LMFDB - Embedded newform 1764.4.f.a.881.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,4,Mod(881,1764)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1764, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1764.881");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1764.f (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$104.079369250$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 290 x^{14} + 1728 x^{13} + 29275 x^{12} - 246984 x^{11} - 955194 x^{10} + \cdots + 7375227456 $ x^16 - 4x^15 - 290x^14 + 1728x^13 + 29275x^12 - 246984x^11 - 955194x^10 + 14344616x^9 - 18123280x^8 - 273588032x^7 + 1239640536x^6 - 1407381792x^5 - 1961185792x^4 + 4297169408x^3 + 2991779296x^2 - 11217342336*x + 7375227456
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{14}\cdot 3^{18}\cdot 7^{4} $
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.1
Root			$4.65022 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1764.881
Dual form			1764.4.f.a.881.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-20.2518 q^{5} +O(q^{10})$	$q-20.2518 q^{5} -17.8438i q^{11} +33.1829i q^{13} -45.8153 q^{17} -40.4250i q^{19} +80.5754i q^{23} +285.136 q^{25} +233.844i q^{29} +225.626i q^{31} -270.912 q^{37} +154.432 q^{41} +367.102 q^{43} -527.705 q^{47} +91.0934i q^{53} +361.370i q^{55} -625.535 q^{59} -91.0300i q^{61} -672.015i q^{65} +863.017 q^{67} -303.596i q^{71} +1153.69i q^{73} -6.96240 q^{79} -815.694 q^{83} +927.844 q^{85} +310.988 q^{89} +818.680i q^{95} -1832.92i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 424 q^{25} - 152 q^{37} + 1408 q^{43} + 3056 q^{67} + 728 q^{79} + 7392 q^{85}+O(q^{100}) $ 16 * q + 424 * q^25 - 152 * q^37 + 1408 * q^43 + 3056 * q^67 + 728 * q^79 + 7392 * q^85

Display 100 coefficients 10 coefficients

Character values

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

$n$	$785$	$883$	$1081$
$\chi(n)$	$-1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−20.2518	−1.81138	−0.905689	−	0.423944i	$-0.860645\pi$
$5$	−20.2518	−1.81138	−0.905689	+	0.423944i	$0.860645\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 17.8438i	− 0.489101i	−0.969637	−	0.244551i	$-0.921360\pi$
$11$	− 17.8438i	− 0.489101i	0.969637	−	0.244551i	$-0.0786404\pi$
$12$	0	0
$13$	33.1829i	0.707945i	0.935256	+	0.353973i	$0.115169\pi$
$13$	33.1829i	0.707945i	−0.935256	+	0.353973i	$0.884831\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−45.8153	−0.653638	−0.326819	−	0.945087i	$-0.605977\pi$
$17$	−45.8153	−0.653638	−0.326819	+	0.945087i	$0.605977\pi$
$18$	0	0
$19$	− 40.4250i	− 0.488112i	−0.969761	−	0.244056i	$-0.921522\pi$
$19$	− 40.4250i	− 0.488112i	0.969761	−	0.244056i	$-0.0784781\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	80.5754i	0.730484i	0.930913	+	0.365242i	$0.119014\pi$
$23$	80.5754i	0.730484i	−0.930913	+	0.365242i	$0.880986\pi$
$24$	0	0
$25$	285.136	2.28109
$26$	0	0
$27$	0	0
$28$	0	0
$29$	233.844i	1.49737i	0.662926	+	0.748685i	$0.269314\pi$
$29$	233.844i	1.49737i	−0.662926	+	0.748685i	$0.730686\pi$
$30$	0	0
$31$	225.626i	1.30721i	0.756834	+	0.653607i	$0.226745\pi$
$31$	225.626i	1.30721i	−0.756834	+	0.653607i	$0.773255\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−270.912	−1.20372	−0.601860	−	0.798601i	$-0.705574\pi$
$37$	−270.912	−1.20372	−0.601860	+	0.798601i	$0.705574\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	154.432	0.588248	0.294124	−	0.955767i	$-0.404972\pi$
$41$	154.432	0.588248	0.294124	+	0.955767i	$0.404972\pi$
$42$	0	0
$43$	367.102	1.30192	0.650960	−	0.759112i	$-0.274366\pi$
$43$	367.102	1.30192	0.650960	+	0.759112i	$0.274366\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−527.705	−1.63774	−0.818869	−	0.573981i	$-0.805398\pi$
$47$	−527.705	−1.63774	−0.818869	+	0.573981i	$0.805398\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	91.0934i	0.236088i	0.993008	+	0.118044i	$0.0376623\pi$
$53$	91.0934i	0.236088i	−0.993008	+	0.118044i	$0.962338\pi$
$54$	0	0
$55$	361.370i	0.885947i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−625.535	−1.38030	−0.690150	−	0.723666i	$-0.742456\pi$
$59$	−625.535	−1.38030	−0.690150	+	0.723666i	$0.742456\pi$
$60$	0	0
$61$	− 91.0300i	− 0.191069i	−0.995426	−	0.0955344i	$-0.969544\pi$
$61$	− 91.0300i	− 0.191069i	0.995426	−	0.0955344i	$-0.0304560\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 672.015i	− 1.28236i
$66$	0	0
$67$	863.017	1.57365	0.786823	−	0.617178i	$-0.211724\pi$
$67$	863.017	1.57365	0.786823	+	0.617178i	$0.211724\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 303.596i	− 0.507468i	−0.967274	−	0.253734i	$-0.918341\pi$
$71$	− 303.596i	− 0.507468i	0.967274	−	0.253734i	$-0.0816587\pi$
$72$	0	0
$73$	1153.69i	1.84972i	0.380309	+	0.924859i	$0.375817\pi$
$73$	1153.69i	1.84972i	−0.380309	+	0.924859i	$0.624183\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−6.96240	−0.00991558	−0.00495779	−	0.999988i	$-0.501578\pi$
$79$	−6.96240	−0.00991558	−0.00495779	+	0.999988i	$0.501578\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−815.694	−1.07872	−0.539362	−	0.842074i	$-0.681334\pi$
$83$	−815.694	−1.07872	−0.539362	+	0.842074i	$0.681334\pi$
$84$	0	0
$85$	927.844	1.18399
$86$	0	0
$87$	0	0
$88$	0	0
$89$	310.988	0.370389	0.185194	−	0.982702i	$-0.440709\pi$
$89$	310.988	0.370389	0.185194	+	0.982702i	$0.440709\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	818.680i	0.884156i
$96$	0	0
$97$	− 1832.92i	− 1.91861i	−0.282370	−	0.959305i	$-0.591121\pi$
$97$	− 1832.92i	− 1.91861i	0.282370	−	0.959305i	$-0.408879\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1442.11	1.42074	0.710371	−	0.703828i	$-0.248527\pi$
$101$	1442.11	1.42074	0.710371	+	0.703828i	$0.248527\pi$
$102$	0	0
$103$	1071.98i	1.02549i	0.858542	+	0.512744i	$0.171371\pi$
$103$	1071.98i	1.02549i	−0.858542	+	0.512744i	$0.828629\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	833.838i	0.753366i	0.926342	+	0.376683i	$0.122935\pi$
$107$	833.838i	0.753366i	−0.926342	+	0.376683i	$0.877065\pi$
$108$	0	0
$109$	−1845.03	−1.62130	−0.810650	−	0.585531i	$-0.800886\pi$
$109$	−1845.03	−1.62130	−0.810650	+	0.585531i	$0.800886\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 1388.90i	− 1.15626i	−0.815946	−	0.578128i	$-0.803784\pi$
$113$	− 1388.90i	− 1.15626i	0.815946	−	0.578128i	$-0.196216\pi$
$114$	0	0
$115$	− 1631.80i	− 1.32318i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1012.60	0.760780
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−3243.04	−2.32053
$126$	0	0
$127$	−838.992	−0.586208	−0.293104	−	0.956080i	$-0.594688\pi$
$127$	−838.992	−0.586208	−0.293104	+	0.956080i	$0.594688\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−390.075	−0.260161	−0.130080	−	0.991503i	$-0.541524\pi$
$131$	−390.075	−0.260161	−0.130080	+	0.991503i	$0.541524\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 1376.08i	− 0.858148i	−0.903269	−	0.429074i	$-0.858840\pi$
$137$	− 1376.08i	− 0.858148i	0.903269	−	0.429074i	$-0.141160\pi$
$138$	0	0
$139$	− 953.705i	− 0.581958i	−0.956729	−	0.290979i	$-0.906019\pi$
$139$	− 953.705i	− 0.581958i	0.956729	−	0.290979i	$-0.0939810\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	592.110	0.346257
$144$	0	0
$145$	− 4735.76i	− 2.71230i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 2539.12i	− 1.39606i	−0.716069	−	0.698029i	$-0.754061\pi$
$149$	− 2539.12i	− 1.39606i	0.716069	−	0.698029i	$-0.245939\pi$
$150$	0	0
$151$	−1030.00	−0.555102	−0.277551	−	0.960711i	$-0.589523\pi$
$151$	−1030.00	−0.555102	−0.277551	+	0.960711i	$0.589523\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 4569.33i	− 2.36786i
$156$	0	0
$157$	− 1799.92i	− 0.914963i	−0.889219	−	0.457481i	$-0.848752\pi$
$157$	− 1799.92i	− 0.914963i	0.889219	−	0.457481i	$-0.151248\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	2624.05	1.26093	0.630466	−	0.776217i	$-0.282864\pi$
$163$	2624.05	1.26093	0.630466	+	0.776217i	$0.282864\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	714.138	0.330908	0.165454	−	0.986217i	$-0.447091\pi$
$167$	714.138	0.330908	0.165454	+	0.986217i	$0.447091\pi$
$168$	0	0
$169$	1095.89	0.498813
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−747.570	−0.328536	−0.164268	−	0.986416i	$-0.552526\pi$
$173$	−747.570	−0.328536	−0.164268	+	0.986416i	$0.552526\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 4385.04i	− 1.83102i	−0.402292	−	0.915511i	$-0.631786\pi$
$179$	− 4385.04i	− 1.83102i	0.402292	−	0.915511i	$-0.368214\pi$
$180$	0	0
$181$	− 2143.42i	− 0.880215i	−0.897945	−	0.440107i	$-0.854940\pi$
$181$	− 2143.42i	− 0.880215i	0.897945	−	0.440107i	$-0.145060\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	5486.46	2.18039
$186$	0	0
$187$	817.520i	0.319695i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4653.67i	1.76297i	0.472209	+	0.881486i	$0.343457\pi$
$191$	4653.67i	1.76297i	−0.472209	+	0.881486i	$0.656543\pi$
$192$	0	0
$193$	−2105.39	−0.785228	−0.392614	−	0.919703i	$-0.628429\pi$
$193$	−2105.39	−0.785228	−0.392614	+	0.919703i	$0.628429\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 707.475i	− 0.255866i	−0.991783	−	0.127933i	$-0.959166\pi$
$197$	− 707.475i	− 0.255866i	0.991783	−	0.127933i	$-0.0408342\pi$
$198$	0	0
$199$	1287.81i	0.458747i	0.973338	+	0.229374i	$0.0736677\pi$
$199$	1287.81i	0.458747i	−0.973338	+	0.229374i	$0.926332\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−3127.52	−1.06554
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−721.336	−0.238736
$210$	0	0
$211$	1193.83	0.389509	0.194755	−	0.980852i	$-0.437609\pi$
$211$	1193.83	0.389509	0.194755	+	0.980852i	$0.437609\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−7434.49	−2.35827
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 1520.29i	− 0.462740i
$222$	0	0
$223$	420.089i	0.126149i	0.998009	+	0.0630745i	$0.0200906\pi$
$223$	420.089i	0.126149i	−0.998009	+	0.0630745i	$0.979909\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−967.657	−0.282932	−0.141466	−	0.989943i	$-0.545182\pi$
$227$	−967.657	−0.282932	−0.141466	+	0.989943i	$0.545182\pi$
$228$	0	0
$229$	− 3485.69i	− 1.00585i	−0.864329	−	0.502927i	$-0.832256\pi$
$229$	− 3485.69i	− 1.00585i	0.864329	−	0.502927i	$-0.167744\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 2095.72i	− 0.589249i	−0.955613	−	0.294625i	$-0.904805\pi$
$233$	− 2095.72i	− 0.589249i	0.955613	−	0.294625i	$-0.0951946\pi$
$234$	0	0
$235$	10687.0	2.96656
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 6892.95i	− 1.86555i	−0.360453	−	0.932777i	$-0.617378\pi$
$239$	− 6892.95i	− 1.86555i	0.360453	−	0.932777i	$-0.382622\pi$
$240$	0	0
$241$	2712.38i	0.724978i	0.931988	+	0.362489i	$0.118073\pi$
$241$	2712.38i	0.724978i	−0.931988	+	0.362489i	$0.881927\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1341.42	0.345557
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−6334.19	−1.59287	−0.796435	−	0.604724i	$-0.793284\pi$
$251$	−6334.19	−1.59287	−0.796435	+	0.604724i	$0.793284\pi$
$252$	0	0
$253$	1437.77	0.357281
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−4791.36	−1.16294	−0.581472	−	0.813566i	$-0.697523\pi$
$257$	−4791.36	−1.16294	−0.581472	+	0.813566i	$0.697523\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 231.068i	− 0.0541759i	−0.999633	−	0.0270879i	$-0.991377\pi$
$263$	− 231.068i	− 0.0541759i	0.999633	−	0.0270879i	$-0.00862341\pi$
$264$	0	0
$265$	− 1844.81i	− 0.427644i
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−4713.73	−1.06841	−0.534203	−	0.845356i	$-0.679388\pi$
$269$	−4713.73	−1.06841	−0.534203	+	0.845356i	$0.679388\pi$
$270$	0	0
$271$	− 3754.51i	− 0.841588i	−0.907156	−	0.420794i	$-0.861752\pi$
$271$	− 3754.51i	− 0.841588i	0.907156	−	0.420794i	$-0.138248\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 5087.91i	− 1.11568i
$276$	0	0
$277$	−1514.24	−0.328455	−0.164227	−	0.986423i	$-0.552513\pi$
$277$	−1514.24	−0.328455	−0.164227	+	0.986423i	$0.552513\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	4418.40i	0.938005i	0.883197	+	0.469003i	$0.155387\pi$
$281$	4418.40i	0.938005i	−0.883197	+	0.469003i	$0.844613\pi$
$282$	0	0
$283$	4872.75i	1.02352i	0.859130	+	0.511758i	$0.171006\pi$
$283$	4872.75i	1.02352i	−0.859130	+	0.511758i	$0.828994\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−2813.95	−0.572757
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−340.271	−0.0678458	−0.0339229	−	0.999424i	$-0.510800\pi$
$293$	−340.271	−0.0678458	−0.0339229	+	0.999424i	$0.510800\pi$
$294$	0	0
$295$	12668.2	2.50025
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2673.73	−0.517143
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1843.52i	0.346098i
$306$	0	0
$307$	− 7302.13i	− 1.35751i	−0.734366	−	0.678754i	$-0.762520\pi$
$307$	− 7302.13i	− 1.35751i	0.734366	−	0.678754i	$-0.237480\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	745.076	0.135850	0.0679251	−	0.997690i	$-0.478362\pi$
$311$	745.076	0.135850	0.0679251	+	0.997690i	$0.478362\pi$
$312$	0	0
$313$	796.445i	0.143827i	0.997411	+	0.0719133i	$0.0229105\pi$
$313$	796.445i	0.143827i	−0.997411	+	0.0719133i	$0.977090\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 1125.24i	− 0.199369i	−0.995019	−	0.0996843i	$-0.968217\pi$
$317$	− 1125.24i	− 0.199369i	0.995019	−	0.0996843i	$-0.0317833\pi$
$318$	0	0
$319$	4172.67	0.732365
$320$	0	0
$321$	0	0
$322$	0	0
$323$	1852.09i	0.319049i
$324$	0	0
$325$	9461.65i	1.61489i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	22.2433	0.00369366	0.00184683	−	0.999998i	$-0.499412\pi$
$331$	22.2433	0.00369366	0.00184683	+	0.999998i	$0.499412\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−17477.7	−2.85047
$336$	0	0
$337$	9570.35	1.54697	0.773487	−	0.633812i	$-0.218511\pi$
$337$	9570.35	1.54697	0.773487	+	0.633812i	$0.218511\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	4026.03	0.639359
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	11462.9i	1.77338i	0.462369	+	0.886688i	$0.346999\pi$
$347$	11462.9i	1.77338i	−0.462369	+	0.886688i	$0.653001\pi$
$348$	0	0
$349$	− 1575.13i	− 0.241590i	−0.992677	−	0.120795i	$-0.961456\pi$
$349$	− 1575.13i	− 0.241590i	0.992677	−	0.120795i	$-0.0385443\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	4633.48	0.698627	0.349313	−	0.937006i	$-0.386415\pi$
$353$	4633.48	0.698627	0.349313	+	0.937006i	$0.386415\pi$
$354$	0	0
$355$	6148.37i	0.919215i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	2381.73i	0.350147i	0.984555	+	0.175073i	$0.0560163\pi$
$359$	2381.73i	0.350147i	−0.984555	+	0.175073i	$0.943984\pi$
$360$	0	0
$361$	5224.82	0.761746
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 23364.4i	− 3.35054i
$366$	0	0
$367$	1144.04i	0.162720i	0.996685	+	0.0813601i	$0.0259264\pi$
$367$	1144.04i	0.162720i	−0.996685	+	0.0813601i	$0.974074\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−410.451	−0.0569769	−0.0284884	−	0.999594i	$-0.509069\pi$
$373$	−410.451	−0.0569769	−0.0284884	+	0.999594i	$0.509069\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−7759.63	−1.06006
$378$	0	0
$379$	−6400.00	−0.867403	−0.433702	−	0.901057i	$-0.642793\pi$
$379$	−6400.00	−0.867403	−0.433702	+	0.901057i	$0.642793\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	3484.39	0.464866	0.232433	−	0.972612i	$-0.425331\pi$
$383$	3484.39	0.464866	0.232433	+	0.972612i	$0.425331\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	4205.05i	0.548084i	0.961718	+	0.274042i	$0.0883607\pi$
$389$	4205.05i	0.548084i	−0.961718	+	0.274042i	$0.911639\pi$
$390$	0	0
$391$	− 3691.59i	− 0.477472i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	141.001	0.0179609
$396$	0	0
$397$	− 5426.37i	− 0.686000i	−0.939335	−	0.343000i	$-0.888557\pi$
$397$	− 5426.37i	− 0.686000i	0.939335	−	0.343000i	$-0.111443\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 8496.15i	− 1.05805i	−0.848607	−	0.529024i	$-0.822558\pi$
$401$	− 8496.15i	− 1.05805i	0.848607	−	0.529024i	$-0.177442\pi$
$402$	0	0
$403$	−7486.93	−0.925436
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4834.10i	0.588741i
$408$	0	0
$409$	− 6444.87i	− 0.779165i	−0.920992	−	0.389582i	$-0.872619\pi$
$409$	− 6444.87i	− 0.779165i	0.920992	−	0.389582i	$-0.127381\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	16519.3	1.95397
$416$	0	0
$417$	0	0
$418$	0	0
$419$	14198.4	1.65546	0.827728	−	0.561130i	$-0.189633\pi$
$419$	14198.4	1.65546	0.827728	+	0.561130i	$0.189633\pi$
$420$	0	0
$421$	6986.18	0.808754	0.404377	−	0.914592i	$-0.367488\pi$
$421$	6986.18	0.808754	0.404377	+	0.914592i	$0.367488\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−13063.6	−1.49101
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 13976.6i	− 1.56201i	−0.624522	−	0.781007i	$-0.714706\pi$
$431$	− 13976.6i	− 1.56201i	0.624522	−	0.781007i	$-0.285294\pi$
$432$	0	0
$433$	4418.70i	0.490414i	0.969471	+	0.245207i	$0.0788559\pi$
$433$	4418.70i	0.490414i	−0.969471	+	0.245207i	$0.921144\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3257.26	0.356558
$438$	0	0
$439$	2883.44i	0.313483i	0.987640	+	0.156742i	$0.0500990\pi$
$439$	2883.44i	0.313483i	−0.987640	+	0.156742i	$0.949901\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 12717.6i	− 1.36395i	−0.731374	−	0.681976i	$-0.761121\pi$
$443$	− 12717.6i	− 1.36395i	0.731374	−	0.681976i	$-0.238879\pi$
$444$	0	0
$445$	−6298.06	−0.670914
$446$	0	0
$447$	0	0
$448$	0	0
$449$	5375.10i	0.564959i	0.959273	+	0.282479i	$0.0911569\pi$
$449$	5375.10i	0.564959i	−0.959273	+	0.282479i	$0.908843\pi$
$450$	0	0
$451$	− 2755.65i	− 0.287713i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−4121.18	−0.421840	−0.210920	−	0.977503i	$-0.567646\pi$
$457$	−4121.18	−0.421840	−0.210920	+	0.977503i	$0.567646\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	6019.06	0.608103	0.304052	−	0.952656i	$-0.401660\pi$
$461$	6019.06	0.608103	0.304052	+	0.952656i	$0.401660\pi$
$462$	0	0
$463$	−3263.90	−0.327616	−0.163808	−	0.986492i	$-0.552378\pi$
$463$	−3263.90	−0.327616	−0.163808	+	0.986492i	$0.552378\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1321.92	0.130988	0.0654938	−	0.997853i	$-0.479138\pi$
$467$	1321.92	0.130988	0.0654938	+	0.997853i	$0.479138\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 6550.51i	− 0.636771i
$474$	0	0
$475$	− 11526.6i	− 1.11343i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−3775.38	−0.360129	−0.180064	−	0.983655i	$-0.557631\pi$
$479$	−3775.38	−0.360129	−0.180064	+	0.983655i	$0.557631\pi$
$480$	0	0
$481$	− 8989.65i	− 0.852168i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	37120.0i	3.47533i
$486$	0	0
$487$	−12239.6	−1.13887	−0.569434	−	0.822037i	$-0.692838\pi$
$487$	−12239.6	−1.13887	−0.569434	+	0.822037i	$0.692838\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 13018.0i	− 1.19652i	−0.801301	−	0.598261i	$-0.795858\pi$
$491$	− 13018.0i	− 1.19652i	0.801301	−	0.598261i	$-0.204142\pi$
$492$	0	0
$493$	− 10713.6i	− 0.978738i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−17207.3	−1.54370	−0.771850	−	0.635804i	$-0.780669\pi$
$499$	−17207.3	−1.54370	−0.771850	+	0.635804i	$0.780669\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−19760.5	−1.75164	−0.875821	−	0.482635i	$-0.839680\pi$
$503$	−19760.5	−1.75164	−0.875821	+	0.482635i	$0.839680\pi$
$504$	0	0
$505$	−29205.3	−2.57350
$506$	0	0
$507$	0	0
$508$	0	0
$509$	4703.65	0.409598	0.204799	−	0.978804i	$-0.434346\pi$
$509$	4703.65	0.409598	0.204799	+	0.978804i	$0.434346\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 21709.5i	− 1.85754i
$516$	0	0
$517$	9416.26i	0.801019i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	15942.2	1.34058	0.670290	−	0.742099i	$-0.266170\pi$
$521$	15942.2	1.34058	0.670290	+	0.742099i	$0.266170\pi$
$522$	0	0
$523$	− 20048.1i	− 1.67618i	−0.545532	−	0.838090i	$-0.683672\pi$
$523$	− 20048.1i	− 1.67618i	0.545532	−	0.838090i	$-0.316328\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 10337.1i	− 0.854445i
$528$	0	0
$529$	5674.60	0.466393
$530$	0	0
$531$	0	0
$532$	0	0
$533$	5124.50i	0.416448i
$534$	0	0
$535$	− 16886.7i	− 1.36463i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	3863.69	0.307048	0.153524	−	0.988145i	$-0.450938\pi$
$541$	3863.69	0.307048	0.153524	+	0.988145i	$0.450938\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	37365.2	2.93678
$546$	0	0
$547$	−18674.5	−1.45972	−0.729859	−	0.683598i	$-0.760414\pi$
$547$	−18674.5	−1.45972	−0.729859	+	0.683598i	$0.760414\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	9453.14	0.730885
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1719.51i	0.130804i	0.997859	+	0.0654020i	$0.0208330\pi$
$557$	1719.51i	0.130804i	−0.997859	+	0.0654020i	$0.979167\pi$
$558$	0	0
$559$	12181.5i	0.921689i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	13192.7	0.987581	0.493790	−	0.869581i	$-0.335611\pi$
$563$	13192.7	0.987581	0.493790	+	0.869581i	$0.335611\pi$
$564$	0	0
$565$	28127.8i	2.09441i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	24716.9i	1.82107i	0.413437	+	0.910533i	$0.364328\pi$
$569$	24716.9i	1.82107i	−0.413437	+	0.910533i	$0.635672\pi$
$570$	0	0
$571$	2065.45	0.151377	0.0756887	−	0.997131i	$-0.475884\pi$
$571$	2065.45	0.151377	0.0756887	+	0.997131i	$0.475884\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	22974.9i	1.66630i
$576$	0	0
$577$	7805.33i	0.563154i	0.959539	+	0.281577i	$0.0908575\pi$
$577$	7805.33i	0.563154i	−0.959539	+	0.281577i	$0.909142\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	1625.45	0.115471
$584$	0	0
$585$	0	0
$586$	0	0
$587$	6297.78	0.442823	0.221412	−	0.975180i	$-0.428934\pi$
$587$	6297.78	0.442823	0.221412	+	0.975180i	$0.428934\pi$
$588$	0	0
$589$	9120.93	0.638067
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−18231.2	−1.26251	−0.631253	−	0.775577i	$-0.717459\pi$
$593$	−18231.2	−1.26251	−0.631253	+	0.775577i	$0.717459\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 8269.45i	− 0.564074i	−0.959403	−	0.282037i	$-0.908990\pi$
$599$	− 8269.45i	− 0.564074i	0.959403	−	0.282037i	$-0.0910101\pi$
$600$	0	0
$601$	11106.1i	0.753788i	0.926256	+	0.376894i	$0.123008\pi$
$601$	11106.1i	0.753788i	−0.926256	+	0.376894i	$0.876992\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−20507.0	−1.37806
$606$	0	0
$607$	− 12025.4i	− 0.804113i	−0.915615	−	0.402056i	$-0.868296\pi$
$607$	− 12025.4i	− 0.804113i	0.915615	−	0.402056i	$-0.131704\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 17510.8i	− 1.15943i
$612$	0	0
$613$	−4055.49	−0.267210	−0.133605	−	0.991035i	$-0.542655\pi$
$613$	−4055.49	−0.267210	−0.133605	+	0.991035i	$0.542655\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	7236.59i	0.472178i	0.971731	+	0.236089i	$0.0758658\pi$
$617$	7236.59i	0.472178i	−0.971731	+	0.236089i	$0.924134\pi$
$618$	0	0
$619$	− 15939.1i	− 1.03497i	−0.855693	−	0.517484i	$-0.826869\pi$
$619$	− 15939.1i	− 1.03497i	0.855693	−	0.517484i	$-0.173131\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	30035.5	1.92227
$626$	0	0
$627$	0	0
$628$	0	0
$629$	12411.9	0.786798
$630$	0	0
$631$	25310.1	1.59680	0.798399	−	0.602129i	$-0.205681\pi$
$631$	25310.1	1.59680	0.798399	+	0.602129i	$0.205681\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	16991.1	1.06184
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	24934.5i	1.53643i	0.640191	+	0.768216i	$0.278855\pi$
$641$	24934.5i	1.53643i	−0.640191	+	0.768216i	$0.721145\pi$
$642$	0	0
$643$	− 12466.1i	− 0.764563i	−0.924046	−	0.382282i	$-0.875138\pi$
$643$	− 12466.1i	− 0.764563i	0.924046	−	0.382282i	$-0.124862\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−24679.2	−1.49960	−0.749799	−	0.661665i	$-0.769850\pi$
$647$	−24679.2	−1.49960	−0.749799	+	0.661665i	$0.769850\pi$
$648$	0	0
$649$	11161.9i	0.675107i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 19050.3i	− 1.14165i	−0.821072	−	0.570825i	$-0.806624\pi$
$653$	− 19050.3i	− 1.14165i	0.821072	−	0.570825i	$-0.193376\pi$
$654$	0	0
$655$	7899.74	0.471249
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 5983.49i	− 0.353693i	−0.984238	−	0.176847i	$-0.943410\pi$
$659$	− 5983.49i	− 0.353693i	0.984238	−	0.176847i	$-0.0565897\pi$
$660$	0	0
$661$	13857.2i	0.815406i	0.913115	+	0.407703i	$0.133670\pi$
$661$	13857.2i	0.815406i	−0.913115	+	0.407703i	$0.866330\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−18842.1	−1.09380
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−1624.32	−0.0934519
$672$	0	0
$673$	−21469.4	−1.22970	−0.614849	−	0.788645i	$-0.710783\pi$
$673$	−21469.4	−1.22970	−0.614849	+	0.788645i	$0.710783\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	20442.7	1.16053	0.580265	−	0.814428i	$-0.302949\pi$
$677$	20442.7	1.16053	0.580265	+	0.814428i	$0.302949\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	19582.2i	1.09706i	0.836132	+	0.548529i	$0.184812\pi$
$683$	19582.2i	1.09706i	−0.836132	+	0.548529i	$0.815188\pi$
$684$	0	0
$685$	27868.1i	1.55443i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−3022.75	−0.167137
$690$	0	0
$691$	11705.5i	0.644426i	0.946667	+	0.322213i	$0.104427\pi$
$691$	11705.5i	0.644426i	−0.946667	+	0.322213i	$0.895573\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	19314.3i	1.05415i
$696$	0	0
$697$	−7075.34	−0.384502
$698$	0	0
$699$	0	0
$700$	0	0
$701$	17998.6i	0.969752i	0.874583	+	0.484876i	$0.161135\pi$
$701$	17998.6i	0.969752i	−0.874583	+	0.484876i	$0.838865\pi$
$702$	0	0
$703$	10951.6i	0.587551i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−23783.1	−1.25979	−0.629895	−	0.776680i	$-0.716902\pi$
$709$	−23783.1	−1.25979	−0.629895	+	0.776680i	$0.716902\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−18179.9	−0.954899
$714$	0	0
$715$	−11991.3	−0.627202
$716$	0	0
$717$	0	0
$718$	0	0
$719$	33027.0	1.71307	0.856536	−	0.516088i	$-0.172612\pi$
$719$	33027.0	1.71307	0.856536	+	0.516088i	$0.172612\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	66677.3i	3.41563i
$726$	0	0
$727$	6816.11i	0.347724i	0.984770	+	0.173862i	$0.0556247\pi$
$727$	6816.11i	0.347724i	−0.984770	+	0.173862i	$0.944375\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−16818.9	−0.850985
$732$	0	0
$733$	23903.9i	1.20452i	0.798301	+	0.602258i	$0.205732\pi$
$733$	23903.9i	1.20452i	−0.798301	+	0.602258i	$0.794268\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 15399.5i	− 0.769672i
$738$	0	0
$739$	9116.14	0.453779	0.226889	−	0.973921i	$-0.427144\pi$
$739$	9116.14	0.453779	0.226889	+	0.973921i	$0.427144\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	32818.5i	1.62045i	0.586121	+	0.810224i	$0.300654\pi$
$743$	32818.5i	1.62045i	−0.586121	+	0.810224i	$0.699346\pi$
$744$	0	0
$745$	51421.7i	2.52879i
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−9240.15	−0.448972	−0.224486	−	0.974477i	$-0.572070\pi$
$751$	−9240.15	−0.448972	−0.224486	+	0.974477i	$0.572070\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	20859.4	1.00550
$756$	0	0
$757$	16610.7	0.797523	0.398761	−	0.917055i	$-0.369440\pi$
$757$	16610.7	0.797523	0.398761	+	0.917055i	$0.369440\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	2242.84	0.106837	0.0534184	−	0.998572i	$-0.482988\pi$
$761$	2242.84	0.106837	0.0534184	+	0.998572i	$0.482988\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 20757.1i	− 0.977178i
$768$	0	0
$769$	9182.73i	0.430608i	0.976547	+	0.215304i	$0.0690743\pi$
$769$	9182.73i	0.430608i	−0.976547	+	0.215304i	$0.930926\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	8544.81	0.397588	0.198794	−	0.980041i	$-0.436298\pi$
$773$	8544.81	0.397588	0.198794	+	0.980041i	$0.436298\pi$
$774$	0	0
$775$	64334.1i	2.98187i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 6242.91i	− 0.287131i
$780$	0	0
$781$	−5417.31	−0.248203
$782$	0	0
$783$	0	0
$784$	0	0
$785$	36451.6i	1.65734i
$786$	0	0
$787$	− 23367.0i	− 1.05838i	−0.848504	−	0.529190i	$-0.822496\pi$
$787$	− 23367.0i	− 1.05838i	0.848504	−	0.529190i	$-0.177504\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	3020.64	0.135266
$794$	0	0
$795$	0	0
$796$	0	0
$797$	8319.18	0.369737	0.184869	−	0.982763i	$-0.440814\pi$
$797$	8319.18	0.369737	0.184869	+	0.982763i	$0.440814\pi$
$798$	0	0
$799$	24177.0	1.07049
$800$	0	0
$801$	0	0
$802$	0	0
$803$	20586.3	0.904699
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	8970.94i	0.389866i	0.980817	+	0.194933i	$0.0624489\pi$
$809$	8970.94i	0.389866i	−0.980817	+	0.194933i	$0.937551\pi$
$810$	0	0
$811$	− 29055.1i	− 1.25803i	−0.777393	−	0.629015i	$-0.783458\pi$
$811$	− 29055.1i	− 1.25803i	0.777393	−	0.629015i	$-0.216542\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−53141.9	−2.28402
$816$	0	0
$817$	− 14840.1i	− 0.635484i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 16009.7i	− 0.680562i	−0.940324	−	0.340281i	$-0.889478\pi$
$821$	− 16009.7i	− 0.680562i	0.940324	−	0.340281i	$-0.110522\pi$
$822$	0	0
$823$	7751.59	0.328315	0.164158	−	0.986434i	$-0.447509\pi$
$823$	7751.59	0.328315	0.164158	+	0.986434i	$0.447509\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	4705.84i	0.197870i	0.995094	+	0.0989348i	$0.0315435\pi$
$827$	4705.84i	0.197870i	−0.995094	+	0.0989348i	$0.968456\pi$
$828$	0	0
$829$	− 39077.8i	− 1.63719i	−0.574374	−	0.818593i	$-0.694755\pi$
$829$	− 39077.8i	− 1.63719i	0.574374	−	0.818593i	$-0.305245\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−14462.6	−0.599399
$836$	0	0
$837$	0	0
$838$	0	0
$839$	24699.3	1.01634	0.508172	−	0.861255i	$-0.330321\pi$
$839$	24699.3	1.01634	0.508172	+	0.861255i	$0.330321\pi$
$840$	0	0
$841$	−30294.0	−1.24212
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−22193.8	−0.903539
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 21828.8i	− 0.879299i
$852$	0	0
$853$	− 4313.27i	− 0.173134i	−0.996246	−	0.0865671i	$-0.972410\pi$
$853$	− 4313.27i	− 0.173134i	0.996246	−	0.0865671i	$-0.0275897\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	26477.1	1.05536	0.527678	−	0.849444i	$-0.323063\pi$
$857$	26477.1	1.05536	0.527678	+	0.849444i	$0.323063\pi$
$858$	0	0
$859$	− 38645.5i	− 1.53500i	−0.641046	−	0.767502i	$-0.721499\pi$
$859$	− 38645.5i	− 1.53500i	0.641046	−	0.767502i	$-0.278501\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	12485.9i	0.492498i	0.969207	+	0.246249i	$0.0791981\pi$
$863$	12485.9i	0.492498i	−0.969207	+	0.246249i	$0.920802\pi$
$864$	0	0
$865$	15139.6	0.595102
$866$	0	0
$867$	0	0
$868$	0	0
$869$	124.236i	0.00484972i
$870$	0	0
$871$	28637.4i	1.11406i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	22852.9	0.879918	0.439959	−	0.898018i	$-0.354993\pi$
$877$	22852.9	0.879918	0.439959	+	0.898018i	$0.354993\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	20304.1	0.776463	0.388231	−	0.921562i	$-0.373086\pi$
$881$	20304.1	0.776463	0.388231	+	0.921562i	$0.373086\pi$
$882$	0	0
$883$	−12868.9	−0.490456	−0.245228	−	0.969465i	$-0.578863\pi$
$883$	−12868.9	−0.490456	−0.245228	+	0.969465i	$0.578863\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	10181.6	0.385418	0.192709	−	0.981256i	$-0.438273\pi$
$887$	10181.6	0.385418	0.192709	+	0.981256i	$0.438273\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	21332.5i	0.799400i
$894$	0	0
$895$	88805.0i	3.31667i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−52761.2	−1.95738
$900$	0	0
$901$	− 4173.47i	− 0.154316i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	43408.1i	1.59440i
$906$	0	0
$907$	9181.32	0.336120	0.168060	−	0.985777i	$-0.446250\pi$
$907$	9181.32	0.336120	0.168060	+	0.985777i	$0.446250\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	18656.7i	0.678511i	0.940694	+	0.339255i	$0.110175\pi$
$911$	18656.7i	0.678511i	−0.940694	+	0.339255i	$0.889825\pi$
$912$	0	0
$913$	14555.1i	0.527605i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−4788.55	−0.171882	−0.0859410	−	0.996300i	$-0.527390\pi$
$919$	−4788.55	−0.171882	−0.0859410	+	0.996300i	$0.527390\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	10074.2	0.359259
$924$	0	0
$925$	−77246.7	−2.74579
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−1397.93	−0.0493697	−0.0246848	−	0.999695i	$-0.507858\pi$
$929$	−1397.93	−0.0493697	−0.0246848	+	0.999695i	$0.507858\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 16556.3i	− 0.579089i
$936$	0	0
$937$	13520.0i	0.471376i	0.971829	+	0.235688i	$0.0757343\pi$
$937$	13520.0i	0.471376i	−0.971829	+	0.235688i	$0.924266\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	41208.9	1.42760	0.713800	−	0.700350i	$-0.246973\pi$
$941$	41208.9	1.42760	0.713800	+	0.700350i	$0.246973\pi$
$942$	0	0
$943$	12443.4i	0.429706i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	7655.29i	0.262686i	0.991337	+	0.131343i	$0.0419289\pi$
$947$	7655.29i	0.262686i	−0.991337	+	0.131343i	$0.958071\pi$
$948$	0	0
$949$	−38282.9	−1.30950
$950$	0	0
$951$	0	0
$952$	0	0
$953$	22854.9i	0.776854i	0.921479	+	0.388427i	$0.126981\pi$
$953$	22854.9i	0.776854i	−0.921479	+	0.388427i	$0.873019\pi$
$954$	0	0
$955$	− 94245.3i	− 3.19341i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−21116.1	−0.708807
$962$	0	0
$963$	0	0
$964$	0	0
$965$	42637.9	1.42234
$966$	0	0
$967$	−11432.2	−0.380180	−0.190090	−	0.981767i	$-0.560878\pi$
$967$	−11432.2	−0.380180	−0.190090	+	0.981767i	$0.560878\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	18807.0	0.621570	0.310785	−	0.950480i	$-0.399408\pi$
$971$	18807.0	0.621570	0.310785	+	0.950480i	$0.399408\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 37064.5i	− 1.21371i	−0.794811	−	0.606857i	$-0.792430\pi$
$977$	− 37064.5i	− 1.21371i	0.794811	−	0.606857i	$-0.207570\pi$
$978$	0	0
$979$	− 5549.20i	− 0.181158i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	47082.9	1.52768	0.763841	−	0.645404i	$-0.223311\pi$
$983$	47082.9	1.52768	0.763841	+	0.645404i	$0.223311\pi$
$984$	0	0
$985$	14327.7i	0.463469i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	29579.4i	0.951032i
$990$	0	0
$991$	−34235.7	−1.09741	−0.548705	−	0.836016i	$-0.684879\pi$
$991$	−34235.7	−1.09741	−0.548705	+	0.836016i	$0.684879\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 26080.6i	− 0.830964i
$996$	0	0
$997$	8642.21i	0.274525i	0.990535	+	0.137263i	$0.0438304\pi$
$997$	8642.21i	0.274525i	−0.990535	+	0.137263i	$0.956170\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.4.f.a.881.1		16
3.2	odd	2	inner	1764.4.f.a.881.16		16
7.2	even	3		252.4.t.a.17.8	yes	16
7.3	odd	6		252.4.t.a.89.1	yes	16
7.4	even	3		1764.4.t.b.1097.8		16
7.5	odd	6		1764.4.t.b.521.1		16
7.6	odd	2	inner	1764.4.f.a.881.15		16
21.2	odd	6		252.4.t.a.17.1	✓	16
21.5	even	6		1764.4.t.b.521.8		16
21.11	odd	6		1764.4.t.b.1097.1		16
21.17	even	6		252.4.t.a.89.8	yes	16
21.20	even	2	inner	1764.4.f.a.881.2		16
28.3	even	6		1008.4.bt.b.593.1		16
28.23	odd	6		1008.4.bt.b.17.8		16
84.23	even	6		1008.4.bt.b.17.1		16
84.59	odd	6		1008.4.bt.b.593.8		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.4.t.a.17.1	✓	16	21.2	odd	6
252.4.t.a.17.8	yes	16	7.2	even	3
252.4.t.a.89.1	yes	16	7.3	odd	6
252.4.t.a.89.8	yes	16	21.17	even	6
1008.4.bt.b.17.1		16	84.23	even	6
1008.4.bt.b.17.8		16	28.23	odd	6
1008.4.bt.b.593.1		16	28.3	even	6
1008.4.bt.b.593.8		16	84.59	odd	6
1764.4.f.a.881.1		16	1.1	even	1	trivial
1764.4.f.a.881.2		16	21.20	even	2	inner
1764.4.f.a.881.15		16	7.6	odd	2	inner
1764.4.f.a.881.16		16	3.2	odd	2	inner
1764.4.t.b.521.1		16	7.5	odd	6
1764.4.t.b.521.8		16	21.5	even	6
1764.4.t.b.1097.1		16	21.11	odd	6
1764.4.t.b.1097.8		16	7.4	even	3

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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1764.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-20.2518 q^{5} +O(q^{10})\)	\(q-20.2518 q^{5} -17.8438i q^{11} +33.1829i q^{13} -45.8153 q^{17} -40.4250i q^{19} +80.5754i q^{23} +285.136 q^{25} +233.844i q^{29} +225.626i q^{31} -270.912 q^{37} +154.432 q^{41} +367.102 q^{43} -527.705 q^{47} +91.0934i q^{53} +361.370i q^{55} -625.535 q^{59} -91.0300i q^{61} -672.015i q^{65} +863.017 q^{67} -303.596i q^{71} +1153.69i q^{73} -6.96240 q^{79} -815.694 q^{83} +927.844 q^{85} +310.988 q^{89} +818.680i q^{95} -1832.92i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 424 q^{25} - 152 q^{37} + 1408 q^{43} + 3056 q^{67} + 728 q^{79} + 7392 q^{85}+O(q^{100}) \) 16 * q + 424 * q^25 - 152 * q^37 + 1408 * q^43 + 3056 * q^67 + 728 * q^79 + 7392 * q^85

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