4.50582i q^{7} -1.26360 q^{11} +4.97755i q^{13} -4.59329i q^{17} -6.38235 q^{19} -6.16729i q^{23} +(-0.418174 + 4.98248i) q^{25} -3.90760 q^{29} -2.17120 q^{31} +(7.41626 - 6.81990i) q^{35} +0.516341i q^{37} -2.23765 q^{41} -5.88515i q^{43} -6.87443i q^{47} -13.3024 q^{49} -6.92003i q^{53} +(-1.91255 - 2.07980i) q^{55} -7.83554 q^{59} +13.2438 q^{61} +(-8.19270 + 7.53390i) q^{65} +3.04258i q^{67} -9.06407 q^{71} +6.90760i q^{73} +5.69355i q^{77} -8.18651 q^{79} +6.59742i q^{83} +(7.56023 - 6.95229i) q^{85} +0.969826 q^{89} +22.4279 q^{91} +(-9.66018 - 10.5049i) q^{95} -12.4909i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 2 q^{5}+O(q^{10}) \) 16 * q + 2 * q^5	\( 16 q + 2 q^{5} - 16 q^{11} + 4 q^{19} + 6 q^{25} - 20 q^{29} + 12 q^{31} + 2 q^{35} + 8 q^{41} - 36 q^{49} + 10 q^{55} + 20 q^{61} - 10 q^{65} + 8 q^{71} - 4 q^{79} - 36 q^{85} - 48 q^{89} - 4 q^{91} + 32 q^{95}+O(q^{100}) \) 16 * q + 2 * q^5 - 16 * q^11 + 4 * q^19 + 6 * q^25 - 20 * q^29 + 12 * q^31 + 2 * q^35 + 8 * q^41 - 36 * q^49 + 10 * q^55 + 20 * q^61 - 10 * q^65 + 8 * q^71 - 4 * q^79 - 36 * q^85 - 48 * q^89 - 4 * q^91 + 32 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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