LMFDB - Embedded newform 1680.3.s.c.1441.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1680,3,Mod(1441,1680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1680, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1680.1441");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1680.s (of order $2$, degree $1$, not 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:	$45.7766844125$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2 x^{11} + 21 x^{10} - 26 x^{9} + 295 x^{8} - 372 x^{7} + 1704 x^{6} - 1074 x^{5} + 5613 x^{4} + \cdots + 441 $ x^12 - 2x^11 + 21x^10 - 26x^9 + 295x^8 - 372x^7 + 1704x^6 - 1074x^5 + 5613x^4 - 5472x^3 + 4950x^2 - 1638*x + 441
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{11} $
Twist minimal:	no (minimal twist has level 105)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1441.3
Root			$1.31896 - 2.28450i$ of defining polynomial
Character	$\chi$	$=$	1680.1441
Dual form			1680.3.s.c.1441.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{3} -2.23607i q^{5} +(6.69736 - 2.03600i) q^{7} -3.00000 q^{9} +O(q^{10})$	$q-1.73205i q^{3} -2.23607i q^{5} +(6.69736 - 2.03600i) q^{7} -3.00000 q^{9} +2.03112 q^{11} -18.0174i q^{13} -3.87298 q^{15} +1.07289i q^{17} -28.7852i q^{19} +(-3.52646 - 11.6002i) q^{21} -24.8710 q^{23} -5.00000 q^{25} +5.19615i q^{27} -38.4300 q^{29} +44.0899i q^{31} -3.51800i q^{33} +(-4.55264 - 14.9758i) q^{35} +37.2832 q^{37} -31.2071 q^{39} +49.9206i q^{41} +9.58871 q^{43} +6.70820i q^{45} -55.6978i q^{47} +(40.7094 - 27.2717i) q^{49} +1.85829 q^{51} +57.4656 q^{53} -4.54171i q^{55} -49.8575 q^{57} -101.697i q^{59} -31.9530i q^{61} +(-20.0921 + 6.10801i) q^{63} -40.2882 q^{65} -95.7318 q^{67} +43.0778i q^{69} +25.8039 q^{71} -95.6803i q^{73} +8.66025i q^{75} +(13.6031 - 4.13536i) q^{77} -28.1212 q^{79} +9.00000 q^{81} +103.374i q^{83} +2.39905 q^{85} +66.5628i q^{87} +29.3629i q^{89} +(-36.6835 - 120.669i) q^{91} +76.3659 q^{93} -64.3658 q^{95} -67.6473i q^{97} -6.09335 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 8 q^{7} - 36 q^{9}+O(q^{10}) $ 12 * q + 8 * q^7 - 36 * q^9	$ 12 q + 8 q^{7} - 36 q^{9} + 16 q^{11} + 36 q^{21} + 64 q^{23} - 60 q^{25} + 104 q^{29} - 60 q^{35} + 32 q^{37} + 24 q^{39} - 152 q^{43} + 60 q^{49} - 24 q^{51} + 176 q^{53} - 240 q^{57} - 24 q^{63} - 240 q^{65} - 168 q^{67} - 32 q^{71} + 8 q^{77} - 120 q^{79} + 108 q^{81} + 120 q^{85} - 24 q^{91} + 48 q^{93} - 48 q^{99}+O(q^{100}) $ 12 * q + 8 * q^7 - 36 * q^9 + 16 * q^11 + 36 * q^21 + 64 * q^23 - 60 * q^25 + 104 * q^29 - 60 * q^35 + 32 * q^37 + 24 * q^39 - 152 * q^43 + 60 * q^49 - 24 * q^51 + 176 * q^53 - 240 * q^57 - 24 * q^63 - 240 * q^65 - 168 * q^67 - 32 * q^71 + 8 * q^77 - 120 * q^79 + 108 * q^81 + 120 * q^85 - 24 * q^91 + 48 * q^93 - 48 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times$.

$n$	$241$	$337$	$421$	$1121$	$1471$
$\chi(n)$	$-1$	$1$	$1$	$1$	$1$

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
$3$		−	1.73205i		−	0.577350i
$3$		−	1.73205i		−	0.577350i
$4$		0			0
$5$		−	2.23607i		−	0.447214i
$5$		−	2.23607i		−	0.447214i
$6$		0			0
$7$	6.69736	−	2.03600i	0.956766	−	0.290857i
$7$	6.69736	−	2.03600i	0.956766	−	0.290857i
$8$		0			0
$9$	−3.00000			−0.333333
$10$		0			0
$11$	2.03112			0.184647			0.0923235	−	0.995729i	$-0.470571\pi$
$11$	2.03112			0.184647			0.0923235	+	0.995729i	$0.470571\pi$
$12$		0			0
$13$		−	18.0174i		−	1.38596i	−0.720958	−	0.692978i	$-0.756298\pi$
$13$		−	18.0174i		−	1.38596i	0.720958	−	0.692978i	$-0.243702\pi$
$14$		0			0
$15$	−3.87298			−0.258199
$16$		0			0
$17$			1.07289i			0.0631109i	0.999502	+	0.0315555i	$0.0100461\pi$
$17$			1.07289i			0.0631109i	−0.999502	+	0.0315555i	$0.989954\pi$
$18$		0			0
$19$		−	28.7852i		−	1.51501i	−0.652828	−	0.757506i	$-0.726417\pi$
$19$		−	28.7852i		−	1.51501i	0.652828	−	0.757506i	$-0.273583\pi$
$20$		0			0
$21$	−3.52646	−	11.6002i	−0.167927	−	0.552389i
$22$		0			0
$23$	−24.8710			−1.08135			−0.540674	−	0.841232i	$-0.681831\pi$
$23$	−24.8710			−1.08135			−0.540674	+	0.841232i	$0.681831\pi$
$24$		0			0
$25$	−5.00000			−0.200000
$26$		0			0
$27$			5.19615i			0.192450i
$28$		0			0
$29$	−38.4300			−1.32517			−0.662587	−	0.748985i	$-0.730542\pi$
$29$	−38.4300			−1.32517			−0.662587	+	0.748985i	$0.730542\pi$
$30$		0			0
$31$			44.0899i			1.42225i	0.703064	+	0.711127i	$0.251815\pi$
$31$			44.0899i			1.42225i	−0.703064	+	0.711127i	$0.748185\pi$
$32$		0			0
$33$		−	3.51800i		−	0.106606i
$34$		0			0
$35$	−4.55264	−	14.9758i	−0.130075	−	0.427879i
$36$		0			0
$37$	37.2832			1.00765			0.503827	−	0.863804i	$-0.331925\pi$
$37$	37.2832			1.00765			0.503827	+	0.863804i	$0.331925\pi$
$38$		0			0
$39$	−31.2071			−0.800182
$40$		0			0
$41$			49.9206i			1.21758i	0.793333	+	0.608788i	$0.208344\pi$
$41$			49.9206i			1.21758i	−0.793333	+	0.608788i	$0.791656\pi$
$42$		0			0
$43$	9.58871			0.222993			0.111497	−	0.993765i	$-0.464436\pi$
$43$	9.58871			0.222993			0.111497	+	0.993765i	$0.464436\pi$
$44$		0			0
$45$			6.70820i			0.149071i
$46$		0			0
$47$		−	55.6978i		−	1.18506i	−0.805549	−	0.592529i	$-0.798129\pi$
$47$		−	55.6978i		−	1.18506i	0.805549	−	0.592529i	$-0.201871\pi$
$48$		0			0
$49$	40.7094	−	27.2717i	0.830804	−	0.556565i
$50$		0			0
$51$	1.85829			0.0364371
$52$		0			0
$53$	57.4656			1.08426			0.542128	−	0.840296i	$-0.317619\pi$
$53$	57.4656			1.08426			0.542128	+	0.840296i	$0.317619\pi$
$54$		0			0
$55$		−	4.54171i		−	0.0825766i
$56$		0			0
$57$	−49.8575			−0.874693
$58$		0			0
$59$		−	101.697i		−	1.72368i	−0.507178	−	0.861841i	$-0.669311\pi$
$59$		−	101.697i		−	1.72368i	0.507178	−	0.861841i	$-0.330689\pi$
$60$		0			0
$61$		−	31.9530i		−	0.523819i	−0.965092	−	0.261910i	$-0.915648\pi$
$61$		−	31.9530i		−	0.523819i	0.965092	−	0.261910i	$-0.0843523\pi$
$62$		0			0
$63$	−20.0921	+	6.10801i	−0.318922	+	0.0969525i
$64$		0			0
$65$	−40.2882			−0.619819
$66$		0			0
$67$	−95.7318			−1.42883			−0.714416	−	0.699721i	$-0.753308\pi$
$67$	−95.7318			−1.42883			−0.714416	+	0.699721i	$0.753308\pi$
$68$		0			0
$69$			43.0778i			0.624316i
$70$		0			0
$71$	25.8039			0.363435			0.181718	−	0.983351i	$-0.441834\pi$
$71$	25.8039			0.363435			0.181718	+	0.983351i	$0.441834\pi$
$72$		0			0
$73$		−	95.6803i		−	1.31069i	−0.755330	−	0.655345i	$-0.772523\pi$
$73$		−	95.6803i		−	1.31069i	0.755330	−	0.655345i	$-0.227477\pi$
$74$		0			0
$75$			8.66025i			0.115470i
$76$		0			0
$77$	13.6031	−	4.13536i	0.176664	−	0.0537059i
$78$		0			0
$79$	−28.1212			−0.355965			−0.177982	−	0.984034i	$-0.556957\pi$
$79$	−28.1212			−0.355965			−0.177982	+	0.984034i	$0.556957\pi$
$80$		0			0
$81$	9.00000			0.111111
$82$		0			0
$83$			103.374i			1.24547i	0.782435	+	0.622733i	$0.213978\pi$
$83$			103.374i			1.24547i	−0.782435	+	0.622733i	$0.786022\pi$
$84$		0			0
$85$	2.39905			0.0282241
$86$		0			0
$87$			66.5628i			0.765090i
$88$		0			0
$89$			29.3629i			0.329920i	0.986300	+	0.164960i	$0.0527496\pi$
$89$			29.3629i			0.329920i	−0.986300	+	0.164960i	$0.947250\pi$
$90$		0			0
$91$	−36.6835	−	120.669i	−0.403116	−	1.32604i
$92$		0			0
$93$	76.3659			0.821138
$94$		0			0
$95$	−64.3658			−0.677534
$96$		0			0
$97$		−	67.6473i		−	0.697395i	−0.937235	−	0.348697i	$-0.886624\pi$
$97$		−	67.6473i		−	0.697395i	0.937235	−	0.348697i	$-0.113376\pi$
$98$		0			0
$99$	−6.09335			−0.0615490
$100$		0			0
$101$			73.3301i			0.726040i	0.931781	+	0.363020i	$0.118254\pi$
$101$			73.3301i			0.726040i	−0.931781	+	0.363020i	$0.881746\pi$
$102$		0			0
$103$			57.3431i			0.556729i	0.960476	+	0.278365i	$0.0897923\pi$
$103$			57.3431i			0.556729i	−0.960476	+	0.278365i	$0.910208\pi$
$104$		0			0
$105$	−25.9388	+	7.88540i	−0.247036	+	0.0750991i
$106$		0			0
$107$	39.8399			0.372336			0.186168	−	0.982518i	$-0.440393\pi$
$107$	39.8399			0.372336			0.186168	+	0.982518i	$0.440393\pi$
$108$		0			0
$109$	−185.052			−1.69773			−0.848863	−	0.528613i	$-0.822712\pi$
$109$	−185.052			−1.69773			−0.848863	+	0.528613i	$0.822712\pi$
$110$		0			0
$111$		−	64.5764i		−	0.581770i
$112$		0			0
$113$	−120.716			−1.06829			−0.534143	−	0.845394i	$-0.679366\pi$
$113$	−120.716			−1.06829			−0.534143	+	0.845394i	$0.679366\pi$
$114$		0			0
$115$			55.6132i			0.483593i
$116$		0			0
$117$			54.0523i			0.461986i
$118$		0			0
$119$	2.18440	+	7.18551i	0.0183563	+	0.0603824i
$120$		0			0
$121$	−116.875			−0.965906
$122$		0			0
$123$	86.4651			0.702968
$124$		0			0
$125$			11.1803i			0.0894427i
$126$		0			0
$127$	83.5096			0.657556			0.328778	−	0.944407i	$-0.393363\pi$
$127$	83.5096			0.657556			0.328778	+	0.944407i	$0.393363\pi$
$128$		0			0
$129$		−	16.6081i		−	0.128745i
$130$		0			0
$131$			198.248i			1.51334i	0.653796	+	0.756671i	$0.273175\pi$
$131$			198.248i			1.51334i	−0.653796	+	0.756671i	$0.726825\pi$
$132$		0			0
$133$	−58.6068	−	192.785i	−0.440653	−	1.44951i
$134$		0			0
$135$	11.6190			0.0860663
$136$		0			0
$137$	−24.1635			−0.176376			−0.0881880	−	0.996104i	$-0.528108\pi$
$137$	−24.1635			−0.176376			−0.0881880	+	0.996104i	$0.528108\pi$
$138$		0			0
$139$		−	61.6370i		−	0.443432i	−0.975111	−	0.221716i	$-0.928834\pi$
$139$		−	61.6370i		−	0.443432i	0.975111	−	0.221716i	$-0.0711658\pi$
$140$		0			0
$141$	−96.4714			−0.684194
$142$		0			0
$143$		−	36.5955i		−	0.255913i
$144$		0			0
$145$			85.9322i			0.592636i
$146$		0			0
$147$	−47.2360	−	70.5107i	−0.321333	−	0.479665i
$148$		0			0
$149$	−28.8271			−0.193470			−0.0967351	−	0.995310i	$-0.530840\pi$
$149$	−28.8271			−0.193470			−0.0967351	+	0.995310i	$0.530840\pi$
$150$		0			0
$151$	−248.311			−1.64445			−0.822223	−	0.569166i	$-0.807266\pi$
$151$	−248.311			−1.64445			−0.822223	+	0.569166i	$0.807266\pi$
$152$		0			0
$153$		−	3.21866i		−	0.0210370i
$154$		0			0
$155$	98.5879			0.636051
$156$		0			0
$157$			41.3848i			0.263597i	0.991277	+	0.131799i	$0.0420752\pi$
$157$			41.3848i			0.263597i	−0.991277	+	0.131799i	$0.957925\pi$
$158$		0			0
$159$		−	99.5334i		−	0.625996i
$160$		0			0
$161$	−166.570	+	50.6374i	−1.03460	+	0.314518i
$162$		0			0
$163$	51.8103			0.317855			0.158927	−	0.987290i	$-0.449196\pi$
$163$	51.8103			0.317855			0.158927	+	0.987290i	$0.449196\pi$
$164$		0			0
$165$	−7.86648			−0.0476756
$166$		0			0
$167$		−	41.9711i		−	0.251324i	−0.992073	−	0.125662i	$-0.959895\pi$
$167$		−	41.9711i		−	0.251324i	0.992073	−	0.125662i	$-0.0401055\pi$
$168$		0			0
$169$	−155.628			−0.920876
$170$		0			0
$171$			86.3557i			0.505004i
$172$		0			0
$173$		−	98.0199i		−	0.566589i	−0.959033	−	0.283295i	$-0.908573\pi$
$173$		−	98.0199i		−	0.566589i	0.959033	−	0.283295i	$-0.0914274\pi$
$174$		0			0
$175$	−33.4868	+	10.1800i	−0.191353	+	0.0581715i
$176$		0			0
$177$	−176.145			−0.995168
$178$		0			0
$179$	−68.5830			−0.383145			−0.191573	−	0.981478i	$-0.561359\pi$
$179$	−68.5830			−0.383145			−0.191573	+	0.981478i	$0.561359\pi$
$180$		0			0
$181$		−	105.124i		−	0.580798i	−0.956906	−	0.290399i	$-0.906212\pi$
$181$		−	105.124i		−	0.580798i	0.956906	−	0.290399i	$-0.0937880\pi$
$182$		0			0
$183$	−55.3442			−0.302427
$184$		0			0
$185$		−	83.3678i		−	0.450637i
$186$		0			0
$187$			2.17916i			0.0116532i
$188$		0			0
$189$	10.5794	+	34.8005i	0.0559755	+	0.184130i
$190$		0			0
$191$	−229.803			−1.20316			−0.601579	−	0.798813i	$-0.705461\pi$
$191$	−229.803			−1.20316			−0.601579	+	0.798813i	$0.705461\pi$
$192$		0			0
$193$	111.530			0.577877			0.288939	−	0.957348i	$-0.406698\pi$
$193$	111.530			0.577877			0.288939	+	0.957348i	$0.406698\pi$
$194$		0			0
$195$			69.7812i			0.357852i
$196$		0			0
$197$	−172.865			−0.877486			−0.438743	−	0.898613i	$-0.644576\pi$
$197$	−172.865			−0.877486			−0.438743	+	0.898613i	$0.644576\pi$
$198$		0			0
$199$		−	117.944i		−	0.592685i	−0.955082	−	0.296343i	$-0.904233\pi$
$199$		−	117.944i		−	0.592685i	0.955082	−	0.296343i	$-0.0957670\pi$
$200$		0			0
$201$			165.812i			0.824937i
$202$		0			0
$203$	−257.380	+	78.2437i	−1.26788	+	0.385437i
$204$		0			0
$205$	111.626			0.544517
$206$		0			0
$207$	74.6130			0.360449
$208$		0			0
$209$		−	58.4662i		−	0.279742i
$210$		0			0
$211$	391.940			1.85754			0.928769	−	0.370660i	$-0.120869\pi$
$211$	391.940			1.85754			0.928769	+	0.370660i	$0.120869\pi$
$212$		0			0
$213$		−	44.6936i		−	0.209829i
$214$		0			0
$215$		−	21.4410i		−	0.0997256i
$216$		0			0
$217$	89.7670	+	295.286i	0.413673	+	1.36076i
$218$		0			0
$219$	−165.723			−0.756727
$220$		0			0
$221$	19.3306			0.0874690
$222$		0			0
$223$			96.2512i			0.431620i	0.976435	+	0.215810i	$0.0692391\pi$
$223$			96.2512i			0.431620i	−0.976435	+	0.215810i	$0.930761\pi$
$224$		0			0
$225$	15.0000			0.0666667
$226$		0			0
$227$			91.6962i			0.403948i	0.979391	+	0.201974i	$0.0647357\pi$
$227$			91.6962i			0.403948i	−0.979391	+	0.201974i	$0.935264\pi$
$228$		0			0
$229$		−	323.972i		−	1.41472i	−0.706852	−	0.707361i	$-0.749885\pi$
$229$		−	323.972i		−	1.41472i	0.706852	−	0.707361i	$-0.250115\pi$
$230$		0			0
$231$	−7.16265	−	23.5613i	−0.0310071	−	0.101997i
$232$		0			0
$233$	182.918			0.785055			0.392528	−	0.919740i	$-0.371601\pi$
$233$	182.918			0.785055			0.392528	+	0.919740i	$0.371601\pi$
$234$		0			0
$235$	−124.544			−0.529974
$236$		0			0
$237$			48.7074i			0.205516i
$238$		0			0
$239$	42.9240			0.179598			0.0897992	−	0.995960i	$-0.471377\pi$
$239$	42.9240			0.179598			0.0897992	+	0.995960i	$0.471377\pi$
$240$		0			0
$241$			99.8942i			0.414499i	0.978288	+	0.207249i	$0.0664511\pi$
$241$			99.8942i			0.414499i	−0.978288	+	0.207249i	$0.933549\pi$
$242$		0			0
$243$		−	15.5885i		−	0.0641500i
$244$		0			0
$245$	−60.9814	−	91.0290i	−0.248904	−	0.371547i
$246$		0			0
$247$	−518.636			−2.09974
$248$		0			0
$249$	179.048			0.719070
$250$		0			0
$251$		−	404.945i		−	1.61333i	−0.591011	−	0.806663i	$-0.701271\pi$
$251$		−	404.945i		−	1.61333i	0.591011	−	0.806663i	$-0.298729\pi$
$252$		0			0
$253$	−50.5159			−0.199668
$254$		0			0
$255$		−	4.15527i		−	0.0162952i
$256$		0			0
$257$		−	102.697i		−	0.399600i	−0.979837	−	0.199800i	$-0.935971\pi$
$257$		−	102.697i		−	0.399600i	0.979837	−	0.199800i	$-0.0640292\pi$
$258$		0			0
$259$	249.699	−	75.9087i	0.964090	−	0.293084i
$260$		0			0
$261$	115.290			0.441725
$262$		0			0
$263$	−281.479			−1.07026			−0.535132	−	0.844769i	$-0.679738\pi$
$263$	−281.479			−1.07026			−0.535132	+	0.844769i	$0.679738\pi$
$264$		0			0
$265$		−	128.497i		−	0.484894i
$266$		0			0
$267$	50.8581			0.190480
$268$		0			0
$269$			176.412i			0.655808i	0.944711	+	0.327904i	$0.106342\pi$
$269$			176.412i			0.655808i	−0.944711	+	0.327904i	$0.893658\pi$
$270$		0			0
$271$		−	306.041i		−	1.12930i	−0.825330	−	0.564651i	$-0.809011\pi$
$271$		−	306.041i		−	1.12930i	0.825330	−	0.564651i	$-0.190989\pi$
$272$		0			0
$273$	−209.005	+	63.5378i	−0.765588	+	0.232739i
$274$		0			0
$275$	−10.1556			−0.0369294
$276$		0			0
$277$	43.7993			0.158120			0.0790601	−	0.996870i	$-0.474808\pi$
$277$	43.7993			0.158120			0.0790601	+	0.996870i	$0.474808\pi$
$278$		0			0
$279$		−	132.270i		−	0.474084i
$280$		0			0
$281$	499.502			1.77759			0.888794	−	0.458307i	$-0.151544\pi$
$281$	499.502			1.77759			0.888794	+	0.458307i	$0.151544\pi$
$282$		0			0
$283$		−	122.672i		−	0.433470i	−0.976231	−	0.216735i	$-0.930459\pi$
$283$		−	122.672i		−	0.433470i	0.976231	−	0.216735i	$-0.0695407\pi$
$284$		0			0
$285$			111.485i			0.391175i
$286$		0			0
$287$	101.639	+	334.337i	0.354141	+	1.16494i
$288$		0			0
$289$	287.849			0.996017
$290$		0			0
$291$	−117.169			−0.402641
$292$		0			0
$293$		−	123.871i		−	0.422769i	−0.977403	−	0.211385i	$-0.932203\pi$
$293$		−	123.871i		−	0.422769i	0.977403	−	0.211385i	$-0.0677972\pi$
$294$		0			0
$295$	−227.402			−0.770854
$296$		0			0
$297$			10.5540i			0.0355353i
$298$		0			0
$299$			448.112i			1.49870i
$300$		0			0
$301$	64.2191	−	19.5226i	0.213352	−	0.0648593i
$302$		0			0
$303$	127.011			0.419180
$304$		0			0
$305$	−71.4490			−0.234259
$306$		0			0
$307$		−	225.577i		−	0.734778i	−0.930067	−	0.367389i	$-0.880252\pi$
$307$		−	225.577i		−	0.734778i	0.930067	−	0.367389i	$-0.119748\pi$
$308$		0			0
$309$	99.3211			0.321428
$310$		0			0
$311$		−	52.9648i		−	0.170305i	−0.996368	−	0.0851525i	$-0.972862\pi$
$311$		−	52.9648i		−	0.170305i	0.996368	−	0.0851525i	$-0.0271377\pi$
$312$		0			0
$313$			374.563i			1.19669i	0.801240	+	0.598343i	$0.204174\pi$
$313$			374.563i			1.19669i	−0.801240	+	0.598343i	$0.795826\pi$
$314$		0			0
$315$	13.6579	+	44.9273i	0.0433585	+	0.142626i
$316$		0			0
$317$	12.0550			0.0380283			0.0190142	−	0.999819i	$-0.493947\pi$
$317$	12.0550			0.0380283			0.0190142	+	0.999819i	$0.493947\pi$
$318$		0			0
$319$	−78.0559			−0.244689
$320$		0			0
$321$		−	69.0048i		−	0.214968i
$322$		0			0
$323$	30.8833			0.0956138
$324$		0			0
$325$			90.0872i			0.277191i
$326$		0			0
$327$			320.520i			0.980183i
$328$		0			0
$329$	−113.401	−	373.028i	−0.344683	−	1.13382i
$330$		0			0
$331$	376.184			1.13651			0.568254	−	0.822853i	$-0.307619\pi$
$331$	376.184			1.13651			0.568254	+	0.822853i	$0.307619\pi$
$332$		0			0
$333$	−111.850			−0.335885
$334$		0			0
$335$			214.063i			0.638993i
$336$		0			0
$337$	−108.973			−0.323363			−0.161682	−	0.986843i	$-0.551692\pi$
$337$	−108.973			−0.323363			−0.161682	+	0.986843i	$0.551692\pi$
$338$		0			0
$339$			209.087i			0.616775i
$340$		0			0
$341$			89.5516i			0.262615i
$342$		0			0
$343$	217.120	−	265.533i	0.633004	−	0.774148i
$344$		0			0
$345$	96.3250			0.279203
$346$		0			0
$347$	206.351			0.594671			0.297335	−	0.954773i	$-0.403902\pi$
$347$	206.351			0.594671			0.297335	+	0.954773i	$0.403902\pi$
$348$		0			0
$349$		−	373.475i		−	1.07013i	−0.844811	−	0.535064i	$-0.820287\pi$
$349$		−	373.475i		−	1.07013i	0.844811	−	0.535064i	$-0.179713\pi$
$350$		0			0
$351$	93.6213			0.266727
$352$		0			0
$353$			646.142i			1.83043i	0.402964	+	0.915216i	$0.367980\pi$
$353$			646.142i			1.83043i	−0.402964	+	0.915216i	$0.632020\pi$
$354$		0			0
$355$		−	57.6992i		−	0.162533i
$356$		0			0
$357$	12.4457	−	3.78349i	0.0348618	−	0.0105980i
$358$		0			0
$359$	−175.261			−0.488192			−0.244096	−	0.969751i	$-0.578491\pi$
$359$	−175.261			−0.488192			−0.244096	+	0.969751i	$0.578491\pi$
$360$		0			0
$361$	−467.590			−1.29526
$362$		0			0
$363$			202.433i			0.557666i
$364$		0			0
$365$	−213.948			−0.586158
$366$		0			0
$367$			106.477i			0.290127i	0.989422	+	0.145063i	$0.0463386\pi$
$367$			106.477i			0.290127i	−0.989422	+	0.145063i	$0.953661\pi$
$368$		0			0
$369$		−	149.762i		−	0.405859i
$370$		0			0
$371$	384.868	−	117.000i	1.03738	−	0.315364i
$372$		0			0
$373$	223.324			0.598723			0.299362	−	0.954140i	$-0.403226\pi$
$373$	223.324			0.598723			0.299362	+	0.954140i	$0.403226\pi$
$374$		0			0
$375$	19.3649			0.0516398
$376$		0			0
$377$			692.411i			1.83663i
$378$		0			0
$379$	−119.075			−0.314183			−0.157092	−	0.987584i	$-0.550212\pi$
$379$	−119.075			−0.314183			−0.157092	+	0.987584i	$0.550212\pi$
$380$		0			0
$381$		−	144.643i		−	0.379640i
$382$		0			0
$383$		−	494.637i		−	1.29148i	−0.763557	−	0.645740i	$-0.776549\pi$
$383$		−	494.637i		−	1.29148i	0.763557	−	0.645740i	$-0.223451\pi$
$384$		0			0
$385$	−9.24694	−	30.4175i	−0.0240180	−	0.0790065i
$386$		0			0
$387$	−28.7661			−0.0743311
$388$		0			0
$389$	−270.578			−0.695574			−0.347787	−	0.937574i	$-0.613067\pi$
$389$	−270.578			−0.695574			−0.347787	+	0.937574i	$0.613067\pi$
$390$		0			0
$391$		−	26.6837i		−	0.0682448i
$392$		0			0
$393$	343.375			0.873728
$394$		0			0
$395$			62.8810i			0.159192i
$396$		0			0
$397$		−	37.1881i		−	0.0936727i	−0.998903	−	0.0468364i	$-0.985086\pi$
$397$		−	37.1881i		−	0.0936727i	0.998903	−	0.0468364i	$-0.0149139\pi$
$398$		0			0
$399$	−333.914	+	101.510i	−0.836877	+	0.254411i
$400$		0			0
$401$	−36.7102			−0.0915467			−0.0457733	−	0.998952i	$-0.514575\pi$
$401$	−36.7102			−0.0915467			−0.0457733	+	0.998952i	$0.514575\pi$
$402$		0			0
$403$	794.386			1.97118
$404$		0			0
$405$		−	20.1246i		−	0.0496904i
$406$		0			0
$407$	75.7266			0.186060
$408$		0			0
$409$		−	495.325i		−	1.21106i	−0.795821	−	0.605532i	$-0.792960\pi$
$409$		−	495.325i		−	1.21106i	0.795821	−	0.605532i	$-0.207040\pi$
$410$		0			0
$411$			41.8524i			0.101831i
$412$		0			0
$413$	−207.056	−	681.104i	−0.501346	−	1.64916i
$414$		0			0
$415$	231.151			0.556989
$416$		0			0
$417$	−106.758			−0.256016
$418$		0			0
$419$		−	269.297i		−	0.642713i	−0.946958	−	0.321357i	$-0.895861\pi$
$419$		−	269.297i		−	0.642713i	0.946958	−	0.321357i	$-0.104139\pi$
$420$		0			0
$421$	756.722			1.79744			0.898719	−	0.438524i	$-0.144499\pi$
$421$	756.722			1.79744			0.898719	+	0.438524i	$0.144499\pi$
$422$		0			0
$423$			167.093i			0.395020i
$424$		0			0
$425$		−	5.36443i		−	0.0126222i
$426$		0			0
$427$	−65.0563	−	214.001i	−0.152357	−	0.501173i
$428$		0			0
$429$	−63.3853			−0.147751
$430$		0			0
$431$	−185.182			−0.429657			−0.214828	−	0.976652i	$-0.568919\pi$
$431$	−185.182			−0.429657			−0.214828	+	0.976652i	$0.568919\pi$
$432$		0			0
$433$		−	642.846i		−	1.48463i	−0.670049	−	0.742317i	$-0.733727\pi$
$433$		−	642.846i		−	1.48463i	0.670049	−	0.742317i	$-0.266273\pi$
$434$		0			0
$435$	148.839			0.342158
$436$		0			0
$437$			715.918i			1.63826i
$438$		0			0
$439$		−	464.239i		−	1.05749i	−0.848780	−	0.528745i	$-0.822663\pi$
$439$		−	464.239i		−	1.05749i	0.848780	−	0.528745i	$-0.177337\pi$
$440$		0			0
$441$	−122.128	+	81.8151i	−0.276935	+	0.185522i
$442$		0			0
$443$	115.228			0.260109			0.130054	−	0.991507i	$-0.458485\pi$
$443$	115.228			0.260109			0.130054	+	0.991507i	$0.458485\pi$
$444$		0			0
$445$	65.6575			0.147545
$446$		0			0
$447$			49.9299i			0.111700i
$448$		0			0
$449$	333.955			0.743774			0.371887	−	0.928278i	$-0.378711\pi$
$449$	333.955			0.743774			0.371887	+	0.928278i	$0.378711\pi$
$450$		0			0
$451$			101.395i			0.224822i
$452$		0			0
$453$			430.088i			0.949421i
$454$		0			0
$455$	−269.825	+	82.0269i	−0.593022	+	0.180279i
$456$		0			0
$457$	354.152			0.774949			0.387474	−	0.921880i	$-0.373348\pi$
$457$	354.152			0.774949			0.387474	+	0.921880i	$0.373348\pi$
$458$		0			0
$459$	−5.57488			−0.0121457
$460$		0			0
$461$			128.906i			0.279623i	0.990178	+	0.139811i	$0.0446497\pi$
$461$			128.906i			0.279623i	−0.990178	+	0.139811i	$0.955350\pi$
$462$		0			0
$463$	302.175			0.652645			0.326322	−	0.945259i	$-0.394190\pi$
$463$	302.175			0.652645			0.326322	+	0.945259i	$0.394190\pi$
$464$		0			0
$465$		−	170.759i		−	0.367224i
$466$		0			0
$467$			808.227i			1.73068i	0.501186	+	0.865339i	$0.332897\pi$
$467$			808.227i			1.73068i	−0.501186	+	0.865339i	$0.667103\pi$
$468$		0			0
$469$	−641.151	+	194.910i	−1.36706	+	0.415587i
$470$		0			0
$471$	71.6805			0.152188
$472$		0			0
$473$	19.4758			0.0411750
$474$		0			0
$475$			143.926i			0.303003i
$476$		0			0
$477$	−172.397			−0.361419
$478$		0			0
$479$		−	48.8836i		−	0.102053i	−0.998697	−	0.0510267i	$-0.983751\pi$
$479$		−	48.8836i		−	0.102053i	0.998697	−	0.0510267i	$-0.0162494\pi$
$480$		0			0
$481$		−	671.748i		−	1.39657i
$482$		0			0
$483$	87.7066	+	288.508i	0.181587	+	0.597325i
$484$		0			0
$485$	−151.264			−0.311884
$486$		0			0
$487$	−334.433			−0.686720			−0.343360	−	0.939204i	$-0.611565\pi$
$487$	−334.433			−0.686720			−0.343360	+	0.939204i	$0.611565\pi$
$488$		0			0
$489$		−	89.7381i		−	0.183514i
$490$		0			0
$491$	549.151			1.11843			0.559217	−	0.829021i	$-0.311102\pi$
$491$	549.151			1.11843			0.559217	+	0.829021i	$0.311102\pi$
$492$		0			0
$493$		−	41.2310i		−	0.0836329i
$494$		0			0
$495$			13.6251i			0.0275255i
$496$		0			0
$497$	172.818	−	52.5368i	0.347722	−	0.105708i
$498$		0			0
$499$	462.450			0.926754			0.463377	−	0.886161i	$-0.346638\pi$
$499$	462.450			0.926754			0.463377	+	0.886161i	$0.346638\pi$
$500$		0			0
$501$	−72.6961			−0.145102
$502$		0			0
$503$		−	676.817i		−	1.34556i	−0.739842	−	0.672781i	$-0.765100\pi$
$503$		−	676.817i		−	1.34556i	0.739842	−	0.672781i	$-0.234900\pi$
$504$		0			0
$505$	163.971			0.324695
$506$		0			0
$507$			269.556i			0.531668i
$508$		0			0
$509$			31.5959i			0.0620744i	0.999518	+	0.0310372i	$0.00988104\pi$
$509$			31.5959i			0.0620744i	−0.999518	+	0.0310372i	$0.990119\pi$
$510$		0			0
$511$	−194.805	−	640.806i	−0.381224	−	1.25402i
$512$		0			0
$513$	149.572			0.291564
$514$		0			0
$515$	128.223			0.248977
$516$		0			0
$517$		−	113.129i		−	0.218817i
$518$		0			0
$519$	−169.775			−0.327120
$520$		0			0
$521$		−	793.420i		−	1.52288i	−0.648236	−	0.761440i	$-0.724493\pi$
$521$		−	793.420i		−	1.52288i	0.648236	−	0.761440i	$-0.275507\pi$
$522$		0			0
$523$		−	593.508i		−	1.13481i	−0.823438	−	0.567407i	$-0.807947\pi$
$523$		−	593.508i		−	1.13481i	0.823438	−	0.567407i	$-0.192053\pi$
$524$		0			0
$525$	17.6323	+	58.0009i	0.0335853	+	0.110478i
$526$		0			0
$527$	−47.3034			−0.0897597
$528$		0			0
$529$	89.5666			0.169313
$530$		0			0
$531$			305.092i			0.574561i
$532$		0			0
$533$	899.442			1.68751
$534$		0			0
$535$		−	89.0848i		−	0.166514i
$536$		0			0
$537$			118.789i			0.221209i
$538$		0			0
$539$	82.6855	−	55.3920i	0.153405	−	0.102768i
$540$		0			0
$541$	217.690			0.402385			0.201192	−	0.979552i	$-0.435518\pi$
$541$	217.690			0.402385			0.201192	+	0.979552i	$0.435518\pi$
$542$		0			0
$543$	−182.081			−0.335324
$544$		0			0
$545$			413.789i			0.759246i
$546$		0			0
$547$	137.891			0.252085			0.126043	−	0.992025i	$-0.459772\pi$
$547$	137.891			0.252085			0.126043	+	0.992025i	$0.459772\pi$
$548$		0			0
$549$			95.8589i			0.174606i
$550$		0			0
$551$			1106.22i			2.00766i
$552$		0			0
$553$	−188.338	+	57.2549i	−0.340575	+	0.103535i
$554$		0			0
$555$	−144.397			−0.260175
$556$		0			0
$557$	−316.337			−0.567930			−0.283965	−	0.958835i	$-0.591650\pi$
$557$	−316.337			−0.567930			−0.283965	+	0.958835i	$0.591650\pi$
$558$		0			0
$559$		−	172.764i		−	0.309059i
$560$		0			0
$561$	3.77441			0.00672800
$562$		0			0
$563$			151.482i			0.269063i	0.990909	+	0.134531i	$0.0429529\pi$
$563$			151.482i			0.269063i	−0.990909	+	0.134531i	$0.957047\pi$
$564$		0			0
$565$			269.930i			0.477752i
$566$		0			0
$567$	60.2763	−	18.3240i	0.106307	−	0.0323175i
$568$		0			0
$569$	−213.993			−0.376086			−0.188043	−	0.982161i	$-0.560214\pi$
$569$	−213.993			−0.376086			−0.188043	+	0.982161i	$0.560214\pi$
$570$		0			0
$571$	204.492			0.358129			0.179065	−	0.983837i	$-0.442693\pi$
$571$	204.492			0.358129			0.179065	+	0.983837i	$0.442693\pi$
$572$		0			0
$573$			398.031i			0.694643i
$574$		0			0
$575$	124.355			0.216270
$576$		0			0
$577$			957.823i			1.66001i	0.557759	+	0.830003i	$0.311661\pi$
$577$			957.823i			1.66001i	−0.557759	+	0.830003i	$0.688339\pi$
$578$		0			0
$579$		−	193.176i		−	0.333638i
$580$		0			0
$581$	210.469	+	692.331i	0.362253	+	1.19162i
$582$		0			0
$583$	116.719			0.200205
$584$		0			0
$585$	120.865			0.206606
$586$		0			0
$587$			981.279i			1.67168i	0.548969	+	0.835842i	$0.315020\pi$
$587$			981.279i			1.67168i	−0.548969	+	0.835842i	$0.684980\pi$
$588$		0			0
$589$	1269.14			2.15473
$590$		0			0
$591$			299.410i			0.506617i
$592$		0			0
$593$			708.384i			1.19458i	0.802026	+	0.597288i	$0.203755\pi$
$593$			708.384i			1.19458i	−0.802026	+	0.597288i	$0.796245\pi$
$594$		0			0
$595$	16.0673	−	4.88446i	0.0270038	−	0.00820918i
$596$		0			0
$597$	−204.286			−0.342187
$598$		0			0
$599$	928.994			1.55091			0.775454	−	0.631404i	$-0.217521\pi$
$599$	928.994			1.55091			0.775454	+	0.631404i	$0.217521\pi$
$600$		0			0
$601$			466.882i			0.776842i	0.921482	+	0.388421i	$0.126979\pi$
$601$			466.882i			0.776842i	−0.921482	+	0.388421i	$0.873021\pi$
$602$		0			0
$603$	287.195			0.476277
$604$		0			0
$605$			261.339i			0.431966i
$606$		0			0
$607$			988.757i			1.62892i	0.580216	+	0.814462i	$0.302968\pi$
$607$			988.757i			1.62892i	−0.580216	+	0.814462i	$0.697032\pi$
$608$		0			0
$609$	135.522	+	445.795i	0.222532	+	0.732012i
$610$		0			0
$611$	−1003.53			−1.64244
$612$		0			0
$613$	469.962			0.766660			0.383330	−	0.923612i	$-0.374777\pi$
$613$	469.962			0.766660			0.383330	+	0.923612i	$0.374777\pi$
$614$		0			0
$615$		−	193.342i		−	0.314377i
$616$		0			0
$617$	1081.72			1.75320			0.876598	−	0.481223i	$-0.159807\pi$
$617$	1081.72			1.75320			0.876598	+	0.481223i	$0.159807\pi$
$618$		0			0
$619$		−	553.956i		−	0.894921i	−0.894304	−	0.447461i	$-0.852328\pi$
$619$		−	553.956i		−	0.894921i	0.894304	−	0.447461i	$-0.147672\pi$
$620$		0			0
$621$		−	129.234i		−	0.208105i
$622$		0			0
$623$	59.7829	+	196.654i	0.0959598	+	0.315657i
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$	−101.266			−0.161509
$628$		0			0
$629$			40.0006i			0.0635940i
$630$		0			0
$631$	77.8822			0.123427			0.0617133	−	0.998094i	$-0.480344\pi$
$631$	77.8822			0.123427			0.0617133	+	0.998094i	$0.480344\pi$
$632$		0			0
$633$		−	678.861i		−	1.07245i
$634$		0			0
$635$		−	186.733i		−	0.294068i
$636$		0			0
$637$	−491.366	−	733.479i	−0.771375	−	1.15146i
$638$		0			0
$639$	−77.4117			−0.121145
$640$		0			0
$641$	−360.619			−0.562589			−0.281294	−	0.959622i	$-0.590764\pi$
$641$	−360.619			−0.562589			−0.281294	+	0.959622i	$0.590764\pi$
$642$		0			0
$643$			793.158i			1.23353i	0.787148	+	0.616764i	$0.211557\pi$
$643$			793.158i			1.23353i	−0.787148	+	0.616764i	$0.788443\pi$
$644$		0			0
$645$	−37.1369			−0.0575766
$646$		0			0
$647$		−	405.244i		−	0.626344i	−0.949696	−	0.313172i	$-0.898608\pi$
$647$		−	405.244i		−	0.626344i	0.949696	−	0.313172i	$-0.101392\pi$
$648$		0			0
$649$		−	206.559i		−	0.318273i
$650$		0			0
$651$	511.450	−	155.481i	0.785638	−	0.238834i
$652$		0			0
$653$	493.508			0.755755			0.377877	−	0.925856i	$-0.376654\pi$
$653$	493.508			0.755755			0.377877	+	0.925856i	$0.376654\pi$
$654$		0			0
$655$	443.295			0.676787
$656$		0			0
$657$			287.041i			0.436896i
$658$		0			0
$659$	1120.88			1.70088			0.850441	−	0.526070i	$-0.176335\pi$
$659$	1120.88			1.70088			0.850441	+	0.526070i	$0.176335\pi$
$660$		0			0
$661$		−	1134.48i		−	1.71631i	−0.513393	−	0.858154i	$-0.671612\pi$
$661$		−	1134.48i		−	1.71631i	0.513393	−	0.858154i	$-0.328388\pi$
$662$		0			0
$663$		−	33.4817i		−	0.0505002i
$664$		0			0
$665$	−431.081	+	131.049i	−0.648242	+	0.197066i
$666$		0			0
$667$	955.794			1.43297
$668$		0			0
$669$	166.712			0.249196
$670$		0			0
$671$		−	64.9002i		−	0.0967216i
$672$		0			0
$673$	−763.267			−1.13413			−0.567063	−	0.823674i	$-0.691920\pi$
$673$	−763.267			−1.13413			−0.567063	+	0.823674i	$0.691920\pi$
$674$		0			0
$675$		−	25.9808i		−	0.0384900i
$676$		0			0
$677$			456.611i			0.674463i	0.941422	+	0.337232i	$0.109491\pi$
$677$			456.611i			0.674463i	−0.941422	+	0.337232i	$0.890509\pi$
$678$		0			0
$679$	−137.730	−	453.058i	−0.202842	−	0.667244i
$680$		0			0
$681$	158.822			0.233219
$682$		0			0
$683$	219.276			0.321049			0.160524	−	0.987032i	$-0.448681\pi$
$683$	219.276			0.321049			0.160524	+	0.987032i	$0.448681\pi$
$684$		0			0
$685$			54.0313i			0.0788777i
$686$		0			0
$687$	−561.135			−0.816791
$688$		0			0
$689$		−	1035.38i		−	1.50273i
$690$		0			0
$691$			396.801i			0.574242i	0.957894	+	0.287121i	$0.0926982\pi$
$691$			396.801i			0.574242i	−0.957894	+	0.287121i	$0.907302\pi$
$692$		0			0
$693$	−40.8094	+	12.4061i	−0.0588880	+	0.0179020i
$694$		0			0
$695$	−137.825			−0.198309
$696$		0			0
$697$	−53.5591			−0.0768424
$698$		0			0
$699$		−	316.823i		−	0.453252i
$700$		0			0
$701$	−493.156			−0.703503			−0.351752	−	0.936093i	$-0.614414\pi$
$701$	−493.156			−0.703503			−0.351752	+	0.936093i	$0.614414\pi$
$702$		0			0
$703$		−	1073.21i		−	1.52661i
$704$		0			0
$705$			215.717i			0.305981i
$706$		0			0
$707$	149.300	+	491.118i	0.211174	+	0.694651i
$708$		0			0
$709$	859.326			1.21202			0.606012	−	0.795455i	$-0.292768\pi$
$709$	859.326			1.21202			0.606012	+	0.795455i	$0.292768\pi$
$710$		0			0
$711$	84.3637			0.118655
$712$		0			0
$713$		−	1096.56i		−	1.53795i
$714$		0			0
$715$	−81.8300			−0.114448
$716$		0			0
$717$		−	74.3466i		−	0.103691i
$718$		0			0
$719$		−	385.539i		−	0.536216i	−0.963389	−	0.268108i	$-0.913602\pi$
$719$		−	385.539i		−	0.536216i	0.963389	−	0.268108i	$-0.0863983\pi$
$720$		0			0
$721$	116.751	+	384.048i	0.161929	+	0.532660i
$722$		0			0
$723$	173.022			0.239311
$724$		0			0
$725$	192.150			0.265035
$726$		0			0
$727$		−	114.722i		−	0.157802i	−0.996882	−	0.0789012i	$-0.974859\pi$
$727$		−	114.722i		−	0.157802i	0.996882	−	0.0789012i	$-0.0251412\pi$
$728$		0			0
$729$	−27.0000			−0.0370370
$730$		0			0
$731$			10.2876i			0.0140733i
$732$		0			0
$733$			140.433i			0.191586i	0.995401	+	0.0957932i	$0.0305388\pi$
$733$			140.433i			0.191586i	−0.995401	+	0.0957932i	$0.969461\pi$
$734$		0			0
$735$	−157.667	+	105.623i	−0.214513	+	0.143705i
$736$		0			0
$737$	−194.442			−0.263830
$738$		0			0
$739$	50.8609			0.0688239			0.0344120	−	0.999408i	$-0.489044\pi$
$739$	50.8609			0.0688239			0.0344120	+	0.999408i	$0.489044\pi$
$740$		0			0
$741$			898.304i			1.21229i
$742$		0			0
$743$	−930.694			−1.25262			−0.626309	−	0.779575i	$-0.715435\pi$
$743$	−930.694			−1.25262			−0.626309	+	0.779575i	$0.715435\pi$
$744$		0			0
$745$			64.4593i			0.0865225i
$746$		0			0
$747$		−	310.121i		−	0.415155i
$748$		0			0
$749$	266.822	−	81.1142i	0.356238	−	0.108297i
$750$		0			0
$751$	446.702			0.594809			0.297404	−	0.954752i	$-0.403879\pi$
$751$	446.702			0.594809			0.297404	+	0.954752i	$0.403879\pi$
$752$		0			0
$753$	−701.385			−0.931455
$754$		0			0
$755$			555.241i			0.735419i
$756$		0			0
$757$	27.2042			0.0359369			0.0179684	−	0.999839i	$-0.494280\pi$
$757$	27.2042			0.0359369			0.0179684	+	0.999839i	$0.494280\pi$
$758$		0			0
$759$			87.4961i			0.115278i
$760$		0			0
$761$		−	333.536i		−	0.438287i	−0.975693	−	0.219143i	$-0.929674\pi$
$761$		−	333.536i		−	0.438287i	0.975693	−	0.219143i	$-0.0703262\pi$
$762$		0			0
$763$	−1239.36	+	376.767i	−1.62433	+	0.493796i
$764$		0			0
$765$	−7.19714			−0.00940802
$766$		0			0
$767$	−1832.32			−2.38895
$768$		0			0
$769$			56.3903i			0.0733294i	0.999328	+	0.0366647i	$0.0116734\pi$
$769$			56.3903i			0.0733294i	−0.999328	+	0.0366647i	$0.988327\pi$
$770$		0			0
$771$	−177.877			−0.230709
$772$		0			0
$773$			296.398i			0.383438i	0.981450	+	0.191719i	$0.0614063\pi$
$773$			296.398i			0.383438i	−0.981450	+	0.191719i	$0.938594\pi$
$774$		0			0
$775$		−	220.449i		−	0.284451i
$776$		0			0
$777$	−131.478	−	432.492i	−0.169212	−	0.556618i
$778$		0			0
$779$	1436.98			1.84464
$780$		0			0
$781$	52.4107			0.0671072
$782$		0			0
$783$		−	199.688i		−	0.255030i
$784$		0			0
$785$	92.5392			0.117884
$786$		0			0
$787$			226.427i			0.287709i	0.989599	+	0.143854i	$0.0459497\pi$
$787$			226.427i			0.287709i	−0.989599	+	0.143854i	$0.954050\pi$
$788$		0			0
$789$			487.537i			0.617917i
$790$		0			0
$791$	−808.481	+	245.779i	−1.02210	+	0.310719i
$792$		0			0
$793$	−575.711			−0.725991
$794$		0			0
$795$	−222.563			−0.279954
$796$		0			0
$797$			305.282i			0.383038i	0.981489	+	0.191519i	$0.0613414\pi$
$797$			305.282i			0.383038i	−0.981489	+	0.191519i	$0.938659\pi$
$798$		0			0
$799$	59.7573			0.0747901
$800$		0			0
$801$		−	88.0887i		−	0.109973i
$802$		0			0
$803$		−	194.338i		−	0.242015i
$804$		0			0
$805$	113.229	+	372.462i	0.140657	+	0.462686i
$806$		0			0
$807$	305.555			0.378631
$808$		0			0
$809$	592.651			0.732573			0.366286	−	0.930502i	$-0.380629\pi$
$809$	592.651			0.732573			0.366286	+	0.930502i	$0.380629\pi$
$810$		0			0
$811$			731.348i			0.901785i	0.892578	+	0.450893i	$0.148894\pi$
$811$			731.348i			0.901785i	−0.892578	+	0.450893i	$0.851106\pi$
$812$		0			0
$813$	−530.078			−0.652003
$814$		0			0
$815$		−	115.851i		−	0.142149i
$816$		0			0
$817$		−	276.013i		−	0.337838i
$818$		0			0
$819$	110.051	+	362.008i	0.134372	+	0.442012i
$820$		0			0
$821$	268.560			0.327114			0.163557	−	0.986534i	$-0.447703\pi$
$821$	268.560			0.327114			0.163557	+	0.986534i	$0.447703\pi$
$822$		0			0
$823$	−1213.35			−1.47430			−0.737151	−	0.675728i	$-0.763830\pi$
$823$	−1213.35			−1.47430			−0.737151	+	0.675728i	$0.763830\pi$
$824$		0			0
$825$			17.5900i			0.0213212i
$826$		0			0
$827$	1238.49			1.49757			0.748785	−	0.662813i	$-0.230637\pi$
$827$	1238.49			1.49757			0.748785	+	0.662813i	$0.230637\pi$
$828$		0			0
$829$			873.056i			1.05314i	0.850131	+	0.526572i	$0.176523\pi$
$829$			873.056i			1.05314i	−0.850131	+	0.526572i	$0.823477\pi$
$830$		0			0
$831$		−	75.8626i		−	0.0912908i
$832$		0			0
$833$	29.2594	+	43.6765i	0.0351253	+	0.0524328i
$834$		0			0
$835$	−93.8503			−0.112396
$836$		0			0
$837$	−229.098			−0.273713
$838$		0			0
$839$		−	100.572i		−	0.119871i	−0.998202	−	0.0599355i	$-0.980910\pi$
$839$		−	100.572i		−	0.119871i	0.998202	−	0.0599355i	$-0.0190895\pi$
$840$		0			0
$841$	635.869			0.756086
$842$		0			0
$843$		−	865.163i		−	1.02629i
$844$		0			0
$845$			347.995i			0.411828i
$846$		0			0
$847$	−782.752	+	237.957i	−0.924146	+	0.280941i
$848$		0			0
$849$	−212.474			−0.250264
$850$		0			0
$851$	−927.271			−1.08963
$852$		0			0
$853$		−	1162.30i		−	1.36260i	−0.732003	−	0.681302i	$-0.761414\pi$
$853$		−	1162.30i		−	1.36260i	0.732003	−	0.681302i	$-0.238586\pi$
$854$		0			0
$855$	193.097			0.225845
$856$		0			0
$857$			87.8213i			0.102475i	0.998686	+	0.0512376i	$0.0163166\pi$
$857$			87.8213i			0.102475i	−0.998686	+	0.0512376i	$0.983683\pi$
$858$		0			0
$859$		−	200.580i		−	0.233504i	−0.993161	−	0.116752i	$-0.962752\pi$
$859$		−	200.580i		−	0.233504i	0.993161	−	0.116752i	$-0.0372482\pi$
$860$		0			0
$861$	579.088	−	176.043i	0.672576	−	0.204464i
$862$		0			0
$863$	617.914			0.716007			0.358004	−	0.933720i	$-0.383458\pi$
$863$	617.914			0.716007			0.358004	+	0.933720i	$0.383458\pi$
$864$		0			0
$865$	−219.179			−0.253386
$866$		0			0
$867$		−	498.569i		−	0.575051i
$868$		0			0
$869$	−57.1175			−0.0657278
$870$		0			0
$871$			1724.84i			1.98030i
$872$		0			0
$873$			202.942i			0.232465i
$874$		0			0
$875$	22.7632	+	74.8788i	0.0260151	+	0.0855758i
$876$		0			0
$877$	−638.649			−0.728220			−0.364110	−	0.931356i	$-0.618627\pi$
$877$	−638.649			−0.728220			−0.364110	+	0.931356i	$0.618627\pi$
$878$		0			0
$879$	−214.551			−0.244086
$880$		0			0
$881$		−	148.009i		−	0.168001i	−0.996466	−	0.0840003i	$-0.973230\pi$
$881$		−	148.009i		−	0.168001i	0.996466	−	0.0840003i	$-0.0267697\pi$
$882$		0			0
$883$	784.505			0.888454			0.444227	−	0.895914i	$-0.353478\pi$
$883$	784.505			0.888454			0.444227	+	0.895914i	$0.353478\pi$
$884$		0			0
$885$			393.872i			0.445053i
$886$		0			0
$887$			1466.40i			1.65321i	0.562783	+	0.826605i	$0.309731\pi$
$887$			1466.40i			1.65321i	−0.562783	+	0.826605i	$0.690269\pi$
$888$		0			0
$889$	559.294	−	170.026i	0.629127	−	0.191255i
$890$		0			0
$891$	18.2800			0.0205163
$892$		0			0
$893$	−1603.27			−1.79538
$894$		0			0
$895$			153.356i			0.171348i
$896$		0			0
$897$	776.152			0.865276
$898$		0			0
$899$		−	1694.38i		−	1.88473i
$900$		0			0
$901$			61.6540i			0.0684284i
$902$		0			0
$903$	−33.8142	−	111.231i	−0.0374465	−	0.123179i
$904$		0			0
$905$	−235.065			−0.259741
$906$		0			0
$907$	−488.454			−0.538538			−0.269269	−	0.963065i	$-0.586782\pi$
$907$	−488.454			−0.538538			−0.269269	+	0.963065i	$0.586782\pi$
$908$		0			0
$909$		−	219.990i		−	0.242013i
$910$		0			0
$911$	−1329.44			−1.45932			−0.729658	−	0.683812i	$-0.760321\pi$
$911$	−1329.44			−1.45932			−0.729658	+	0.683812i	$0.760321\pi$
$912$		0			0
$913$			209.964i			0.229971i
$914$		0			0
$915$			123.753i			0.135250i
$916$		0			0
$917$	403.633	+	1327.74i	0.440167	+	1.44791i
$918$		0			0
$919$	−1369.52			−1.49023			−0.745113	−	0.666938i	$-0.767605\pi$
$919$	−1369.52			−1.49023			−0.745113	+	0.666938i	$0.767605\pi$
$920$		0			0
$921$	−390.710			−0.424224
$922$		0			0
$923$		−	464.920i		−	0.503705i
$924$		0			0
$925$	−186.416			−0.201531
$926$		0			0
$927$		−	172.029i		−	0.185576i
$928$		0			0
$929$		−	909.289i		−	0.978783i	−0.872064	−	0.489391i	$-0.837219\pi$
$929$		−	909.289i		−	0.978783i	0.872064	−	0.489391i	$-0.162781\pi$
$930$		0			0
$931$	−785.022	−	1171.83i	−0.843203	−	1.25868i
$932$		0			0
$933$	−91.7378			−0.0983256
$934$		0			0
$935$	4.87274			0.00521149
$936$		0			0
$937$		−	184.883i		−	0.197313i	−0.995122	−	0.0986566i	$-0.968545\pi$
$937$		−	184.883i		−	0.197313i	0.995122	−	0.0986566i	$-0.0314545\pi$
$938$		0			0
$939$	648.762			0.690907
$940$		0			0
$941$		−	884.454i		−	0.939909i	−0.882691	−	0.469955i	$-0.844270\pi$
$941$		−	884.454i		−	0.939909i	0.882691	−	0.469955i	$-0.155730\pi$
$942$		0			0
$943$		−	1241.58i		−	1.31662i
$944$		0			0
$945$	77.8163	−	23.6562i	0.0823453	−	0.0250330i
$946$		0			0
$947$	651.331			0.687784			0.343892	−	0.939009i	$-0.388255\pi$
$947$	651.331			0.687784			0.343892	+	0.939009i	$0.388255\pi$
$948$		0			0
$949$	−1723.91			−1.81656
$950$		0			0
$951$		−	20.8798i		−	0.0219557i
$952$		0			0
$953$	1622.61			1.70263			0.851316	−	0.524654i	$-0.175805\pi$
$953$	1622.61			1.70263			0.851316	+	0.524654i	$0.175805\pi$
$954$		0			0
$955$			513.855i			0.538069i
$956$		0			0
$957$			135.197i			0.141271i
$958$		0			0
$959$	−161.832	+	49.1970i	−0.168751	+	0.0513003i
$960$		0			0
$961$	−982.915			−1.02280
$962$		0			0
$963$	−119.520			−0.124112
$964$		0			0
$965$		−	249.389i		−	0.258435i
$966$		0			0
$967$	1491.24			1.54213			0.771067	−	0.636754i	$-0.219724\pi$
$967$	1491.24			1.54213			0.771067	+	0.636754i	$0.219724\pi$
$968$		0			0
$969$		−	53.4914i		−	0.0552027i
$970$		0			0
$971$		−	1739.36i		−	1.79130i	−0.444756	−	0.895652i	$-0.646709\pi$
$971$		−	1739.36i		−	1.79130i	0.444756	−	0.895652i	$-0.353291\pi$
$972$		0			0
$973$	−125.493	−	412.806i	−0.128975	−	0.424261i
$974$		0			0
$975$	156.036			0.160036
$976$		0			0
$977$	−62.8973			−0.0643780			−0.0321890	−	0.999482i	$-0.510248\pi$
$977$	−62.8973			−0.0643780			−0.0321890	+	0.999482i	$0.510248\pi$
$978$		0			0
$979$			59.6395i			0.0609188i
$980$		0			0
$981$	555.156			0.565909
$982$		0			0
$983$			1214.68i			1.23569i	0.786299	+	0.617846i	$0.211994\pi$
$983$			1214.68i			1.23569i	−0.786299	+	0.617846i	$0.788006\pi$
$984$		0			0
$985$			386.537i			0.392424i
$986$		0			0
$987$	−646.104	+	196.416i	−0.654614	+	0.199003i
$988$		0			0
$989$	−238.481			−0.241133
$990$		0			0
$991$	−114.216			−0.115253			−0.0576266	−	0.998338i	$-0.518353\pi$
$991$	−114.216			−0.115253			−0.0576266	+	0.998338i	$0.518353\pi$
$992$		0			0
$993$		−	651.570i		−	0.656163i
$994$		0			0
$995$	−263.732			−0.265057
$996$		0			0
$997$			178.000i			0.178536i	0.996008	+	0.0892678i	$0.0284527\pi$
$997$			178.000i			0.178536i	−0.996008	+	0.0892678i	$0.971547\pi$
$998$		0			0
$999$			193.729i			0.193923i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.3.s.c.1441.3		12
4.3	odd	2		105.3.h.a.76.12	yes	12
7.6	odd	2	inner	1680.3.s.c.1441.12		12
12.11	even	2		315.3.h.d.181.2		12
20.3	even	4		525.3.e.c.349.9		24
20.7	even	4		525.3.e.c.349.24		24
20.19	odd	2		525.3.h.d.76.1		12
28.27	even	2		105.3.h.a.76.11	✓	12
84.83	odd	2		315.3.h.d.181.1		12
140.27	odd	4		525.3.e.c.349.10		24
140.83	odd	4		525.3.e.c.349.23		24
140.139	even	2		525.3.h.d.76.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.3.h.a.76.11	✓	12	28.27	even	2
105.3.h.a.76.12	yes	12	4.3	odd	2
315.3.h.d.181.1		12	84.83	odd	2
315.3.h.d.181.2		12	12.11	even	2
525.3.e.c.349.9		24	20.3	even	4
525.3.e.c.349.10		24	140.27	odd	4
525.3.e.c.349.23		24	140.83	odd	4
525.3.e.c.349.24		24	20.7	even	4
525.3.h.d.76.1		12	20.19	odd	2
525.3.h.d.76.2		12	140.139	even	2
1680.3.s.c.1441.3		12	1.1	even	1	trivial
1680.3.s.c.1441.12		12	7.6	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:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1680.s (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.73205i q^{3} -2.23607i q^{5} +(6.69736 - 2.03600i) q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q-1.73205i q^{3} -2.23607i q^{5} +(6.69736 - 2.03600i) q^{7} -3.00000 q^{9} +2.03112 q^{11} -18.0174i q^{13} -3.87298 q^{15} +1.07289i q^{17} -28.7852i q^{19} +(-3.52646 - 11.6002i) q^{21} -24.8710 q^{23} -5.00000 q^{25} +5.19615i q^{27} -38.4300 q^{29} +44.0899i q^{31} -3.51800i q^{33} +(-4.55264 - 14.9758i) q^{35} +37.2832 q^{37} -31.2071 q^{39} +49.9206i q^{41} +9.58871 q^{43} +6.70820i q^{45} -55.6978i q^{47} +(40.7094 - 27.2717i) q^{49} +1.85829 q^{51} +57.4656 q^{53} -4.54171i q^{55} -49.8575 q^{57} -101.697i q^{59} -31.9530i q^{61} +(-20.0921 + 6.10801i) q^{63} -40.2882 q^{65} -95.7318 q^{67} +43.0778i q^{69} +25.8039 q^{71} -95.6803i q^{73} +8.66025i q^{75} +(13.6031 - 4.13536i) q^{77} -28.1212 q^{79} +9.00000 q^{81} +103.374i q^{83} +2.39905 q^{85} +66.5628i q^{87} +29.3629i q^{89} +(-36.6835 - 120.669i) q^{91} +76.3659 q^{93} -64.3658 q^{95} -67.6473i q^{97} -6.09335 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 8 q^{7} - 36 q^{9}+O(q^{10}) \) 12 * q + 8 * q^7 - 36 * q^9	\( 12 q + 8 q^{7} - 36 q^{9} + 16 q^{11} + 36 q^{21} + 64 q^{23} - 60 q^{25} + 104 q^{29} - 60 q^{35} + 32 q^{37} + 24 q^{39} - 152 q^{43} + 60 q^{49} - 24 q^{51} + 176 q^{53} - 240 q^{57} - 24 q^{63} - 240 q^{65} - 168 q^{67} - 32 q^{71} + 8 q^{77} - 120 q^{79} + 108 q^{81} + 120 q^{85} - 24 q^{91} + 48 q^{93} - 48 q^{99}+O(q^{100}) \) 12 * q + 8 * q^7 - 36 * q^9 + 16 * q^11 + 36 * q^21 + 64 * q^23 - 60 * q^25 + 104 * q^29 - 60 * q^35 + 32 * q^37 + 24 * q^39 - 152 * q^43 + 60 * q^49 - 24 * q^51 + 176 * q^53 - 240 * q^57 - 24 * q^63 - 240 * q^65 - 168 * q^67 - 32 * q^71 + 8 * q^77 - 120 * q^79 + 108 * q^81 + 120 * q^85 - 24 * q^91 + 48 * q^93 - 48 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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