LMFDB - Embedded newform 2700.3.p.d.1601.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2700,3,Mod(1601,2700)]

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, 1, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("2700.1601");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2700.p (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:	$73.5696713773$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} + 10 x^{14} - 12 x^{13} + 6 x^{12} - 27 x^{11} - 585 x^{10} + 3618 x^{9} + \cdots + 43046721 $ x^16 - 2x^15 + 10x^14 - 12x^13 + 6x^12 - 27x^11 - 585x^10 + 3618x^9 - 12150x^8 + 32562x^7 - 47385x^6 - 19683x^5 + 39366x^4 - 708588x^3 + 5314410x^2 - 9565938*x + 43046721
Coefficient ring:	$\Z[a_1, \ldots, a_{41}]$
Coefficient ring index:	$ 3^{14} $
Twist minimal:	no (minimal twist has level 900)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1601.7
Root			$0.127146 + 2.99730i$ of defining polynomial
Character	$\chi$	$=$	2700.1601
Dual form			2700.3.p.d.2501.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(4.69530 - 8.13249i) q^{7} +O(q^{10})$	$q+(4.69530 - 8.13249i) q^{7} +(-15.4241 - 8.90510i) q^{11} +(-6.92181 - 11.9889i) q^{13} -30.7147i q^{17} +28.5585 q^{19} +(-10.6734 + 6.16230i) q^{23} +(5.02918 + 2.90360i) q^{29} +(-3.25379 - 5.63573i) q^{31} -66.5560 q^{37} +(33.0070 - 19.0566i) q^{41} +(27.5777 - 47.7660i) q^{43} +(14.2631 + 8.23479i) q^{47} +(-19.5916 - 33.9337i) q^{49} +69.8669i q^{53} +(-91.2592 + 52.6885i) q^{59} +(-33.5580 + 58.1242i) q^{61} +(22.9853 + 39.8118i) q^{67} -31.1942i q^{71} +73.5877 q^{73} +(-144.841 + 83.6241i) q^{77} +(-47.3263 + 81.9715i) q^{79} +(13.4020 + 7.73763i) q^{83} +52.7229i q^{89} -130.000 q^{91} +(38.1126 - 66.0130i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - q^{7}+O(q^{10}) $ 16 * q - q^7	$ 16 q - q^{7} - 10 q^{13} + 2 q^{19} - 27 q^{23} - 9 q^{29} + 8 q^{31} - 22 q^{37} - 54 q^{41} + 44 q^{43} + 108 q^{47} - 45 q^{49} - 9 q^{59} - 55 q^{61} - 28 q^{67} + 86 q^{73} - 342 q^{77} + 11 q^{79} + 306 q^{83} - 134 q^{91} + 41 q^{97}+O(q^{100}) $ 16 * q - q^7 - 10 * q^13 + 2 * q^19 - 27 * q^23 - 9 * q^29 + 8 * q^31 - 22 * q^37 - 54 * q^41 + 44 * q^43 + 108 * q^47 - 45 * q^49 - 9 * q^59 - 55 * q^61 - 28 * q^67 + 86 * q^73 - 342 * q^77 + 11 * q^79 + 306 * q^83 - 134 * q^91 + 41 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$1001$	$1351$	$2377$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$1$	$1$

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
$7$	4.69530	−	8.13249i	0.670757	−	1.16178i	−0.306933	−	0.951731i	$-0.599303\pi$
$7$	4.69530	−	8.13249i	0.670757	−	1.16178i	0.977690	−	0.210053i	$-0.0673638\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−15.4241	−	8.90510i	−1.40219	−	0.809554i	−0.407572	−	0.913173i	$-0.633624\pi$
$11$	−15.4241	−	8.90510i	−1.40219	−	0.809554i	−0.994617	+	0.103619i	$0.966958\pi$
$12$		0			0
$13$	−6.92181	−	11.9889i	−0.532447	−	0.922226i	−0.999282	−	0.0378812i	$-0.987939\pi$
$13$	−6.92181	−	11.9889i	−0.532447	−	0.922226i	0.466835	−	0.884344i	$-0.345394\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	30.7147i		−	1.80675i	−0.428856	−	0.903373i	$-0.641083\pi$
$17$		−	30.7147i		−	1.80675i	0.428856	−	0.903373i	$-0.358917\pi$
$18$		0			0
$19$	28.5585			1.50308			0.751540	−	0.659687i	$-0.229311\pi$
$19$	28.5585			1.50308			0.751540	+	0.659687i	$0.229311\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−10.6734	+	6.16230i	−0.464062	+	0.267926i	−0.713751	−	0.700400i	$-0.753005\pi$
$23$	−10.6734	+	6.16230i	−0.464062	+	0.267926i	0.249689	+	0.968326i	$0.419672\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$	5.02918	+	2.90360i	0.173420	+	0.100124i	0.584198	−	0.811612i	$-0.301409\pi$
$29$	5.02918	+	2.90360i	0.173420	+	0.100124i	−0.410777	+	0.911736i	$0.634743\pi$
$30$		0			0
$31$	−3.25379	−	5.63573i	−0.104961	−	0.181798i	0.808761	−	0.588137i	$-0.200138\pi$
$31$	−3.25379	−	5.63573i	−0.104961	−	0.181798i	−0.913722	+	0.406339i	$0.866805\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−66.5560			−1.79881			−0.899405	−	0.437116i	$-0.856000\pi$
$37$	−66.5560			−1.79881			−0.899405	+	0.437116i	$0.856000\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	33.0070	−	19.0566i	0.805049	−	0.464795i	−0.0401848	−	0.999192i	$-0.512795\pi$
$41$	33.0070	−	19.0566i	0.805049	−	0.464795i	0.845233	+	0.534397i	$0.179461\pi$
$42$		0			0
$43$	27.5777	−	47.7660i	0.641342	−	1.11084i	−0.343792	−	0.939046i	$-0.611711\pi$
$43$	27.5777	−	47.7660i	0.641342	−	1.11084i	0.985133	−	0.171791i	$-0.0549553\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	14.2631	+	8.23479i	0.303470	+	0.175208i	0.644001	−	0.765025i	$-0.277273\pi$
$47$	14.2631	+	8.23479i	0.303470	+	0.175208i	−0.340531	+	0.940233i	$0.610607\pi$
$48$		0			0
$49$	−19.5916	−	33.9337i	−0.399829	−	0.692523i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			69.8669i			1.31824i	0.752036	+	0.659121i	$0.229072\pi$
$53$			69.8669i			1.31824i	−0.752036	+	0.659121i	$0.770928\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−91.2592	+	52.6885i	−1.54677	+	0.893026i	−0.548380	+	0.836229i	$0.684755\pi$
$59$	−91.2592	+	52.6885i	−1.54677	+	0.893026i	−0.998386	+	0.0567963i	$0.981911\pi$
$60$		0			0
$61$	−33.5580	+	58.1242i	−0.550131	+	0.952855i	0.448133	+	0.893967i	$0.352089\pi$
$61$	−33.5580	+	58.1242i	−0.550131	+	0.952855i	−0.998265	+	0.0588885i	$0.981244\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	22.9853	+	39.8118i	0.343065	+	0.594206i	0.985000	−	0.172553i	$-0.0552016\pi$
$67$	22.9853	+	39.8118i	0.343065	+	0.594206i	−0.641935	+	0.766759i	$0.721868\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	31.1942i		−	0.439354i	−0.975573	−	0.219677i	$-0.929500\pi$
$71$		−	31.1942i		−	0.439354i	0.975573	−	0.219677i	$-0.0705004\pi$
$72$		0			0
$73$	73.5877			1.00805			0.504026	−	0.863689i	$-0.331852\pi$
$73$	73.5877			1.00805			0.504026	+	0.863689i	$0.331852\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−144.841	+	83.6241i	−1.88105	+	1.08603i
$78$		0			0
$79$	−47.3263	+	81.9715i	−0.599067	+	1.03761i	0.393892	+	0.919157i	$0.371128\pi$
$79$	−47.3263	+	81.9715i	−0.599067	+	1.03761i	−0.992959	+	0.118458i	$0.962205\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	13.4020	+	7.73763i	0.161470	+	0.0932245i	0.578557	−	0.815642i	$-0.303616\pi$
$83$	13.4020	+	7.73763i	0.161470	+	0.0932245i	−0.417088	+	0.908866i	$0.636949\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$			52.7229i			0.592392i	0.955127	+	0.296196i	$0.0957181\pi$
$89$			52.7229i			0.592392i	−0.955127	+	0.296196i	$0.904282\pi$
$90$		0			0
$91$	−130.000			−1.42857
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	38.1126	−	66.0130i	0.392913	−	0.680546i	−0.599919	−	0.800061i	$-0.704801\pi$
$97$	38.1126	−	66.0130i	0.392913	−	0.680546i	0.992832	+	0.119515i	$0.0381338\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−69.5637	−	40.1626i	−0.688750	−	0.397650i	0.114394	−	0.993435i	$-0.463507\pi$
$101$	−69.5637	−	40.1626i	−0.688750	−	0.397650i	−0.803144	+	0.595786i	$0.796841\pi$
$102$		0			0
$103$	−15.0683	−	26.0991i	−0.146295	−	0.253390i	0.783561	−	0.621315i	$-0.213401\pi$
$103$	−15.0683	−	26.0991i	−0.146295	−	0.253390i	−0.929855	+	0.367926i	$0.880068\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			10.4620i			0.0977760i	0.998804	+	0.0488880i	$0.0155677\pi$
$107$			10.4620i			0.0977760i	−0.998804	+	0.0488880i	$0.984432\pi$
$108$		0			0
$109$	−187.263			−1.71801			−0.859006	−	0.511965i	$-0.828918\pi$
$109$	−187.263			−1.71801			−0.859006	+	0.511965i	$0.828918\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	80.6035	−	46.5365i	0.713305	−	0.411827i	−0.0989784	−	0.995090i	$-0.531557\pi$
$113$	80.6035	−	46.5365i	0.713305	−	0.411827i	0.812284	+	0.583263i	$0.198224\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−249.787	−	144.214i	−2.09905	−	1.21189i
$120$		0			0
$121$	98.1015	+	169.917i	0.810756	+	1.40427i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	48.4899			0.381810			0.190905	−	0.981608i	$-0.438858\pi$
$127$	48.4899			0.381810			0.190905	+	0.981608i	$0.438858\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	54.0490	−	31.2052i	0.412588	−	0.238208i	−0.279313	−	0.960200i	$-0.590107\pi$
$131$	54.0490	−	31.2052i	0.412588	−	0.238208i	0.691901	+	0.721992i	$0.256773\pi$
$132$		0			0
$133$	134.091	−	232.252i	1.00820	−	1.74626i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1.85510	−	1.07104i	−0.0135409	−	0.00781782i	0.493214	−	0.869908i	$-0.335822\pi$
$137$	−1.85510	−	1.07104i	−0.0135409	−	0.00781782i	−0.506755	+	0.862090i	$0.669155\pi$
$138$		0			0
$139$	−7.40371	−	12.8236i	−0.0532641	−	0.0922562i	0.838164	−	0.545418i	$-0.183629\pi$
$139$	−7.40371	−	12.8236i	−0.0532641	−	0.0922562i	−0.891428	+	0.453162i	$0.850296\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			246.558i			1.72418i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	177.328	−	102.380i	1.19012	−	0.687116i	0.231785	−	0.972767i	$-0.425543\pi$
$149$	177.328	−	102.380i	1.19012	−	0.687116i	0.958334	+	0.285651i	$0.0922099\pi$
$150$		0			0
$151$	7.45266	−	12.9084i	0.0493553	−	0.0854860i	−0.840292	−	0.542134i	$-0.817617\pi$
$151$	7.45266	−	12.9084i	0.0493553	−	0.0854860i	0.889648	+	0.456648i	$0.150950\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	9.15362	+	15.8545i	0.0583033	+	0.100984i	0.893704	−	0.448657i	$-0.148098\pi$
$157$	9.15362	+	15.8545i	0.0583033	+	0.100984i	−0.835401	+	0.549642i	$0.814764\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			115.735i			0.718853i
$162$		0			0
$163$	43.2107			0.265096			0.132548	−	0.991177i	$-0.457684\pi$
$163$	43.2107			0.265096			0.132548	+	0.991177i	$0.457684\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−61.7635	+	35.6591i	−0.369841	+	0.213528i	−0.673389	−	0.739288i	$-0.735162\pi$
$167$	−61.7635	+	35.6591i	−0.369841	+	0.213528i	0.303548	+	0.952816i	$0.401829\pi$
$168$		0			0
$169$	−11.3230	+	19.6120i	−0.0670000	+	0.116047i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	145.397	+	83.9449i	0.840444	+	0.485230i	0.857415	−	0.514626i	$-0.172069\pi$
$173$	145.397	+	83.9449i	0.840444	+	0.485230i	−0.0169713	+	0.999856i	$0.505402\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	117.614i		−	0.657063i	−0.944493	−	0.328531i	$-0.893446\pi$
$179$		−	117.614i		−	0.657063i	0.944493	−	0.328531i	$-0.106554\pi$
$180$		0			0
$181$	96.5280			0.533304			0.266652	−	0.963793i	$-0.414083\pi$
$181$	96.5280			0.533304			0.266652	+	0.963793i	$0.414083\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−273.517	+	473.746i	−1.46266	+	2.53340i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−51.9090	−	29.9697i	−0.271775	−	0.156909i	0.357919	−	0.933753i	$-0.383486\pi$
$191$	−51.9090	−	29.9697i	−0.271775	−	0.156909i	−0.629694	+	0.776843i	$0.716820\pi$
$192$		0			0
$193$	30.6963	+	53.1675i	0.159048	+	0.275479i	0.934526	−	0.355896i	$-0.115824\pi$
$193$	30.6963	+	53.1675i	0.159048	+	0.275479i	−0.775478	+	0.631375i	$0.782491\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	289.758i		−	1.47085i	−0.677604	−	0.735427i	$-0.736982\pi$
$197$		−	289.758i		−	1.47085i	0.677604	−	0.735427i	$-0.263018\pi$
$198$		0			0
$199$	−100.347			−0.504254			−0.252127	−	0.967694i	$-0.581130\pi$
$199$	−100.347			−0.504254			−0.252127	+	0.967694i	$0.581130\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	47.2270	−	27.2665i	0.232645	−	0.134318i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−440.489	−	254.316i	−2.10760	−	1.21683i
$210$		0			0
$211$	−106.282	−	184.086i	−0.503706	−	0.872444i	−0.999991	−	0.00428419i	$-0.998636\pi$
$211$	−106.282	−	184.086i	−0.503706	−	0.872444i	0.496285	−	0.868160i	$-0.334697\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−61.1100			−0.281613
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−368.236	+	212.601i	−1.66623	+	0.961996i
$222$		0			0
$223$	−111.359	+	192.880i	−0.499369	+	0.864932i	−1.00000		0.000728529i	$-0.999768\pi$
$223$	−111.359	+	192.880i	−0.499369	+	0.864932i	0.500631	+	0.865661i	$0.333101\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−117.812	−	68.0187i	−0.518995	−	0.299642i	0.217528	−	0.976054i	$-0.430201\pi$
$227$	−117.812	−	68.0187i	−0.518995	−	0.299642i	−0.736523	+	0.676412i	$0.763534\pi$
$228$		0			0
$229$	197.762	+	342.533i	0.863589	+	1.49578i	0.868442	+	0.495791i	$0.165122\pi$
$229$	197.762	+	342.533i	0.863589	+	1.49578i	−0.00485333	+	0.999988i	$0.501545\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	403.347i		−	1.73110i	−0.500821	−	0.865551i	$-0.666969\pi$
$233$		−	403.347i		−	1.73110i	0.500821	−	0.865551i	$-0.333031\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−370.505	+	213.911i	−1.55023	+	0.895025i	−0.552106	+	0.833774i	$0.686176\pi$
$239$	−370.505	+	213.911i	−1.55023	+	0.895025i	−0.998122	+	0.0612505i	$0.980491\pi$
$240$		0			0
$241$	−27.4457	+	47.5373i	−0.113882	+	0.197250i	−0.917332	−	0.398122i	$-0.869662\pi$
$241$	−27.4457	+	47.5373i	−0.113882	+	0.197250i	0.803450	+	0.595372i	$0.202995\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−197.677	−	342.386i	−0.800311	−	1.38618i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	404.670i		−	1.61223i	−0.591759	−	0.806115i	$-0.701566\pi$
$251$		−	404.670i		−	1.61223i	0.591759	−	0.806115i	$-0.298434\pi$
$252$		0			0
$253$	219.504			0.867603
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−77.2834	+	44.6196i	−0.300713	+	0.173617i	−0.642763	−	0.766065i	$-0.722212\pi$
$257$	−77.2834	+	44.6196i	−0.300713	+	0.173617i	0.342050	+	0.939682i	$0.388879\pi$
$258$		0			0
$259$	−312.500	+	541.266i	−1.20656	+	2.08983i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	291.278	+	168.170i	1.10752	+	0.639428i	0.938186	−	0.346130i	$-0.112505\pi$
$263$	291.278	+	168.170i	1.10752	+	0.639428i	0.169335	+	0.985558i	$0.445838\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$			225.504i			0.838305i	0.907916	+	0.419153i	$0.137673\pi$
$269$			225.504i			0.838305i	−0.907916	+	0.419153i	$0.862327\pi$
$270$		0			0
$271$	−196.656			−0.725669			−0.362834	−	0.931854i	$-0.618191\pi$
$271$	−196.656			−0.725669			−0.362834	+	0.931854i	$0.618191\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−89.0481	+	154.236i	−0.321473	+	0.556808i	−0.980792	−	0.195055i	$-0.937511\pi$
$277$	−89.0481	+	154.236i	−0.321473	+	0.556808i	0.659319	+	0.751863i	$0.270845\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−74.3930	−	42.9508i	−0.264744	−	0.152850i	0.361753	−	0.932274i	$-0.382178\pi$
$281$	−74.3930	−	42.9508i	−0.264744	−	0.152850i	−0.626497	+	0.779424i	$0.715512\pi$
$282$		0			0
$283$	−60.9884	−	105.635i	−0.215507	−	0.373269i	0.737922	−	0.674886i	$-0.235807\pi$
$283$	−60.9884	−	105.635i	−0.215507	−	0.373269i	−0.953429	+	0.301617i	$0.902474\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	357.905i		−	1.24706i
$288$		0			0
$289$	−654.391			−2.26433
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−77.1800	+	44.5599i	−0.263413	+	0.152081i	−0.625890	−	0.779911i	$-0.715264\pi$
$293$	−77.1800	+	44.5599i	−0.263413	+	0.152081i	0.362478	+	0.931992i	$0.381931\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	147.759	+	85.3086i	0.494177	+	0.285313i
$300$		0			0
$301$	−258.971	−	448.551i	−0.860368	−	1.49020i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−602.803			−1.96353			−0.981764	−	0.190105i	$-0.939117\pi$
$307$	−602.803			−1.96353			−0.981764	+	0.190105i	$0.939117\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−142.632	+	82.3487i	−0.458624	+	0.264787i	−0.711466	−	0.702721i	$-0.751968\pi$
$311$	−142.632	+	82.3487i	−0.458624	+	0.264787i	0.252841	+	0.967508i	$0.418635\pi$
$312$		0			0
$313$	7.33340	−	12.7018i	0.0234294	−	0.0405809i	−0.854073	−	0.520153i	$-0.825875\pi$
$313$	7.33340	−	12.7018i	0.0234294	−	0.0405809i	0.877502	+	0.479572i	$0.159208\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−408.868	−	236.060i	−1.28981	−	0.744670i	−0.311186	−	0.950349i	$-0.600726\pi$
$317$	−408.868	−	236.060i	−1.28981	−	0.744670i	−0.978619	+	0.205679i	$0.934060\pi$
$318$		0			0
$319$	−51.7137	−	89.5707i	−0.162112	−	0.280786i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	877.166i		−	2.71568i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	133.939	−	77.3296i	0.407109	−	0.235044i
$330$		0			0
$331$	−241.860	+	418.914i	−0.730695	+	1.26560i	0.225892	+	0.974152i	$0.427470\pi$
$331$	−241.860	+	418.914i	−0.730695	+	1.26560i	−0.956587	+	0.291448i	$0.905863\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	181.488	+	314.346i	0.538540	+	0.932778i	0.998983	+	0.0450890i	$0.0143571\pi$
$337$	181.488	+	314.346i	0.538540	+	0.932778i	−0.460443	+	0.887689i	$0.652310\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			115.901i			0.339886i
$342$		0			0
$343$	92.1855			0.268762
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−318.851	+	184.089i	−0.918879	+	0.530515i	−0.883277	−	0.468851i	$-0.844668\pi$
$347$	−318.851	+	184.089i	−0.918879	+	0.530515i	−0.0356015	+	0.999366i	$0.511335\pi$
$348$		0			0
$349$	−199.861	+	346.170i	−0.572668	+	0.991890i	0.423623	+	0.905839i	$0.360758\pi$
$349$	−199.861	+	346.170i	−0.572668	+	0.991890i	−0.996291	+	0.0860512i	$0.972575\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	480.996	+	277.703i	1.36259	+	0.786694i	0.989969	−	0.141288i	$-0.0451242\pi$
$353$	480.996	+	277.703i	1.36259	+	0.786694i	0.372626	+	0.927982i	$0.378458\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	670.647i		−	1.86810i	−0.357147	−	0.934048i	$-0.616251\pi$
$359$		−	670.647i		−	1.86810i	0.357147	−	0.934048i	$-0.383749\pi$
$360$		0			0
$361$	454.590			1.25925
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	47.1109	−	81.5985i	0.128368	−	0.222339i	−0.794677	−	0.607033i	$-0.792360\pi$
$367$	47.1109	−	81.5985i	0.128368	−	0.222339i	0.923044	+	0.384694i	$0.125693\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	568.192	+	328.046i	1.53151	+	0.884220i
$372$		0			0
$373$	96.2886	+	166.777i	0.258146	+	0.447123i	0.965745	−	0.259492i	$-0.0835550\pi$
$373$	96.2886	+	166.777i	0.258146	+	0.447123i	−0.707599	+	0.706614i	$0.750222\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	80.3927i		−	0.213243i
$378$		0			0
$379$	184.387			0.486510			0.243255	−	0.969962i	$-0.421785\pi$
$379$	184.387			0.486510			0.243255	+	0.969962i	$0.421785\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	575.841	−	332.462i	1.50350	−	0.868047i	0.503510	−	0.863990i	$-0.332042\pi$
$383$	575.841	−	332.462i	1.50350	−	0.868047i	0.999992	−	0.00405754i	$-0.00129156\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−539.718	−	311.606i	−1.38745	−	0.801044i	−0.394422	−	0.918929i	$-0.629055\pi$
$389$	−539.718	−	311.606i	−1.38745	−	0.801044i	−0.993027	+	0.117885i	$0.962388\pi$
$390$		0			0
$391$	189.273	+	327.830i	0.484074	+	0.838441i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	247.371			0.623100			0.311550	−	0.950230i	$-0.399152\pi$
$397$	247.371			0.623100			0.311550	+	0.950230i	$0.399152\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−369.248	+	213.185i	−0.920818	+	0.531635i	−0.883896	−	0.467684i	$-0.845089\pi$
$401$	−369.248	+	213.185i	−0.920818	+	0.531635i	−0.0369222	+	0.999318i	$0.511755\pi$
$402$		0			0
$403$	−45.0443	+	78.0189i	−0.111772	+	0.193595i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	1026.56	+	592.687i	2.52227	+	1.45623i
$408$		0			0
$409$	−87.0543	−	150.782i	−0.212847	−	0.368661i	0.739758	−	0.672873i	$-0.234940\pi$
$409$	−87.0543	−	150.782i	−0.212847	−	0.368661i	−0.952604	+	0.304212i	$0.901607\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			989.553i			2.39601i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	265.100	−	153.055i	0.632696	−	0.365287i	−0.149099	−	0.988822i	$-0.547637\pi$
$419$	265.100	−	153.055i	0.632696	−	0.365287i	0.781796	+	0.623535i	$0.214304\pi$
$420$		0			0
$421$	140.251	−	242.922i	0.333138	−	0.577012i	−0.649987	−	0.759945i	$-0.725226\pi$
$421$	140.251	−	242.922i	0.333138	−	0.577012i	0.983125	+	0.182933i	$0.0585591\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	315.130	+	545.820i	0.738008	+	1.27827i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	675.034i		−	1.56620i	−0.621893	−	0.783102i	$-0.713636\pi$
$431$		−	675.034i		−	1.56620i	0.621893	−	0.783102i	$-0.286364\pi$
$432$		0			0
$433$	−273.497			−0.631633			−0.315817	−	0.948820i	$-0.602278\pi$
$433$	−273.497			−0.631633			−0.315817	+	0.948820i	$0.602278\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−304.817	+	175.986i	−0.697522	+	0.402715i
$438$		0			0
$439$	262.426	−	454.535i	0.597781	−	1.03539i	−0.395367	−	0.918523i	$-0.629382\pi$
$439$	262.426	−	454.535i	0.597781	−	1.03539i	0.993148	−	0.116864i	$-0.0372842\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−76.6862	−	44.2748i	−0.173106	−	0.0999431i	0.410943	−	0.911661i	$-0.365199\pi$
$443$	−76.6862	−	44.2748i	−0.173106	−	0.0999431i	−0.584050	+	0.811718i	$0.698533\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	46.0938i		−	0.102659i	−0.998682	−	0.0513294i	$-0.983654\pi$
$449$		−	46.0938i		−	0.102659i	0.998682	−	0.0513294i	$-0.0163458\pi$
$450$		0			0
$451$	−678.803			−1.50511
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	94.6301	−	163.904i	0.207068	−	0.358652i	−0.743722	−	0.668489i	$-0.766941\pi$
$457$	94.6301	−	163.904i	0.207068	−	0.358652i	0.950790	+	0.309837i	$0.100275\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	362.663	+	209.384i	0.786689	+	0.454195i	0.838795	−	0.544447i	$-0.183260\pi$
$461$	362.663	+	209.384i	0.786689	+	0.454195i	−0.0521069	+	0.998642i	$0.516594\pi$
$462$		0			0
$463$	−156.514	−	271.090i	−0.338043	−	0.585507i	0.646022	−	0.763319i	$-0.276431\pi$
$463$	−156.514	−	271.090i	−0.338043	−	0.585507i	−0.984064	+	0.177812i	$0.943098\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	148.419i		−	0.317814i	−0.987294	−	0.158907i	$-0.949203\pi$
$467$		−	148.419i		−	0.317814i	0.987294	−	0.158907i	$-0.0507971\pi$
$468$		0			0
$469$	431.692			0.920452
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−850.721	+	491.164i	−1.79856	+	1.03840i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−273.219	−	157.743i	−0.570395	−	0.329317i	0.186912	−	0.982377i	$-0.440152\pi$
$479$	−273.219	−	157.743i	−0.570395	−	0.329317i	−0.757307	+	0.653059i	$0.773485\pi$
$480$		0			0
$481$	460.688	+	797.935i	0.957771	+	1.65891i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	492.691			1.01169			0.505843	−	0.862626i	$-0.331182\pi$
$487$	492.691			1.01169			0.505843	+	0.862626i	$0.331182\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−719.318	+	415.298i	−1.46501	+	0.845822i	−0.999236	−	0.0390843i	$-0.987556\pi$
$491$	−719.318	+	415.298i	−1.46501	+	0.845822i	−0.465770	+	0.884906i	$0.654223\pi$
$492$		0			0
$493$	89.1831	−	154.470i	0.180899	−	0.313326i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−253.686	−	146.466i	−0.510435	−	0.294700i
$498$		0			0
$499$	−14.1379	−	24.4876i	−0.0283325	−	0.0490733i	0.851512	−	0.524336i	$-0.175686\pi$
$499$	−14.1379	−	24.4876i	−0.0283325	−	0.0490733i	−0.879844	+	0.475263i	$0.842353\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	31.5709i		−	0.0627652i	−0.999507	−	0.0313826i	$-0.990009\pi$
$503$		−	31.5709i		−	0.0627652i	0.999507	−	0.0313826i	$-0.00999103\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−204.163	+	117.874i	−0.401107	+	0.231579i	−0.686961	−	0.726694i	$-0.741056\pi$
$509$	−204.163	+	117.874i	−0.401107	+	0.231579i	0.285855	+	0.958273i	$0.407723\pi$
$510$		0			0
$511$	345.516	−	598.452i	0.676157	−	1.17114i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−146.663	−	254.028i	−0.283681	−	0.491350i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	582.254i		−	1.11757i	−0.829312	−	0.558785i	$-0.811268\pi$
$521$		−	582.254i		−	1.11757i	0.829312	−	0.558785i	$-0.188732\pi$
$522$		0			0
$523$	201.726			0.385709			0.192855	−	0.981227i	$-0.438225\pi$
$523$	201.726			0.385709			0.192855	+	0.981227i	$0.438225\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−173.100	+	99.9391i	−0.328462	+	0.189638i
$528$		0			0
$529$	−188.552	+	326.582i	−0.356431	+	0.617357i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−456.937	−	263.812i	−0.857292	−	0.494958i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$			697.860i			1.29473i
$540$		0			0
$541$	390.540			0.721886			0.360943	−	0.932588i	$-0.382455\pi$
$541$	390.540			0.721886			0.360943	+	0.932588i	$0.382455\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	321.098	−	556.158i	0.587017	−	1.01674i	−0.407604	−	0.913159i	$-0.633636\pi$
$547$	321.098	−	556.158i	0.587017	−	1.01674i	0.994621	−	0.103584i	$-0.0330310\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	143.626	+	82.9225i	0.260664	+	0.150495i
$552$		0			0
$553$	444.422	+	769.761i	0.803656	+	1.39197i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	145.052i		−	0.260417i	−0.991487	−	0.130208i	$-0.958435\pi$
$557$		−	145.052i		−	0.260417i	0.991487	−	0.130208i	$-0.0415646\pi$
$558$		0			0
$559$	−763.551			−1.36592
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−732.570	+	422.949i	−1.30119	+	0.751242i	−0.980608	−	0.195979i	$-0.937212\pi$
$563$	−732.570	+	422.949i	−1.30119	+	0.751242i	−0.320581	+	0.947221i	$0.603878\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	725.474	+	418.853i	1.27500	+	0.736121i	0.975924	−	0.218109i	$-0.0699888\pi$
$569$	725.474	+	418.853i	1.27500	+	0.736121i	0.299074	+	0.954230i	$0.403322\pi$
$570$		0			0
$571$	−348.675	−	603.922i	−0.610639	−	1.05766i	−0.991133	−	0.132874i	$-0.957579\pi$
$571$	−348.675	−	603.922i	−0.610639	−	1.05766i	0.380494	−	0.924783i	$-0.375754\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−713.999			−1.23743			−0.618717	−	0.785614i	$-0.712347\pi$
$577$	−713.999			−1.23743			−0.618717	+	0.785614i	$0.712347\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	125.852	−	72.6610i	0.216614	−	0.125062i
$582$		0			0
$583$	622.171	−	1077.63i	1.06719	−	1.84843i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	706.988	+	408.180i	1.20441	+	0.695366i	0.961532	−	0.274692i	$-0.0885758\pi$
$587$	706.988	+	408.180i	1.20441	+	0.695366i	0.242876	+	0.970057i	$0.421909\pi$
$588$		0			0
$589$	−92.9235	−	160.948i	−0.157765	−	0.273257i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	64.6676i		−	0.109052i	−0.998512	−	0.0545258i	$-0.982635\pi$
$593$		−	64.6676i		−	0.109052i	0.998512	−	0.0545258i	$-0.0173647\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	204.565	−	118.105i	0.341510	−	0.197171i	−0.319430	−	0.947610i	$-0.603491\pi$
$599$	204.565	−	118.105i	0.341510	−	0.197171i	0.660940	+	0.750439i	$0.270158\pi$
$600$		0			0
$601$	84.0789	−	145.629i	0.139898	−	0.242311i	−0.787560	−	0.616238i	$-0.788656\pi$
$601$	84.0789	−	145.629i	0.139898	−	0.242311i	0.927458	+	0.373927i	$0.121989\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−55.0163	−	95.2911i	−0.0906365	−	0.156987i	0.817143	−	0.576435i	$-0.195557\pi$
$607$	−55.0163	−	95.2911i	−0.0906365	−	0.156987i	−0.907779	+	0.419449i	$0.862223\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	227.999i		−	0.373157i
$612$		0			0
$613$	387.013			0.631343			0.315671	−	0.948869i	$-0.397770\pi$
$613$	387.013			0.631343			0.315671	+	0.948869i	$0.397770\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−333.200	+	192.373i	−0.540032	+	0.311787i	−0.745092	−	0.666962i	$-0.767594\pi$
$617$	−333.200	+	192.373i	−0.540032	+	0.311787i	0.205060	+	0.978749i	$0.434261\pi$
$618$		0			0
$619$	403.446	−	698.789i	0.651771	−	1.12890i	−0.330922	−	0.943658i	$-0.607360\pi$
$619$	403.446	−	698.789i	0.651771	−	1.12890i	0.982693	−	0.185242i	$-0.0593068\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	428.768	+	247.549i	0.688232	+	0.397351i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$			2044.24i			3.24999i
$630$		0			0
$631$	−753.299			−1.19382			−0.596909	−	0.802309i	$-0.703605\pi$
$631$	−753.299			−1.19382			−0.596909	+	0.802309i	$0.703605\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−271.219	+	469.765i	−0.425775	+	0.737464i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	570.820	+	329.563i	0.890515	+	0.514139i	0.874111	−	0.485726i	$-0.161445\pi$
$641$	570.820	+	329.563i	0.890515	+	0.514139i	0.0164041	+	0.999865i	$0.494778\pi$
$642$		0			0
$643$	−452.382	−	783.548i	−0.703549	−	1.21858i	−0.967213	−	0.253968i	$-0.918264\pi$
$643$	−452.382	−	783.548i	−0.703549	−	1.21858i	0.263664	−	0.964615i	$-0.415069\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	360.764i		−	0.557594i	−0.960350	−	0.278797i	$-0.910064\pi$
$647$		−	360.764i		−	0.557594i	0.960350	−	0.278797i	$-0.0899357\pi$
$648$		0			0
$649$	1876.79			2.89181
$650$		0			0
$651$		0			0
$652$		0			0
$653$	365.501	−	211.022i	0.559726	−	0.323158i	−0.193310	−	0.981138i	$-0.561922\pi$
$653$	365.501	−	211.022i	0.559726	−	0.323158i	0.753035	+	0.657980i	$0.228589\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−520.928	−	300.758i	−0.790483	−	0.456385i	0.0496496	−	0.998767i	$-0.484190\pi$
$659$	−520.928	−	300.758i	−0.790483	−	0.456385i	−0.840133	+	0.542381i	$0.817523\pi$
$660$		0			0
$661$	465.379	+	806.060i	0.704053	+	1.21946i	0.967032	+	0.254654i	$0.0819617\pi$
$661$	465.379	+	806.060i	0.704053	+	1.21946i	−0.262979	+	0.964802i	$0.584705\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−71.5714			−0.107303
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1035.20	−	597.675i	1.54278	−	0.890722i
$672$		0			0
$673$	29.2343	−	50.6352i	0.0434387	−	0.0752381i	−0.843489	−	0.537147i	$-0.819502\pi$
$673$	29.2343	−	50.6352i	0.0434387	−	0.0752381i	0.886927	+	0.461909i	$0.152835\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	549.689	+	317.363i	0.811948	+	0.468778i	0.847632	−	0.530585i	$-0.178028\pi$
$677$	549.689	+	317.363i	0.811948	+	0.468778i	−0.0356841	+	0.999363i	$0.511361\pi$
$678$		0			0
$679$	−357.900	−	619.901i	−0.527099	−	0.912962i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	387.682i		−	0.567617i	−0.958881	−	0.283808i	$-0.908402\pi$
$683$		−	387.682i		−	0.567617i	0.958881	−	0.283808i	$-0.0915980\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	837.629	−	483.605i	1.21572	−	0.701895i
$690$		0			0
$691$	−483.665	+	837.732i	−0.699949	+	1.21235i	0.268535	+	0.963270i	$0.413461\pi$
$691$	−483.665	+	837.732i	−0.699949	+	1.21235i	−0.968484	+	0.249077i	$0.919873\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−585.317	−	1013.80i	−0.839766	−	1.45452i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			895.039i			1.27680i	0.769703	+	0.638402i	$0.220404\pi$
$701$			895.039i			1.27680i	−0.769703	+	0.638402i	$0.779596\pi$
$702$		0			0
$703$	−1900.74			−2.70376
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−653.245	+	377.151i	−0.923967	+	0.533453i
$708$		0			0
$709$	624.531	−	1081.72i	0.880861	−	1.52570i	0.0304761	−	0.999535i	$-0.490298\pi$
$709$	624.531	−	1081.72i	0.880861	−	1.52570i	0.850385	−	0.526161i	$-0.176369\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	69.4581	+	40.1017i	0.0974167	+	0.0562436i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$			418.228i			0.581681i	0.956772	+	0.290840i	$0.0939348\pi$
$719$			418.228i			0.581681i	−0.956772	+	0.290840i	$0.906065\pi$
$720$		0			0
$721$	−283.001			−0.392512
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	258.116	−	447.070i	0.355043	−	0.614952i	−0.632082	−	0.774901i	$-0.717800\pi$
$727$	258.116	−	447.070i	0.355043	−	0.614952i	0.987125	+	0.159949i	$0.0511329\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−1467.12	−	847.040i	−2.00700	−	1.15874i
$732$		0			0
$733$	−4.12563	−	7.14580i	−0.00562842	−	0.00974871i	0.863197	−	0.504866i	$-0.168458\pi$
$733$	−4.12563	−	7.14580i	−0.00562842	−	0.00974871i	−0.868826	+	0.495118i	$0.835125\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	818.747i		−	1.11092i
$738$		0			0
$739$	294.686			0.398763			0.199381	−	0.979922i	$-0.436107\pi$
$739$	294.686			0.398763			0.199381	+	0.979922i	$0.436107\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−438.540	+	253.191i	−0.590229	+	0.340769i	−0.765188	−	0.643807i	$-0.777354\pi$
$743$	−438.540	+	253.191i	−0.590229	+	0.340769i	0.174959	+	0.984576i	$0.444021\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	85.0824	+	49.1223i	0.113595	+	0.0655839i
$750$		0			0
$751$	−395.768	−	685.491i	−0.526989	−	0.912771i	−0.999505	−	0.0314493i	$-0.989988\pi$
$751$	−395.768	−	685.491i	−0.526989	−	0.912771i	0.472517	−	0.881322i	$-0.343346\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	957.457			1.26480			0.632402	−	0.774640i	$-0.282069\pi$
$757$	957.457			1.26480			0.632402	+	0.774640i	$0.282069\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	378.389	−	218.463i	0.497225	−	0.287073i	−0.230342	−	0.973110i	$-0.573984\pi$
$761$	378.389	−	218.463i	0.497225	−	0.287073i	0.727567	+	0.686037i	$0.240651\pi$
$762$		0			0
$763$	−879.257	+	1522.92i	−1.15237	+	1.99596i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1263.36	+	729.400i	1.64714	+	0.950978i
$768$		0			0
$769$	302.360	+	523.703i	0.393186	+	0.681018i	0.992868	−	0.119221i	$-0.0380396\pi$
$769$	302.360	+	523.703i	0.393186	+	0.681018i	−0.599682	+	0.800238i	$0.704706\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	596.625i		−	0.771831i	−0.922534	−	0.385915i	$-0.873886\pi$
$773$		−	596.625i		−	0.771831i	0.922534	−	0.385915i	$-0.126114\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	942.631	−	544.228i	1.21005	−	0.698625i
$780$		0			0
$781$	−277.787	+	481.141i	−0.355681	+	0.616058i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	272.220	+	471.499i	0.345896	+	0.599109i	0.985516	−	0.169582i	$-0.0542418\pi$
$787$	272.220	+	471.499i	0.345896	+	0.599109i	−0.639620	+	0.768691i	$0.720908\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	874.010i		−	1.10494i
$792$		0			0
$793$	929.129			1.17166
$794$		0			0
$795$		0			0
$796$		0			0
$797$	200.065	−	115.508i	0.251023	−	0.144928i	−0.369210	−	0.929346i	$-0.620372\pi$
$797$	200.065	−	115.508i	0.251023	−	0.144928i	0.620232	+	0.784418i	$0.287038\pi$
$798$		0			0
$799$	252.929	−	438.086i	0.316557	−	0.548293i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1135.02	−	655.306i	−1.41348	−	0.816072i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	75.6693i		−	0.0935344i	−0.998906	−	0.0467672i	$-0.985108\pi$
$809$		−	75.6693i		−	0.0935344i	0.998906	−	0.0467672i	$-0.0148919\pi$
$810$		0			0
$811$	1184.72			1.46081			0.730404	−	0.683015i	$-0.239332\pi$
$811$	1184.72			1.46081			0.730404	+	0.683015i	$0.239332\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	787.579	−	1364.13i	0.963988	−	1.66968i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1282.37	+	740.377i	1.56196	+	0.901799i	0.997059	+	0.0766424i	$0.0244200\pi$
$821$	1282.37	+	740.377i	1.56196	+	0.901799i	0.564904	+	0.825157i	$0.308913\pi$
$822$		0			0
$823$	−380.688	−	659.371i	−0.462562	−	0.801180i	0.536526	−	0.843884i	$-0.319736\pi$
$823$	−380.688	−	659.371i	−0.462562	−	0.801180i	−0.999088	+	0.0427034i	$0.986403\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	1019.60i		−	1.23290i	−0.787396	−	0.616448i	$-0.788571\pi$
$827$		−	1019.60i		−	1.23290i	0.787396	−	0.616448i	$-0.211429\pi$
$828$		0			0
$829$	58.7988			0.0709274			0.0354637	−	0.999371i	$-0.488709\pi$
$829$	58.7988			0.0709274			0.0354637	+	0.999371i	$0.488709\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−1042.26	+	601.750i	−1.25121	+	0.722388i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	1005.77	+	580.682i	1.19877	+	0.692112i	0.960282	−	0.279032i	$-0.0900136\pi$
$839$	1005.77	+	580.682i	1.19877	+	0.692112i	0.238492	+	0.971144i	$0.423347\pi$
$840$		0			0
$841$	−403.638	−	699.122i	−0.479950	−	0.831298i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	1842.46			2.17528
$848$		0			0
$849$		0			0
$850$		0			0
$851$	710.380	−	410.138i	0.834759	−	0.481948i
$852$		0			0
$853$	−695.272	+	1204.25i	−0.815090	+	1.41178i	0.0941731	+	0.995556i	$0.469979\pi$
$853$	−695.272	+	1204.25i	−0.815090	+	1.41178i	−0.909263	+	0.416222i	$0.863354\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	914.904	+	528.220i	1.06757	+	0.616359i	0.927515	−	0.373785i	$-0.121940\pi$
$857$	914.904	+	528.220i	1.06757	+	0.616359i	0.140050	+	0.990144i	$0.455274\pi$
$858$		0			0
$859$	537.750	+	931.410i	0.626018	+	1.08430i	0.988343	+	0.152244i	$0.0486498\pi$
$859$	537.750	+	931.410i	0.626018	+	1.08430i	−0.362325	+	0.932052i	$0.618017\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	66.3453i		−	0.0768776i	−0.999261	−	0.0384388i	$-0.987762\pi$
$863$		−	66.3453i		−	0.0768776i	0.999261	−	0.0384388i	$-0.0122385\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	1459.93	−	842.890i	1.68001	−	0.969954i
$870$		0			0
$871$	318.200	−	551.139i	0.365328	−	0.632766i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−211.603	−	366.507i	−0.241280	−	0.417909i	0.719799	−	0.694182i	$-0.244234\pi$
$877$	−211.603	−	366.507i	−0.241280	−	0.417909i	−0.961079	+	0.276273i	$0.910901\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			525.156i			0.596090i	0.954552	+	0.298045i	$0.0963346\pi$
$881$			525.156i			0.596090i	−0.954552	+	0.298045i	$0.903665\pi$
$882$		0			0
$883$	−44.5609			−0.0504653			−0.0252327	−	0.999682i	$-0.508033\pi$
$883$	−44.5609			−0.0504653			−0.0252327	+	0.999682i	$0.508033\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1223.72	+	706.516i	−1.37962	+	0.796523i	−0.992113	−	0.125345i	$-0.959996\pi$
$887$	−1223.72	+	706.516i	−1.37962	+	0.796523i	−0.387505	+	0.921868i	$0.626663\pi$
$888$		0			0
$889$	227.674	−	394.344i	0.256102	−	0.443581i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	407.333	+	235.174i	0.456140	+	0.263352i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	37.7908i		−	0.0420365i
$900$		0			0
$901$	2145.94			2.38173
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−150.861	+	261.298i	−0.166329	+	0.288090i	−0.937126	−	0.348990i	$-0.886525\pi$
$907$	−150.861	+	261.298i	−0.166329	+	0.288090i	0.770797	+	0.637080i	$0.219858\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−243.600	−	140.643i	−0.267399	−	0.154383i	0.360306	−	0.932834i	$-0.382672\pi$
$911$	−243.600	−	140.643i	−0.267399	−	0.154383i	−0.627705	+	0.778451i	$0.716006\pi$
$912$		0			0
$913$	−137.809	−	238.692i	−0.150941	−	0.261437i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	586.071i		−	0.639117i
$918$		0			0
$919$	309.762			0.337064			0.168532	−	0.985696i	$-0.446097\pi$
$919$	309.762			0.337064			0.168532	+	0.985696i	$0.446097\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−373.985	+	215.920i	−0.405184	+	0.233933i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	416.036	+	240.198i	0.447832	+	0.258556i	0.706914	−	0.707300i	$-0.250087\pi$
$929$	416.036	+	240.198i	0.447832	+	0.258556i	−0.259082	+	0.965855i	$0.583420\pi$
$930$		0			0
$931$	−559.507	−	969.095i	−0.600975	−	1.04092i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−918.055			−0.979781			−0.489891	−	0.871784i	$-0.662963\pi$
$937$	−918.055			−0.979781			−0.489891	+	0.871784i	$0.662963\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1228.87	+	709.489i	−1.30592	+	0.753973i	−0.981413	−	0.191910i	$-0.938532\pi$
$941$	−1228.87	+	709.489i	−1.30592	+	0.753973i	−0.324507	+	0.945883i	$0.605198\pi$
$942$		0			0
$943$	−234.865	+	406.798i	−0.249061	+	0.431387i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	232.422	+	134.189i	0.245430	+	0.141699i	0.617670	−	0.786438i	$-0.288077\pi$
$947$	232.422	+	134.189i	0.245430	+	0.141699i	−0.372240	+	0.928136i	$0.621410\pi$
$948$		0			0
$949$	−509.361	−	882.238i	−0.536734	−	0.929651i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			476.437i			0.499934i	0.968254	+	0.249967i	$0.0804198\pi$
$953$			476.437i			0.499934i	−0.968254	+	0.249967i	$0.919580\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−17.4205	+	10.0577i	−0.0181652	+	0.0104877i
$960$		0			0
$961$	459.326	−	795.575i	0.477966	−	0.827862i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	506.040	+	876.486i	0.523309	+	0.906398i	0.999632	+	0.0271272i	$0.00863592\pi$
$967$	506.040	+	876.486i	0.523309	+	0.906398i	−0.476323	+	0.879270i	$0.658031\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			772.488i			0.795560i	0.917481	+	0.397780i	$0.130219\pi$
$971$			772.488i			0.795560i	−0.917481	+	0.397780i	$0.869781\pi$
$972$		0			0
$973$	−139.050			−0.142909
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1407.33	−	812.523i	1.44046	−	0.831651i	0.442581	−	0.896728i	$-0.354063\pi$
$977$	1407.33	−	812.523i	1.44046	−	0.831651i	0.997880	+	0.0650777i	$0.0207295\pi$
$978$		0			0
$979$	469.502	−	813.202i	0.479573	−	0.830645i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−698.536	−	403.300i	−0.710616	−	0.410275i	0.100673	−	0.994920i	$-0.467900\pi$
$983$	−698.536	−	403.300i	−0.710616	−	0.410275i	−0.811289	+	0.584645i	$0.801234\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$			679.768i			0.687329i
$990$		0			0
$991$	434.414			0.438359			0.219180	−	0.975685i	$-0.429662\pi$
$991$	434.414			0.438359			0.219180	+	0.975685i	$0.429662\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−883.384	+	1530.07i	−0.886042	+	1.53467i	−0.0415279	+	0.999137i	$0.513223\pi$
$997$	−883.384	+	1530.07i	−0.886042	+	1.53467i	−0.844514	+	0.535533i	$0.820111\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2700.3.p.d.1601.7		16
3.2	odd	2		900.3.p.e.101.7	yes	16
5.2	odd	4		2700.3.u.d.2249.14		32
5.3	odd	4		2700.3.u.d.2249.3		32
5.4	even	2		2700.3.p.e.1601.2		16
9.4	even	3		900.3.p.e.401.7	yes	16
9.5	odd	6	inner	2700.3.p.d.2501.7		16
15.2	even	4		900.3.u.d.749.5		32
15.8	even	4		900.3.u.d.749.12		32
15.14	odd	2		900.3.p.d.101.2	✓	16
45.4	even	6		900.3.p.d.401.2	yes	16
45.13	odd	12		900.3.u.d.149.5		32
45.14	odd	6		2700.3.p.e.2501.2		16
45.22	odd	12		900.3.u.d.149.12		32
45.23	even	12		2700.3.u.d.449.14		32
45.32	even	12		2700.3.u.d.449.3		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
900.3.p.d.101.2	✓	16	15.14	odd	2
900.3.p.d.401.2	yes	16	45.4	even	6
900.3.p.e.101.7	yes	16	3.2	odd	2
900.3.p.e.401.7	yes	16	9.4	even	3
900.3.u.d.149.5		32	45.13	odd	12
900.3.u.d.149.12		32	45.22	odd	12
900.3.u.d.749.5		32	15.2	even	4
900.3.u.d.749.12		32	15.8	even	4
2700.3.p.d.1601.7		16	1.1	even	1	trivial
2700.3.p.d.2501.7		16	9.5	odd	6	inner
2700.3.p.e.1601.2		16	5.4	even	2
2700.3.p.e.2501.2		16	45.14	odd	6
2700.3.u.d.449.3		32	45.32	even	12
2700.3.u.d.449.14		32	45.23	even	12
2700.3.u.d.2249.3		32	5.3	odd	4
2700.3.u.d.2249.14		32	5.2	odd	4

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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2700.p (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(4.69530 - 8.13249i) q^{7} +O(q^{10})\)	\(q+(4.69530 - 8.13249i) q^{7} +(-15.4241 - 8.90510i) q^{11} +(-6.92181 - 11.9889i) q^{13} -30.7147i q^{17} +28.5585 q^{19} +(-10.6734 + 6.16230i) q^{23} +(5.02918 + 2.90360i) q^{29} +(-3.25379 - 5.63573i) q^{31} -66.5560 q^{37} +(33.0070 - 19.0566i) q^{41} +(27.5777 - 47.7660i) q^{43} +(14.2631 + 8.23479i) q^{47} +(-19.5916 - 33.9337i) q^{49} +69.8669i q^{53} +(-91.2592 + 52.6885i) q^{59} +(-33.5580 + 58.1242i) q^{61} +(22.9853 + 39.8118i) q^{67} -31.1942i q^{71} +73.5877 q^{73} +(-144.841 + 83.6241i) q^{77} +(-47.3263 + 81.9715i) q^{79} +(13.4020 + 7.73763i) q^{83} +52.7229i q^{89} -130.000 q^{91} +(38.1126 - 66.0130i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - q^{7}+O(q^{10}) \) 16 * q - q^7	\( 16 q - q^{7} - 10 q^{13} + 2 q^{19} - 27 q^{23} - 9 q^{29} + 8 q^{31} - 22 q^{37} - 54 q^{41} + 44 q^{43} + 108 q^{47} - 45 q^{49} - 9 q^{59} - 55 q^{61} - 28 q^{67} + 86 q^{73} - 342 q^{77} + 11 q^{79} + 306 q^{83} - 134 q^{91} + 41 q^{97}+O(q^{100}) \) 16 * q - q^7 - 10 * q^13 + 2 * q^19 - 27 * q^23 - 9 * q^29 + 8 * q^31 - 22 * q^37 - 54 * q^41 + 44 * q^43 + 108 * q^47 - 45 * q^49 - 9 * q^59 - 55 * q^61 - 28 * q^67 + 86 * q^73 - 342 * q^77 + 11 * q^79 + 306 * q^83 - 134 * q^91 + 41 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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