LMFDB - Embedded newform 1764.3.z.l.901.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,3,Mod(325,1764)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1764, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([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("1764.325");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1764.z (of order $6$, degree $2$, 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:	$48.0655186332$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.339738624.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 $ x^8 - 4x^6 + 14x^4 - 8*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 7^{2} $
Twist minimal:	no (minimal twist has level 588)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			901.1
Root			$-0.662827 + 0.382683i$ of defining polynomial
Character	$\chi$	$=$	1764.901
Dual form			1764.3.z.l.325.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.65891 - 2.68982i) q^{5} +O(q^{10})$	$q+(-4.65891 - 2.68982i) q^{5} +(-4.29579 - 7.44053i) q^{11} +21.0158i q^{13} +(-4.75120 + 2.74311i) q^{17} +(-6.27088 - 3.62049i) q^{19} +(14.0278 - 24.2969i) q^{23} +(1.97027 + 3.41261i) q^{25} -40.3447 q^{29} +(35.0828 - 20.2550i) q^{31} +(-33.3185 + 57.7093i) q^{37} +33.6357i q^{41} +0.932907 q^{43} +(74.1789 + 42.8272i) q^{47} +(-22.2977 - 38.6207i) q^{53} +46.2197i q^{55} +(-55.1615 + 31.8475i) q^{59} +(27.7201 + 16.0042i) q^{61} +(56.5288 - 97.9108i) q^{65} +(23.8671 + 41.3390i) q^{67} -14.9676 q^{71} +(121.501 - 70.1488i) q^{73} +(61.1537 - 105.921i) q^{79} +33.1852i q^{83} +29.5139 q^{85} +(-31.2848 - 18.0623i) q^{89} +(19.4769 + 33.7351i) q^{95} -16.2175i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 48 q^{17} - 96 q^{19} - 8 q^{23} - 36 q^{25} - 80 q^{29} + 48 q^{31} - 64 q^{37} - 112 q^{43} + 264 q^{47} - 72 q^{53} - 168 q^{59} - 144 q^{61} + 120 q^{65} + 32 q^{67} - 224 q^{71} + 336 q^{73} + 216 q^{79} - 96 q^{85} - 96 q^{89} - 136 q^{95}+O(q^{100}) $ 8 * q - 48 * q^17 - 96 * q^19 - 8 * q^23 - 36 * q^25 - 80 * q^29 + 48 * q^31 - 64 * q^37 - 112 * q^43 + 264 * q^47 - 72 * q^53 - 168 * q^59 - 144 * q^61 + 120 * q^65 + 32 * q^67 - 224 * q^71 + 336 * q^73 + 216 * q^79 - 96 * q^85 - 96 * q^89 - 136 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$.

$n$	$785$	$883$	$1081$
$\chi(n)$	$1$	$1$	$e\left(\frac{5}{6}\right)$

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
$5$	−4.65891	−	2.68982i	−0.931781	−	0.537964i	−0.0444067	−	0.999014i	$-0.514140\pi$
$5$	−4.65891	−	2.68982i	−0.931781	−	0.537964i	−0.887374	+	0.461049i	$0.847473\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−4.29579	−	7.44053i	−0.390527	−	0.676412i	0.601992	−	0.798502i	$-0.294374\pi$
$11$	−4.29579	−	7.44053i	−0.390527	−	0.676412i	−0.992519	+	0.122090i	$0.961040\pi$
$12$		0			0
$13$			21.0158i			1.61660i	0.588769	+	0.808301i	$0.299613\pi$
$13$			21.0158i			1.61660i	−0.588769	+	0.808301i	$0.700387\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−4.75120	+	2.74311i	−0.279483	+	0.161359i	−0.633189	−	0.773997i	$-0.718254\pi$
$17$	−4.75120	+	2.74311i	−0.279483	+	0.161359i	0.353707	+	0.935356i	$0.384921\pi$
$18$		0			0
$19$	−6.27088	−	3.62049i	−0.330046	−	0.190552i	0.325815	−	0.945433i	$-0.394361\pi$
$19$	−6.27088	−	3.62049i	−0.330046	−	0.190552i	−0.655862	+	0.754881i	$0.727695\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	14.0278	−	24.2969i	0.609905	−	1.05639i	−0.381350	−	0.924431i	$-0.624541\pi$
$23$	14.0278	−	24.2969i	0.609905	−	1.05639i	0.991256	−	0.131956i	$-0.0421258\pi$
$24$		0			0
$25$	1.97027	+	3.41261i	0.0788107	+	0.136504i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−40.3447			−1.39120			−0.695599	−	0.718431i	$-0.744861\pi$
$29$	−40.3447			−1.39120			−0.695599	+	0.718431i	$0.744861\pi$
$30$		0			0
$31$	35.0828	−	20.2550i	1.13170	−	0.653388i	0.187340	−	0.982295i	$-0.440014\pi$
$31$	35.0828	−	20.2550i	1.13170	−	0.653388i	0.944362	+	0.328907i	$0.106680\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−33.3185	+	57.7093i	−0.900500	+	1.55971i	−0.0736541	+	0.997284i	$0.523466\pi$
$37$	−33.3185	+	57.7093i	−0.900500	+	1.55971i	−0.826846	+	0.562428i	$0.809867\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			33.6357i			0.820383i	0.911999	+	0.410191i	$0.134538\pi$
$41$			33.6357i			0.820383i	−0.911999	+	0.410191i	$0.865462\pi$
$42$		0			0
$43$	0.932907			0.0216955			0.0108478	−	0.999941i	$-0.496547\pi$
$43$	0.932907			0.0216955			0.0108478	+	0.999941i	$0.496547\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	74.1789	+	42.8272i	1.57827	+	0.911217i	0.995100	+	0.0988687i	$0.0315224\pi$
$47$	74.1789	+	42.8272i	1.57827	+	0.911217i	0.583173	+	0.812348i	$0.301811\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−22.2977	−	38.6207i	−0.420711	−	0.728693i	0.575298	−	0.817944i	$-0.304886\pi$
$53$	−22.2977	−	38.6207i	−0.420711	−	0.728693i	−0.996009	+	0.0892508i	$0.971553\pi$
$54$		0			0
$55$			46.2197i			0.840358i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−55.1615	+	31.8475i	−0.934940	+	0.539788i	−0.888371	−	0.459127i	$-0.848162\pi$
$59$	−55.1615	+	31.8475i	−0.934940	+	0.539788i	−0.0465695	+	0.998915i	$0.514829\pi$
$60$		0			0
$61$	27.7201	+	16.0042i	0.454427	+	0.262364i	0.709698	−	0.704506i	$-0.248831\pi$
$61$	27.7201	+	16.0042i	0.454427	+	0.262364i	−0.255271	+	0.966870i	$0.582165\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	56.5288	−	97.9108i	0.869674	−	1.50632i
$66$		0			0
$67$	23.8671	+	41.3390i	0.356225	+	0.617000i	0.987327	−	0.158700i	$-0.0507304\pi$
$67$	23.8671	+	41.3390i	0.356225	+	0.617000i	−0.631102	+	0.775700i	$0.717397\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−14.9676			−0.210811			−0.105405	−	0.994429i	$-0.533614\pi$
$71$	−14.9676			−0.210811			−0.105405	+	0.994429i	$0.533614\pi$
$72$		0			0
$73$	121.501	−	70.1488i	1.66440	−	0.960942i	0.693825	−	0.720144i	$-0.255924\pi$
$73$	121.501	−	70.1488i	1.66440	−	0.960942i	0.970575	−	0.240798i	$-0.0774091\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	61.1537	−	105.921i	0.774098	−	1.34078i	−0.161202	−	0.986921i	$-0.551537\pi$
$79$	61.1537	−	105.921i	0.774098	−	1.34078i	0.935300	−	0.353856i	$-0.115130\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			33.1852i			0.399822i	0.979814	+	0.199911i	$0.0640652\pi$
$83$			33.1852i			0.399822i	−0.979814	+	0.199911i	$0.935935\pi$
$84$		0			0
$85$	29.5139			0.347222
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−31.2848	−	18.0623i	−0.351515	−	0.202947i	0.313838	−	0.949477i	$-0.398385\pi$
$89$	−31.2848	−	18.0623i	−0.351515	−	0.202947i	−0.665352	+	0.746530i	$0.731719\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	19.4769	+	33.7351i	0.205021	+	0.355106i
$96$		0			0
$97$		−	16.2175i		−	0.167191i	−0.996500	−	0.0835956i	$-0.973360\pi$
$97$		−	16.2175i		−	0.167191i	0.996500	−	0.0835956i	$-0.0266404\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	103.459	−	59.7322i	1.02435	−	0.591407i	0.108987	−	0.994043i	$-0.465239\pi$
$101$	103.459	−	59.7322i	1.02435	−	0.591407i	0.915360	+	0.402636i	$0.131906\pi$
$102$		0			0
$103$	−7.73523	−	4.46594i	−0.0750994	−	0.0433586i	0.461980	−	0.886890i	$-0.347139\pi$
$103$	−7.73523	−	4.46594i	−0.0750994	−	0.0433586i	−0.537079	+	0.843532i	$0.680472\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	86.9279	−	150.564i	0.812410	−	1.40714i	−0.0987623	−	0.995111i	$-0.531488\pi$
$107$	86.9279	−	150.564i	0.812410	−	1.40714i	0.911173	−	0.412025i	$-0.135178\pi$
$108$		0			0
$109$	80.1573	+	138.837i	0.735388	+	1.27373i	0.954553	+	0.298042i	$0.0963334\pi$
$109$	80.1573	+	138.837i	0.735388	+	1.27373i	−0.219165	+	0.975688i	$0.570333\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	81.4420			0.720725			0.360363	−	0.932812i	$-0.382653\pi$
$113$	81.4420			0.720725			0.360363	+	0.932812i	$0.382653\pi$
$114$		0			0
$115$	−130.709	+	75.4646i	−1.13660	+	0.656214i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	23.5923	−	40.8631i	0.194978	−	0.337711i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			113.292i			0.906339i
$126$		0			0
$127$	117.172			0.922613			0.461307	−	0.887241i	$-0.347381\pi$
$127$	117.172			0.922613			0.461307	+	0.887241i	$0.347381\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	211.440	+	122.075i	1.61404	+	0.931869i	0.988420	+	0.151745i	$0.0484893\pi$
$131$	211.440	+	122.075i	1.61404	+	0.931869i	0.625625	+	0.780124i	$0.284844\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	122.706	+	212.533i	0.895663	+	1.55133i	0.832981	+	0.553301i	$0.186632\pi$
$137$	122.706	+	212.533i	0.895663	+	1.55133i	0.0626820	+	0.998034i	$0.480035\pi$
$138$		0			0
$139$			17.1371i			0.123288i	0.998098	+	0.0616441i	$0.0196344\pi$
$139$			17.1371i			0.123288i	−0.998098	+	0.0616441i	$0.980366\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	156.369	−	90.2797i	1.09349	−	0.631326i
$144$		0			0
$145$	187.962	+	108.520i	1.29629	+	0.748414i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	67.1209	−	116.257i	0.450476	−	0.780247i	−0.547940	−	0.836518i	$-0.684588\pi$
$149$	67.1209	−	116.257i	0.450476	−	0.780247i	0.998416	+	0.0562707i	$0.0179210\pi$
$150$		0			0
$151$	−99.5047	−	172.347i	−0.658971	−	1.14137i	−0.980882	−	0.194602i	$-0.937659\pi$
$151$	−99.5047	−	172.347i	−0.658971	−	1.14137i	0.321911	−	0.946770i	$-0.395675\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−217.930			−1.40600
$156$		0			0
$157$	−66.7004	+	38.5095i	−0.424843	+	0.245283i	−0.697147	−	0.716928i	$-0.745548\pi$
$157$	−66.7004	+	38.5095i	−0.424843	+	0.245283i	0.272304	+	0.962211i	$0.412214\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	115.190	−	199.514i	0.706684	−	1.22401i	−0.259396	−	0.965771i	$-0.583523\pi$
$163$	115.190	−	199.514i	0.706684	−	1.22401i	0.966080	−	0.258242i	$-0.0831432\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			229.231i			1.37264i	0.727298	+	0.686321i	$0.240776\pi$
$167$			229.231i			1.37264i	−0.727298	+	0.686321i	$0.759224\pi$
$168$		0			0
$169$	−272.665			−1.61340
$170$		0			0
$171$		0			0
$172$		0			0
$173$	97.4044	+	56.2365i	0.563031	+	0.325066i	0.754361	−	0.656459i	$-0.227947\pi$
$173$	97.4044	+	56.2365i	0.563031	+	0.325066i	−0.191330	+	0.981526i	$0.561280\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	52.7470	+	91.3605i	0.294676	+	0.510394i	0.974909	−	0.222602i	$-0.0714550\pi$
$179$	52.7470	+	91.3605i	0.294676	+	0.510394i	−0.680234	+	0.732995i	$0.738122\pi$
$180$		0			0
$181$		−	15.2683i		−	0.0843553i	−0.999110	−	0.0421776i	$-0.986570\pi$
$181$		−	15.2683i		−	0.0843553i	0.999110	−	0.0421776i	$-0.0134295\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	310.456	−	179.242i	1.67814	−	0.968874i
$186$		0			0
$187$	40.8204	+	23.5677i	0.218291	+	0.126030i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−153.093	+	265.165i	−0.801534	+	1.38830i	0.117073	+	0.993123i	$0.462649\pi$
$191$	−153.093	+	265.165i	−0.801534	+	1.38830i	−0.918606	+	0.395174i	$0.870684\pi$
$192$		0			0
$193$	0.920499	+	1.59435i	0.00476943	+	0.00826089i	0.868400	−	0.495864i	$-0.165149\pi$
$193$	0.920499	+	1.59435i	0.00476943	+	0.00826089i	−0.863631	+	0.504125i	$0.831815\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	255.334			1.29611			0.648056	−	0.761593i	$-0.275582\pi$
$197$	255.334			1.29611			0.648056	+	0.761593i	$0.275582\pi$
$198$		0			0
$199$	31.7384	−	18.3242i	0.159489	−	0.0920812i	−0.418131	−	0.908387i	$-0.637315\pi$
$199$	31.7384	−	18.3242i	0.159489	−	0.0920812i	0.577620	+	0.816305i	$0.303981\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	90.4740	−	156.706i	0.441337	−	0.764417i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			62.2116i			0.297663i
$210$		0			0
$211$	−126.571			−0.599862			−0.299931	−	0.953961i	$-0.596964\pi$
$211$	−126.571			−0.599862			−0.299931	+	0.953961i	$0.596964\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−4.34633	−	2.50935i	−0.0202155	−	0.0116714i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−57.6487	−	99.8505i	−0.260854	−	0.451812i
$222$		0			0
$223$			212.193i			0.951536i	0.879571	+	0.475768i	$0.157830\pi$
$223$			212.193i			0.951536i	−0.879571	+	0.475768i	$0.842170\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−91.3560	+	52.7444i	−0.402449	+	0.232354i	−0.687540	−	0.726146i	$-0.741310\pi$
$227$	−91.3560	+	52.7444i	−0.402449	+	0.232354i	0.285091	+	0.958500i	$0.407976\pi$
$228$		0			0
$229$	6.05426	+	3.49543i	0.0264378	+	0.0152639i	0.513161	−	0.858293i	$-0.328474\pi$
$229$	6.05426	+	3.49543i	0.0264378	+	0.0152639i	−0.486723	+	0.873556i	$0.661808\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	168.023	−	291.024i	0.721128	−	1.24903i	−0.239419	−	0.970916i	$-0.576957\pi$
$233$	168.023	−	291.024i	0.721128	−	1.24903i	0.960548	−	0.278115i	$-0.0897096\pi$
$234$		0			0
$235$	−230.395	−	399.056i	−0.980404	−	1.69811i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	232.382			0.972309			0.486155	−	0.873873i	$-0.338399\pi$
$239$	232.382			0.972309			0.486155	+	0.873873i	$0.338399\pi$
$240$		0			0
$241$	55.1958	−	31.8673i	0.229028	−	0.132229i	−0.381095	−	0.924536i	$-0.624453\pi$
$241$	55.1958	−	31.8673i	0.229028	−	0.132229i	0.610124	+	0.792306i	$0.291120\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	76.0876	−	131.788i	0.308047	−	0.533553i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			415.450i			1.65518i	0.561334	+	0.827589i	$0.310288\pi$
$251$			415.450i			1.65518i	−0.561334	+	0.827589i	$0.689712\pi$
$252$		0			0
$253$	−241.043			−0.952737
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−230.014	−	132.799i	−0.894998	−	0.516727i	−0.0194240	−	0.999811i	$-0.506183\pi$
$257$	−230.014	−	132.799i	−0.894998	−	0.516727i	−0.875574	+	0.483084i	$0.839517\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	78.1866	+	135.423i	0.297288	+	0.514917i	0.975514	−	0.219935i	$-0.0705846\pi$
$263$	78.1866	+	135.423i	0.297288	+	0.514917i	−0.678227	+	0.734853i	$0.737251\pi$
$264$		0			0
$265$			239.907i			0.905310i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	288.266	−	166.431i	1.07162	−	0.618701i	0.142998	−	0.989723i	$-0.454326\pi$
$269$	288.266	−	166.431i	1.07162	−	0.618701i	0.928624	+	0.371022i	$0.120992\pi$
$270$		0			0
$271$	−181.076	−	104.544i	−0.668178	−	0.385773i	0.127208	−	0.991876i	$-0.459398\pi$
$271$	−181.076	−	104.544i	−0.668178	−	0.385773i	−0.795386	+	0.606103i	$0.792732\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	16.9277	−	29.3197i	0.0615554	−	0.106617i
$276$		0			0
$277$	131.616	+	227.965i	0.475146	+	0.822978i	0.999595	−	0.0284646i	$-0.00906180\pi$
$277$	131.616	+	227.965i	0.475146	+	0.822978i	−0.524448	+	0.851442i	$0.675728\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−391.519			−1.39331			−0.696653	−	0.717408i	$-0.745328\pi$
$281$	−391.519			−1.39331			−0.696653	+	0.717408i	$0.745328\pi$
$282$		0			0
$283$	−94.7376	+	54.6968i	−0.334762	+	0.193275i	−0.657953	−	0.753059i	$-0.728578\pi$
$283$	−94.7376	+	54.6968i	−0.334762	+	0.193275i	0.323191	+	0.946334i	$0.395244\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−129.451	+	224.215i	−0.447926	+	0.775831i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			35.3685i			0.120712i	0.998177	+	0.0603558i	$0.0192235\pi$
$293$			35.3685i			0.120712i	−0.998177	+	0.0603558i	$0.980776\pi$
$294$		0			0
$295$	342.656			1.16155
$296$		0			0
$297$		0			0
$298$		0			0
$299$	510.619	+	294.806i	1.70776	+	0.985974i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−86.0968	−	149.124i	−0.282284	−	0.488931i
$306$		0			0
$307$		−	125.621i		−	0.409189i	−0.978847	−	0.204594i	$-0.934412\pi$
$307$		−	125.621i		−	0.409189i	0.978847	−	0.204594i	$-0.0655875\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−273.435	+	157.868i	−0.879211	+	0.507613i	−0.870398	−	0.492348i	$-0.836139\pi$
$311$	−273.435	+	157.868i	−0.879211	+	0.507613i	−0.00881288	+	0.999961i	$0.502805\pi$
$312$		0			0
$313$	−44.6909	−	25.8023i	−0.142782	−	0.0824355i	0.426907	−	0.904296i	$-0.359603\pi$
$313$	−44.6909	−	25.8023i	−0.142782	−	0.0824355i	−0.569689	+	0.821860i	$0.692936\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−2.49696	+	4.32486i	−0.00787683	+	0.0136431i	−0.869937	−	0.493163i	$-0.835841\pi$
$317$	−2.49696	+	4.32486i	−0.00787683	+	0.0136431i	0.862060	+	0.506806i	$0.169174\pi$
$318$		0			0
$319$	173.313	+	300.186i	0.543300	+	0.941023i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	39.7256			0.122990
$324$		0			0
$325$	−71.7187	+	41.4068i	−0.220673	+	0.127406i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−227.390	+	393.851i	−0.686980	+	1.18988i	0.285831	+	0.958280i	$0.407731\pi$
$331$	−227.390	+	393.851i	−0.686980	+	1.18988i	−0.972810	+	0.231603i	$0.925603\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	256.792i		−	0.766545i
$336$		0			0
$337$	−183.824			−0.545471			−0.272736	−	0.962089i	$-0.587928\pi$
$337$	−183.824			−0.545471			−0.272736	+	0.962089i	$0.587928\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−301.417	−	174.023i	−0.883920	−	0.510331i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	72.3882	+	125.380i	0.208611	+	0.361326i	0.951277	−	0.308337i	$-0.0997723\pi$
$347$	72.3882	+	125.380i	0.208611	+	0.361326i	−0.742666	+	0.669662i	$0.766439\pi$
$348$		0			0
$349$		−	187.069i		−	0.536015i	−0.963417	−	0.268007i	$-0.913635\pi$
$349$		−	187.069i		−	0.536015i	0.963417	−	0.268007i	$-0.0863651\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−192.778	+	111.300i	−0.546113	+	0.315299i	−0.747553	−	0.664202i	$-0.768771\pi$
$353$	−192.778	+	111.300i	−0.546113	+	0.315299i	0.201440	+	0.979501i	$0.435438\pi$
$354$		0			0
$355$	69.7324	+	40.2600i	0.196429	+	0.113409i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	31.4928	−	54.5472i	0.0877237	−	0.151942i	−0.818825	−	0.574043i	$-0.805374\pi$
$359$	31.4928	−	54.5472i	0.0877237	−	0.151942i	0.906549	+	0.422101i	$0.138707\pi$
$360$		0			0
$361$	−154.284	−	267.228i	−0.427380	−	0.740243i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−754.750			−2.06781
$366$		0			0
$367$	473.803	−	273.550i	1.29102	−	0.745368i	0.312181	−	0.950023i	$-0.398940\pi$
$367$	473.803	−	273.550i	1.29102	−	0.745368i	0.978834	+	0.204654i	$0.0656071\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	16.2208	−	28.0953i	0.0434875	−	0.0753226i	−0.843462	−	0.537188i	$-0.819486\pi$
$373$	16.2208	−	28.0953i	0.0434875	−	0.0753226i	0.886950	+	0.461866i	$0.152820\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	847.878i		−	2.24901i
$378$		0			0
$379$	508.859			1.34263			0.671317	−	0.741170i	$-0.265729\pi$
$379$	508.859			1.34263			0.671317	+	0.741170i	$0.265729\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−22.7625	−	13.1419i	−0.0594321	−	0.0343131i	0.469990	−	0.882672i	$-0.344258\pi$
$383$	−22.7625	−	13.1419i	−0.0594321	−	0.0343131i	−0.529422	+	0.848359i	$0.677591\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−180.064	−	311.880i	−0.462890	−	0.801749i	0.536214	−	0.844082i	$-0.319854\pi$
$389$	−180.064	−	311.880i	−0.462890	−	0.801749i	−0.999104	+	0.0423337i	$0.986521\pi$
$390$		0			0
$391$			153.919i			0.393656i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−569.819	+	328.985i	−1.44258	+	0.832874i
$396$		0			0
$397$	494.494	+	285.496i	1.24558	+	0.719134i	0.970224	−	0.242210i	$-0.0778722\pi$
$397$	494.494	+	285.496i	1.24558	+	0.719134i	0.275352	+	0.961343i	$0.411206\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	34.2007	−	59.2373i	0.0852885	−	0.147724i	−0.820226	−	0.572040i	$-0.806152\pi$
$401$	34.2007	−	59.2373i	0.0852885	−	0.147724i	0.905514	+	0.424316i	$0.139485\pi$
$402$		0			0
$403$	425.677	+	737.293i	1.05627	+	1.82951i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	572.518			1.40668
$408$		0			0
$409$	13.3064	−	7.68246i	0.0325340	−	0.0187835i	−0.483645	−	0.875264i	$-0.660687\pi$
$409$	13.3064	−	7.68246i	0.0325340	−	0.0187835i	0.516179	+	0.856481i	$0.327354\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	89.2622	−	154.607i	0.215090	−	0.372546i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	366.079i		−	0.873696i	−0.899535	−	0.436848i	$-0.856095\pi$
$419$		−	366.079i		−	0.873696i	0.899535	−	0.436848i	$-0.143905\pi$
$420$		0			0
$421$	607.135			1.44213			0.721063	−	0.692870i	$-0.243654\pi$
$421$	607.135			1.44213			0.721063	+	0.692870i	$0.243654\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−18.7223	−	10.8093i	−0.0440525	−	0.0254337i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−293.177	−	507.797i	−0.680224	−	1.17818i	−0.974912	−	0.222590i	$-0.928549\pi$
$431$	−293.177	−	507.797i	−0.680224	−	1.17818i	0.294688	−	0.955594i	$-0.404784\pi$
$432$		0			0
$433$		−	518.769i		−	1.19808i	−0.800719	−	0.599040i	$-0.795549\pi$
$433$		−	518.769i		−	1.19808i	0.800719	−	0.599040i	$-0.204451\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−175.933	+	101.575i	−0.402594	+	0.232438i
$438$		0			0
$439$	−191.845	−	110.762i	−0.437005	−	0.252305i	0.265321	−	0.964160i	$-0.414522\pi$
$439$	−191.845	−	110.762i	−0.437005	−	0.252305i	−0.702326	+	0.711855i	$0.747855\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−113.208	+	196.081i	−0.255548	+	0.442622i	−0.965044	−	0.262087i	$-0.915589\pi$
$443$	−113.208	+	196.081i	−0.255548	+	0.442622i	0.709496	+	0.704709i	$0.248922\pi$
$444$		0			0
$445$	97.1686	+	168.301i	0.218356	+	0.378205i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	378.422			0.842810			0.421405	−	0.906873i	$-0.361537\pi$
$449$	378.422			0.842810			0.421405	+	0.906873i	$0.361537\pi$
$450$		0			0
$451$	250.268	−	144.492i	0.554917	−	0.320382i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−219.077	+	379.452i	−0.479380	+	0.830311i	−0.999720	−	0.0236483i	$-0.992472\pi$
$457$	−219.077	+	379.452i	−0.479380	+	0.830311i	0.520340	+	0.853959i	$0.325805\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			46.3981i			0.100647i	0.998733	+	0.0503234i	$0.0160252\pi$
$461$			46.3981i			0.100647i	−0.998733	+	0.0503234i	$0.983975\pi$
$462$		0			0
$463$	367.455			0.793639			0.396820	−	0.917897i	$-0.370114\pi$
$463$	367.455			0.793639			0.396820	+	0.917897i	$0.370114\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−458.530	−	264.733i	−0.981863	−	0.566879i	−0.0790311	−	0.996872i	$-0.525183\pi$
$467$	−458.530	−	264.733i	−0.981863	−	0.566879i	−0.902832	+	0.429993i	$0.858516\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−4.00758	−	6.94133i	−0.00847268	−	0.0146751i
$474$		0			0
$475$		−	28.5334i		−	0.0600702i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−250.940	+	144.880i	−0.523883	+	0.302464i	−0.738522	−	0.674229i	$-0.764476\pi$
$479$	−250.940	+	144.880i	−0.523883	+	0.302464i	0.214639	+	0.976694i	$0.431143\pi$
$480$		0			0
$481$	−1212.81	−	700.216i	−2.52143	−	1.45575i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−43.6223	+	75.5560i	−0.0899428	+	0.155786i
$486$		0			0
$487$	−219.818	−	380.736i	−0.451372	−	0.781799i	0.547100	−	0.837067i	$-0.315732\pi$
$487$	−219.818	−	380.736i	−0.451372	−	0.781799i	−0.998472	+	0.0552685i	$0.982399\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	320.561			0.652874			0.326437	−	0.945219i	$-0.394152\pi$
$491$	320.561			0.652874			0.326437	+	0.945219i	$0.394152\pi$
$492$		0			0
$493$	191.686	−	110.670i	0.388815	−	0.224483i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−385.986	+	668.548i	−0.773520	+	1.33978i	0.162103	+	0.986774i	$0.448172\pi$
$499$	−385.986	+	668.548i	−0.773520	+	1.33978i	−0.935623	+	0.353002i	$0.885161\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			101.632i			0.202052i	0.994884	+	0.101026i	$0.0322126\pi$
$503$			101.632i			0.202052i	−0.994884	+	0.101026i	$0.967787\pi$
$504$		0			0
$505$	−642.675			−1.27262
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−524.298	−	302.704i	−1.03006	−	0.594703i	−0.113055	−	0.993589i	$-0.536063\pi$
$509$	−524.298	−	302.704i	−1.03006	−	0.594703i	−0.917000	+	0.398886i	$0.869397\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	24.0252	+	41.6128i	0.0466508	+	0.0808015i
$516$		0			0
$517$		−	735.907i		−	1.42342i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−331.942	+	191.647i	−0.637124	+	0.367844i	−0.783506	−	0.621384i	$-0.786571\pi$
$521$	−331.942	+	191.647i	−0.637124	+	0.367844i	0.146382	+	0.989228i	$0.453237\pi$
$522$		0			0
$523$	388.061	+	224.047i	0.741991	+	0.428389i	0.822793	−	0.568341i	$-0.192415\pi$
$523$	388.061	+	224.047i	0.741991	+	0.428389i	−0.0808018	+	0.996730i	$0.525748\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−111.124	+	192.472i	−0.210861	+	0.365221i
$528$		0			0
$529$	−129.059	−	223.538i	−0.243969	−	0.422566i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−706.882			−1.32623
$534$		0			0
$535$	−809.978	+	467.641i	−1.51398	+	0.874095i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	135.629	−	234.916i	0.250700	−	0.434226i	−0.713019	−	0.701145i	$-0.752672\pi$
$541$	135.629	−	234.916i	0.250700	−	0.434226i	0.963719	+	0.266920i	$0.0860058\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	862.435i		−	1.58245i
$546$		0			0
$547$	−590.544			−1.07961			−0.539803	−	0.841791i	$-0.681501\pi$
$547$	−590.544			−1.07961			−0.539803	+	0.841791i	$0.681501\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	252.997	+	146.068i	0.459159	+	0.265096i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	94.1473	+	163.068i	0.169026	+	0.292761i	0.938078	−	0.346425i	$-0.112605\pi$
$557$	94.1473	+	163.068i	0.169026	+	0.292761i	−0.769052	+	0.639186i	$0.779271\pi$
$558$		0			0
$559$			19.6058i			0.0350730i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	176.805	−	102.078i	0.314041	−	0.181312i	−0.334692	−	0.942327i	$-0.608632\pi$
$563$	176.805	−	102.078i	0.314041	−	0.181312i	0.648733	+	0.761016i	$0.275299\pi$
$564$		0			0
$565$	−379.430	−	219.064i	−0.671558	−	0.387724i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−330.456	+	572.367i	−0.580767	+	1.00592i	0.414622	+	0.909994i	$0.363914\pi$
$569$	−330.456	+	572.367i	−0.580767	+	1.00592i	−0.995389	+	0.0959241i	$0.969419\pi$
$570$		0			0
$571$	266.989	+	462.438i	0.467581	+	0.809874i	0.999314	−	0.0370381i	$-0.0117923\pi$
$571$	266.989	+	462.438i	0.467581	+	0.809874i	−0.531733	+	0.846912i	$0.678459\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	110.554			0.192268
$576$		0			0
$577$	46.2750	−	26.7169i	0.0801993	−	0.0463031i	−0.459364	−	0.888248i	$-0.651923\pi$
$577$	46.2750	−	26.7169i	0.0801993	−	0.0463031i	0.539563	+	0.841945i	$0.318589\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−191.573	+	331.813i	−0.328598	+	0.569148i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			413.063i			0.703684i	0.936059	+	0.351842i	$0.114445\pi$
$587$			413.063i			0.703684i	−0.936059	+	0.351842i	$0.885555\pi$
$588$		0			0
$589$	−293.333			−0.498018
$590$		0			0
$591$		0			0
$592$		0			0
$593$	106.047	+	61.2261i	0.178831	+	0.103248i	0.586743	−	0.809773i	$-0.300410\pi$
$593$	106.047	+	61.2261i	0.178831	+	0.103248i	−0.407912	+	0.913021i	$0.633743\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−322.568	−	558.704i	−0.538511	−	0.932728i	−0.998985	−	0.0450546i	$-0.985654\pi$
$599$	−322.568	−	558.704i	−0.538511	−	0.932728i	0.460474	−	0.887673i	$-0.347680\pi$
$600$		0			0
$601$			683.488i			1.13725i	0.822596	+	0.568626i	$0.192525\pi$
$601$			683.488i			1.13725i	−0.822596	+	0.568626i	$0.807475\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−219.829	+	126.918i	−0.363353	+	0.209782i
$606$		0			0
$607$	222.788	+	128.627i	0.367031	+	0.211905i	0.672161	−	0.740405i	$-0.265366\pi$
$607$	222.788	+	128.627i	0.367031	+	0.211905i	−0.305130	+	0.952311i	$0.598700\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−900.049	+	1558.93i	−1.47307	+	2.55144i
$612$		0			0
$613$	−442.856	−	767.050i	−0.722441	−	1.25130i	−0.960019	−	0.279936i	$-0.909687\pi$
$613$	−442.856	−	767.050i	−0.722441	−	1.25130i	0.237578	−	0.971369i	$-0.423647\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−46.0724			−0.0746717			−0.0373358	−	0.999303i	$-0.511887\pi$
$617$	−46.0724			−0.0746717			−0.0373358	+	0.999303i	$0.511887\pi$
$618$		0			0
$619$	−949.446	+	548.163i	−1.53384	+	0.885562i	−0.534658	+	0.845069i	$0.679560\pi$
$619$	−949.446	+	548.163i	−1.53384	+	0.885562i	−0.999180	+	0.0404930i	$0.987107\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	353.993	−	613.134i	0.566388	−	0.981014i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	365.585i		−	0.581217i
$630$		0			0
$631$	606.319			0.960886			0.480443	−	0.877026i	$-0.340476\pi$
$631$	606.319			0.960886			0.480443	+	0.877026i	$0.340476\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−545.893	−	315.171i	−0.859674	−	0.496333i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	469.592	+	813.357i	0.732593	+	1.26889i	0.955771	+	0.294111i	$0.0950234\pi$
$641$	469.592	+	813.357i	0.732593	+	1.26889i	−0.223178	+	0.974778i	$0.571643\pi$
$642$		0			0
$643$			992.960i			1.54426i	0.635464	+	0.772131i	$0.280809\pi$
$643$			992.960i			1.54426i	−0.635464	+	0.772131i	$0.719191\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	229.656	−	132.592i	0.354955	−	0.204933i	−0.311910	−	0.950112i	$-0.600969\pi$
$647$	229.656	−	132.592i	0.354955	−	0.204933i	0.666865	+	0.745178i	$0.267636\pi$
$648$		0			0
$649$	473.925	+	273.620i	0.730238	+	0.421603i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−567.071	+	982.196i	−0.868409	+	1.50413i	−0.00478648	+	0.999989i	$0.501524\pi$
$653$	−567.071	+	982.196i	−0.868409	+	1.50413i	−0.863622	+	0.504139i	$0.831810\pi$
$654$		0			0
$655$	−656.719	−	1137.47i	−1.00262	−	1.73660i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	924.147			1.40235			0.701174	−	0.712990i	$-0.252660\pi$
$659$	924.147			1.40235			0.701174	+	0.712990i	$0.252660\pi$
$660$		0			0
$661$	−202.968	+	117.184i	−0.307062	+	0.177282i	−0.645611	−	0.763666i	$-0.723397\pi$
$661$	−202.968	+	117.184i	−0.307062	+	0.177282i	0.338549	+	0.940949i	$0.390064\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−565.948	+	980.251i	−0.848498	+	1.46964i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	275.003i		−	0.409840i
$672$		0			0
$673$	−465.127			−0.691125			−0.345563	−	0.938396i	$-0.612312\pi$
$673$	−465.127			−0.691125			−0.345563	+	0.938396i	$0.612312\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−1122.31	−	647.964i	−1.65776	−	0.957110i	−0.973744	−	0.227647i	$-0.926897\pi$
$677$	−1122.31	−	647.964i	−1.65776	−	0.957110i	−0.684020	−	0.729463i	$-0.739770\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	8.13060	+	14.0826i	0.0119042	+	0.0206188i	0.871916	−	0.489655i	$-0.162877\pi$
$683$	8.13060	+	14.0826i	0.0119042	+	0.0206188i	−0.860012	+	0.510274i	$0.829544\pi$
$684$		0			0
$685$		−	1320.23i		−	1.92734i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	811.647	−	468.604i	1.17801	−	0.680123i
$690$		0			0
$691$	−59.3542	−	34.2682i	−0.0858962	−	0.0495922i	0.456437	−	0.889756i	$-0.349125\pi$
$691$	−59.3542	−	34.2682i	−0.0858962	−	0.0495922i	−0.542333	+	0.840164i	$0.682459\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	46.0956	−	79.8400i	0.0663246	−	0.114878i
$696$		0			0
$697$	−92.2664	−	159.810i	−0.132376	−	0.229283i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−923.360			−1.31720			−0.658602	−	0.752491i	$-0.728852\pi$
$701$	−923.360			−1.31720			−0.658602	+	0.752491i	$0.728852\pi$
$702$		0			0
$703$	417.872	−	241.259i	0.594413	−	0.343185i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	456.431	−	790.562i	0.643768	−	1.11504i	−0.340817	−	0.940130i	$-0.610704\pi$
$709$	456.431	−	790.562i	0.643768	−	1.11504i	0.984585	−	0.174909i	$-0.0559631\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	1136.54i		−	1.59402i
$714$		0			0
$715$	−971.345			−1.35852
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−472.001	−	272.510i	−0.656469	−	0.379013i	0.134461	−	0.990919i	$-0.457070\pi$
$719$	−472.001	−	272.510i	−0.656469	−	0.379013i	−0.790930	+	0.611906i	$0.790403\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−79.4899	−	137.681i	−0.109641	−	0.189904i
$726$		0			0
$727$		−	750.292i		−	1.03204i	−0.856577	−	0.516019i	$-0.827413\pi$
$727$		−	750.292i		−	1.03204i	0.856577	−	0.516019i	$-0.172587\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−4.43243	+	2.55907i	−0.00606352	+	0.00350078i
$732$		0			0
$733$	−802.458	−	463.299i	−1.09476	−	0.632059i	−0.159919	−	0.987130i	$-0.551123\pi$
$733$	−802.458	−	463.299i	−1.09476	−	0.632059i	−0.934839	+	0.355071i	$0.884457\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	205.056	−	355.167i	0.278231	−	0.481910i
$738$		0			0
$739$	−244.073	−	422.748i	−0.330275	−	0.572054i	0.652290	−	0.757969i	$-0.273808\pi$
$739$	−244.073	−	422.748i	−0.330275	−	0.572054i	−0.982566	+	0.185915i	$0.940475\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	1091.72			1.46935			0.734674	−	0.678421i	$-0.237335\pi$
$743$	1091.72			1.46935			0.734674	+	0.678421i	$0.237335\pi$
$744$		0			0
$745$	−625.420	+	361.086i	−0.839490	+	0.484680i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	240.222	−	416.076i	0.319869	−	0.554029i	−0.660591	−	0.750746i	$-0.729694\pi$
$751$	240.222	−	416.076i	0.319869	−	0.554029i	0.980460	+	0.196716i	$0.0630278\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			1070.60i			1.41801i
$756$		0			0
$757$	941.400			1.24359			0.621796	−	0.783179i	$-0.286403\pi$
$757$	941.400			1.24359			0.621796	+	0.783179i	$0.286403\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1032.60	+	596.172i	1.35690	+	0.783407i	0.989205	−	0.146540i	$-0.0468136\pi$
$761$	1032.60	+	596.172i	1.35690	+	0.783407i	0.367695	+	0.929946i	$0.380147\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−669.301	−	1159.26i	−0.872622	−	1.51143i
$768$		0			0
$769$			908.294i			1.18114i	0.806988	+	0.590568i	$0.201096\pi$
$769$			908.294i			1.18114i	−0.806988	+	0.590568i	$0.798904\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	1030.03	−	594.690i	1.33251	−	0.769328i	0.346830	−	0.937928i	$-0.387258\pi$
$773$	1030.03	−	594.690i	1.33251	−	0.769328i	0.985685	+	0.168600i	$0.0539246\pi$
$774$		0			0
$775$	138.245	+	79.8158i	0.178381	+	0.102988i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	121.778	−	210.925i	0.156326	−	0.270764i
$780$		0			0
$781$	64.2976	+	111.367i	0.0823272	+	0.142595i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	414.334			0.527814
$786$		0			0
$787$	−915.492	+	528.559i	−1.16327	+	0.671613i	−0.952085	−	0.305834i	$-0.901065\pi$
$787$	−915.492	+	528.559i	−1.16327	+	0.671613i	−0.211183	+	0.977447i	$0.567731\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−336.341	+	582.560i	−0.424138	+	0.734628i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			1037.94i			1.30231i	0.758945	+	0.651154i	$0.225715\pi$
$797$			1037.94i			1.30231i	−0.758945	+	0.651154i	$0.774285\pi$
$798$		0			0
$799$	−469.919			−0.588133
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1043.89	−	602.689i	−1.29999	−	0.750547i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	543.681	+	941.684i	0.672041	+	1.16401i	0.977324	+	0.211748i	$0.0679155\pi$
$809$	543.681	+	941.684i	0.672041	+	1.16401i	−0.305283	+	0.952262i	$0.598751\pi$
$810$		0			0
$811$		−	926.995i		−	1.14303i	−0.820593	−	0.571514i	$-0.806356\pi$
$811$		−	926.995i		−	1.14303i	0.820593	−	0.571514i	$-0.193644\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−1073.31	+	619.678i	−1.31695	+	0.760342i
$816$		0			0
$817$	−5.85015	−	3.37758i	−0.00716052	−	0.00413413i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−215.150	+	372.651i	−0.262058	+	0.453898i	−0.966789	−	0.255577i	$-0.917735\pi$
$821$	−215.150	+	372.651i	−0.262058	+	0.453898i	0.704730	+	0.709475i	$0.251068\pi$
$822$		0			0
$823$	522.368	+	904.767i	0.634712	+	1.09935i	0.986576	+	0.163302i	$0.0522144\pi$
$823$	522.368	+	904.767i	0.634712	+	1.09935i	−0.351865	+	0.936051i	$0.614452\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	610.313			0.737984			0.368992	−	0.929433i	$-0.379703\pi$
$827$	610.313			0.737984			0.368992	+	0.929433i	$0.379703\pi$
$828$		0			0
$829$	1224.47	−	706.950i	1.47705	−	0.852775i	0.477385	−	0.878694i	$-0.341585\pi$
$829$	1224.47	−	706.950i	1.47705	−	0.852775i	0.999664	+	0.0259193i	$0.00825131\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	616.591	−	1067.97i	0.738433	−	1.27900i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			46.3303i			0.0552209i	0.999619	+	0.0276105i	$0.00878980\pi$
$839$			46.3303i			0.0552209i	−0.999619	+	0.0276105i	$0.991210\pi$
$840$		0			0
$841$	786.696			0.935430
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1270.32	+	733.420i	1.50334	+	0.867953i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	934.772	+	1619.07i	1.09844	+	1.90255i
$852$		0			0
$853$		−	773.111i		−	0.906343i	−0.891423	−	0.453172i	$-0.850292\pi$
$853$		−	773.111i		−	0.906343i	0.891423	−	0.453172i	$-0.149708\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	379.457	−	219.080i	0.442774	−	0.255636i	−0.262000	−	0.965068i	$-0.584382\pi$
$857$	379.457	−	219.080i	0.442774	−	0.255636i	0.704774	+	0.709432i	$0.251049\pi$
$858$		0			0
$859$	62.6499	+	36.1710i	0.0729336	+	0.0421082i	0.536023	−	0.844203i	$-0.319926\pi$
$859$	62.6499	+	36.1710i	0.0729336	+	0.0421082i	−0.463090	+	0.886311i	$0.653259\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	213.247	−	369.355i	0.247100	−	0.427990i	−0.715620	−	0.698490i	$-0.753856\pi$
$863$	213.247	−	369.355i	0.247100	−	0.427990i	0.962720	+	0.270500i	$0.0871890\pi$
$864$		0			0
$865$	−302.532	−	524.001i	−0.349748	−	0.605781i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1050.82			−1.20922
$870$		0			0
$871$	−868.773	+	501.586i	−0.997443	+	0.575874i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−208.243	+	360.687i	−0.237449	+	0.411273i	−0.959982	−	0.280063i	$-0.909645\pi$
$877$	−208.243	+	360.687i	−0.237449	+	0.411273i	0.722533	+	0.691337i	$0.242978\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	1249.01i		−	1.41771i	−0.705353	−	0.708857i	$-0.749211\pi$
$881$		−	1249.01i		−	1.41771i	0.705353	−	0.708857i	$-0.250789\pi$
$882$		0			0
$883$	81.5906			0.0924016			0.0462008	−	0.998932i	$-0.485289\pi$
$883$	81.5906			0.0924016			0.0462008	+	0.998932i	$0.485289\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	794.777	+	458.865i	0.896028	+	0.517322i	0.875910	−	0.482475i	$-0.160262\pi$
$887$	794.777	+	458.865i	0.896028	+	0.517322i	0.0201189	+	0.999798i	$0.493596\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−310.111	−	537.128i	−0.347269	−	0.601487i
$894$		0			0
$895$		−	567.520i		−	0.634100i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−1415.40	+	817.184i	−1.57442	+	0.908992i
$900$		0			0
$901$	211.882	+	122.330i	0.235163	+	0.135771i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−41.0690	+	71.1336i	−0.0453801	+	0.0786007i
$906$		0			0
$907$	725.700	+	1256.95i	0.800110	+	1.38583i	0.919543	+	0.392989i	$0.128559\pi$
$907$	725.700	+	1256.95i	0.800110	+	1.38583i	−0.119433	+	0.992842i	$0.538108\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	562.064			0.616975			0.308487	−	0.951228i	$-0.400177\pi$
$911$	562.064			0.616975			0.308487	+	0.951228i	$0.400177\pi$
$912$		0			0
$913$	246.916	−	142.557i	0.270444	−	0.156141i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−269.079	+	466.059i	−0.292795	+	0.507137i	−0.974470	−	0.224519i	$-0.927919\pi$
$919$	−269.079	+	466.059i	−0.292795	+	0.507137i	0.681674	+	0.731656i	$0.261252\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	314.556i		−	0.340797i
$924$		0			0
$925$	−262.586			−0.283876
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−716.272	−	413.540i	−0.771014	−	0.445145i	0.0622221	−	0.998062i	$-0.480181\pi$
$929$	−716.272	−	413.540i	−0.771014	−	0.445145i	−0.833236	+	0.552917i	$0.813515\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−126.786	−	219.599i	−0.135600	−	0.234865i
$936$		0			0
$937$			1501.10i			1.60203i	0.598644	+	0.801016i	$0.295707\pi$
$937$			1501.10i			1.60203i	−0.598644	+	0.801016i	$0.704293\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−663.958	+	383.336i	−0.705587	+	0.407371i	−0.809425	−	0.587223i	$-0.800221\pi$
$941$	−663.958	+	383.336i	−0.705587	+	0.407371i	0.103838	+	0.994594i	$0.466888\pi$
$942$		0			0
$943$	817.243	+	471.836i	0.866642	+	0.500356i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	681.117	−	1179.73i	0.719237	−	1.24576i	−0.242065	−	0.970260i	$-0.577825\pi$
$947$	681.117	−	1179.73i	0.719237	−	1.24576i	0.961302	−	0.275495i	$-0.0888418\pi$
$948$		0			0
$949$	1474.23	+	2553.45i	1.55346	+	2.69067i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1431.37			−1.50196			−0.750980	−	0.660325i	$-0.770419\pi$
$953$	−1431.37			−1.50196			−0.750980	+	0.660325i	$0.770419\pi$
$954$		0			0
$955$	1426.49	−	823.585i	1.49371	−	0.862393i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	340.034	−	588.955i	0.353833	−	0.612857i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	9.90391i		−	0.0102631i
$966$		0			0
$967$	−426.276			−0.440823			−0.220411	−	0.975407i	$-0.570740\pi$
$967$	−426.276			−0.440823			−0.220411	+	0.975407i	$0.570740\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	1496.84	+	864.203i	1.54155	+	0.890013i	0.998741	+	0.0501561i	$0.0159719\pi$
$971$	1496.84	+	864.203i	1.54155	+	0.890013i	0.542807	+	0.839857i	$0.317361\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−356.160	−	616.887i	−0.364545	−	0.631410i	0.624158	−	0.781298i	$-0.285442\pi$
$977$	−356.160	−	616.887i	−0.364545	−	0.631410i	−0.988703	+	0.149888i	$0.952109\pi$
$978$		0			0
$979$			310.368i			0.317025i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	902.887	−	521.282i	0.918502	−	0.530297i	0.0353451	−	0.999375i	$-0.488747\pi$
$983$	902.887	−	521.282i	0.918502	−	0.530297i	0.883157	+	0.469078i	$0.155414\pi$
$984$		0			0
$985$	−1189.58	−	686.803i	−1.20769	−	0.697262i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	13.0867	−	22.6668i	0.0132322	−	0.0229189i
$990$		0			0
$991$	−698.706	−	1210.19i	−0.705052	−	1.22119i	−0.966673	−	0.256015i	$-0.917590\pi$
$991$	−698.706	−	1210.19i	−0.705052	−	1.22119i	0.261621	−	0.965171i	$-0.415743\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−197.155			−0.198146
$996$		0			0
$997$	621.773	−	358.981i	0.623644	−	0.360061i	−0.154642	−	0.987971i	$-0.549422\pi$
$997$	621.773	−	358.981i	0.623644	−	0.360061i	0.778286	+	0.627909i	$0.216089\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.3.z.l.901.1		8
3.2	odd	2		588.3.m.f.313.4		8
7.2	even	3		1764.3.d.h.685.7		8
7.3	odd	6	inner	1764.3.z.l.325.1		8
7.4	even	3		1764.3.z.m.325.4		8
7.5	odd	6		1764.3.d.h.685.2		8
7.6	odd	2		1764.3.z.m.901.4		8
21.2	odd	6		588.3.d.c.97.5	yes	8
21.5	even	6		588.3.d.c.97.4	✓	8
21.11	odd	6		588.3.m.e.325.1		8
21.17	even	6		588.3.m.f.325.4		8
21.20	even	2		588.3.m.e.313.1		8
84.23	even	6		2352.3.f.j.97.1		8
84.47	odd	6		2352.3.f.j.97.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
588.3.d.c.97.4	✓	8	21.5	even	6
588.3.d.c.97.5	yes	8	21.2	odd	6
588.3.m.e.313.1		8	21.20	even	2
588.3.m.e.325.1		8	21.11	odd	6
588.3.m.f.313.4		8	3.2	odd	2
588.3.m.f.325.4		8	21.17	even	6
1764.3.d.h.685.2		8	7.5	odd	6
1764.3.d.h.685.7		8	7.2	even	3
1764.3.z.l.325.1		8	7.3	odd	6	inner
1764.3.z.l.901.1		8	1.1	even	1	trivial
1764.3.z.m.325.4		8	7.4	even	3
1764.3.z.m.901.4		8	7.6	odd	2
2352.3.f.j.97.1		8	84.23	even	6
2352.3.f.j.97.8		8	84.47	odd	6

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:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1764.z (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-4.65891 - 2.68982i) q^{5} +O(q^{10})\)	\(q+(-4.65891 - 2.68982i) q^{5} +(-4.29579 - 7.44053i) q^{11} +21.0158i q^{13} +(-4.75120 + 2.74311i) q^{17} +(-6.27088 - 3.62049i) q^{19} +(14.0278 - 24.2969i) q^{23} +(1.97027 + 3.41261i) q^{25} -40.3447 q^{29} +(35.0828 - 20.2550i) q^{31} +(-33.3185 + 57.7093i) q^{37} +33.6357i q^{41} +0.932907 q^{43} +(74.1789 + 42.8272i) q^{47} +(-22.2977 - 38.6207i) q^{53} +46.2197i q^{55} +(-55.1615 + 31.8475i) q^{59} +(27.7201 + 16.0042i) q^{61} +(56.5288 - 97.9108i) q^{65} +(23.8671 + 41.3390i) q^{67} -14.9676 q^{71} +(121.501 - 70.1488i) q^{73} +(61.1537 - 105.921i) q^{79} +33.1852i q^{83} +29.5139 q^{85} +(-31.2848 - 18.0623i) q^{89} +(19.4769 + 33.7351i) q^{95} -16.2175i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 48 q^{17} - 96 q^{19} - 8 q^{23} - 36 q^{25} - 80 q^{29} + 48 q^{31} - 64 q^{37} - 112 q^{43} + 264 q^{47} - 72 q^{53} - 168 q^{59} - 144 q^{61} + 120 q^{65} + 32 q^{67} - 224 q^{71} + 336 q^{73} + 216 q^{79} - 96 q^{85} - 96 q^{89} - 136 q^{95}+O(q^{100}) \) 8 * q - 48 * q^17 - 96 * q^19 - 8 * q^23 - 36 * q^25 - 80 * q^29 + 48 * q^31 - 64 * q^37 - 112 * q^43 + 264 * q^47 - 72 * q^53 - 168 * q^59 - 144 * q^61 + 120 * q^65 + 32 * q^67 - 224 * q^71 + 336 * q^73 + 216 * q^79 - 96 * q^85 - 96 * q^89 - 136 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more