LMFDB - Embedded newform 2700.2.j.j.1457.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2700,2,Mod(593,2700)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2700, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 2, 3]))

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

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

chi := DirichletCharacter("2700.593");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2700.j (of order $4$, degree $2$, 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:	$21.5596085457$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.33973862400.8
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 44x^{4} + 100 $ x^8 + 44*x^4 + 100
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 540)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1457.3
Root			$0.880486 - 0.880486i$ of defining polynomial
Character	$\chi$	$=$	2700.1457
Dual form			2700.2.j.j.593.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.22474 + 2.22474i) q^{7} +O(q^{10})$	$q+(2.22474 + 2.22474i) q^{7} -5.67868i q^{11} +(-1.22474 + 1.22474i) q^{13} +(-5.67868 + 5.67868i) q^{17} +5.89898i q^{19} +(-1.27626 - 1.27626i) q^{23} +5.67868 q^{29} -4.44949 q^{31} +(3.67423 + 3.67423i) q^{37} +8.23119i q^{41} +(-5.44949 + 5.44949i) q^{43} +2.89898i q^{49} +(1.27626 + 1.27626i) q^{53} +2.55251 q^{59} -6.79796 q^{61} +(-1.32577 - 1.32577i) q^{67} -8.23119i q^{71} +(3.77526 - 3.77526i) q^{73} +(12.6336 - 12.6336i) q^{77} +15.2474i q^{79} +(6.95494 + 6.95494i) q^{83} -2.55251 q^{89} -5.44949 q^{91} +(7.22474 + 7.22474i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{7}+O(q^{10}) $ 8 * q + 8 * q^7	$ 8 q + 8 q^{7} - 16 q^{31} - 24 q^{43} + 24 q^{61} - 40 q^{67} + 40 q^{73} - 24 q^{91} + 48 q^{97}+O(q^{100}) $ 8 * q + 8 * q^7 - 16 * q^31 - 24 * q^43 + 24 * q^61 - 40 * q^67 + 40 * q^73 - 24 * q^91 + 48 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2700\mathbb{Z}\right)^\times$.

$n$	$1001$	$1351$	$2377$
$\chi(n)$	$-1$	$1$	$e\left(\frac{1}{4}\right)$

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
$5$		0			0
$5$		0			0
$6$		0			0
$7$	2.22474	+	2.22474i	0.840875	+	0.840875i	0.988973	−	0.148098i	$-0.0473152\pi$
$7$	2.22474	+	2.22474i	0.840875	+	0.840875i	−0.148098	+	0.988973i	$0.547315\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	5.67868i		−	1.71219i	−0.516821	−	0.856094i	$-0.672885\pi$
$11$		−	5.67868i		−	1.71219i	0.516821	−	0.856094i	$-0.327115\pi$
$12$		0			0
$13$	−1.22474	+	1.22474i	−0.339683	+	0.339683i	−0.856248	−	0.516565i	$-0.827210\pi$
$13$	−1.22474	+	1.22474i	−0.339683	+	0.339683i	0.516565	+	0.856248i	$0.327210\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−5.67868	+	5.67868i	−1.37728	+	1.37728i	−0.528102	+	0.849181i	$0.677096\pi$
$17$	−5.67868	+	5.67868i	−1.37728	+	1.37728i	−0.849181	+	0.528102i	$0.822904\pi$
$18$		0			0
$19$			5.89898i			1.35332i	0.736296	+	0.676659i	$0.236573\pi$
$19$			5.89898i			1.35332i	−0.736296	+	0.676659i	$0.763427\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−1.27626	−	1.27626i	−0.266118	−	0.266118i	0.561416	−	0.827534i	$-0.310257\pi$
$23$	−1.27626	−	1.27626i	−0.266118	−	0.266118i	−0.827534	+	0.561416i	$0.810257\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$	5.67868			1.05451			0.527253	−	0.849709i	$-0.323222\pi$
$29$	5.67868			1.05451			0.527253	+	0.849709i	$0.323222\pi$
$30$		0			0
$31$	−4.44949			−0.799152			−0.399576	−	0.916700i	$-0.630843\pi$
$31$	−4.44949			−0.799152			−0.399576	+	0.916700i	$0.630843\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	3.67423	+	3.67423i	0.604040	+	0.604040i	0.941382	−	0.337342i	$-0.109528\pi$
$37$	3.67423	+	3.67423i	0.604040	+	0.604040i	−0.337342	+	0.941382i	$0.609528\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			8.23119i			1.28550i	0.766077	+	0.642748i	$0.222206\pi$
$41$			8.23119i			1.28550i	−0.766077	+	0.642748i	$0.777794\pi$
$42$		0			0
$43$	−5.44949	+	5.44949i	−0.831039	+	0.831039i	−0.987659	−	0.156620i	$-0.949940\pi$
$43$	−5.44949	+	5.44949i	−0.831039	+	0.831039i	0.156620	+	0.987659i	$0.449940\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$47$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$48$		0			0
$49$			2.89898i			0.414140i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	1.27626	+	1.27626i	0.175307	+	0.175307i	0.789307	−	0.613999i	$-0.210440\pi$
$53$	1.27626	+	1.27626i	0.175307	+	0.175307i	−0.613999	+	0.789307i	$0.710440\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	2.55251			0.332309			0.166154	−	0.986100i	$-0.446865\pi$
$59$	2.55251			0.332309			0.166154	+	0.986100i	$0.446865\pi$
$60$		0			0
$61$	−6.79796			−0.870389			−0.435195	−	0.900336i	$-0.643320\pi$
$61$	−6.79796			−0.870389			−0.435195	+	0.900336i	$0.643320\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−1.32577	−	1.32577i	−0.161968	−	0.161968i	0.621470	−	0.783438i	$-0.286536\pi$
$67$	−1.32577	−	1.32577i	−0.161968	−	0.161968i	−0.783438	+	0.621470i	$0.786536\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	8.23119i		−	0.976863i	−0.872602	−	0.488431i	$-0.837569\pi$
$71$		−	8.23119i		−	0.976863i	0.872602	−	0.488431i	$-0.162431\pi$
$72$		0			0
$73$	3.77526	−	3.77526i	0.441860	−	0.441860i	−0.450777	−	0.892637i	$-0.648853\pi$
$73$	3.77526	−	3.77526i	0.441860	−	0.441860i	0.892637	+	0.450777i	$0.148853\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	12.6336	−	12.6336i	1.43973	−	1.43973i
$78$		0			0
$79$			15.2474i			1.71547i	0.514090	+	0.857736i	$0.328130\pi$
$79$			15.2474i			1.71547i	−0.514090	+	0.857736i	$0.671870\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	6.95494	+	6.95494i	0.763404	+	0.763404i	0.976936	−	0.213532i	$-0.0684969\pi$
$83$	6.95494	+	6.95494i	0.763404	+	0.763404i	−0.213532	+	0.976936i	$0.568497\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−2.55251			−0.270566			−0.135283	−	0.990807i	$-0.543194\pi$
$89$	−2.55251			−0.270566			−0.135283	+	0.990807i	$0.543194\pi$
$90$		0			0
$91$	−5.44949			−0.571262
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	7.22474	+	7.22474i	0.733562	+	0.733562i	0.971323	−	0.237762i	$-0.0764138\pi$
$97$	7.22474	+	7.22474i	0.733562	+	0.733562i	−0.237762	+	0.971323i	$0.576414\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			8.80486i			0.876116i	0.898947	+	0.438058i	$0.144334\pi$
$101$			8.80486i			0.876116i	−0.898947	+	0.438058i	$0.855666\pi$
$102$		0			0
$103$	3.77526	−	3.77526i	0.371987	−	0.371987i	−0.496214	−	0.868200i	$-0.665277\pi$
$103$	3.77526	−	3.77526i	0.371987	−	0.371987i	0.868200	+	0.496214i	$0.165277\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	4.40243	−	4.40243i	0.425599	−	0.425599i	−0.461527	−	0.887126i	$-0.652698\pi$
$107$	4.40243	−	4.40243i	0.425599	−	0.425599i	0.887126	+	0.461527i	$0.152698\pi$
$108$		0			0
$109$			3.34847i			0.320725i	0.987058	+	0.160363i	$0.0512663\pi$
$109$			3.34847i			0.320725i	−0.987058	+	0.160363i	$0.948734\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−12.6336	−	12.6336i	−1.18847	−	1.18847i	−0.977490	−	0.210981i	$-0.932334\pi$
$113$	−12.6336	−	12.6336i	−1.18847	−	1.18847i	−0.210981	−	0.977490i	$-0.567666\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−25.2672			−2.31624
$120$		0			0
$121$	−21.2474			−1.93159
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−0.101021	−	0.101021i	−0.00896412	−	0.00896412i	0.702611	−	0.711575i	$-0.252018\pi$
$127$	−0.101021	−	0.101021i	−0.00896412	−	0.00896412i	−0.711575	+	0.702611i	$0.752018\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			11.3574i			0.992298i	0.868237	+	0.496149i	$0.165253\pi$
$131$			11.3574i			0.992298i	−0.868237	+	0.496149i	$0.834747\pi$
$132$		0			0
$133$	−13.1237	+	13.1237i	−1.13797	+	1.13797i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−6.95494	+	6.95494i	−0.594201	+	0.594201i	−0.938763	−	0.344563i	$-0.888027\pi$
$137$	−6.95494	+	6.95494i	−0.594201	+	0.594201i	0.344563	+	0.938763i	$0.388027\pi$
$138$		0			0
$139$			5.24745i			0.445083i	0.974923	+	0.222541i	$0.0714353\pi$
$139$			5.24745i			0.445083i	−0.974923	+	0.222541i	$0.928565\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	6.95494	+	6.95494i	0.581601	+	0.581601i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−13.9099			−1.13954			−0.569771	−	0.821804i	$-0.692968\pi$
$149$	−13.9099			−1.13954			−0.569771	+	0.821804i	$0.692968\pi$
$150$		0			0
$151$	15.4495			1.25726			0.628631	−	0.777704i	$-0.283616\pi$
$151$	15.4495			1.25726			0.628631	+	0.777704i	$0.283616\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	1.34847	+	1.34847i	0.107620	+	0.107620i	0.758866	−	0.651247i	$-0.225754\pi$
$157$	1.34847	+	1.34847i	0.107620	+	0.107620i	−0.651247	+	0.758866i	$0.725754\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	5.67868i		−	0.447543i
$162$		0			0
$163$	−6.22474	+	6.22474i	−0.487560	+	0.487560i	−0.907535	−	0.419976i	$-0.862039\pi$
$163$	−6.22474	+	6.22474i	−0.487560	+	0.487560i	0.419976	+	0.907535i	$0.362039\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−8.23119	+	8.23119i	−0.636949	+	0.636949i	−0.949802	−	0.312853i	$-0.898715\pi$
$167$	−8.23119	+	8.23119i	−0.636949	+	0.636949i	0.312853	+	0.949802i	$0.398715\pi$
$168$		0			0
$169$			10.0000i			0.769231i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−11.3574	−	11.3574i	−0.863485	−	0.863485i	0.128256	−	0.991741i	$-0.459062\pi$
$173$	−11.3574	−	11.3574i	−0.863485	−	0.863485i	−0.991741	+	0.128256i	$0.959062\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	8.23119			0.615228			0.307614	−	0.951511i	$-0.400469\pi$
$179$	8.23119			0.615228			0.307614	+	0.951511i	$0.400469\pi$
$180$		0			0
$181$	−13.2474			−0.984675			−0.492338	−	0.870404i	$-0.663857\pi$
$181$	−13.2474			−0.984675			−0.492338	+	0.870404i	$0.663857\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	32.2474	+	32.2474i	2.35817	+	2.35817i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			2.55251i			0.184693i	0.995727	+	0.0923466i	$0.0294368\pi$
$191$			2.55251i			0.184693i	−0.995727	+	0.0923466i	$0.970563\pi$
$192$		0			0
$193$	−5.57321	+	5.57321i	−0.401169	+	0.401169i	−0.878645	−	0.477476i	$-0.841552\pi$
$193$	−5.57321	+	5.57321i	−0.401169	+	0.401169i	0.477476	+	0.878645i	$0.341552\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	13.9099	−	13.9099i	0.991038	−	0.991038i	−0.00892246	−	0.999960i	$-0.502840\pi$
$197$	13.9099	−	13.9099i	0.991038	−	0.991038i	0.999960	+	0.00892246i	$0.00284014\pi$
$198$		0			0
$199$			15.2474i			1.08086i	0.841388	+	0.540431i	$0.181739\pi$
$199$			15.2474i			1.08086i	−0.841388	+	0.540431i	$0.818261\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	12.6336	+	12.6336i	0.886706	+	0.886706i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	33.4984			2.31714
$210$		0			0
$211$	11.2474			0.774306			0.387153	−	0.922015i	$-0.373458\pi$
$211$	11.2474			0.774306			0.387153	+	0.922015i	$0.373458\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−9.89898	−	9.89898i	−0.671987	−	0.671987i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	13.9099i		−	0.935680i
$222$		0			0
$223$	6.79796	−	6.79796i	0.455225	−	0.455225i	−0.441859	−	0.897084i	$-0.645681\pi$
$223$	6.79796	−	6.79796i	0.455225	−	0.455225i	0.897084	+	0.441859i	$0.145681\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	9.50745	−	9.50745i	0.631032	−	0.631032i	−0.317295	−	0.948327i	$-0.602775\pi$
$227$	9.50745	−	9.50745i	0.631032	−	0.631032i	0.948327	+	0.317295i	$0.102775\pi$
$228$		0			0
$229$			13.3485i			0.882092i	0.897485	+	0.441046i	$0.145392\pi$
$229$			13.3485i			0.882092i	−0.897485	+	0.441046i	$0.854608\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	3.12617	+	3.12617i	0.204802	+	0.204802i	0.802054	−	0.597252i	$-0.203741\pi$
$233$	3.12617	+	3.12617i	0.204802	+	0.204802i	−0.597252	+	0.802054i	$0.703741\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	17.0361			1.10197			0.550985	−	0.834515i	$-0.314252\pi$
$239$	17.0361			1.10197			0.550985	+	0.834515i	$0.314252\pi$
$240$		0			0
$241$	19.0000			1.22390			0.611949	−	0.790897i	$-0.290386\pi$
$241$	19.0000			1.22390			0.611949	+	0.790897i	$0.290386\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−7.22474	−	7.22474i	−0.459700	−	0.459700i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			10.7837i			0.680661i	0.940306	+	0.340331i	$0.110539\pi$
$251$			10.7837i			0.680661i	−0.940306	+	0.340331i	$0.889461\pi$
$252$		0			0
$253$	−7.24745	+	7.24745i	−0.455643	+	0.455643i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−14.4835	+	14.4835i	−0.903458	+	0.903458i	−0.995734	−	0.0922751i	$-0.970586\pi$
$257$	−14.4835	+	14.4835i	−0.903458	+	0.903458i	0.0922751	+	0.995734i	$0.470586\pi$
$258$		0			0
$259$			16.3485i			1.01584i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−2.55251	−	2.55251i	−0.157395	−	0.157395i	0.624017	−	0.781411i	$-0.285500\pi$
$263$	−2.55251	−	2.55251i	−0.157395	−	0.157395i	−0.781411	+	0.624017i	$0.785500\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	2.55251			0.155629			0.0778146	−	0.996968i	$-0.475206\pi$
$269$	2.55251			0.155629			0.0778146	+	0.996968i	$0.475206\pi$
$270$		0			0
$271$	−21.0000			−1.27566			−0.637830	−	0.770178i	$-0.720168\pi$
$271$	−21.0000			−1.27566			−0.637830	+	0.770178i	$0.720168\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	4.10102	+	4.10102i	0.246406	+	0.246406i	0.819494	−	0.573088i	$-0.194255\pi$
$277$	4.10102	+	4.10102i	0.246406	+	0.246406i	−0.573088	+	0.819494i	$0.694255\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	20.1622i		−	1.20278i	−0.798957	−	0.601389i	$-0.794614\pi$
$281$		−	20.1622i		−	1.20278i	0.798957	−	0.601389i	$-0.205386\pi$
$282$		0			0
$283$	16.7980	−	16.7980i	0.998535	−	0.998535i	−0.00146391	−	0.999999i	$-0.500466\pi$
$283$	16.7980	−	16.7980i	0.998535	−	0.998535i	0.999999	+	0.00146391i	$0.000465978\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−18.3123	+	18.3123i	−1.08094	+	1.08094i
$288$		0			0
$289$		−	47.4949i		−	2.79382i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	20.8648	+	20.8648i	1.21894	+	1.21894i	0.968005	+	0.250930i	$0.0807366\pi$
$293$	20.8648	+	20.8648i	1.21894	+	1.21894i	0.250930	+	0.968005i	$0.419263\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	3.12617			0.180791
$300$		0			0
$301$	−24.2474			−1.39760
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	1.34847	+	1.34847i	0.0769612	+	0.0769612i	0.744540	−	0.667578i	$-0.232669\pi$
$307$	1.34847	+	1.34847i	0.0769612	+	0.0769612i	−0.667578	+	0.744540i	$0.732669\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	25.2672i		−	1.43277i	−0.697703	−	0.716387i	$-0.745795\pi$
$311$		−	25.2672i		−	1.43277i	0.697703	−	0.716387i	$-0.254205\pi$
$312$		0			0
$313$	−16.2247	+	16.2247i	−0.917077	+	0.917077i	−0.996816	−	0.0797390i	$-0.974591\pi$
$313$	−16.2247	+	16.2247i	−0.917077	+	0.917077i	0.0797390	+	0.996816i	$0.474591\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−10.7837	+	10.7837i	−0.605673	+	0.605673i	−0.941812	−	0.336139i	$-0.890879\pi$
$317$	−10.7837	+	10.7837i	−0.605673	+	0.605673i	0.336139	+	0.941812i	$0.390879\pi$
$318$		0			0
$319$		−	32.2474i		−	1.80551i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−33.4984	−	33.4984i	−1.86390	−	1.86390i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	31.2474			1.71752			0.858758	−	0.512382i	$-0.171237\pi$
$331$	31.2474			1.71752			0.858758	+	0.512382i	$0.171237\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−23.5732	−	23.5732i	−1.28411	−	1.28411i	−0.938304	−	0.345810i	$-0.887604\pi$
$337$	−23.5732	−	23.5732i	−1.28411	−	1.28411i	−0.345810	−	0.938304i	$-0.612396\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			25.2672i			1.36830i
$342$		0			0
$343$	9.12372	−	9.12372i	0.492635	−	0.492635i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−12.6336	+	12.6336i	−0.678208	+	0.678208i	−0.959595	−	0.281386i	$-0.909206\pi$
$347$	−12.6336	+	12.6336i	−0.678208	+	0.678208i	0.281386	+	0.959595i	$0.409206\pi$
$348$		0			0
$349$			13.0000i			0.695874i	0.937518	+	0.347937i	$0.113118\pi$
$349$			13.0000i			0.695874i	−0.937518	+	0.347937i	$0.886882\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	3.12617	+	3.12617i	0.166389	+	0.166389i	0.785390	−	0.619001i	$-0.212462\pi$
$353$	3.12617	+	3.12617i	0.166389	+	0.166389i	−0.619001	+	0.785390i	$0.712462\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−19.5886			−1.03384			−0.516922	−	0.856032i	$-0.672922\pi$
$359$	−19.5886			−1.03384			−0.516922	+	0.856032i	$0.672922\pi$
$360$		0			0
$361$	−15.7980			−0.831472
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	1.42679	+	1.42679i	0.0744776	+	0.0744776i	0.743364	−	0.668887i	$-0.233229\pi$
$367$	1.42679	+	1.42679i	0.0744776	+	0.0744776i	−0.668887	+	0.743364i	$0.733229\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			5.67868i			0.294823i
$372$		0			0
$373$	11.6742	−	11.6742i	0.604469	−	0.604469i	−0.337026	−	0.941495i	$-0.609421\pi$
$373$	11.6742	−	11.6742i	0.604469	−	0.604469i	0.941495	+	0.337026i	$0.109421\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−6.95494	+	6.95494i	−0.358198	+	0.358198i
$378$		0			0
$379$			13.0000i			0.667765i	0.942615	+	0.333883i	$0.108359\pi$
$379$			13.0000i			0.667765i	−0.942615	+	0.333883i	$0.891641\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−20.1622	−	20.1622i	−1.03024	−	1.03024i	−0.999528	−	0.0307133i	$-0.990222\pi$
$383$	−20.1622	−	20.1622i	−1.03024	−	1.03024i	−0.0307133	−	0.999528i	$-0.509778\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−34.0721			−1.72752			−0.863762	−	0.503900i	$-0.831898\pi$
$389$	−34.0721			−1.72752			−0.863762	+	0.503900i	$0.831898\pi$
$390$		0			0
$391$	14.4949			0.733038
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−18.1464	−	18.1464i	−0.910743	−	0.910743i	0.0855875	−	0.996331i	$-0.472723\pi$
$397$	−18.1464	−	18.1464i	−0.910743	−	0.910743i	−0.996331	+	0.0855875i	$0.972723\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	30.9459i		−	1.54537i	−0.634792	−	0.772683i	$-0.718914\pi$
$401$		−	30.9459i		−	1.54537i	0.634792	−	0.772683i	$-0.281086\pi$
$402$		0			0
$403$	5.44949	−	5.44949i	0.271458	−	0.271458i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	20.8648	−	20.8648i	1.03423	−	1.03423i
$408$		0			0
$409$		−	21.4949i		−	1.06285i	−0.847104	−	0.531427i	$-0.821656\pi$
$409$		−	21.4949i		−	1.06285i	0.847104	−	0.531427i	$-0.178344\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	5.67868	+	5.67868i	0.279430	+	0.279430i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	6.25235			0.305447			0.152724	−	0.988269i	$-0.451196\pi$
$419$	6.25235			0.305447			0.152724	+	0.988269i	$0.451196\pi$
$420$		0			0
$421$	−13.2474			−0.645641			−0.322821	−	0.946460i	$-0.604631\pi$
$421$	−13.2474			−0.645641			−0.322821	+	0.946460i	$0.604631\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−15.1237	−	15.1237i	−0.731888	−	0.731888i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			14.4835i			0.697648i	0.937188	+	0.348824i	$0.113419\pi$
$431$			14.4835i			0.697648i	−0.937188	+	0.348824i	$0.886581\pi$
$432$		0			0
$433$	16.7980	−	16.7980i	0.807258	−	0.807258i	−0.176960	−	0.984218i	$-0.556626\pi$
$433$	16.7980	−	16.7980i	0.807258	−	0.807258i	0.984218	+	0.176960i	$0.0566262\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	7.52860	−	7.52860i	0.360142	−	0.360142i
$438$		0			0
$439$		−	21.1464i		−	1.00926i	−0.863335	−	0.504632i	$-0.831628\pi$
$439$		−	21.1464i		−	1.00926i	0.863335	−	0.504632i	$-0.168372\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	13.9099	+	13.9099i	0.660878	+	0.660878i	0.955587	−	0.294709i	$-0.0952227\pi$
$443$	13.9099	+	13.9099i	0.660878	+	0.660878i	−0.294709	+	0.955587i	$0.595223\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	2.55251			0.120460			0.0602302	−	0.998185i	$-0.480817\pi$
$449$	2.55251			0.120460			0.0602302	+	0.998185i	$0.480817\pi$
$450$		0			0
$451$	46.7423			2.20101
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	7.65153	+	7.65153i	0.357924	+	0.357924i	0.863047	−	0.505124i	$-0.168553\pi$
$457$	7.65153	+	7.65153i	0.357924	+	0.357924i	−0.505124	+	0.863047i	$0.668553\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			31.5196i			1.46801i	0.679142	+	0.734007i	$0.262352\pi$
$461$			31.5196i			1.46801i	−0.679142	+	0.734007i	$0.737648\pi$
$462$		0			0
$463$	21.0227	−	21.0227i	0.977008	−	0.977008i	−0.0227337	−	0.999742i	$-0.507237\pi$
$463$	21.0227	−	21.0227i	0.977008	−	0.977008i	0.999742	+	0.0227337i	$0.00723697\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	22.1411	−	22.1411i	1.02457	−	1.02457i	0.0248760	−	0.999691i	$-0.492081\pi$
$467$	22.1411	−	22.1411i	1.02457	−	1.02457i	0.999691	−	0.0248760i	$-0.00791911\pi$
$468$		0			0
$469$		−	5.89898i		−	0.272390i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	30.9459	+	30.9459i	1.42290	+	1.42290i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	19.5886			0.895024			0.447512	−	0.894278i	$-0.352310\pi$
$479$	19.5886			0.895024			0.447512	+	0.894278i	$0.352310\pi$
$480$		0			0
$481$	−9.00000			−0.410365
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−11.3258	−	11.3258i	−0.513219	−	0.513219i	0.402292	−	0.915511i	$-0.368214\pi$
$487$	−11.3258	−	11.3258i	−0.513219	−	0.513219i	−0.915511	+	0.402292i	$0.868214\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	1.97885i		−	0.0893041i	−0.999003	−	0.0446520i	$-0.985782\pi$
$491$		−	1.97885i		−	0.0893041i	0.999003	−	0.0446520i	$-0.0142179\pi$
$492$		0			0
$493$	−32.2474	+	32.2474i	−1.45235	+	1.45235i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	18.3123	−	18.3123i	0.821419	−	0.821419i
$498$		0			0
$499$		−	8.89898i		−	0.398373i	−0.979962	−	0.199187i	$-0.936170\pi$
$499$		−	8.89898i		−	0.398373i	0.979962	−	0.199187i	$-0.0638300\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	3.82877	+	3.82877i	0.170716	+	0.170716i	0.787294	−	0.616578i	$-0.211481\pi$
$503$	3.82877	+	3.82877i	0.170716	+	0.170716i	−0.616578	+	0.787294i	$0.711481\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−22.1411			−0.981386			−0.490693	−	0.871333i	$-0.663256\pi$
$509$	−22.1411			−0.981386			−0.490693	+	0.871333i	$0.663256\pi$
$510$		0			0
$511$	16.7980			0.743098
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	3.12617i		−	0.136960i	−0.997652	−	0.0684801i	$-0.978185\pi$
$521$		−	3.12617i		−	0.136960i	0.997652	−	0.0684801i	$-0.0218150\pi$
$522$		0			0
$523$	3.92168	−	3.92168i	0.171483	−	0.171483i	−0.616148	−	0.787631i	$-0.711307\pi$
$523$	3.92168	−	3.92168i	0.171483	−	0.171483i	0.787631	+	0.616148i	$0.211307\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	25.2672	−	25.2672i	1.10066	−	1.10066i
$528$		0			0
$529$		−	19.7423i		−	0.858363i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−10.0811	−	10.0811i	−0.436661	−	0.436661i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	16.4624			0.709085
$540$		0			0
$541$	15.4495			0.664225			0.332113	−	0.943240i	$-0.392239\pi$
$541$	15.4495			0.664225			0.332113	+	0.943240i	$0.392239\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	6.42679	+	6.42679i	0.274790	+	0.274790i	0.831025	−	0.556235i	$-0.187755\pi$
$547$	6.42679	+	6.42679i	0.274790	+	0.274790i	−0.556235	+	0.831025i	$0.687755\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			33.4984i			1.42708i
$552$		0			0
$553$	−33.9217	+	33.9217i	−1.44250	+	1.44250i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	20.1622	−	20.1622i	0.854301	−	0.854301i	−0.136359	−	0.990660i	$-0.543540\pi$
$557$	20.1622	−	20.1622i	0.854301	−	0.854301i	0.990660	+	0.136359i	$0.0435400\pi$
$558$		0			0
$559$		−	13.3485i		−	0.564580i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−17.0361	−	17.0361i	−0.717984	−	0.717984i	0.250208	−	0.968192i	$-0.419501\pi$
$563$	−17.0361	−	17.0361i	−0.717984	−	0.717984i	−0.968192	+	0.250208i	$0.919501\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−8.80486			−0.369119			−0.184559	−	0.982821i	$-0.559086\pi$
$569$	−8.80486			−0.369119			−0.184559	+	0.982821i	$0.559086\pi$
$570$		0			0
$571$	−4.55051			−0.190433			−0.0952165	−	0.995457i	$-0.530354\pi$
$571$	−4.55051			−0.190433			−0.0952165	+	0.995457i	$0.530354\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−0.0227038	−	0.0227038i	−0.000945173	−	0.000945173i	0.706634	−	0.707579i	$-0.250213\pi$
$577$	−0.0227038	−	0.0227038i	−0.000945173	−	0.000945173i	−0.707579	+	0.706634i	$0.750213\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			30.9459i			1.28385i
$582$		0			0
$583$	7.24745	−	7.24745i	0.300159	−	0.300159i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−20.8648	+	20.8648i	−0.861183	+	0.861183i	−0.991476	−	0.130292i	$-0.958408\pi$
$587$	−20.8648	+	20.8648i	−0.861183	+	0.861183i	0.130292	+	0.991476i	$0.458408\pi$
$588$		0			0
$589$		−	26.2474i		−	1.08151i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−4.40243	−	4.40243i	−0.180786	−	0.180786i	0.610912	−	0.791698i	$-0.290803\pi$
$593$	−4.40243	−	4.40243i	−0.180786	−	0.180786i	−0.791698	+	0.610912i	$0.790803\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	13.3362			0.544903			0.272451	−	0.962170i	$-0.412166\pi$
$599$	13.3362			0.544903			0.272451	+	0.962170i	$0.412166\pi$
$600$		0			0
$601$	30.0454			1.22558			0.612789	−	0.790247i	$-0.290048\pi$
$601$	30.0454			1.22558			0.612789	+	0.790247i	$0.290048\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	3.67423	+	3.67423i	0.149133	+	0.149133i	0.777730	−	0.628598i	$-0.216371\pi$
$607$	3.67423	+	3.67423i	0.149133	+	0.149133i	−0.628598	+	0.777730i	$0.716371\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	28.7753	−	28.7753i	1.16222	−	1.16222i	0.178233	−	0.983988i	$-0.442962\pi$
$613$	28.7753	−	28.7753i	1.16222	−	1.16222i	0.983988	−	0.178233i	$-0.0570382\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−10.0811	+	10.0811i	−0.405850	+	0.405850i	−0.880289	−	0.474438i	$-0.842651\pi$
$617$	−10.0811	+	10.0811i	−0.405850	+	0.405850i	0.474438	+	0.880289i	$0.342651\pi$
$618$		0			0
$619$		−	16.3485i		−	0.657100i	−0.944486	−	0.328550i	$-0.893440\pi$
$619$		−	16.3485i		−	0.657100i	0.944486	−	0.328550i	$-0.106560\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−5.67868	−	5.67868i	−0.227512	−	0.227512i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−41.7296			−1.66387
$630$		0			0
$631$	13.4949			0.537223			0.268612	−	0.963249i	$-0.413435\pi$
$631$	13.4949			0.537223			0.268612	+	0.963249i	$0.413435\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−3.55051	−	3.55051i	−0.140676	−	0.140676i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	39.1771i		−	1.54740i	−0.633550	−	0.773702i	$-0.718403\pi$
$641$		−	39.1771i		−	1.54740i	0.633550	−	0.773702i	$-0.281597\pi$
$642$		0			0
$643$	17.4495	−	17.4495i	0.688141	−	0.688141i	−0.273680	−	0.961821i	$-0.588241\pi$
$643$	17.4495	−	17.4495i	0.688141	−	0.688141i	0.961821	+	0.273680i	$0.0882409\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−15.1861	+	15.1861i	−0.597028	+	0.597028i	−0.939521	−	0.342492i	$-0.888729\pi$
$647$	−15.1861	+	15.1861i	−0.597028	+	0.597028i	0.342492	+	0.939521i	$0.388729\pi$
$648$		0			0
$649$		−	14.4949i		−	0.568974i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−10.7837	−	10.7837i	−0.421999	−	0.421999i	0.463893	−	0.885891i	$-0.346452\pi$
$653$	−10.7837	−	10.7837i	−0.421999	−	0.421999i	−0.885891	+	0.463893i	$0.846452\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−10.2100			−0.397727			−0.198863	−	0.980027i	$-0.563725\pi$
$659$	−10.2100			−0.397727			−0.198863	+	0.980027i	$0.563725\pi$
$660$		0			0
$661$	−1.00000			−0.0388955			−0.0194477	−	0.999811i	$-0.506191\pi$
$661$	−1.00000			−0.0388955			−0.0194477	+	0.999811i	$0.506191\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−7.24745	−	7.24745i	−0.280622	−	0.280622i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			38.6035i			1.49027i
$672$		0			0
$673$	28.2702	−	28.2702i	1.08973	−	1.08973i	0.0941790	−	0.995555i	$-0.469977\pi$
$673$	28.2702	−	28.2702i	1.08973	−	1.08973i	0.995555	−	0.0941790i	$-0.0300226\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−22.7147	+	22.7147i	−0.872998	+	0.872998i	−0.992798	−	0.119800i	$-0.961775\pi$
$677$	−22.7147	+	22.7147i	−0.872998	+	0.872998i	0.119800	+	0.992798i	$0.461775\pi$
$678$		0			0
$679$			32.1464i			1.23367i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	27.1172	+	27.1172i	1.03761	+	1.03761i	0.999265	+	0.0383449i	$0.0122085\pi$
$683$	27.1172	+	27.1172i	1.03761	+	1.03761i	0.0383449	+	0.999265i	$0.487791\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−3.12617			−0.119098
$690$		0			0
$691$	24.2474			0.922416			0.461208	−	0.887292i	$-0.347416\pi$
$691$	24.2474			0.922416			0.461208	+	0.887292i	$0.347416\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−46.7423	−	46.7423i	−1.77049	−	1.77049i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	47.4083i		−	1.79059i	−0.445476	−	0.895294i	$-0.646966\pi$
$701$		−	47.4083i		−	1.79059i	0.445476	−	0.895294i	$-0.353034\pi$
$702$		0			0
$703$	−21.6742	+	21.6742i	−0.817459	+	0.817459i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−19.5886	+	19.5886i	−0.736704	+	0.736704i
$708$		0			0
$709$		−	6.34847i		−	0.238422i	−0.992869	−	0.119211i	$-0.961964\pi$
$709$		−	6.34847i		−	0.238422i	0.992869	−	0.119211i	$-0.0380365\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	5.67868	+	5.67868i	0.212668	+	0.212668i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	46.0031			1.71563			0.857814	−	0.513961i	$-0.171822\pi$
$719$	46.0031			1.71563			0.857814	+	0.513961i	$0.171822\pi$
$720$		0			0
$721$	16.7980			0.625589
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	4.10102	+	4.10102i	0.152098	+	0.152098i	0.779055	−	0.626956i	$-0.215699\pi$
$727$	4.10102	+	4.10102i	0.152098	+	0.152098i	−0.626956	+	0.779055i	$0.715699\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	61.8919i		−	2.28915i
$732$		0			0
$733$	7.44949	−	7.44949i	0.275153	−	0.275153i	−0.556017	−	0.831171i	$-0.687671\pi$
$733$	7.44949	−	7.44949i	0.275153	−	0.275153i	0.831171	+	0.556017i	$0.187671\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−7.52860	+	7.52860i	−0.277320	+	0.277320i
$738$		0			0
$739$			1.75255i			0.0644686i	0.999480	+	0.0322343i	$0.0102623\pi$
$739$			1.75255i			0.0644686i	−0.999480	+	0.0322343i	$0.989738\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−13.3362	−	13.3362i	−0.489258	−	0.489258i	0.418814	−	0.908072i	$-0.362446\pi$
$743$	−13.3362	−	13.3362i	−0.489258	−	0.489258i	−0.908072	+	0.418814i	$0.862446\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	19.5886			0.715750
$750$		0			0
$751$	9.00000			0.328415			0.164207	−	0.986426i	$-0.447493\pi$
$751$	9.00000			0.328415			0.164207	+	0.986426i	$0.447493\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	31.7196	+	31.7196i	1.15287	+	1.15287i	0.985975	+	0.166895i	$0.0533742\pi$
$757$	31.7196	+	31.7196i	1.15287	+	1.15287i	0.166895	+	0.985975i	$0.446626\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			36.0509i			1.30685i	0.756993	+	0.653423i	$0.226668\pi$
$761$			36.0509i			1.30685i	−0.756993	+	0.653423i	$0.773332\pi$
$762$		0			0
$763$	−7.44949	+	7.44949i	−0.269690	+	0.269690i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−3.12617	+	3.12617i	−0.112880	+	0.112880i
$768$		0			0
$769$		−	43.7423i		−	1.57739i	−0.614785	−	0.788695i	$-0.710757\pi$
$769$		−	43.7423i		−	1.57739i	0.614785	−	0.788695i	$-0.289243\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	6.95494	+	6.95494i	0.250152	+	0.250152i	0.821033	−	0.570881i	$-0.193398\pi$
$773$	6.95494	+	6.95494i	0.250152	+	0.250152i	−0.570881	+	0.821033i	$0.693398\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−48.5556			−1.73969
$780$		0			0
$781$	−46.7423			−1.67257
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	9.47219	+	9.47219i	0.337647	+	0.337647i	0.855481	−	0.517834i	$-0.173261\pi$
$787$	9.47219	+	9.47219i	0.337647	+	0.337647i	−0.517834	+	0.855481i	$0.673261\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	56.2132i		−	1.99871i
$792$		0			0
$793$	8.32577	−	8.32577i	0.295657	−	0.295657i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	10.7837	−	10.7837i	0.381978	−	0.381978i	−0.489836	−	0.871815i	$-0.662943\pi$
$797$	10.7837	−	10.7837i	0.381978	−	0.381978i	0.871815	+	0.489836i	$0.162943\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−21.4385	−	21.4385i	−0.756548	−	0.756548i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	25.2672			0.888349			0.444175	−	0.895940i	$-0.353497\pi$
$809$	25.2672			0.888349			0.444175	+	0.895940i	$0.353497\pi$
$810$		0			0
$811$	−24.4495			−0.858538			−0.429269	−	0.903177i	$-0.641229\pi$
$811$	−24.4495			−0.858538			−0.429269	+	0.903177i	$0.641229\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−32.1464	−	32.1464i	−1.12466	−	1.12466i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	8.80486i		−	0.307292i	−0.988126	−	0.153646i	$-0.950899\pi$
$821$		−	8.80486i		−	0.307292i	0.988126	−	0.153646i	$-0.0491015\pi$
$822$		0			0
$823$	−8.97730	+	8.97730i	−0.312929	+	0.312929i	−0.846043	−	0.533114i	$-0.821022\pi$
$823$	−8.97730	+	8.97730i	−0.312929	+	0.312929i	0.533114	+	0.846043i	$0.321022\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−13.9099	+	13.9099i	−0.483694	+	0.483694i	−0.906309	−	0.422615i	$-0.861112\pi$
$827$	−13.9099	+	13.9099i	−0.483694	+	0.483694i	0.422615	+	0.906309i	$0.361112\pi$
$828$		0			0
$829$		−	24.7526i		−	0.859692i	−0.902902	−	0.429846i	$-0.858568\pi$
$829$		−	24.7526i		−	0.859692i	0.902902	−	0.429846i	$-0.141432\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−16.4624	−	16.4624i	−0.570388	−	0.570388i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	5.10502			0.176245			0.0881224	−	0.996110i	$-0.471913\pi$
$839$	5.10502			0.176245			0.0881224	+	0.996110i	$0.471913\pi$
$840$		0			0
$841$	3.24745			0.111981
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−47.2702	−	47.2702i	−1.62422	−	1.62422i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	9.37852i		−	0.321492i
$852$		0			0
$853$	−23.4722	+	23.4722i	−0.803673	+	0.803673i	−0.983668	−	0.179995i	$-0.942392\pi$
$853$	−23.4722	+	23.4722i	−0.803673	+	0.803673i	0.179995	+	0.983668i	$0.442392\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−6.95494	+	6.95494i	−0.237576	+	0.237576i	−0.815846	−	0.578270i	$-0.803728\pi$
$857$	−6.95494	+	6.95494i	−0.237576	+	0.237576i	0.578270	+	0.815846i	$0.303728\pi$
$858$		0			0
$859$		−	19.2474i		−	0.656714i	−0.944554	−	0.328357i	$-0.893505\pi$
$859$		−	19.2474i		−	0.656714i	0.944554	−	0.328357i	$-0.106495\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	33.4984	+	33.4984i	1.14030	+	1.14030i	0.988395	+	0.151905i	$0.0485408\pi$
$863$	33.4984	+	33.4984i	1.14030	+	1.14030i	0.151905	+	0.988395i	$0.451459\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	86.5854			2.93721
$870$		0			0
$871$	3.24745			0.110036
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−7.77526	−	7.77526i	−0.262552	−	0.262552i	0.563538	−	0.826090i	$-0.309440\pi$
$877$	−7.77526	−	7.77526i	−0.262552	−	0.262552i	−0.826090	+	0.563538i	$0.809440\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	5.67868i		−	0.191320i	−0.995414	−	0.0956599i	$-0.969504\pi$
$881$		−	5.67868i		−	0.191320i	0.995414	−	0.0956599i	$-0.0304961\pi$
$882$		0			0
$883$	8.92168	−	8.92168i	0.300239	−	0.300239i	−0.540869	−	0.841107i	$-0.681904\pi$
$883$	8.92168	−	8.92168i	0.300239	−	0.300239i	0.841107	+	0.540869i	$0.181904\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	27.8198	−	27.8198i	0.934096	−	0.934096i	−0.0638627	−	0.997959i	$-0.520342\pi$
$887$	27.8198	−	27.8198i	0.934096	−	0.934096i	0.997959	+	0.0638627i	$0.0203420\pi$
$888$		0			0
$889$		−	0.449490i		−	0.0150754i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−25.2672			−0.842710
$900$		0			0
$901$	−14.4949			−0.482895
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−6.32577	−	6.32577i	−0.210044	−	0.210044i	0.594242	−	0.804286i	$-0.297452\pi$
$907$	−6.32577	−	6.32577i	−0.210044	−	0.210044i	−0.804286	+	0.594242i	$0.797452\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			36.6246i			1.21343i	0.794920	+	0.606714i	$0.207513\pi$
$911$			36.6246i			1.21343i	−0.794920	+	0.606714i	$0.792487\pi$
$912$		0			0
$913$	39.4949	−	39.4949i	1.30709	−	1.30709i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−25.2672	+	25.2672i	−0.834398	+	0.834398i
$918$		0			0
$919$			33.3485i			1.10006i	0.835143	+	0.550032i	$0.185385\pi$
$919$			33.3485i			1.10006i	−0.835143	+	0.550032i	$0.814615\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	10.0811	+	10.0811i	0.331824	+	0.331824i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	32.0933			1.05295			0.526473	−	0.850192i	$-0.323514\pi$
$929$	32.0933			1.05295			0.526473	+	0.850192i	$0.323514\pi$
$930$		0			0
$931$	−17.1010			−0.560463
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	12.2247	+	12.2247i	0.399365	+	0.399365i	0.878009	−	0.478644i	$-0.158872\pi$
$937$	12.2247	+	12.2247i	0.399365	+	0.399365i	−0.478644	+	0.878009i	$0.658872\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	3.12617i		−	0.101910i	−0.998701	−	0.0509552i	$-0.983773\pi$
$941$		−	3.12617i		−	0.101910i	0.998701	−	0.0509552i	$-0.0162266\pi$
$942$		0			0
$943$	10.5051	−	10.5051i	0.342093	−	0.342093i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−22.1411	+	22.1411i	−0.719488	+	0.719488i	−0.968500	−	0.249012i	$-0.919894\pi$
$947$	−22.1411	+	22.1411i	−0.719488	+	0.719488i	0.249012	+	0.968500i	$0.419894\pi$
$948$		0			0
$949$			9.24745i			0.300185i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−20.1622	−	20.1622i	−0.653119	−	0.653119i	0.300624	−	0.953743i	$-0.402805\pi$
$953$	−20.1622	−	20.1622i	−0.653119	−	0.653119i	−0.953743	+	0.300624i	$0.902805\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−30.9459			−0.999296
$960$		0			0
$961$	−11.2020			−0.361356
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	38.6742	+	38.6742i	1.24368	+	1.24368i	0.958463	+	0.285216i	$0.0920655\pi$
$967$	38.6742	+	38.6742i	1.24368	+	1.24368i	0.285216	+	0.958463i	$0.407935\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			50.5345i			1.62173i	0.585234	+	0.810864i	$0.301003\pi$
$971$			50.5345i			1.62173i	−0.585234	+	0.810864i	$0.698997\pi$
$972$		0			0
$973$	−11.6742	+	11.6742i	−0.374259	+	0.374259i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−0.702591	+	0.702591i	−0.0224779	+	0.0224779i	−0.718256	−	0.695779i	$-0.755060\pi$
$977$	−0.702591	+	0.702591i	−0.0224779	+	0.0224779i	0.695779	+	0.718256i	$0.255060\pi$
$978$		0			0
$979$			14.4949i			0.463259i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	38.4745	+	38.4745i	1.22715	+	1.22715i	0.965038	+	0.262109i	$0.0844179\pi$
$983$	38.4745	+	38.4745i	1.22715	+	1.22715i	0.262109	+	0.965038i	$0.415582\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	13.9099			0.442308
$990$		0			0
$991$	−9.04541			−0.287337			−0.143668	−	0.989626i	$-0.545890\pi$
$991$	−9.04541			−0.287337			−0.143668	+	0.989626i	$0.545890\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	9.89898	+	9.89898i	0.313504	+	0.313504i	0.846265	−	0.532762i	$-0.178846\pi$
$997$	9.89898	+	9.89898i	0.313504	+	0.313504i	−0.532762	+	0.846265i	$0.678846\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2700.2.j.j.1457.3		8
3.2	odd	2	inner	2700.2.j.j.1457.4		8
5.2	odd	4		540.2.j.a.53.1	✓	8
5.3	odd	4	inner	2700.2.j.j.593.3		8
5.4	even	2		540.2.j.a.377.4	yes	8
15.2	even	4		540.2.j.a.53.4	yes	8
15.8	even	4	inner	2700.2.j.j.593.4		8
15.14	odd	2		540.2.j.a.377.1	yes	8
20.7	even	4		2160.2.w.e.593.1		8
20.19	odd	2		2160.2.w.e.1457.4		8
45.2	even	12		1620.2.x.d.53.2		16
45.4	even	6		1620.2.x.d.917.2		16
45.7	odd	12		1620.2.x.d.53.3		16
45.14	odd	6		1620.2.x.d.917.3		16
45.22	odd	12		1620.2.x.d.593.3		16
45.29	odd	6		1620.2.x.d.377.3		16
45.32	even	12		1620.2.x.d.593.2		16
45.34	even	6		1620.2.x.d.377.2		16
60.47	odd	4		2160.2.w.e.593.4		8
60.59	even	2		2160.2.w.e.1457.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
540.2.j.a.53.1	✓	8	5.2	odd	4
540.2.j.a.53.4	yes	8	15.2	even	4
540.2.j.a.377.1	yes	8	15.14	odd	2
540.2.j.a.377.4	yes	8	5.4	even	2
1620.2.x.d.53.2		16	45.2	even	12
1620.2.x.d.53.3		16	45.7	odd	12
1620.2.x.d.377.2		16	45.34	even	6
1620.2.x.d.377.3		16	45.29	odd	6
1620.2.x.d.593.2		16	45.32	even	12
1620.2.x.d.593.3		16	45.22	odd	12
1620.2.x.d.917.2		16	45.4	even	6
1620.2.x.d.917.3		16	45.14	odd	6
2160.2.w.e.593.1		8	20.7	even	4
2160.2.w.e.593.4		8	60.47	odd	4
2160.2.w.e.1457.1		8	60.59	even	2
2160.2.w.e.1457.4		8	20.19	odd	2
2700.2.j.j.593.3		8	5.3	odd	4	inner
2700.2.j.j.593.4		8	15.8	even	4	inner
2700.2.j.j.1457.3		8	1.1	even	1	trivial
2700.2.j.j.1457.4		8	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2700.j (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(2.22474 + 2.22474i) q^{7} +O(q^{10})\)	\(q+(2.22474 + 2.22474i) q^{7} -5.67868i q^{11} +(-1.22474 + 1.22474i) q^{13} +(-5.67868 + 5.67868i) q^{17} +5.89898i q^{19} +(-1.27626 - 1.27626i) q^{23} +5.67868 q^{29} -4.44949 q^{31} +(3.67423 + 3.67423i) q^{37} +8.23119i q^{41} +(-5.44949 + 5.44949i) q^{43} +2.89898i q^{49} +(1.27626 + 1.27626i) q^{53} +2.55251 q^{59} -6.79796 q^{61} +(-1.32577 - 1.32577i) q^{67} -8.23119i q^{71} +(3.77526 - 3.77526i) q^{73} +(12.6336 - 12.6336i) q^{77} +15.2474i q^{79} +(6.95494 + 6.95494i) q^{83} -2.55251 q^{89} -5.44949 q^{91} +(7.22474 + 7.22474i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{7}+O(q^{10}) \) 8 * q + 8 * q^7	\( 8 q + 8 q^{7} - 16 q^{31} - 24 q^{43} + 24 q^{61} - 40 q^{67} + 40 q^{73} - 24 q^{91} + 48 q^{97}+O(q^{100}) \) 8 * q + 8 * q^7 - 16 * q^31 - 24 * q^43 + 24 * q^61 - 40 * q^67 + 40 * q^73 - 24 * q^91 + 48 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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