LMFDB - Embedded newform 2700.2.i.c.1801.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2700,2,Mod(901,2700)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2700, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("2700.901");

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.i (of order $3$, 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:	$21.5596085457$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	6.0.954288.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 $ x^6 - x^5 - 2x^4 + 3x^3 - 6x^2 - 9x + 27
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{3} $
Twist minimal:	no (minimal twist has level 180)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1801.2
Root			$1.71903 + 0.211943i$ of defining polynomial
Character	$\chi$	$=$	2700.1801
Dual form			2700.2.i.c.901.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.36710 - 2.36788i) q^{7} +(2.76210 - 4.78410i) q^{11} +(-1.76210 - 3.05205i) q^{13} -5.52420 q^{17} +7.52420 q^{19} +(-0.367095 - 0.635828i) q^{23} +(-2.23419 + 3.86973i) q^{29} +(-3.76210 - 6.51615i) q^{31} -6.05582 q^{37} +(-0.527909 - 0.914365i) q^{41} +(-1.76210 + 3.05205i) q^{43} +(-0.604996 + 1.04788i) q^{47} +(-0.237900 - 0.412055i) q^{49} +1.46838 q^{53} +(-0.734191 - 1.27166i) q^{59} +(-4.52791 + 7.84257i) q^{61} +(-2.12920 - 3.68787i) q^{67} -10.0558 q^{71} -8.00000 q^{73} +(-7.55211 - 13.0806i) q^{77} +(-1.00000 + 1.73205i) q^{79} +(2.63290 - 4.56032i) q^{83} +3.00000 q^{89} -9.63583 q^{91} +(4.73419 - 8.19986i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 3 q^{7} + 6 q^{13} + 12 q^{19} + 3 q^{23} - 3 q^{29} - 6 q^{31} - 24 q^{37} + 3 q^{41} + 6 q^{43} - 15 q^{47} - 18 q^{49} - 12 q^{53} + 6 q^{59} - 21 q^{61} + 9 q^{67} - 48 q^{71} - 48 q^{73} - 6 q^{77}+ \cdots + 18 q^{97}+O(q^{100}) $ 6 * q + 3 * q^7 + 6 * q^13 + 12 * q^19 + 3 * q^23 - 3 * q^29 - 6 * q^31 - 24 * q^37 + 3 * q^41 + 6 * q^43 - 15 * q^47 - 18 * q^49 - 12 * q^53 + 6 * q^59 - 21 * q^61 + 9 * q^67 - 48 * q^71 - 48 * q^73 - 6 * q^77 - 6 * q^79 + 21 * q^83 + 18 * q^89 + 18 * q^97

Character values

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

$n$	$1001$	$1351$	$2377$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$	$1$

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
$5$		0			0
$5$		0			0
$6$		0			0
$7$	1.36710	−	2.36788i	0.516714	−	0.894974i	−0.483098	−	0.875566i	$-0.660489\pi$
$7$	1.36710	−	2.36788i	0.516714	−	0.894974i	0.999812	−	0.0194079i	$-0.00617810\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.76210	−	4.78410i	0.832804	−	1.44246i	−0.0630012	−	0.998013i	$-0.520067\pi$
$11$	2.76210	−	4.78410i	0.832804	−	1.44246i	0.895806	−	0.444446i	$-0.146599\pi$
$12$		0			0
$13$	−1.76210	−	3.05205i	−0.488719	−	0.846485i	0.511197	−	0.859463i	$-0.329202\pi$
$13$	−1.76210	−	3.05205i	−0.488719	−	0.846485i	−0.999916	+	0.0129781i	$0.995869\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−5.52420			−1.33982			−0.669908	−	0.742444i	$-0.733666\pi$
$17$	−5.52420			−1.33982			−0.669908	+	0.742444i	$0.733666\pi$
$18$		0			0
$19$	7.52420			1.72617			0.863085	−	0.505059i	$-0.168529\pi$
$19$	7.52420			1.72617			0.863085	+	0.505059i	$0.168529\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−0.367095	−	0.635828i	−0.0765447	−	0.132579i	0.825212	−	0.564823i	$-0.191055\pi$
$23$	−0.367095	−	0.635828i	−0.0765447	−	0.132579i	−0.901757	+	0.432243i	$0.857722\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2.23419	+	3.86973i	−0.414879	+	0.718591i	−0.995416	−	0.0956427i	$-0.969509\pi$
$29$	−2.23419	+	3.86973i	−0.414879	+	0.718591i	0.580537	+	0.814234i	$0.302843\pi$
$30$		0			0
$31$	−3.76210	−	6.51615i	−0.675693	−	1.17033i	−0.976266	−	0.216576i	$-0.930511\pi$
$31$	−3.76210	−	6.51615i	−0.675693	−	1.17033i	0.300573	−	0.953759i	$-0.402822\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−6.05582			−0.995570			−0.497785	−	0.867300i	$-0.665853\pi$
$37$	−6.05582			−0.995570			−0.497785	+	0.867300i	$0.665853\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−0.527909	−	0.914365i	−0.0824455	−	0.142800i	0.821854	−	0.569698i	$-0.192940\pi$
$41$	−0.527909	−	0.914365i	−0.0824455	−	0.142800i	−0.904300	+	0.426898i	$0.859606\pi$
$42$		0			0
$43$	−1.76210	+	3.05205i	−0.268718	+	0.465433i	−0.968531	−	0.248893i	$-0.919933\pi$
$43$	−1.76210	+	3.05205i	−0.268718	+	0.465433i	0.699813	+	0.714326i	$0.253267\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−0.604996	+	1.04788i	−0.0882477	+	0.152849i	−0.906771	−	0.421625i	$-0.861460\pi$
$47$	−0.604996	+	1.04788i	−0.0882477	+	0.152849i	0.818523	+	0.574474i	$0.194793\pi$
$48$		0			0
$49$	−0.237900	−	0.412055i	−0.0339857	−	0.0588650i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	1.46838			0.201698			0.100849	−	0.994902i	$-0.467844\pi$
$53$	1.46838			0.201698			0.100849	+	0.994902i	$0.467844\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−0.734191	−	1.27166i	−0.0955835	−	0.165556i	0.814268	−	0.580488i	$-0.197138\pi$
$59$	−0.734191	−	1.27166i	−0.0955835	−	0.165556i	−0.909852	+	0.414933i	$0.863805\pi$
$60$		0			0
$61$	−4.52791	+	7.84257i	−0.579739	+	1.00414i	0.415770	+	0.909470i	$0.363512\pi$
$61$	−4.52791	+	7.84257i	−0.579739	+	1.00414i	−0.995509	+	0.0946680i	$0.969821\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−2.12920	−	3.68787i	−0.260123	−	0.450546i	0.706152	−	0.708061i	$-0.250430\pi$
$67$	−2.12920	−	3.68787i	−0.260123	−	0.450546i	−0.966274	+	0.257515i	$0.917096\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−10.0558			−1.19341			−0.596703	−	0.802462i	$-0.703523\pi$
$71$	−10.0558			−1.19341			−0.596703	+	0.802462i	$0.703523\pi$
$72$		0			0
$73$	−8.00000			−0.936329			−0.468165	−	0.883641i	$-0.655085\pi$
$73$	−8.00000			−0.936329			−0.468165	+	0.883641i	$0.655085\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−7.55211	−	13.0806i	−0.860643	−	1.49068i
$78$		0			0
$79$	−1.00000	+	1.73205i	−0.112509	+	0.194871i	−0.916781	−	0.399390i	$-0.869222\pi$
$79$	−1.00000	+	1.73205i	−0.112509	+	0.194871i	0.804272	+	0.594261i	$0.202555\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	2.63290	−	4.56032i	0.288999	−	0.500561i	−0.684572	−	0.728945i	$-0.740011\pi$
$83$	2.63290	−	4.56032i	0.288999	−	0.500561i	0.973571	+	0.228384i	$0.0733443\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	3.00000			0.317999			0.159000	−	0.987279i	$-0.449173\pi$
$89$	3.00000			0.317999			0.159000	+	0.987279i	$0.449173\pi$
$90$		0			0
$91$	−9.63583			−1.01011
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	4.73419	−	8.19986i	0.480684	−	0.832570i	−0.519070	−	0.854732i	$-0.673722\pi$
$97$	4.73419	−	8.19986i	0.480684	−	0.832570i	0.999754	+	0.0221621i	$0.00705499\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	6.73419	−	11.6640i	0.670077	−	1.16061i	−0.307805	−	0.951450i	$-0.599594\pi$
$101$	6.73419	−	11.6640i	0.670077	−	1.16061i	0.977882	−	0.209158i	$-0.0670723\pi$
$102$		0			0
$103$	9.28630	+	16.0843i	0.915006	+	1.58484i	0.806892	+	0.590699i	$0.201148\pi$
$103$	9.28630	+	16.0843i	0.915006	+	1.58484i	0.108114	+	0.994138i	$0.465519\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	19.2510			1.86106			0.930531	−	0.366213i	$-0.119346\pi$
$107$	19.2510			1.86106			0.930531	+	0.366213i	$0.119346\pi$
$108$		0			0
$109$	−0.524200			−0.0502092			−0.0251046	−	0.999685i	$-0.507992\pi$
$109$	−0.524200			−0.0502092			−0.0251046	+	0.999685i	$0.507992\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	3.73419	+	6.46781i	0.351283	+	0.608440i	0.986475	−	0.163915i	$-0.0524121\pi$
$113$	3.73419	+	6.46781i	0.351283	+	0.608440i	−0.635191	+	0.772355i	$0.719079\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−7.55211	+	13.0806i	−0.692301	+	1.19910i
$120$		0			0
$121$	−9.75839	−	16.9020i	−0.887126	−	1.53655i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−8.25839			−0.732814			−0.366407	−	0.930455i	$-0.619412\pi$
$127$	−8.25839			−0.732814			−0.366407	+	0.930455i	$0.619412\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−8.04840	−	13.9402i	−0.703192	−	1.21796i	−0.967340	−	0.253482i	$-0.918424\pi$
$131$	−8.04840	−	13.9402i	−0.703192	−	1.21796i	0.264148	−	0.964482i	$-0.414909\pi$
$132$		0			0
$133$	10.2863	−	17.8164i	0.891935	−	1.54488i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	9.25839	−	16.0360i	0.790998	−	1.37005i	−0.134352	−	0.990934i	$-0.542895\pi$
$137$	9.25839	−	16.0360i	0.790998	−	1.37005i	0.925350	−	0.379115i	$-0.123771\pi$
$138$		0			0
$139$	2.00000	+	3.46410i	0.169638	+	0.293821i	0.938293	−	0.345843i	$-0.112407\pi$
$139$	2.00000	+	3.46410i	0.169638	+	0.293821i	−0.768655	+	0.639664i	$0.779074\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−19.4684			−1.62803
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	6.78630	+	11.7542i	0.555955	+	0.962943i	0.997829	+	0.0658653i	$0.0209808\pi$
$149$	6.78630	+	11.7542i	0.555955	+	0.962943i	−0.441873	+	0.897078i	$0.645686\pi$
$150$		0			0
$151$	6.23048	−	10.7915i	0.507029	−	0.878201i	−0.492937	−	0.870065i	$-0.664077\pi$
$151$	6.23048	−	10.7915i	0.507029	−	0.878201i	0.999967	−	0.00813598i	$-0.00258979\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−0.790009	−	1.36834i	−0.0630496	−	0.109205i	0.832778	−	0.553608i	$-0.186749\pi$
$157$	−0.790009	−	1.36834i	−0.0630496	−	0.109205i	−0.895827	+	0.444403i	$0.853416\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−2.00742			−0.158207
$162$		0			0
$163$	4.47580			0.350572			0.175286	−	0.984518i	$-0.443915\pi$
$163$	4.47580			0.350572			0.175286	+	0.984518i	$0.443915\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−8.39500	−	14.5406i	−0.649625	−	1.12518i	−0.983212	−	0.182464i	$-0.941593\pi$
$167$	−8.39500	−	14.5406i	−0.649625	−	1.12518i	0.333588	−	0.942719i	$-0.391741\pi$
$168$		0			0
$169$	0.290009	−	0.502310i	0.0223084	−	0.0386392i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	11.5242	−	19.9605i	0.876169	−	1.51757i	0.0206561	−	0.999787i	$-0.493424\pi$
$173$	11.5242	−	19.9605i	0.876169	−	1.51757i	0.855513	−	0.517782i	$-0.173242\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−12.9926			−0.971111			−0.485556	−	0.874206i	$-0.661383\pi$
$179$	−12.9926			−0.971111			−0.485556	+	0.874206i	$0.661383\pi$
$180$		0			0
$181$	−24.5652			−1.82592			−0.912958	−	0.408054i	$-0.866207\pi$
$181$	−24.5652			−1.82592			−0.912958	+	0.408054i	$0.866207\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−15.2584	+	26.4283i	−1.11580	+	1.93263i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	7.55211	−	13.0806i	0.546451	−	0.946482i	−0.452063	−	0.891986i	$-0.649312\pi$
$191$	7.55211	−	13.0806i	0.546451	−	0.946482i	0.998514	−	0.0544954i	$-0.0173550\pi$
$192$		0			0
$193$	−7.28630	−	12.6202i	−0.524479	−	0.908425i	−0.999594	−	0.0285008i	$-0.990927\pi$
$193$	−7.28630	−	12.6202i	−0.524479	−	0.908425i	0.475114	−	0.879924i	$-0.342407\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−4.53162			−0.322864			−0.161432	−	0.986884i	$-0.551611\pi$
$197$	−4.53162			−0.322864			−0.161432	+	0.986884i	$0.551611\pi$
$198$		0			0
$199$	−7.41256			−0.525463			−0.262731	−	0.964869i	$-0.584623\pi$
$199$	−7.41256			−0.525463			−0.262731	+	0.964869i	$0.584623\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	6.10870	+	10.5806i	0.428747	+	0.742612i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	20.7826	−	35.9965i	1.43756	−	2.48993i
$210$		0			0
$211$	6.70628	+	11.6156i	0.461680	+	0.799652i	0.999045	−	0.0436972i	$-0.0139137\pi$
$211$	6.70628	+	11.6156i	0.461680	+	0.799652i	−0.537365	+	0.843350i	$0.680580\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−20.5726			−1.39656
$218$		0			0
$219$		0			0
$220$		0			0
$221$	9.73419	+	16.8601i	0.654793	+	1.13413i
$222$		0			0
$223$	−4.39500	+	7.61237i	−0.294311	+	0.509762i	−0.974824	−	0.222974i	$-0.928424\pi$
$223$	−4.39500	+	7.61237i	−0.294311	+	0.509762i	0.680513	+	0.732736i	$0.261757\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−2.02791	+	3.51244i	−0.134597	+	0.233129i	−0.925443	−	0.378886i	$-0.876307\pi$
$227$	−2.02791	+	3.51244i	−0.134597	+	0.233129i	0.790846	+	0.612015i	$0.209641\pi$
$228$		0			0
$229$	−1.29001	−	2.23436i	−0.0852462	−	0.147651i	0.820250	−	0.572005i	$-0.193834\pi$
$229$	−1.29001	−	2.23436i	−0.0852462	−	0.147651i	−0.905496	+	0.424355i	$0.860501\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−1.94418			−0.127368			−0.0636838	−	0.997970i	$-0.520285\pi$
$233$	−1.94418			−0.127368			−0.0636838	+	0.997970i	$0.520285\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	3.73419	+	6.46781i	0.241545	+	0.418368i	0.961154	−	0.276011i	$-0.0890126\pi$
$239$	3.73419	+	6.46781i	0.241545	+	0.418368i	−0.719610	+	0.694379i	$0.755679\pi$
$240$		0			0
$241$	−1.20628	+	2.08934i	−0.0777035	+	0.134586i	−0.902259	−	0.431195i	$-0.858092\pi$
$241$	−1.20628	+	2.08934i	−0.0777035	+	0.134586i	0.824555	+	0.565781i	$0.191425\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−13.2584	−	22.9642i	−0.843611	−	1.46118i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	6.99258			0.441368			0.220684	−	0.975345i	$-0.429171\pi$
$251$	6.99258			0.441368			0.220684	+	0.975345i	$0.429171\pi$
$252$		0			0
$253$	−4.05582			−0.254987
$254$		0			0
$255$		0			0
$256$		0			0
$257$	14.0484	+	24.3325i	0.876315	+	1.51782i	0.855355	+	0.518042i	$0.173339\pi$
$257$	14.0484	+	24.3325i	0.876315	+	1.51782i	0.0209598	+	0.999780i	$0.493328\pi$
$258$		0			0
$259$	−8.27888	+	14.3394i	−0.514425	+	0.891010i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−3.81792	+	6.61283i	−0.235423	+	0.407764i	−0.959395	−	0.282064i	$-0.908981\pi$
$263$	−3.81792	+	6.61283i	−0.235423	+	0.407764i	0.723973	+	0.689829i	$0.242314\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	18.6210			1.13534			0.567671	−	0.823255i	$-0.307845\pi$
$269$	18.6210			1.13534			0.567671	+	0.823255i	$0.307845\pi$
$270$		0			0
$271$	6.57260			0.399257			0.199628	−	0.979872i	$-0.436026\pi$
$271$	6.57260			0.399257			0.199628	+	0.979872i	$0.436026\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	0.762100	−	1.32000i	0.0457901	−	0.0793108i	−0.842222	−	0.539131i	$-0.818753\pi$
$277$	0.762100	−	1.32000i	0.0457901	−	0.0793108i	0.888012	+	0.459820i	$0.152086\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	7.26210	−	12.5783i	0.433221	−	0.750360i	−0.563928	−	0.825824i	$-0.690710\pi$
$281$	7.26210	−	12.5783i	0.433221	−	0.750360i	0.997149	+	0.0754640i	$0.0240438\pi$
$282$		0			0
$283$	−8.36710	−	14.4922i	−0.497372	−	0.861474i	0.502623	−	0.864506i	$-0.332368\pi$
$283$	−8.36710	−	14.4922i	−0.497372	−	0.861474i	−0.999995	+	0.00303167i	$0.999035\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−2.88681			−0.170403
$288$		0			0
$289$	13.5168			0.795105
$290$		0			0
$291$		0			0
$292$		0			0
$293$	10.2305	+	17.7197i	0.597671	+	1.03520i	0.993164	+	0.116728i	$0.0372405\pi$
$293$	10.2305	+	17.7197i	0.597671	+	1.03520i	−0.395493	+	0.918469i	$0.629426\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1.29372	+	2.24078i	−0.0748176	+	0.129588i
$300$		0			0
$301$	4.81792	+	8.34488i	0.277700	+	0.480991i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	18.2026			1.03888			0.519438	−	0.854508i	$-0.326141\pi$
$307$	18.2026			1.03888			0.519438	+	0.854508i	$0.326141\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	1.55211	+	2.68833i	0.0880120	+	0.152441i	0.906671	−	0.421839i	$-0.138615\pi$
$311$	1.55211	+	2.68833i	0.0880120	+	0.152441i	−0.818659	+	0.574280i	$0.805282\pi$
$312$		0			0
$313$	−9.55211	+	16.5447i	−0.539917	+	0.935164i	0.458991	+	0.888441i	$0.348211\pi$
$313$	−9.55211	+	16.5447i	−0.539917	+	0.935164i	−0.998908	+	0.0467228i	$0.985122\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−2.50371	+	4.33655i	−0.140622	+	0.243565i	−0.927731	−	0.373249i	$-0.878244\pi$
$317$	−2.50371	+	4.33655i	−0.140622	+	0.243565i	0.787109	+	0.616814i	$0.211577\pi$
$318$		0			0
$319$	12.3421	+	21.3772i	0.691026	+	1.19689i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−41.5652			−2.31275
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	1.65417	+	2.86511i	0.0911976	+	0.157959i
$330$		0			0
$331$	1.74161	−	3.01656i	0.0957275	−	0.165805i	−0.814184	−	0.580606i	$-0.802816\pi$
$331$	1.74161	−	3.01656i	0.0957275	−	0.165805i	0.909912	+	0.414801i	$0.136149\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	5.79001	+	10.0286i	0.315402	+	0.546292i	0.979523	−	0.201333i	$-0.0645274\pi$
$337$	5.79001	+	10.0286i	0.315402	+	0.546292i	−0.664121	+	0.747625i	$0.731194\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−41.5652			−2.25088
$342$		0			0
$343$	17.8384			0.963183
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−13.2305	−	22.9159i	−0.710249	−	1.23019i	−0.964763	−	0.263119i	$-0.915249\pi$
$347$	−13.2305	−	22.9159i	−0.710249	−	1.23019i	0.254514	−	0.967069i	$-0.418085\pi$
$348$		0			0
$349$	13.0168	−	22.5457i	0.696772	−	1.20685i	−0.272807	−	0.962069i	$-0.587952\pi$
$349$	13.0168	−	22.5457i	0.696772	−	1.20685i	0.969580	−	0.244776i	$-0.0787145\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	12.0000	−	20.7846i	0.638696	−	1.10625i	−0.347024	−	0.937856i	$-0.612808\pi$
$353$	12.0000	−	20.7846i	0.638696	−	1.10625i	0.985719	−	0.168397i	$-0.0538590\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−6.04098			−0.318831			−0.159415	−	0.987212i	$-0.550961\pi$
$359$	−6.04098			−0.318831			−0.159415	+	0.987212i	$0.550961\pi$
$360$		0			0
$361$	37.6136			1.97966
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	11.2305	−	19.4518i	0.586226	−	1.01537i	−0.408495	−	0.912761i	$-0.633946\pi$
$367$	11.2305	−	19.4518i	0.586226	−	1.01537i	0.994721	−	0.102613i	$-0.0327204\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	2.00742	−	3.47695i	0.104220	−	0.180514i
$372$		0			0
$373$	−1.26581	−	2.19245i	−0.0655411	−	0.113521i	0.831393	−	0.555685i	$-0.187544\pi$
$373$	−1.26581	−	2.19245i	−0.0655411	−	0.113521i	−0.896934	+	0.442165i	$0.854211\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	15.7475			0.811036
$378$		0			0
$379$	−21.0484			−1.08118			−0.540592	−	0.841285i	$-0.681800\pi$
$379$	−21.0484			−1.08118			−0.540592	+	0.841285i	$0.681800\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	3.81792	+	6.61283i	0.195086	+	0.337900i	0.946929	−	0.321443i	$-0.104168\pi$
$383$	3.81792	+	6.61283i	0.195086	+	0.337900i	−0.751842	+	0.659343i	$0.770835\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−9.46467	+	16.3933i	−0.479878	+	0.831173i	−0.999734	−	0.0230811i	$-0.992652\pi$
$389$	−9.46467	+	16.3933i	−0.479878	+	0.831173i	0.519856	+	0.854254i	$0.325986\pi$
$390$		0			0
$391$	2.02791	+	3.51244i	0.102556	+	0.177632i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	29.1600			1.46350			0.731750	−	0.681573i	$-0.238704\pi$
$397$	29.1600			1.46350			0.731750	+	0.681573i	$0.238704\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	16.3142	+	28.2570i	0.814693	+	1.41109i	0.909548	+	0.415598i	$0.136428\pi$
$401$	16.3142	+	28.2570i	0.814693	+	1.41109i	−0.0948557	+	0.995491i	$0.530239\pi$
$402$		0			0
$403$	−13.2584	+	22.9642i	−0.660447	+	1.14393i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−16.7268	+	28.9716i	−0.829115	+	1.43607i
$408$		0			0
$409$	−16.1042	−	27.8933i	−0.796302	−	1.37924i	−0.922009	−	0.387169i	$-0.873453\pi$
$409$	−16.1042	−	27.8933i	−0.796302	−	1.37924i	0.125707	−	0.992067i	$-0.459880\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−4.01484			−0.197557
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−10.7063	−	18.5438i	−0.523036	−	0.905925i	−0.999641	−	0.0268073i	$-0.991466\pi$
$419$	−10.7063	−	18.5438i	−0.523036	−	0.905925i	0.476605	−	0.879118i	$-0.341867\pi$
$420$		0			0
$421$	0.944182	−	1.63537i	0.0460166	−	0.0797031i	−0.842100	−	0.539322i	$-0.818681\pi$
$421$	0.944182	−	1.63537i	0.0460166	−	0.0797031i	0.888116	+	0.459619i	$0.152014\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	12.3802	+	21.4431i	0.599118	+	1.03770i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	2.11164			0.101714			0.0508569	−	0.998706i	$-0.483805\pi$
$431$	2.11164			0.101714			0.0508569	+	0.998706i	$0.483805\pi$
$432$		0			0
$433$	9.52420			0.457704			0.228852	−	0.973461i	$-0.426503\pi$
$433$	9.52420			0.457704			0.228852	+	0.973461i	$0.426503\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−2.76210	−	4.78410i	−0.132129	−	0.228854i
$438$		0			0
$439$	7.20257	−	12.4752i	0.343760	−	0.595410i	−0.641368	−	0.767234i	$-0.721633\pi$
$439$	7.20257	−	12.4752i	0.343760	−	0.595410i	0.985128	+	0.171824i	$0.0549660\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	9.60500	−	16.6363i	0.456347	−	0.790416i	−0.542417	−	0.840109i	$-0.682491\pi$
$443$	9.60500	−	16.6363i	0.456347	−	0.790416i	0.998765	+	0.0496927i	$0.0158242\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−33.5800			−1.58474			−0.792369	−	0.610041i	$-0.791153\pi$
$449$	−33.5800			−1.58474			−0.792369	+	0.610041i	$0.791153\pi$
$450$		0			0
$451$	−5.83255			−0.274644
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1.08373	−	1.87707i	0.0506946	−	0.0878056i	−0.839565	−	0.543260i	$-0.817190\pi$
$457$	1.08373	−	1.87707i	0.0506946	−	0.0878056i	0.890259	+	0.455454i	$0.150523\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−12.8700	+	22.2915i	−0.599417	+	1.03822i	0.393490	+	0.919329i	$0.371267\pi$
$461$	−12.8700	+	22.2915i	−0.599417	+	1.03822i	−0.992907	+	0.118892i	$0.962066\pi$
$462$		0			0
$463$	6.02791	+	10.4406i	0.280141	+	0.485218i	0.971419	−	0.237371i	$-0.0762855\pi$
$463$	6.02791	+	10.4406i	0.280141	+	0.485218i	−0.691279	+	0.722588i	$0.742952\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−14.4610			−0.669174			−0.334587	−	0.942365i	$-0.608597\pi$
$467$	−14.4610			−0.669174			−0.334587	+	0.942365i	$0.608597\pi$
$468$		0			0
$469$	−11.6433			−0.537635
$470$		0			0
$471$		0			0
$472$		0			0
$473$	9.73419	+	16.8601i	0.447579	+	0.775229i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−9.99258	+	17.3077i	−0.456573	+	0.790807i	−0.998777	−	0.0494395i	$-0.984257\pi$
$479$	−9.99258	+	17.3077i	−0.456573	+	0.790807i	0.542204	+	0.840247i	$0.317590\pi$
$480$		0			0
$481$	10.6710	+	18.4826i	0.486554	+	0.842736i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	18.1526			0.822574			0.411287	−	0.911506i	$-0.365079\pi$
$487$	18.1526			0.822574			0.411287	+	0.911506i	$0.365079\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	19.7900	+	34.2773i	0.893111	+	1.54691i	0.836126	+	0.548538i	$0.184815\pi$
$491$	19.7900	+	34.2773i	0.893111	+	1.54691i	0.0569849	+	0.998375i	$0.481851\pi$
$492$		0			0
$493$	12.3421	−	21.3772i	0.555861	−	0.962779i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−13.7473	+	23.8110i	−0.616649	+	1.06807i
$498$		0			0
$499$	21.3142	+	36.9173i	0.954155	+	1.65264i	0.736290	+	0.676666i	$0.236576\pi$
$499$	21.3142	+	36.9173i	0.954155	+	1.65264i	0.217865	+	0.975979i	$0.430091\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	21.1952			0.945045			0.472523	−	0.881319i	$-0.343343\pi$
$503$	21.1952			0.945045			0.472523	+	0.881319i	$0.343343\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−11.0168	−	19.0816i	−0.488310	−	0.845778i	0.511599	−	0.859224i	$-0.329053\pi$
$509$	−11.0168	−	19.0816i	−0.488310	−	0.845778i	−0.999910	+	0.0134460i	$0.995720\pi$
$510$		0			0
$511$	−10.9368	+	18.9430i	−0.483814	+	0.837990i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	3.34212	+	5.78872i	0.146986	+	0.254587i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−1.40515			−0.0615606			−0.0307803	−	0.999526i	$-0.509799\pi$
$521$	−1.40515			−0.0615606			−0.0307803	+	0.999526i	$0.509799\pi$
$522$		0			0
$523$	11.8532			0.518306			0.259153	−	0.965836i	$-0.416557\pi$
$523$	11.8532			0.518306			0.259153	+	0.965836i	$0.416557\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	20.7826	+	35.9965i	0.905304	+	1.56803i
$528$		0			0
$529$	11.2305	−	19.4518i	0.488282	−	0.845729i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−1.86046	+	3.22240i	−0.0805853	+	0.139578i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−2.62842			−0.113214
$540$		0			0
$541$	−8.98516			−0.386302			−0.193151	−	0.981169i	$-0.561871\pi$
$541$	−8.98516			−0.386302			−0.193151	+	0.981169i	$0.561871\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−3.10129	+	5.37159i	−0.132601	+	0.229672i	−0.924679	−	0.380749i	$-0.875666\pi$
$547$	−3.10129	+	5.37159i	−0.132601	+	0.229672i	0.792077	+	0.610421i	$0.209000\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−16.8105	+	29.1166i	−0.716151	+	1.24041i
$552$		0			0
$553$	2.73419	+	4.73576i	0.116270	+	0.201385i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	6.00000			0.254228			0.127114	−	0.991888i	$-0.459429\pi$
$557$	6.00000			0.254228			0.127114	+	0.991888i	$0.459429\pi$
$558$		0			0
$559$	12.4200			0.525309
$560$		0			0
$561$		0			0
$562$		0			0
$563$	2.13661	+	3.70072i	0.0900475	+	0.155967i	0.907531	−	0.419985i	$-0.137965\pi$
$563$	2.13661	+	3.70072i	0.0900475	+	0.155967i	−0.817483	+	0.575952i	$0.804631\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	8.04840	−	13.9402i	0.337406	−	0.584405i	−0.646538	−	0.762882i	$-0.723784\pi$
$569$	8.04840	−	13.9402i	0.337406	−	0.584405i	0.983944	+	0.178477i	$0.0571170\pi$
$570$		0			0
$571$	−5.46838	−	9.47152i	−0.228845	−	0.396371i	0.728621	−	0.684917i	$-0.240161\pi$
$571$	−5.46838	−	9.47152i	−0.228845	−	0.396371i	−0.957466	+	0.288546i	$0.906828\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	37.6620			1.56789			0.783944	−	0.620831i	$-0.213205\pi$
$577$	37.6620			1.56789			0.783944	+	0.620831i	$0.213205\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−7.19886	−	12.4688i	−0.298659	−	0.517293i
$582$		0			0
$583$	4.05582	−	7.02488i	0.167975	−	0.290941i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−10.8987	+	18.8771i	−0.449838	+	0.779142i	−0.998375	−	0.0569839i	$-0.981852\pi$
$587$	−10.8987	+	18.8771i	−0.449838	+	0.779142i	0.548537	+	0.836126i	$0.315185\pi$
$588$		0			0
$589$	−28.3068	−	49.0288i	−1.16636	−	2.02020i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	28.2643			1.16067			0.580337	−	0.814377i	$-0.302921\pi$
$593$	28.2643			1.16067			0.580337	+	0.814377i	$0.302921\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	7.05582	+	12.2210i	0.288293	+	0.499338i	0.973402	−	0.229102i	$-0.0735790\pi$
$599$	7.05582	+	12.2210i	0.288293	+	0.499338i	−0.685109	+	0.728440i	$0.740246\pi$
$600$		0			0
$601$	14.1042	−	24.4292i	0.575323	−	0.996489i	−0.420683	−	0.907207i	$-0.638210\pi$
$601$	14.1042	−	24.4292i	0.575323	−	0.996489i	0.996006	−	0.0892812i	$-0.0284570\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	14.1850	+	24.5692i	0.575752	+	0.997232i	0.995959	+	0.0898036i	$0.0286239\pi$
$607$	14.1850	+	24.5692i	0.575752	+	0.997232i	−0.420208	+	0.907428i	$0.638043\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	4.26425			0.172513
$612$		0			0
$613$	−24.5726			−0.992478			−0.496239	−	0.868186i	$-0.665286\pi$
$613$	−24.5726			−0.992478			−0.496239	+	0.868186i	$0.665286\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	13.3142	+	23.0609i	0.536010	+	0.928396i	0.999114	+	0.0420923i	$0.0134023\pi$
$617$	13.3142	+	23.0609i	0.536010	+	0.928396i	−0.463104	+	0.886304i	$0.653264\pi$
$618$		0			0
$619$	−15.2863	+	26.4766i	−0.614408	+	1.06419i	0.376080	+	0.926587i	$0.377272\pi$
$619$	−15.2863	+	26.4766i	−0.614408	+	1.06419i	−0.990488	+	0.137599i	$0.956061\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	4.10129	−	7.10364i	0.164315	−	0.284601i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$	33.4535			1.33388
$630$		0			0
$631$	27.4684			1.09350			0.546750	−	0.837296i	$-0.315865\pi$
$631$	27.4684			1.09350			0.546750	+	0.837296i	$0.315865\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−0.838408	+	1.45216i	−0.0332189	+	0.0575369i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	6.26952	−	10.8591i	0.247631	−	0.428910i	−0.715237	−	0.698882i	$-0.753681\pi$
$641$	6.26952	−	10.8591i	0.247631	−	0.428910i	0.962868	+	0.269972i	$0.0870146\pi$
$642$		0			0
$643$	−5.12920	−	8.88403i	−0.202276	−	0.350352i	0.746986	−	0.664840i	$-0.231500\pi$
$643$	−5.12920	−	8.88403i	−0.202276	−	0.350352i	−0.949261	+	0.314488i	$0.898167\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	16.7900			0.660083			0.330042	−	0.943966i	$-0.392937\pi$
$647$	16.7900			0.660083			0.330042	+	0.943966i	$0.392937\pi$
$648$		0			0
$649$	−8.11164			−0.318410
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−25.0131	−	43.3239i	−0.978837	−	1.69540i	−0.666643	−	0.745377i	$-0.732269\pi$
$653$	−25.0131	−	43.3239i	−0.978837	−	1.69540i	−0.312194	−	0.950018i	$-0.601064\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	2.78259	−	4.81959i	0.108394	−	0.187744i	−0.806726	−	0.590926i	$-0.798762\pi$
$659$	2.78259	−	4.81959i	0.108394	−	0.187744i	0.915120	+	0.403182i	$0.132096\pi$
$660$		0			0
$661$	19.6284	+	33.9974i	0.763457	+	1.32235i	0.941059	+	0.338244i	$0.109833\pi$
$661$	19.6284	+	33.9974i	0.763457	+	1.32235i	−0.177602	+	0.984102i	$0.556834\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	3.28065			0.127027
$668$		0			0
$669$		0			0
$670$		0			0
$671$	25.0131	+	43.3239i	0.965619	+	1.67250i
$672$		0			0
$673$	7.97209	−	13.8081i	0.307302	−	0.532262i	−0.670470	−	0.741937i	$-0.733907\pi$
$673$	7.97209	−	13.8081i	0.307302	−	0.532262i	0.977771	+	0.209675i	$0.0672406\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	10.4479	−	18.0963i	0.401545	−	0.695497i	−0.592368	−	0.805668i	$-0.701807\pi$
$677$	10.4479	−	18.0963i	0.401545	−	0.695497i	0.993913	+	0.110171i	$0.0351400\pi$
$678$		0			0
$679$	−12.9442	−	22.4200i	−0.496752	−	0.860400i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−19.6358			−0.751344			−0.375672	−	0.926753i	$-0.622588\pi$
$683$	−19.6358			−0.751344			−0.375672	+	0.926753i	$0.622588\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−2.58744	−	4.48157i	−0.0985734	−	0.170734i
$690$		0			0
$691$	21.5726	−	37.3648i	0.820660	−	1.42143i	−0.0845309	−	0.996421i	$-0.526939\pi$
$691$	21.5726	−	37.3648i	0.820660	−	1.42143i	0.905191	−	0.425005i	$-0.139728\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	2.91627	+	5.05113i	0.110462	+	0.191325i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−27.0410			−1.02132			−0.510662	−	0.859782i	$-0.670600\pi$
$701$	−27.0410			−1.02132			−0.510662	+	0.859782i	$0.670600\pi$
$702$		0			0
$703$	−45.5652			−1.71852
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−18.4126	−	31.8915i	−0.692476	−	1.19940i
$708$		0			0
$709$	−21.7510	+	37.6738i	−0.816875	+	1.41487i	0.0910989	+	0.995842i	$0.470962\pi$
$709$	−21.7510	+	37.6738i	−0.816875	+	1.41487i	−0.907974	+	0.419027i	$0.862371\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−2.76210	+	4.78410i	−0.103441	+	0.179166i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	4.40515			0.164284			0.0821421	−	0.996621i	$-0.473824\pi$
$719$	4.40515			0.164284			0.0821421	+	0.996621i	$0.473824\pi$
$720$		0			0
$721$	50.7810			1.89118
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	6.24083	−	10.8094i	0.231460	−	0.400900i	−0.726778	−	0.686872i	$-0.758983\pi$
$727$	6.24083	−	10.8094i	0.231460	−	0.400900i	0.958238	+	0.285972i	$0.0923166\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	9.73419	−	16.8601i	0.360032	−	0.623594i
$732$		0			0
$733$	19.5168	+	33.8041i	0.720869	+	1.24858i	0.960652	+	0.277755i	$0.0895903\pi$
$733$	19.5168	+	33.8041i	0.720869	+	1.24858i	−0.239783	+	0.970826i	$0.577076\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−23.5242			−0.866525
$738$		0			0
$739$	−12.2935			−0.452224			−0.226112	−	0.974101i	$-0.572602\pi$
$739$	−12.2935			−0.452224			−0.226112	+	0.974101i	$0.572602\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	20.4155	+	35.3607i	0.748972	+	1.29726i	0.948316	+	0.317328i	$0.102786\pi$
$743$	20.4155	+	35.3607i	0.748972	+	1.29726i	−0.199344	+	0.979930i	$0.563881\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	26.3179	−	45.5840i	0.961636	−	1.66560i
$750$		0			0
$751$	7.54469	+	13.0678i	0.275310	+	0.476850i	0.970213	−	0.242253i	$-0.0778863\pi$
$751$	7.54469	+	13.0678i	0.275310	+	0.476850i	−0.694904	+	0.719103i	$0.744553\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−34.4610			−1.25251			−0.626253	−	0.779620i	$-0.715412\pi$
$757$	−34.4610			−1.25251			−0.626253	+	0.779620i	$0.715412\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−4.17837	−	7.23716i	−0.151466	−	0.262347i	0.780301	−	0.625405i	$-0.215066\pi$
$761$	−4.17837	−	7.23716i	−0.151466	−	0.262347i	−0.931767	+	0.363058i	$0.881733\pi$
$762$		0			0
$763$	−0.716631	+	1.24124i	−0.0259438	+	0.0449359i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−2.58744	+	4.48157i	−0.0934269	+	0.161820i
$768$		0			0
$769$	9.17837	+	15.8974i	0.330981	+	0.573275i	0.982704	−	0.185181i	$-0.0592872\pi$
$769$	9.17837	+	15.8974i	0.330981	+	0.573275i	−0.651724	+	0.758456i	$0.725954\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−10.0968			−0.363157			−0.181578	−	0.983376i	$-0.558121\pi$
$773$	−10.0968			−0.363157			−0.181578	+	0.983376i	$0.558121\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−3.97209	−	6.87986i	−0.142315	−	0.246497i
$780$		0			0
$781$	−27.7752	+	48.1080i	−0.993874	+	1.72144i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−7.28630	−	12.6202i	−0.259729	−	0.449863i	0.706441	−	0.707772i	$-0.250300\pi$
$787$	−7.28630	−	12.6202i	−0.259729	−	0.449863i	−0.966169	+	0.257909i	$0.916966\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	20.4200			0.726051
$792$		0			0
$793$	31.9145			1.13332
$794$		0			0
$795$		0			0
$796$		0			0
$797$	1.03533	+	1.79324i	0.0366732	+	0.0635198i	0.883779	−	0.467904i	$-0.154991\pi$
$797$	1.03533	+	1.79324i	0.0366732	+	0.0635198i	−0.847106	+	0.531424i	$0.821657\pi$
$798$		0			0
$799$	3.34212	−	5.78872i	0.118236	−	0.204790i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−22.0968	+	38.2728i	−0.779779	+	1.35062i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−3.37158			−0.118539			−0.0592693	−	0.998242i	$-0.518877\pi$
$809$	−3.37158			−0.118539			−0.0592693	+	0.998242i	$0.518877\pi$
$810$		0			0
$811$	−12.9368			−0.454271			−0.227136	−	0.973863i	$-0.572936\pi$
$811$	−12.9368			−0.454271			−0.227136	+	0.973863i	$0.572936\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−13.2584	+	22.9642i	−0.463852	+	0.803416i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−10.9963	+	19.0461i	−0.383773	+	0.664715i	−0.991598	−	0.129356i	$-0.958709\pi$
$821$	−10.9963	+	19.0461i	−0.383773	+	0.664715i	0.607825	+	0.794071i	$0.292042\pi$
$822$		0			0
$823$	7.45082	+	12.9052i	0.259719	+	0.449847i	0.966167	−	0.257919i	$-0.0830366\pi$
$823$	7.45082	+	12.9052i	0.259719	+	0.449847i	−0.706447	+	0.707766i	$0.749703\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	56.5785			1.96743			0.983713	−	0.179747i	$-0.0575279\pi$
$827$	56.5785			1.96743			0.983713	+	0.179747i	$0.0575279\pi$
$828$		0			0
$829$	27.0558			0.939687			0.469844	−	0.882750i	$-0.344310\pi$
$829$	27.0558			0.939687			0.469844	+	0.882750i	$0.344310\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	1.31421	+	2.27628i	0.0455346	+	0.0788683i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−17.2863	+	29.9407i	−0.596789	+	1.03367i	0.396502	+	0.918034i	$0.370224\pi$
$839$	−17.2863	+	29.9407i	−0.596789	+	1.03367i	−0.993292	+	0.115636i	$0.963109\pi$
$840$		0			0
$841$	4.51678	+	7.82329i	0.155751	+	0.269769i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−53.3626			−1.83356
$848$		0			0
$849$		0			0
$850$		0			0
$851$	2.22306	+	3.85046i	0.0762056	+	0.131992i
$852$		0			0
$853$	8.48887	−	14.7032i	0.290653	−	0.503427i	−0.683311	−	0.730127i	$-0.739461\pi$
$853$	8.48887	−	14.7032i	0.290653	−	0.503427i	0.973964	+	0.226701i	$0.0727940\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1.70628	+	2.95537i	−0.0582855	+	0.100953i	−0.893696	−	0.448673i	$-0.851897\pi$
$857$	−1.70628	+	2.95537i	−0.0582855	+	0.100953i	0.835410	+	0.549627i	$0.185230\pi$
$858$		0			0
$859$	4.91627	+	8.51524i	0.167741	+	0.290536i	0.937625	−	0.347647i	$-0.113019\pi$
$859$	4.91627	+	8.51524i	0.167741	+	0.290536i	−0.769884	+	0.638184i	$0.779686\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−48.9836			−1.66742			−0.833711	−	0.552202i	$-0.813788\pi$
$863$	−48.9836			−1.66742			−0.833711	+	0.552202i	$0.813788\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	5.52420	+	9.56819i	0.187396	+	0.324579i
$870$		0			0
$871$	−7.50371	+	12.9968i	−0.254253	+	0.440380i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−6.33470	−	10.9720i	−0.213908	−	0.370499i	0.739027	−	0.673676i	$-0.235286\pi$
$877$	−6.33470	−	10.9720i	−0.213908	−	0.370499i	−0.952934	+	0.303178i	$0.901952\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	26.0894			0.878974			0.439487	−	0.898249i	$-0.355160\pi$
$881$	26.0894			0.878974			0.439487	+	0.898249i	$0.355160\pi$
$882$		0			0
$883$	−2.69321			−0.0906337			−0.0453169	−	0.998973i	$-0.514430\pi$
$883$	−2.69321			−0.0906337			−0.0453169	+	0.998973i	$0.514430\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	12.2789	+	21.2676i	0.412284	+	0.714098i	0.995139	−	0.0984788i	$-0.0313977\pi$
$887$	12.2789	+	21.2676i	0.412284	+	0.714098i	−0.582855	+	0.812576i	$0.698064\pi$
$888$		0			0
$889$	−11.2900	+	19.5549i	−0.378655	+	0.655849i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−4.55211	+	7.88448i	−0.152330	+	0.263844i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	33.6210			1.12132
$900$		0			0
$901$	−8.11164			−0.270238
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	17.1850	−	29.7653i	0.570619	−	0.988341i	−0.425884	−	0.904778i	$-0.640037\pi$
$907$	17.1850	−	29.7653i	0.570619	−	0.988341i	0.996502	−	0.0835631i	$-0.0266300\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	18.7752	−	32.5196i	0.622049	−	1.07742i	−0.367054	−	0.930199i	$-0.619634\pi$
$911$	18.7752	−	32.5196i	0.622049	−	1.07742i	0.989104	−	0.147221i	$-0.0470330\pi$
$912$		0			0
$913$	−14.5447	−	25.1921i	−0.481359	−	0.833738i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−44.0117			−1.45340
$918$		0			0
$919$	10.1116			0.333552			0.166776	−	0.985995i	$-0.446664\pi$
$919$	10.1116			0.333552			0.166776	+	0.985995i	$0.446664\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	17.7194	+	30.6908i	0.583240	+	1.01020i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	12.4126	−	21.4992i	0.407243	−	0.705366i	−0.587337	−	0.809343i	$-0.699824\pi$
$929$	12.4126	−	21.4992i	0.407243	−	0.705366i	0.994580	+	0.103977i	$0.0331569\pi$
$930$		0			0
$931$	−1.79001	−	3.10039i	−0.0586652	−	0.101611i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−1.56518			−0.0511322			−0.0255661	−	0.999673i	$-0.508139\pi$
$937$	−1.56518			−0.0511322			−0.0255661	+	0.999673i	$0.508139\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−5.23419	−	9.06588i	−0.170630	−	0.295539i	0.768010	−	0.640437i	$-0.221247\pi$
$941$	−5.23419	−	9.06588i	−0.170630	−	0.295539i	−0.938640	+	0.344898i	$0.887914\pi$
$942$		0			0
$943$	−0.387586	+	0.671318i	−0.0126215	+	0.0218611i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1.59758	−	2.76709i	0.0519143	−	0.0899182i	−0.838900	−	0.544285i	$-0.816801\pi$
$947$	1.59758	−	2.76709i	0.0519143	−	0.0899182i	0.890815	+	0.454367i	$0.150134\pi$
$948$		0			0
$949$	14.0968	+	24.4164i	0.457601	+	0.792589i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−33.9293			−1.09908			−0.549540	−	0.835468i	$-0.685197\pi$
$953$	−33.9293			−1.09908			−0.549540	+	0.835468i	$0.685197\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−25.3142	−	43.8455i	−0.817438	−	1.41584i
$960$		0			0
$961$	−12.8068	+	22.1820i	−0.413122	+	0.715549i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	19.2129	+	33.2778i	0.617846	+	1.07014i	0.989878	+	0.141921i	$0.0453278\pi$
$967$	19.2129	+	33.2778i	0.617846	+	1.07014i	−0.372032	+	0.928220i	$0.621339\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−29.7326			−0.954166			−0.477083	−	0.878858i	$-0.658306\pi$
$971$	−29.7326			−0.954166			−0.477083	+	0.878858i	$0.658306\pi$
$972$		0			0
$973$	10.9368			0.350617
$974$		0			0
$975$		0			0
$976$		0			0
$977$	16.4889	+	28.5596i	0.527526	+	0.913701i	0.999485	+	0.0320812i	$0.0102135\pi$
$977$	16.4889	+	28.5596i	0.527526	+	0.913701i	−0.471960	+	0.881620i	$0.656453\pi$
$978$		0			0
$979$	8.28630	−	14.3523i	0.264831	−	0.458701i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	3.68872	−	6.38905i	0.117652	−	0.203779i	−0.801185	−	0.598417i	$-0.795797\pi$
$983$	3.68872	−	6.38905i	0.117652	−	0.203779i	0.918837	+	0.394638i	$0.129130\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	2.58744			0.0822757
$990$		0			0
$991$	−4.51678			−0.143480			−0.0717401	−	0.997423i	$-0.522855\pi$
$991$	−4.51678			−0.143480			−0.0717401	+	0.997423i	$0.522855\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−24.3142	+	42.1134i	−0.770039	+	1.33375i	0.167502	+	0.985872i	$0.446430\pi$
$997$	−24.3142	+	42.1134i	−0.770039	+	1.33375i	−0.937541	+	0.347874i	$0.886904\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2700.2.i.c.1801.2		6
3.2	odd	2		900.2.i.c.601.3		6
5.2	odd	4		2700.2.s.c.2449.4		12
5.3	odd	4		2700.2.s.c.2449.3		12
5.4	even	2		540.2.i.b.181.2		6
9.2	odd	6		8100.2.a.v.1.2		3
9.4	even	3	inner	2700.2.i.c.901.2		6
9.5	odd	6		900.2.i.c.301.3		6
9.7	even	3		8100.2.a.u.1.2		3
15.2	even	4		900.2.s.c.349.3		12
15.8	even	4		900.2.s.c.349.4		12
15.14	odd	2		180.2.i.b.61.1	✓	6
20.19	odd	2		2160.2.q.i.721.2		6
45.2	even	12		8100.2.d.p.649.3		6
45.4	even	6		540.2.i.b.361.2		6
45.7	odd	12		8100.2.d.o.649.3		6
45.13	odd	12		2700.2.s.c.1549.4		12
45.14	odd	6		180.2.i.b.121.1	yes	6
45.22	odd	12		2700.2.s.c.1549.3		12
45.23	even	12		900.2.s.c.49.3		12
45.29	odd	6		1620.2.a.i.1.2		3
45.32	even	12		900.2.s.c.49.4		12
45.34	even	6		1620.2.a.j.1.2		3
45.38	even	12		8100.2.d.p.649.4		6
45.43	odd	12		8100.2.d.o.649.4		6
60.59	even	2		720.2.q.k.241.3		6
180.59	even	6		720.2.q.k.481.3		6
180.79	odd	6		6480.2.a.bw.1.2		3
180.119	even	6		6480.2.a.bt.1.2		3
180.139	odd	6		2160.2.q.i.1441.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.2.i.b.61.1	✓	6	15.14	odd	2
180.2.i.b.121.1	yes	6	45.14	odd	6
540.2.i.b.181.2		6	5.4	even	2
540.2.i.b.361.2		6	45.4	even	6
720.2.q.k.241.3		6	60.59	even	2
720.2.q.k.481.3		6	180.59	even	6
900.2.i.c.301.3		6	9.5	odd	6
900.2.i.c.601.3		6	3.2	odd	2
900.2.s.c.49.3		12	45.23	even	12
900.2.s.c.49.4		12	45.32	even	12
900.2.s.c.349.3		12	15.2	even	4
900.2.s.c.349.4		12	15.8	even	4
1620.2.a.i.1.2		3	45.29	odd	6
1620.2.a.j.1.2		3	45.34	even	6
2160.2.q.i.721.2		6	20.19	odd	2
2160.2.q.i.1441.2		6	180.139	odd	6
2700.2.i.c.901.2		6	9.4	even	3	inner
2700.2.i.c.1801.2		6	1.1	even	1	trivial
2700.2.s.c.1549.3		12	45.22	odd	12
2700.2.s.c.1549.4		12	45.13	odd	12
2700.2.s.c.2449.3		12	5.3	odd	4
2700.2.s.c.2449.4		12	5.2	odd	4
6480.2.a.bt.1.2		3	180.119	even	6
6480.2.a.bw.1.2		3	180.79	odd	6
8100.2.a.u.1.2		3	9.7	even	3
8100.2.a.v.1.2		3	9.2	odd	6
8100.2.d.o.649.3		6	45.7	odd	12
8100.2.d.o.649.4		6	45.43	odd	12
8100.2.d.p.649.3		6	45.2	even	12
8100.2.d.p.649.4		6	45.38	even	12

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.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.36710 - 2.36788i) q^{7} +(2.76210 - 4.78410i) q^{11} +(-1.76210 - 3.05205i) q^{13} -5.52420 q^{17} +7.52420 q^{19} +(-0.367095 - 0.635828i) q^{23} +(-2.23419 + 3.86973i) q^{29} +(-3.76210 - 6.51615i) q^{31} -6.05582 q^{37} +(-0.527909 - 0.914365i) q^{41} +(-1.76210 + 3.05205i) q^{43} +(-0.604996 + 1.04788i) q^{47} +(-0.237900 - 0.412055i) q^{49} +1.46838 q^{53} +(-0.734191 - 1.27166i) q^{59} +(-4.52791 + 7.84257i) q^{61} +(-2.12920 - 3.68787i) q^{67} -10.0558 q^{71} -8.00000 q^{73} +(-7.55211 - 13.0806i) q^{77} +(-1.00000 + 1.73205i) q^{79} +(2.63290 - 4.56032i) q^{83} +3.00000 q^{89} -9.63583 q^{91} +(4.73419 - 8.19986i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{7} + 6 q^{13} + 12 q^{19} + 3 q^{23} - 3 q^{29} - 6 q^{31} - 24 q^{37} + 3 q^{41} + 6 q^{43} - 15 q^{47} - 18 q^{49} - 12 q^{53} + 6 q^{59} - 21 q^{61} + 9 q^{67} - 48 q^{71} - 48 q^{73} - 6 q^{77}+ \cdots + 18 q^{97}+O(q^{100}) \) 6 * q + 3 * q^7 + 6 * q^13 + 12 * q^19 + 3 * q^23 - 3 * q^29 - 6 * q^31 - 24 * q^37 + 3 * q^41 + 6 * q^43 - 15 * q^47 - 18 * q^49 - 12 * q^53 + 6 * q^59 - 21 * q^61 + 9 * q^67 - 48 * q^71 - 48 * q^73 - 6 * q^77 - 6 * q^79 + 21 * q^83 + 18 * q^89 + 18 * q^97

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