LMFDB - Embedded newform 1728.2.r.d.289.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,2,Mod(289,1728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1728, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 4]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1728.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1728.r (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:	$13.7981494693$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.12960000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 $ x^8 - 3x^6 + 8x^4 - 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 576)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			289.1
Root			$0.535233 + 0.309017i$ of defining polynomial
Character	$\chi$	$=$	1728.289
Dual form			1728.2.r.d.1441.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.35410 + 1.93649i) q^{5} +(-1.93649 + 3.35410i) q^{7} +O(q^{10})$	$q+(-3.35410 + 1.93649i) q^{5} +(-1.93649 + 3.35410i) q^{7} +(-2.59808 - 1.50000i) q^{11} +(-3.35410 + 1.93649i) q^{13} +2.00000i q^{19} +(1.93649 + 3.35410i) q^{23} +(5.00000 - 8.66025i) q^{25} +(3.35410 + 1.93649i) q^{29} +(-1.93649 - 3.35410i) q^{31} -15.0000i q^{35} -7.74597i q^{37} +(4.50000 + 7.79423i) q^{41} +(-6.06218 - 3.50000i) q^{43} +(1.93649 - 3.35410i) q^{47} +(-4.00000 - 6.92820i) q^{49} -7.74597i q^{53} +11.6190 q^{55} +(-2.59808 + 1.50000i) q^{59} +(10.0623 + 5.80948i) q^{61} +(7.50000 - 12.9904i) q^{65} +(-9.52628 + 5.50000i) q^{67} -7.74597 q^{71} +8.00000 q^{73} +(10.0623 - 5.80948i) q^{77} +(1.93649 - 3.35410i) q^{79} +(2.59808 + 1.50000i) q^{83} +12.0000 q^{89} -15.0000i q^{91} +(-3.87298 - 6.70820i) q^{95} +(-0.500000 + 0.866025i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 40 q^{25} + 36 q^{41} - 32 q^{49} + 60 q^{65} + 64 q^{73} + 96 q^{89} - 4 q^{97}+O(q^{100}) $ 8 * q + 40 * q^25 + 36 * q^41 - 32 * q^49 + 60 * q^65 + 64 * q^73 + 96 * q^89 - 4 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$.

$n$	$325$	$703$	$1217$
$\chi(n)$	$-1$	$1$	$e\left(\frac{2}{3}\right)$

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
$5$	−3.35410	+	1.93649i	−1.50000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$5$	−3.35410	+	1.93649i	−1.50000	+	0.866025i	−1.00000			$\pi$
$6$		0			0
$7$	−1.93649	+	3.35410i	−0.731925	+	1.26773i	0.224134	+	0.974558i	$0.428045\pi$
$7$	−1.93649	+	3.35410i	−0.731925	+	1.26773i	−0.956059	+	0.293173i	$0.905289\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.59808	−	1.50000i	−0.783349	−	0.452267i	0.0542666	−	0.998526i	$-0.482718\pi$
$11$	−2.59808	−	1.50000i	−0.783349	−	0.452267i	−0.837616	+	0.546259i	$0.816051\pi$
$12$		0			0
$13$	−3.35410	+	1.93649i	−0.930261	+	0.537086i	−0.886894	−	0.461973i	$-0.847142\pi$
$13$	−3.35410	+	1.93649i	−0.930261	+	0.537086i	−0.0433666	+	0.999059i	$0.513808\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$			2.00000i			0.458831i	0.973329	+	0.229416i	$0.0736815\pi$
$19$			2.00000i			0.458831i	−0.973329	+	0.229416i	$0.926318\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.93649	+	3.35410i	0.403786	+	0.699379i	0.994179	−	0.107737i	$-0.0343606\pi$
$23$	1.93649	+	3.35410i	0.403786	+	0.699379i	−0.590393	+	0.807116i	$0.701027\pi$
$24$		0			0
$25$	5.00000	−	8.66025i	1.00000	−	1.73205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3.35410	+	1.93649i	0.622841	+	0.359597i	0.777974	−	0.628296i	$-0.216247\pi$
$29$	3.35410	+	1.93649i	0.622841	+	0.359597i	−0.155133	+	0.987894i	$0.549581\pi$
$30$		0			0
$31$	−1.93649	−	3.35410i	−0.347804	−	0.602414i	0.638055	−	0.769991i	$-0.279739\pi$
$31$	−1.93649	−	3.35410i	−0.347804	−	0.602414i	−0.985859	+	0.167576i	$0.946406\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	15.0000i		−	2.53546i
$36$		0			0
$37$		−	7.74597i		−	1.27343i	−0.771100	−	0.636715i	$-0.780293\pi$
$37$		−	7.74597i		−	1.27343i	0.771100	−	0.636715i	$-0.219707\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	4.50000	+	7.79423i	0.702782	+	1.21725i	0.967486	+	0.252924i	$0.0813924\pi$
$41$	4.50000	+	7.79423i	0.702782	+	1.21725i	−0.264704	+	0.964330i	$0.585274\pi$
$42$		0			0
$43$	−6.06218	−	3.50000i	−0.924473	−	0.533745i	−0.0394140	−	0.999223i	$-0.512549\pi$
$43$	−6.06218	−	3.50000i	−0.924473	−	0.533745i	−0.885059	+	0.465478i	$0.845882\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.93649	−	3.35410i	0.282466	−	0.489246i	−0.689525	−	0.724262i	$-0.742181\pi$
$47$	1.93649	−	3.35410i	0.282466	−	0.489246i	0.971992	+	0.235016i	$0.0755141\pi$
$48$		0			0
$49$	−4.00000	−	6.92820i	−0.571429	−	0.989743i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	7.74597i		−	1.06399i	−0.846747	−	0.531995i	$-0.821442\pi$
$53$		−	7.74597i		−	1.06399i	0.846747	−	0.531995i	$-0.178558\pi$
$54$		0			0
$55$	11.6190			1.56670
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−2.59808	+	1.50000i	−0.338241	+	0.195283i	−0.659494	−	0.751710i	$-0.729229\pi$
$59$	−2.59808	+	1.50000i	−0.338241	+	0.195283i	0.321253	+	0.946993i	$0.395896\pi$
$60$		0			0
$61$	10.0623	+	5.80948i	1.28835	+	0.743827i	0.978359	−	0.206914i	$-0.0663419\pi$
$61$	10.0623	+	5.80948i	1.28835	+	0.743827i	0.309987	+	0.950741i	$0.399675\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	7.50000	−	12.9904i	0.930261	−	1.61126i
$66$		0			0
$67$	−9.52628	+	5.50000i	−1.16382	+	0.671932i	−0.952217	−	0.305424i	$-0.901202\pi$
$67$	−9.52628	+	5.50000i	−1.16382	+	0.671932i	−0.211604	+	0.977356i	$0.567869\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−7.74597			−0.919277			−0.459639	−	0.888106i	$-0.652021\pi$
$71$	−7.74597			−0.919277			−0.459639	+	0.888106i	$0.652021\pi$
$72$		0			0
$73$	8.00000			0.936329			0.468165	−	0.883641i	$-0.344915\pi$
$73$	8.00000			0.936329			0.468165	+	0.883641i	$0.344915\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	10.0623	−	5.80948i	1.14671	−	0.662051i
$78$		0			0
$79$	1.93649	−	3.35410i	0.217872	−	0.377366i	−0.736285	−	0.676672i	$-0.763422\pi$
$79$	1.93649	−	3.35410i	0.217872	−	0.377366i	0.954157	+	0.299306i	$0.0967550\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	2.59808	+	1.50000i	0.285176	+	0.164646i	0.635764	−	0.771883i	$-0.280685\pi$
$83$	2.59808	+	1.50000i	0.285176	+	0.164646i	−0.350588	+	0.936530i	$0.614018\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	12.0000			1.27200			0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000			1.27200			0.635999	+	0.771690i	$0.280588\pi$
$90$		0			0
$91$		−	15.0000i		−	1.57243i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−3.87298	−	6.70820i	−0.397360	−	0.688247i
$96$		0			0
$97$	−0.500000	+	0.866025i	−0.0507673	+	0.0879316i	−0.890292	−	0.455389i	$-0.849500\pi$
$97$	−0.500000	+	0.866025i	−0.0507673	+	0.0879316i	0.839525	+	0.543321i	$0.182833\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−3.35410	−	1.93649i	−0.333746	−	0.192688i	0.323757	−	0.946140i	$-0.395054\pi$
$101$	−3.35410	−	1.93649i	−0.333746	−	0.192688i	−0.657503	+	0.753452i	$0.728387\pi$
$102$		0			0
$103$	−5.80948	−	10.0623i	−0.572425	−	0.991468i	−0.996316	−	0.0857555i	$-0.972670\pi$
$103$	−5.80948	−	10.0623i	−0.572425	−	0.991468i	0.423892	−	0.905713i	$-0.360664\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	6.00000i		−	0.580042i	−0.957020	−	0.290021i	$-0.906338\pi$
$107$		−	6.00000i		−	0.580042i	0.957020	−	0.290021i	$-0.0936623\pi$
$108$		0			0
$109$			7.74597i			0.741929i	0.928647	+	0.370965i	$0.120973\pi$
$109$			7.74597i			0.741929i	−0.928647	+	0.370965i	$0.879027\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1.50000	+	2.59808i	0.141108	+	0.244406i	0.927914	−	0.372794i	$-0.121600\pi$
$113$	1.50000	+	2.59808i	0.141108	+	0.244406i	−0.786806	+	0.617200i	$0.788267\pi$
$114$		0			0
$115$	−12.9904	−	7.50000i	−1.21136	−	0.699379i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−1.00000	−	1.73205i	−0.0909091	−	0.157459i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			19.3649i			1.73205i
$126$		0			0
$127$	−15.4919			−1.37469			−0.687343	−	0.726333i	$-0.741223\pi$
$127$	−15.4919			−1.37469			−0.687343	+	0.726333i	$0.741223\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−2.59808	+	1.50000i	−0.226995	+	0.131056i	−0.609185	−	0.793028i	$-0.708503\pi$
$131$	−2.59808	+	1.50000i	−0.226995	+	0.131056i	0.382190	+	0.924084i	$0.375170\pi$
$132$		0			0
$133$	−6.70820	−	3.87298i	−0.581675	−	0.335830i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	10.5000	−	18.1865i	0.897076	−	1.55378i	0.0658609	−	0.997829i	$-0.479021\pi$
$137$	10.5000	−	18.1865i	0.897076	−	1.55378i	0.831215	−	0.555952i	$-0.187646\pi$
$138$		0			0
$139$	−0.866025	+	0.500000i	−0.0734553	+	0.0424094i	−0.536278	−	0.844042i	$-0.680170\pi$
$139$	−0.866025	+	0.500000i	−0.0734553	+	0.0424094i	0.462822	+	0.886451i	$0.346837\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	11.6190			0.971625
$144$		0			0
$145$	−15.0000			−1.24568
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−10.0623	+	5.80948i	−0.824336	+	0.475931i	−0.851909	−	0.523689i	$-0.824555\pi$
$149$	−10.0623	+	5.80948i	−0.824336	+	0.475931i	0.0275733	+	0.999620i	$0.491222\pi$
$150$		0			0
$151$	−5.80948	+	10.0623i	−0.472768	+	0.818859i	−0.999514	−	0.0311639i	$-0.990079\pi$
$151$	−5.80948	+	10.0623i	−0.472768	+	0.818859i	0.526746	+	0.850023i	$0.323412\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	12.9904	+	7.50000i	1.04341	+	0.602414i
$156$		0			0
$157$	3.35410	−	1.93649i	0.267686	−	0.154549i	−0.360149	−	0.932895i	$-0.617274\pi$
$157$	3.35410	−	1.93649i	0.267686	−	0.154549i	0.627836	+	0.778346i	$0.283941\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−15.0000			−1.18217
$162$		0			0
$163$			2.00000i			0.156652i	0.996928	+	0.0783260i	$0.0249575\pi$
$163$			2.00000i			0.156652i	−0.996928	+	0.0783260i	$0.975042\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−1.93649	−	3.35410i	−0.149850	−	0.259548i	0.781322	−	0.624128i	$-0.214546\pi$
$167$	−1.93649	−	3.35410i	−0.149850	−	0.259548i	−0.931172	+	0.364580i	$0.881212\pi$
$168$		0			0
$169$	1.00000	−	1.73205i	0.0769231	−	0.133235i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−16.7705	−	9.68246i	−1.27504	−	0.736144i	−0.299106	−	0.954220i	$-0.596689\pi$
$173$	−16.7705	−	9.68246i	−1.27504	−	0.736144i	−0.975932	+	0.218076i	$0.930022\pi$
$174$		0			0
$175$	19.3649	+	33.5410i	1.46385	+	2.53546i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	18.0000i		−	1.34538i	−0.739923	−	0.672692i	$-0.765138\pi$
$179$		−	18.0000i		−	1.34538i	0.739923	−	0.672692i	$-0.234862\pi$
$180$		0			0
$181$			7.74597i			0.575753i	0.957668	+	0.287877i	$0.0929493\pi$
$181$			7.74597i			0.575753i	−0.957668	+	0.287877i	$0.907051\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	15.0000	+	25.9808i	1.10282	+	1.91014i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−13.5554	+	23.4787i	−0.980837	+	1.69886i	−0.321692	+	0.946844i	$0.604252\pi$
$191$	−13.5554	+	23.4787i	−0.980837	+	1.69886i	−0.659145	+	0.752016i	$0.729082\pi$
$192$		0			0
$193$	−2.50000	−	4.33013i	−0.179954	−	0.311689i	0.761911	−	0.647682i	$-0.224262\pi$
$193$	−2.50000	−	4.33013i	−0.179954	−	0.311689i	−0.941865	+	0.335993i	$0.890928\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	−7.74597			−0.549097			−0.274549	−	0.961573i	$-0.588528\pi$
$199$	−7.74597			−0.549097			−0.274549	+	0.961573i	$0.588528\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−12.9904	+	7.50000i	−0.911746	+	0.526397i
$204$		0			0
$205$	−30.1869	−	17.4284i	−2.10835	−	1.21725i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	3.00000	−	5.19615i	0.207514	−	0.359425i
$210$		0			0
$211$	0.866025	−	0.500000i	0.0596196	−	0.0344214i	−0.469894	−	0.882723i	$-0.655708\pi$
$211$	0.866025	−	0.500000i	0.0596196	−	0.0344214i	0.529514	+	0.848301i	$0.322374\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	27.1109			1.84895
$216$		0			0
$217$	15.0000			1.01827
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	13.5554	−	23.4787i	0.907740	−	1.57225i	0.0905425	−	0.995893i	$-0.471140\pi$
$223$	13.5554	−	23.4787i	0.907740	−	1.57225i	0.817197	−	0.576358i	$-0.195527\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	18.1865	+	10.5000i	1.20708	+	0.696909i	0.962121	−	0.272623i	$-0.0878913\pi$
$227$	18.1865	+	10.5000i	1.20708	+	0.696909i	0.244962	+	0.969533i	$0.421225\pi$
$228$		0			0
$229$	3.35410	−	1.93649i	0.221645	−	0.127967i	−0.385067	−	0.922889i	$-0.625822\pi$
$229$	3.35410	−	1.93649i	0.221645	−	0.127967i	0.606712	+	0.794922i	$0.292488\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−18.0000			−1.17922			−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000			−1.17922			−0.589610	+	0.807688i	$0.700718\pi$
$234$		0			0
$235$			15.0000i			0.978492i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	9.68246	+	16.7705i	0.626306	+	1.08479i	0.988287	+	0.152608i	$0.0487673\pi$
$239$	9.68246	+	16.7705i	0.626306	+	1.08479i	−0.361981	+	0.932186i	$0.617899\pi$
$240$		0			0
$241$	−5.50000	+	9.52628i	−0.354286	+	0.613642i	−0.986996	−	0.160748i	$-0.948609\pi$
$241$	−5.50000	+	9.52628i	−0.354286	+	0.613642i	0.632709	+	0.774389i	$0.281943\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	26.8328	+	15.4919i	1.71429	+	0.989743i
$246$		0			0
$247$	−3.87298	−	6.70820i	−0.246432	−	0.426833i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			6.00000i			0.378717i	0.981908	+	0.189358i	$0.0606408\pi$
$251$			6.00000i			0.378717i	−0.981908	+	0.189358i	$0.939359\pi$
$252$		0			0
$253$		−	11.6190i		−	0.730477i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−10.5000	−	18.1865i	−0.654972	−	1.13444i	−0.981901	−	0.189396i	$-0.939347\pi$
$257$	−10.5000	−	18.1865i	−0.654972	−	1.13444i	0.326929	−	0.945049i	$-0.393986\pi$
$258$		0			0
$259$	25.9808	+	15.0000i	1.61437	+	0.932055i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	5.80948	−	10.0623i	0.358228	−	0.620468i	−0.629437	−	0.777051i	$-0.716715\pi$
$263$	5.80948	−	10.0623i	0.358228	−	0.620468i	0.987665	+	0.156583i	$0.0500479\pi$
$264$		0			0
$265$	15.0000	+	25.9808i	0.921443	+	1.59599i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	23.2379i		−	1.41684i	−0.705791	−	0.708420i	$-0.749408\pi$
$269$		−	23.2379i		−	1.41684i	0.705791	−	0.708420i	$-0.250592\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−25.9808	+	15.0000i	−1.56670	+	0.904534i
$276$		0			0
$277$	16.7705	+	9.68246i	1.00764	+	0.581763i	0.910500	−	0.413508i	$-0.135697\pi$
$277$	16.7705	+	9.68246i	1.00764	+	0.581763i	0.0971418	+	0.995271i	$0.469030\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−13.5000	+	23.3827i	−0.805342	+	1.39489i	0.110717	+	0.993852i	$0.464685\pi$
$281$	−13.5000	+	23.3827i	−0.805342	+	1.39489i	−0.916060	+	0.401042i	$0.868648\pi$
$282$		0			0
$283$	19.9186	−	11.5000i	1.18404	−	0.683604i	0.227092	−	0.973873i	$-0.427078\pi$
$283$	19.9186	−	11.5000i	1.18404	−	0.683604i	0.956945	+	0.290269i	$0.0937449\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−34.8569			−2.05753
$288$		0			0
$289$	−17.0000			−1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$	23.4787	−	13.5554i	1.37164	−	0.791917i	0.380506	−	0.924778i	$-0.375750\pi$
$293$	23.4787	−	13.5554i	1.37164	−	0.791917i	0.991135	+	0.132861i	$0.0424164\pi$
$294$		0			0
$295$	5.80948	−	10.0623i	0.338241	−	0.585850i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−12.9904	−	7.50000i	−0.751253	−	0.433736i
$300$		0			0
$301$	23.4787	−	13.5554i	1.35329	−	0.781323i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−45.0000			−2.57669
$306$		0			0
$307$			10.0000i			0.570730i	0.958419	+	0.285365i	$0.0921148\pi$
$307$			10.0000i			0.570730i	−0.958419	+	0.285365i	$0.907885\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−13.5554	−	23.4787i	−0.768659	−	1.33136i	−0.938290	−	0.345848i	$-0.887591\pi$
$311$	−13.5554	−	23.4787i	−0.768659	−	1.33136i	0.169632	−	0.985507i	$-0.445742\pi$
$312$		0			0
$313$	8.50000	−	14.7224i	0.480448	−	0.832161i	−0.519300	−	0.854592i	$-0.673807\pi$
$313$	8.50000	−	14.7224i	0.480448	−	0.832161i	0.999748	+	0.0224310i	$0.00714060\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−10.0623	−	5.80948i	−0.565155	−	0.326293i	0.190057	−	0.981773i	$-0.439133\pi$
$317$	−10.0623	−	5.80948i	−0.565155	−	0.326293i	−0.755212	+	0.655480i	$0.772466\pi$
$318$		0			0
$319$	−5.80948	−	10.0623i	−0.325268	−	0.563381i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$			38.7298i			2.14834i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	7.50000	+	12.9904i	0.413488	+	0.716183i
$330$		0			0
$331$	16.4545	+	9.50000i	0.904420	+	0.522167i	0.878632	−	0.477500i	$-0.158457\pi$
$331$	16.4545	+	9.50000i	0.904420	+	0.522167i	0.0257885	+	0.999667i	$0.491790\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	21.3014	−	36.8951i	1.16382	−	2.01580i
$336$		0			0
$337$	2.50000	+	4.33013i	0.136184	+	0.235877i	0.926049	−	0.377403i	$-0.123183\pi$
$337$	2.50000	+	4.33013i	0.136184	+	0.235877i	−0.789865	+	0.613280i	$0.789850\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			11.6190i			0.629201i
$342$		0			0
$343$	3.87298			0.209121
$344$		0			0
$345$		0			0
$346$		0			0
$347$	12.9904	−	7.50000i	0.697360	−	0.402621i	−0.109003	−	0.994041i	$-0.534766\pi$
$347$	12.9904	−	7.50000i	0.697360	−	0.402621i	0.806363	+	0.591420i	$0.201433\pi$
$348$		0			0
$349$	16.7705	+	9.68246i	0.897705	+	0.518290i	0.876455	−	0.481484i	$-0.159902\pi$
$349$	16.7705	+	9.68246i	0.897705	+	0.518290i	0.0212500	+	0.999774i	$0.493235\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1.50000	+	2.59808i	−0.0798369	+	0.138282i	−0.903179	−	0.429263i	$-0.858773\pi$
$353$	−1.50000	+	2.59808i	−0.0798369	+	0.138282i	0.823343	+	0.567545i	$0.192107\pi$
$354$		0			0
$355$	25.9808	−	15.0000i	1.37892	−	0.796117i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−30.9839			−1.63527			−0.817633	−	0.575740i	$-0.804714\pi$
$359$	−30.9839			−1.63527			−0.817633	+	0.575740i	$0.804714\pi$
$360$		0			0
$361$	15.0000			0.789474
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−26.8328	+	15.4919i	−1.40449	+	0.810885i
$366$		0			0
$367$	−5.80948	+	10.0623i	−0.303252	+	0.525248i	−0.976871	−	0.213831i	$-0.931406\pi$
$367$	−5.80948	+	10.0623i	−0.303252	+	0.525248i	0.673619	+	0.739079i	$0.264739\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	25.9808	+	15.0000i	1.34885	+	0.778761i
$372$		0			0
$373$	−3.35410	+	1.93649i	−0.173669	+	0.100268i	−0.584315	−	0.811527i	$-0.698637\pi$
$373$	−3.35410	+	1.93649i	−0.173669	+	0.100268i	0.410646	+	0.911795i	$0.365303\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−15.0000			−0.772539
$378$		0			0
$379$			16.0000i			0.821865i	0.911666	+	0.410932i	$0.134797\pi$
$379$			16.0000i			0.821865i	−0.911666	+	0.410932i	$0.865203\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−1.93649	−	3.35410i	−0.0989501	−	0.171387i	0.812300	−	0.583240i	$-0.198215\pi$
$383$	−1.93649	−	3.35410i	−0.0989501	−	0.171387i	−0.911250	+	0.411853i	$0.864882\pi$
$384$		0			0
$385$	−22.5000	+	38.9711i	−1.14671	+	1.98615i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−3.35410	−	1.93649i	−0.170060	−	0.0981840i	0.412554	−	0.910933i	$-0.364637\pi$
$389$	−3.35410	−	1.93649i	−0.170060	−	0.0981840i	−0.582614	+	0.812749i	$0.697970\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$			15.0000i			0.754732i
$396$		0			0
$397$		−	7.74597i		−	0.388759i	−0.980926	−	0.194379i	$-0.937731\pi$
$397$		−	7.74597i		−	0.388759i	0.980926	−	0.194379i	$-0.0622693\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−1.50000	−	2.59808i	−0.0749064	−	0.129742i	0.826139	−	0.563466i	$-0.190532\pi$
$401$	−1.50000	−	2.59808i	−0.0749064	−	0.129742i	−0.901046	+	0.433724i	$0.857199\pi$
$402$		0			0
$403$	12.9904	+	7.50000i	0.647097	+	0.373602i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−11.6190	+	20.1246i	−0.575930	+	0.997540i
$408$		0			0
$409$	−11.5000	−	19.9186i	−0.568638	−	0.984911i	−0.996701	−	0.0811615i	$-0.974137\pi$
$409$	−11.5000	−	19.9186i	−0.568638	−	0.984911i	0.428063	−	0.903749i	$-0.359196\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	11.6190i		−	0.571731i
$414$		0			0
$415$	−11.6190			−0.570352
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−28.5788	+	16.5000i	−1.39617	+	0.806078i	−0.993989	−	0.109483i	$-0.965080\pi$
$419$	−28.5788	+	16.5000i	−1.39617	+	0.806078i	−0.402179	+	0.915561i	$0.631747\pi$
$420$		0			0
$421$	−3.35410	−	1.93649i	−0.163469	−	0.0943788i	0.416034	−	0.909349i	$-0.363420\pi$
$421$	−3.35410	−	1.93649i	−0.163469	−	0.0943788i	−0.579503	+	0.814970i	$0.696753\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−38.9711	+	22.5000i	−1.88595	+	1.08885i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	15.4919			0.746220			0.373110	−	0.927787i	$-0.378291\pi$
$431$	15.4919			0.746220			0.373110	+	0.927787i	$0.378291\pi$
$432$		0			0
$433$	−20.0000			−0.961139			−0.480569	−	0.876957i	$-0.659570\pi$
$433$	−20.0000			−0.961139			−0.480569	+	0.876957i	$0.659570\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−6.70820	+	3.87298i	−0.320897	+	0.185270i
$438$		0			0
$439$	−5.80948	+	10.0623i	−0.277271	+	0.480248i	−0.970706	−	0.240272i	$-0.922763\pi$
$439$	−5.80948	+	10.0623i	−0.277271	+	0.480248i	0.693434	+	0.720520i	$0.256097\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	12.9904	+	7.50000i	0.617192	+	0.356336i	0.775775	−	0.631010i	$-0.217359\pi$
$443$	12.9904	+	7.50000i	0.617192	+	0.356336i	−0.158583	+	0.987346i	$0.550693\pi$
$444$		0			0
$445$	−40.2492	+	23.2379i	−1.90800	+	1.10158i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−12.0000			−0.566315			−0.283158	−	0.959073i	$-0.591382\pi$
$449$	−12.0000			−0.566315			−0.283158	+	0.959073i	$0.591382\pi$
$450$		0			0
$451$		−	27.0000i		−	1.27138i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	29.0474	+	50.3115i	1.36176	+	2.35864i
$456$		0			0
$457$	−14.5000	+	25.1147i	−0.678281	+	1.17482i	0.297217	+	0.954810i	$0.403942\pi$
$457$	−14.5000	+	25.1147i	−0.678281	+	1.17482i	−0.975498	+	0.220008i	$0.929392\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	30.1869	+	17.4284i	1.40594	+	0.811723i	0.994994	−	0.0999348i	$-0.0318634\pi$
$461$	30.1869	+	17.4284i	1.40594	+	0.811723i	0.410951	+	0.911657i	$0.365197\pi$
$462$		0			0
$463$	−9.68246	−	16.7705i	−0.449982	−	0.779392i	0.548402	−	0.836215i	$-0.315236\pi$
$463$	−9.68246	−	16.7705i	−0.449982	−	0.779392i	−0.998384	+	0.0568230i	$0.981903\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			6.00000i			0.277647i	0.990317	+	0.138823i	$0.0443321\pi$
$467$			6.00000i			0.277647i	−0.990317	+	0.138823i	$0.955668\pi$
$468$		0			0
$469$		−	42.6028i		−	1.96722i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	10.5000	+	18.1865i	0.482791	+	0.836218i
$474$		0			0
$475$	17.3205	+	10.0000i	0.794719	+	0.458831i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	5.80948	−	10.0623i	0.265442	−	0.459758i	−0.702238	−	0.711943i	$-0.747816\pi$
$479$	5.80948	−	10.0623i	0.265442	−	0.459758i	0.967679	+	0.252184i	$0.0811489\pi$
$480$		0			0
$481$	15.0000	+	25.9808i	0.683941	+	1.18462i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	3.87298i		−	0.175863i
$486$		0			0
$487$	30.9839			1.40401			0.702007	−	0.712171i	$-0.252288\pi$
$487$	30.9839			1.40401			0.702007	+	0.712171i	$0.252288\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	18.1865	−	10.5000i	0.820747	−	0.473858i	−0.0299272	−	0.999552i	$-0.509528\pi$
$491$	18.1865	−	10.5000i	0.820747	−	0.473858i	0.850674	+	0.525694i	$0.176194\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	15.0000	−	25.9808i	0.672842	−	1.16540i
$498$		0			0
$499$	6.06218	−	3.50000i	0.271380	−	0.156682i	−0.358134	−	0.933670i	$-0.616587\pi$
$499$	6.06218	−	3.50000i	0.271380	−	0.156682i	0.629515	+	0.776989i	$0.283254\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	7.74597			0.345376			0.172688	−	0.984977i	$-0.444755\pi$
$503$	7.74597			0.345376			0.172688	+	0.984977i	$0.444755\pi$
$504$		0			0
$505$	15.0000			0.667491
$506$		0			0
$507$		0			0
$508$		0			0
$509$	3.35410	−	1.93649i	0.148668	−	0.0858335i	−0.423821	−	0.905746i	$-0.639311\pi$
$509$	3.35410	−	1.93649i	0.148668	−	0.0858335i	0.572489	+	0.819913i	$0.305978\pi$
$510$		0			0
$511$	−15.4919	+	26.8328i	−0.685323	+	1.18701i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	38.9711	+	22.5000i	1.71727	+	0.991468i
$516$		0			0
$517$	−10.0623	+	5.80948i	−0.442540	+	0.255500i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	18.0000			0.788594			0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000			0.788594			0.394297	+	0.918983i	$0.370988\pi$
$522$		0			0
$523$		−	8.00000i		−	0.349816i	−0.984585	−	0.174908i	$-0.944037\pi$
$523$		−	8.00000i		−	0.349816i	0.984585	−	0.174908i	$-0.0559627\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	4.00000	−	6.92820i	0.173913	−	0.301226i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−30.1869	−	17.4284i	−1.30754	−	0.754909i
$534$		0			0
$535$	11.6190	+	20.1246i	0.502331	+	0.870063i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			24.0000i			1.03375i
$540$		0			0
$541$		−	30.9839i		−	1.33210i	−0.745907	−	0.666050i	$-0.767984\pi$
$541$		−	30.9839i		−	1.33210i	0.745907	−	0.666050i	$-0.232016\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−15.0000	−	25.9808i	−0.642529	−	1.11289i
$546$		0			0
$547$	−35.5070	−	20.5000i	−1.51817	−	0.876517i	−0.999771	−	0.0213785i	$-0.993195\pi$
$547$	−35.5070	−	20.5000i	−1.51817	−	0.876517i	−0.518400	−	0.855138i	$-0.673472\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−3.87298	+	6.70820i	−0.164995	+	0.285779i
$552$		0			0
$553$	7.50000	+	12.9904i	0.318932	+	0.552407i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			23.2379i			0.984621i	0.870420	+	0.492311i	$0.163848\pi$
$557$			23.2379i			0.984621i	−0.870420	+	0.492311i	$0.836152\pi$
$558$		0			0
$559$	27.1109			1.14667
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−28.5788	+	16.5000i	−1.20445	+	0.695392i	−0.961542	−	0.274656i	$-0.911436\pi$
$563$	−28.5788	+	16.5000i	−1.20445	+	0.695392i	−0.242912	+	0.970048i	$0.578103\pi$
$564$		0			0
$565$	−10.0623	−	5.80948i	−0.423324	−	0.244406i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−4.50000	+	7.79423i	−0.188650	+	0.326751i	−0.944800	−	0.327647i	$-0.893744\pi$
$569$	−4.50000	+	7.79423i	−0.188650	+	0.326751i	0.756151	+	0.654398i	$0.227078\pi$
$570$		0			0
$571$	35.5070	−	20.5000i	1.48592	−	0.857898i	0.486052	−	0.873930i	$-0.338437\pi$
$571$	35.5070	−	20.5000i	1.48592	−	0.857898i	0.999871	+	0.0160316i	$0.00510324\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	38.7298			1.61515
$576$		0			0
$577$	−44.0000			−1.83174			−0.915872	−	0.401470i	$-0.868499\pi$
$577$	−44.0000			−1.83174			−0.915872	+	0.401470i	$0.868499\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−10.0623	+	5.80948i	−0.417455	+	0.241018i
$582$		0			0
$583$	−11.6190	+	20.1246i	−0.481208	+	0.833476i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−38.9711	−	22.5000i	−1.60851	−	0.928674i	−0.989704	−	0.143132i	$-0.954283\pi$
$587$	−38.9711	−	22.5000i	−1.60851	−	0.928674i	−0.618808	−	0.785543i	$-0.712384\pi$
$588$		0			0
$589$	6.70820	−	3.87298i	0.276407	−	0.159583i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	12.0000			0.492781			0.246390	−	0.969171i	$-0.420755\pi$
$593$	12.0000			0.492781			0.246390	+	0.969171i	$0.420755\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−9.68246	−	16.7705i	−0.395615	−	0.685224i	0.597565	−	0.801821i	$-0.296135\pi$
$599$	−9.68246	−	16.7705i	−0.395615	−	0.685224i	−0.993179	+	0.116596i	$0.962802\pi$
$600$		0			0
$601$	−8.50000	+	14.7224i	−0.346722	+	0.600541i	−0.985665	−	0.168714i	$-0.946039\pi$
$601$	−8.50000	+	14.7224i	−0.346722	+	0.600541i	0.638943	+	0.769254i	$0.279372\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	6.70820	+	3.87298i	0.272727	+	0.157459i
$606$		0			0
$607$	−13.5554	−	23.4787i	−0.550198	−	0.952972i	−0.998260	−	0.0589688i	$-0.981219\pi$
$607$	−13.5554	−	23.4787i	−0.550198	−	0.952972i	0.448061	−	0.894003i	$-0.352115\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			15.0000i			0.606835i
$612$		0			0
$613$		−	23.2379i		−	0.938570i	−0.883047	−	0.469285i	$-0.844512\pi$
$613$		−	23.2379i		−	0.938570i	0.883047	−	0.469285i	$-0.155488\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−19.5000	−	33.7750i	−0.785040	−	1.35973i	−0.928975	−	0.370143i	$-0.879309\pi$
$617$	−19.5000	−	33.7750i	−0.785040	−	1.35973i	0.143934	−	0.989587i	$-0.454025\pi$
$618$		0			0
$619$	−14.7224	−	8.50000i	−0.591744	−	0.341644i	0.174042	−	0.984738i	$-0.444317\pi$
$619$	−14.7224	−	8.50000i	−0.591744	−	0.341644i	−0.765787	+	0.643094i	$0.777650\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−23.2379	+	40.2492i	−0.931007	+	1.61255i
$624$		0			0
$625$	−12.5000	−	21.6506i	−0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−15.4919			−0.616724			−0.308362	−	0.951269i	$-0.599781\pi$
$631$	−15.4919			−0.616724			−0.308362	+	0.951269i	$0.599781\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	51.9615	−	30.0000i	2.06203	−	1.19051i
$636$		0			0
$637$	26.8328	+	15.4919i	1.06315	+	0.613813i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−10.5000	+	18.1865i	−0.414725	+	0.718325i	−0.995400	−	0.0958109i	$-0.969456\pi$
$641$	−10.5000	+	18.1865i	−0.414725	+	0.718325i	0.580674	+	0.814136i	$0.302789\pi$
$642$		0			0
$643$	−6.06218	+	3.50000i	−0.239069	+	0.138027i	−0.614749	−	0.788723i	$-0.710743\pi$
$643$	−6.06218	+	3.50000i	−0.239069	+	0.138027i	0.375680	+	0.926750i	$0.377409\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	15.4919			0.609051			0.304525	−	0.952504i	$-0.401502\pi$
$647$	15.4919			0.609051			0.304525	+	0.952504i	$0.401502\pi$
$648$		0			0
$649$	9.00000			0.353281
$650$		0			0
$651$		0			0
$652$		0			0
$653$	10.0623	−	5.80948i	0.393768	−	0.227342i	−0.290023	−	0.957020i	$-0.593663\pi$
$653$	10.0623	−	5.80948i	0.393768	−	0.227342i	0.683792	+	0.729677i	$0.260330\pi$
$654$		0			0
$655$	5.80948	−	10.0623i	0.226995	−	0.393167i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−23.3827	−	13.5000i	−0.910860	−	0.525885i	−0.0301523	−	0.999545i	$-0.509599\pi$
$659$	−23.3827	−	13.5000i	−0.910860	−	0.525885i	−0.880708	+	0.473660i	$0.842933\pi$
$660$		0			0
$661$	−10.0623	+	5.80948i	−0.391378	+	0.225962i	−0.682757	−	0.730645i	$-0.739219\pi$
$661$	−10.0623	+	5.80948i	−0.391378	+	0.225962i	0.291379	+	0.956608i	$0.405886\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	30.0000			1.16335
$666$		0			0
$667$			15.0000i			0.580802i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−17.4284	−	30.1869i	−0.672817	−	1.16535i
$672$		0			0
$673$	17.5000	−	30.3109i	0.674575	−	1.16840i	−0.302017	−	0.953302i	$-0.597660\pi$
$673$	17.5000	−	30.3109i	0.674575	−	1.16840i	0.976593	−	0.215096i	$-0.0690066\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−16.7705	−	9.68246i	−0.644543	−	0.372127i	0.141819	−	0.989893i	$-0.454705\pi$
$677$	−16.7705	−	9.68246i	−0.644543	−	0.372127i	−0.786362	+	0.617765i	$0.788038\pi$
$678$		0			0
$679$	−1.93649	−	3.35410i	−0.0743157	−	0.128719i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	18.0000i		−	0.688751i	−0.938832	−	0.344375i	$-0.888091\pi$
$683$		−	18.0000i		−	0.688751i	0.938832	−	0.344375i	$-0.111909\pi$
$684$		0			0
$685$			81.3327i			3.10756i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	15.0000	+	25.9808i	0.571454	+	0.989788i
$690$		0			0
$691$	0.866025	+	0.500000i	0.0329452	+	0.0190209i	0.516382	−	0.856358i	$-0.327278\pi$
$691$	0.866025	+	0.500000i	0.0329452	+	0.0190209i	−0.483437	+	0.875379i	$0.660612\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	1.93649	−	3.35410i	0.0734553	−	0.127228i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	23.2379i		−	0.877683i	−0.898564	−	0.438842i	$-0.855389\pi$
$701$		−	23.2379i		−	0.877683i	0.898564	−	0.438842i	$-0.144611\pi$
$702$		0			0
$703$	15.4919			0.584289
$704$		0			0
$705$		0			0
$706$		0			0
$707$	12.9904	−	7.50000i	0.488554	−	0.282067i
$708$		0			0
$709$	3.35410	+	1.93649i	0.125966	+	0.0727265i	0.561659	−	0.827369i	$-0.310163\pi$
$709$	3.35410	+	1.93649i	0.125966	+	0.0727265i	−0.435693	+	0.900095i	$0.643497\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	7.50000	−	12.9904i	0.280877	−	0.486494i
$714$		0			0
$715$	−38.9711	+	22.5000i	−1.45744	+	0.841452i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−46.4758			−1.73325			−0.866627	−	0.498956i	$-0.833717\pi$
$719$	−46.4758			−1.73325			−0.866627	+	0.498956i	$0.833717\pi$
$720$		0			0
$721$	45.0000			1.67589
$722$		0			0
$723$		0			0
$724$		0			0
$725$	33.5410	−	19.3649i	1.24568	−	0.719195i
$726$		0			0
$727$	17.4284	−	30.1869i	0.646385	−	1.11957i	−0.337595	−	0.941291i	$-0.609614\pi$
$727$	17.4284	−	30.1869i	0.646385	−	1.11957i	0.983980	−	0.178279i	$-0.0570531\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−3.35410	+	1.93649i	−0.123887	+	0.0715260i	−0.560663	−	0.828044i	$-0.689454\pi$
$733$	−3.35410	+	1.93649i	−0.123887	+	0.0715260i	0.436776	+	0.899570i	$0.356120\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	33.0000			1.21557
$738$		0			0
$739$		−	32.0000i		−	1.17714i	−0.808447	−	0.588570i	$-0.799691\pi$
$739$		−	32.0000i		−	1.17714i	0.808447	−	0.588570i	$-0.200309\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−25.1744	−	43.6033i	−0.923559	−	1.59965i	−0.793863	−	0.608097i	$-0.791933\pi$
$743$	−25.1744	−	43.6033i	−0.923559	−	1.59965i	−0.129696	−	0.991554i	$-0.541400\pi$
$744$		0			0
$745$	22.5000	−	38.9711i	0.824336	−	1.42779i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	20.1246	+	11.6190i	0.735337	+	0.424547i
$750$		0			0
$751$	−13.5554	−	23.4787i	−0.494645	−	0.856750i	0.505336	−	0.862923i	$-0.331369\pi$
$751$	−13.5554	−	23.4787i	−0.494645	−	0.856750i	−0.999981	+	0.00617232i	$0.998035\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	45.0000i		−	1.63772i
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	10.5000	+	18.1865i	0.380625	+	0.659261i	0.991152	−	0.132734i	$-0.0423756\pi$
$761$	10.5000	+	18.1865i	0.380625	+	0.659261i	−0.610527	+	0.791995i	$0.709042\pi$
$762$		0			0
$763$	−25.9808	−	15.0000i	−0.940567	−	0.543036i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	5.80948	−	10.0623i	0.209768	−	0.363329i
$768$		0			0
$769$	−11.5000	−	19.9186i	−0.414701	−	0.718283i	0.580696	−	0.814120i	$-0.302780\pi$
$769$	−11.5000	−	19.9186i	−0.414701	−	0.718283i	−0.995397	+	0.0958377i	$0.969447\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			38.7298i			1.39302i	0.717549	+	0.696508i	$0.245264\pi$
$773$			38.7298i			1.39302i	−0.717549	+	0.696508i	$0.754736\pi$
$774$		0			0
$775$	−38.7298			−1.39122
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−15.5885	+	9.00000i	−0.558514	+	0.322458i
$780$		0			0
$781$	20.1246	+	11.6190i	0.720115	+	0.415759i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−7.50000	+	12.9904i	−0.267686	+	0.463647i
$786$		0			0
$787$	−21.6506	+	12.5000i	−0.771762	+	0.445577i	−0.833503	−	0.552515i	$-0.813668\pi$
$787$	−21.6506	+	12.5000i	−0.771762	+	0.445577i	0.0617409	+	0.998092i	$0.480335\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−11.6190			−0.413122
$792$		0			0
$793$	−45.0000			−1.59800
$794$		0			0
$795$		0			0
$796$		0			0
$797$	16.7705	−	9.68246i	0.594042	−	0.342970i	−0.172652	−	0.984983i	$-0.555234\pi$
$797$	16.7705	−	9.68246i	0.594042	−	0.342970i	0.766694	+	0.642013i	$0.221900\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−20.7846	−	12.0000i	−0.733473	−	0.423471i
$804$		0			0
$805$	50.3115	−	29.0474i	1.77325	−	1.02379i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	12.0000			0.421898			0.210949	−	0.977497i	$-0.432345\pi$
$809$	12.0000			0.421898			0.210949	+	0.977497i	$0.432345\pi$
$810$		0			0
$811$			34.0000i			1.19390i	0.802278	+	0.596951i	$0.203621\pi$
$811$			34.0000i			1.19390i	−0.802278	+	0.596951i	$0.796379\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−3.87298	−	6.70820i	−0.135665	−	0.234978i
$816$		0			0
$817$	7.00000	−	12.1244i	0.244899	−	0.424178i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−36.8951	−	21.3014i	−1.28765	−	0.743424i	−0.309414	−	0.950927i	$-0.600133\pi$
$821$	−36.8951	−	21.3014i	−1.28765	−	0.743424i	−0.978234	+	0.207503i	$0.933466\pi$
$822$		0			0
$823$	−1.93649	−	3.35410i	−0.0675019	−	0.116917i	0.830299	−	0.557318i	$-0.188170\pi$
$823$	−1.93649	−	3.35410i	−0.0675019	−	0.116917i	−0.897801	+	0.440401i	$0.854836\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	6.00000i		−	0.208640i	−0.994544	−	0.104320i	$-0.966733\pi$
$827$		−	6.00000i		−	0.208640i	0.994544	−	0.104320i	$-0.0332667\pi$
$828$		0			0
$829$		−	7.74597i		−	0.269029i	−0.990912	−	0.134514i	$-0.957053\pi$
$829$		−	7.74597i		−	0.269029i	0.990912	−	0.134514i	$-0.0429474\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	12.9904	+	7.50000i	0.449551	+	0.259548i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	25.1744	−	43.6033i	0.869117	−	1.50535i	0.00621569	−	0.999981i	$-0.498021\pi$
$839$	25.1744	−	43.6033i	0.869117	−	1.50535i	0.862901	−	0.505373i	$-0.168645\pi$
$840$		0			0
$841$	−7.00000	−	12.1244i	−0.241379	−	0.418081i
$842$		0			0
$843$		0			0
$844$		0			0
$845$			7.74597i			0.266469i
$846$		0			0
$847$	7.74597			0.266155
$848$		0			0
$849$		0			0
$850$		0			0
$851$	25.9808	−	15.0000i	0.890609	−	0.514193i
$852$		0			0
$853$	10.0623	+	5.80948i	0.344527	+	0.198913i	0.662272	−	0.749263i	$-0.269592\pi$
$853$	10.0623	+	5.80948i	0.344527	+	0.198913i	−0.317745	+	0.948176i	$0.602926\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	10.5000	−	18.1865i	0.358673	−	0.621240i	−0.629066	−	0.777352i	$-0.716563\pi$
$857$	10.5000	−	18.1865i	0.358673	−	0.621240i	0.987739	+	0.156112i	$0.0498959\pi$
$858$		0			0
$859$	11.2583	−	6.50000i	0.384129	−	0.221777i	−0.295484	−	0.955348i	$-0.595481\pi$
$859$	11.2583	−	6.50000i	0.384129	−	0.221777i	0.679613	+	0.733571i	$0.262148\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	15.4919			0.527352			0.263676	−	0.964611i	$-0.415065\pi$
$863$	15.4919			0.527352			0.263676	+	0.964611i	$0.415065\pi$
$864$		0			0
$865$	75.0000			2.55008
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−10.0623	+	5.80948i	−0.341340	+	0.197073i
$870$		0			0
$871$	21.3014	−	36.8951i	0.721771	−	1.25014i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−64.9519	−	37.5000i	−2.19578	−	1.26773i
$876$		0			0
$877$	23.4787	−	13.5554i	0.792820	−	0.457735i	−0.0481345	−	0.998841i	$-0.515328\pi$
$877$	23.4787	−	13.5554i	0.792820	−	0.457735i	0.840954	+	0.541106i	$0.181994\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	12.0000			0.404290			0.202145	−	0.979356i	$-0.435209\pi$
$881$	12.0000			0.404290			0.202145	+	0.979356i	$0.435209\pi$
$882$		0			0
$883$			50.0000i			1.68263i	0.540542	+	0.841317i	$0.318219\pi$
$883$			50.0000i			1.68263i	−0.540542	+	0.841317i	$0.681781\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	17.4284	+	30.1869i	0.585189	+	1.01358i	0.994852	+	0.101340i	$0.0323131\pi$
$887$	17.4284	+	30.1869i	0.585189	+	1.01358i	−0.409663	+	0.912237i	$0.634354\pi$
$888$		0			0
$889$	30.0000	−	51.9615i	1.00617	−	1.74273i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	6.70820	+	3.87298i	0.224481	+	0.129604i
$894$		0			0
$895$	34.8569	+	60.3738i	1.16514	+	2.01807i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	15.0000i		−	0.500278i
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−15.0000	−	25.9808i	−0.498617	−	0.863630i
$906$		0			0
$907$	−16.4545	−	9.50000i	−0.546362	−	0.315442i	0.201291	−	0.979531i	$-0.435486\pi$
$907$	−16.4545	−	9.50000i	−0.546362	−	0.315442i	−0.747653	+	0.664089i	$0.768820\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−9.68246	+	16.7705i	−0.320794	+	0.555632i	−0.980652	−	0.195759i	$-0.937283\pi$
$911$	−9.68246	+	16.7705i	−0.320794	+	0.555632i	0.659858	+	0.751390i	$0.270616\pi$
$912$		0			0
$913$	−4.50000	−	7.79423i	−0.148928	−	0.257951i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	11.6190i		−	0.383692i
$918$		0			0
$919$	−46.4758			−1.53310			−0.766548	−	0.642188i	$-0.778027\pi$
$919$	−46.4758			−1.53310			−0.766548	+	0.642188i	$0.778027\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	25.9808	−	15.0000i	0.855167	−	0.493731i
$924$		0			0
$925$	−67.0820	−	38.7298i	−2.20564	−	1.27343i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−19.5000	+	33.7750i	−0.639774	+	1.10812i	0.345708	+	0.938342i	$0.387639\pi$
$929$	−19.5000	+	33.7750i	−0.639774	+	1.10812i	−0.985482	+	0.169779i	$0.945695\pi$
$930$		0			0
$931$	13.8564	−	8.00000i	0.454125	−	0.262189i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	4.00000			0.130674			0.0653372	−	0.997863i	$-0.479188\pi$
$937$	4.00000			0.130674			0.0653372	+	0.997863i	$0.479188\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−3.35410	+	1.93649i	−0.109341	+	0.0631278i	−0.553673	−	0.832734i	$-0.686774\pi$
$941$	−3.35410	+	1.93649i	−0.109341	+	0.0631278i	0.444332	+	0.895862i	$0.353441\pi$
$942$		0			0
$943$	−17.4284	+	30.1869i	−0.567548	+	0.983021i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−44.1673	−	25.5000i	−1.43524	−	0.828639i	−0.437730	−	0.899106i	$-0.644217\pi$
$947$	−44.1673	−	25.5000i	−1.43524	−	0.828639i	−0.997514	+	0.0704677i	$0.977551\pi$
$948$		0			0
$949$	−26.8328	+	15.4919i	−0.871030	+	0.502889i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$		−	105.000i		−	3.39772i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	40.6663	+	70.4361i	1.31318	+	2.27450i
$960$		0			0
$961$	8.00000	−	13.8564i	0.258065	−	0.446981i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	16.7705	+	9.68246i	0.539862	+	0.311689i
$966$		0			0
$967$	29.0474	+	50.3115i	0.934101	+	1.61791i	0.776230	+	0.630450i	$0.217130\pi$
$967$	29.0474	+	50.3115i	0.934101	+	1.61791i	0.157871	+	0.987460i	$0.449537\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			24.0000i			0.770197i	0.922876	+	0.385098i	$0.125832\pi$
$971$			24.0000i			0.770197i	−0.922876	+	0.385098i	$0.874168\pi$
$972$		0			0
$973$		−	3.87298i		−	0.124162i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−4.50000	−	7.79423i	−0.143968	−	0.249359i	0.785020	−	0.619471i	$-0.212653\pi$
$977$	−4.50000	−	7.79423i	−0.143968	−	0.249359i	−0.928987	+	0.370111i	$0.879319\pi$
$978$		0			0
$979$	−31.1769	−	18.0000i	−0.996419	−	0.575282i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−17.4284	+	30.1869i	−0.555880	+	0.962813i	0.441954	+	0.897038i	$0.354285\pi$
$983$	−17.4284	+	30.1869i	−0.555880	+	0.962813i	−0.997834	+	0.0657754i	$0.979048\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	27.1109i		−	0.862076i
$990$		0			0
$991$	46.4758			1.47635			0.738176	−	0.674608i	$-0.235687\pi$
$991$	46.4758			1.47635			0.738176	+	0.674608i	$0.235687\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	25.9808	−	15.0000i	0.823646	−	0.475532i
$996$		0			0
$997$	−50.3115	−	29.0474i	−1.59338	−	0.919940i	−0.992721	−	0.120436i	$-0.961571\pi$
$997$	−50.3115	−	29.0474i	−1.59338	−	0.919940i	−0.600661	−	0.799504i	$-0.705096\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.2.r.d.289.1		8
3.2	odd	2		576.2.r.d.97.4	yes	8
4.3	odd	2	inner	1728.2.r.d.289.2		8
8.3	odd	2	inner	1728.2.r.d.289.4		8
8.5	even	2	inner	1728.2.r.d.289.3		8
9.2	odd	6		5184.2.d.h.2593.4		4
9.4	even	3	inner	1728.2.r.d.1441.3		8
9.5	odd	6		576.2.r.d.481.1	yes	8
9.7	even	3		5184.2.d.g.2593.2		4
12.11	even	2		576.2.r.d.97.2	yes	8
24.5	odd	2		576.2.r.d.97.1	✓	8
24.11	even	2		576.2.r.d.97.3	yes	8
36.7	odd	6		5184.2.d.g.2593.1		4
36.11	even	6		5184.2.d.h.2593.3		4
36.23	even	6		576.2.r.d.481.3	yes	8
36.31	odd	6	inner	1728.2.r.d.1441.4		8
72.5	odd	6		576.2.r.d.481.4	yes	8
72.11	even	6		5184.2.d.h.2593.1		4
72.13	even	6	inner	1728.2.r.d.1441.1		8
72.29	odd	6		5184.2.d.h.2593.2		4
72.43	odd	6		5184.2.d.g.2593.3		4
72.59	even	6		576.2.r.d.481.2	yes	8
72.61	even	6		5184.2.d.g.2593.4		4
72.67	odd	6	inner	1728.2.r.d.1441.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.r.d.97.1	✓	8	24.5	odd	2
576.2.r.d.97.2	yes	8	12.11	even	2
576.2.r.d.97.3	yes	8	24.11	even	2
576.2.r.d.97.4	yes	8	3.2	odd	2
576.2.r.d.481.1	yes	8	9.5	odd	6
576.2.r.d.481.2	yes	8	72.59	even	6
576.2.r.d.481.3	yes	8	36.23	even	6
576.2.r.d.481.4	yes	8	72.5	odd	6
1728.2.r.d.289.1		8	1.1	even	1	trivial
1728.2.r.d.289.2		8	4.3	odd	2	inner
1728.2.r.d.289.3		8	8.5	even	2	inner
1728.2.r.d.289.4		8	8.3	odd	2	inner
1728.2.r.d.1441.1		8	72.13	even	6	inner
1728.2.r.d.1441.2		8	72.67	odd	6	inner
1728.2.r.d.1441.3		8	9.4	even	3	inner
1728.2.r.d.1441.4		8	36.31	odd	6	inner
5184.2.d.g.2593.1		4	36.7	odd	6
5184.2.d.g.2593.2		4	9.7	even	3
5184.2.d.g.2593.3		4	72.43	odd	6
5184.2.d.g.2593.4		4	72.61	even	6
5184.2.d.h.2593.1		4	72.11	even	6
5184.2.d.h.2593.2		4	72.29	odd	6
5184.2.d.h.2593.3		4	36.11	even	6
5184.2.d.h.2593.4		4	9.2	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:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1728.r (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-3.35410 + 1.93649i) q^{5} +(-1.93649 + 3.35410i) q^{7} +O(q^{10})\)	\(q+(-3.35410 + 1.93649i) q^{5} +(-1.93649 + 3.35410i) q^{7} +(-2.59808 - 1.50000i) q^{11} +(-3.35410 + 1.93649i) q^{13} +2.00000i q^{19} +(1.93649 + 3.35410i) q^{23} +(5.00000 - 8.66025i) q^{25} +(3.35410 + 1.93649i) q^{29} +(-1.93649 - 3.35410i) q^{31} -15.0000i q^{35} -7.74597i q^{37} +(4.50000 + 7.79423i) q^{41} +(-6.06218 - 3.50000i) q^{43} +(1.93649 - 3.35410i) q^{47} +(-4.00000 - 6.92820i) q^{49} -7.74597i q^{53} +11.6190 q^{55} +(-2.59808 + 1.50000i) q^{59} +(10.0623 + 5.80948i) q^{61} +(7.50000 - 12.9904i) q^{65} +(-9.52628 + 5.50000i) q^{67} -7.74597 q^{71} +8.00000 q^{73} +(10.0623 - 5.80948i) q^{77} +(1.93649 - 3.35410i) q^{79} +(2.59808 + 1.50000i) q^{83} +12.0000 q^{89} -15.0000i q^{91} +(-3.87298 - 6.70820i) q^{95} +(-0.500000 + 0.866025i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 40 q^{25} + 36 q^{41} - 32 q^{49} + 60 q^{65} + 64 q^{73} + 96 q^{89} - 4 q^{97}+O(q^{100}) \) 8 * q + 40 * q^25 + 36 * q^41 - 32 * q^49 + 60 * q^65 + 64 * q^73 + 96 * q^89 - 4 * q^97

Introduction

L-functions

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