LMFDB - Embedded newform 507.3.c.c.170.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [507,3,Mod(170,507)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(507, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("507.170");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 507 = 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	507.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$13.8147494031$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 13 $ x^2 + 13
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			170.2
Root			$3.60555i$ of defining polynomial
Character	$\chi$	$=$	507.170
Dual form			507.3.c.c.170.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.60555i q^{2} -3.00000 q^{3} -9.00000 q^{4} -7.21110i q^{5} -10.8167i q^{6} -18.0278i q^{8} +9.00000 q^{9} +O(q^{10})$	$q+3.60555i q^{2} -3.00000 q^{3} -9.00000 q^{4} -7.21110i q^{5} -10.8167i q^{6} -18.0278i q^{8} +9.00000 q^{9} +26.0000 q^{10} +7.21110i q^{11} +27.0000 q^{12} +21.6333i q^{15} +29.0000 q^{16} +32.4500i q^{18} +64.8999i q^{20} -26.0000 q^{22} +54.0833i q^{24} -27.0000 q^{25} -27.0000 q^{27} -78.0000 q^{30} +32.4500i q^{32} -21.6333i q^{33} -81.0000 q^{36} -130.000 q^{40} +79.3221i q^{41} +70.0000 q^{43} -64.8999i q^{44} -64.8999i q^{45} +93.7443i q^{47} -87.0000 q^{48} -49.0000 q^{49} -97.3499i q^{50} -97.3499i q^{54} +52.0000 q^{55} +7.21110i q^{59} -194.700i q^{60} -70.0000 q^{61} -1.00000 q^{64} +78.0000 q^{66} +79.3221i q^{71} -162.250i q^{72} +81.0000 q^{75} +50.0000 q^{79} -209.122i q^{80} +81.0000 q^{81} -286.000 q^{82} +165.855i q^{83} +252.389i q^{86} +130.000 q^{88} -79.3221i q^{89} +234.000 q^{90} -338.000 q^{94} -97.3499i q^{96} -176.672i q^{98} +64.8999i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} - 18 q^{4} + 18 q^{9}+O(q^{10}) $ 2 * q - 6 * q^3 - 18 * q^4 + 18 * q^9	$ 2 q - 6 q^{3} - 18 q^{4} + 18 q^{9} + 52 q^{10} + 54 q^{12} + 58 q^{16} - 52 q^{22} - 54 q^{25} - 54 q^{27} - 156 q^{30} - 162 q^{36} - 260 q^{40} + 140 q^{43} - 174 q^{48} - 98 q^{49} + 104 q^{55} - 140 q^{61} - 2 q^{64} + 156 q^{66} + 162 q^{75} + 100 q^{79} + 162 q^{81} - 572 q^{82} + 260 q^{88} + 468 q^{90} - 676 q^{94}+O(q^{100}) $ 2 * q - 6 * q^3 - 18 * q^4 + 18 * q^9 + 52 * q^10 + 54 * q^12 + 58 * q^16 - 52 * q^22 - 54 * q^25 - 54 * q^27 - 156 * q^30 - 162 * q^36 - 260 * q^40 + 140 * q^43 - 174 * q^48 - 98 * q^49 + 104 * q^55 - 140 * q^61 - 2 * q^64 + 156 * q^66 + 162 * q^75 + 100 * q^79 + 162 * q^81 - 572 * q^82 + 260 * q^88 + 468 * q^90 - 676 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/507\mathbb{Z}\right)^\times$.

$n$	$170$	$340$
$\chi(n)$	$-1$	$1$

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 $

$2$	3.60555i	1.80278i	0.433013	+	0.901388i	$0.357451\pi$
$2$	3.60555i	1.80278i	−0.433013	+	0.901388i	$0.642549\pi$
$3$	−3.00000	−1.00000
$3$	−3.00000	−1.00000
$4$	−9.00000	−2.25000
$5$	− 7.21110i	− 1.44222i	−0.692820	−	0.721110i	$-0.743632\pi$
$5$	− 7.21110i	− 1.44222i	0.692820	−	0.721110i	$-0.256368\pi$
$6$	− 10.8167i	− 1.80278i
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	− 18.0278i	− 2.25347i
$9$	9.00000	1.00000
$10$	26.0000	2.60000
$11$	7.21110i	0.655555i	0.944755	+	0.327777i	$0.106300\pi$
$11$	7.21110i	0.655555i	−0.944755	+	0.327777i	$0.893700\pi$
$12$	27.0000	2.25000
$13$	0	0
$13$	0	0
$14$	0	0
$15$	21.6333i	1.44222i
$16$	29.0000	1.81250
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	32.4500i	1.80278i
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	64.8999i	3.24500i
$21$	0	0
$22$	−26.0000	−1.18182
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	54.0833i	2.25347i
$25$	−27.0000	−1.08000
$26$	0	0
$27$	−27.0000	−1.00000
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	−78.0000	−2.60000
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	32.4500i	1.01406i
$33$	− 21.6333i	− 0.655555i
$34$	0	0
$35$	0	0
$36$	−81.0000	−2.25000
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	0	0
$39$	0	0
$40$	−130.000	−3.25000
$41$	79.3221i	1.93469i	0.253471	+	0.967343i	$0.418428\pi$
$41$	79.3221i	1.93469i	−0.253471	+	0.967343i	$0.581572\pi$
$42$	0	0
$43$	70.0000	1.62791	0.813953	−	0.580930i	$-0.197311\pi$
$43$	70.0000	1.62791	0.813953	+	0.580930i	$0.197311\pi$
$44$	− 64.8999i	− 1.47500i
$45$	− 64.8999i	− 1.44222i
$46$	0	0
$47$	93.7443i	1.99456i	0.0737043	+	0.997280i	$0.476518\pi$
$47$	93.7443i	1.99456i	−0.0737043	+	0.997280i	$0.523482\pi$
$48$	−87.0000	−1.81250
$49$	−49.0000	−1.00000
$50$	− 97.3499i	− 1.94700i
$51$	0	0
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	− 97.3499i	− 1.80278i
$55$	52.0000	0.945455
$56$	0	0
$57$	0	0
$58$	0	0
$59$	7.21110i	0.122222i	0.998131	+	0.0611110i	$0.0194644\pi$
$59$	7.21110i	0.122222i	−0.998131	+	0.0611110i	$0.980536\pi$
$60$	− 194.700i	− 3.24500i
$61$	−70.0000	−1.14754	−0.573770	−	0.819016i	$-0.694520\pi$
$61$	−70.0000	−1.14754	−0.573770	+	0.819016i	$0.694520\pi$
$62$	0	0
$63$	0	0
$64$	−1.00000	−0.0156250
$65$	0	0
$66$	78.0000	1.18182
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	79.3221i	1.11721i	0.829433	+	0.558607i	$0.188664\pi$
$71$	79.3221i	1.11721i	−0.829433	+	0.558607i	$0.811336\pi$
$72$	− 162.250i	− 2.25347i
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	0	0
$75$	81.0000	1.08000
$76$	0	0
$77$	0	0
$78$	0	0
$79$	50.0000	0.632911	0.316456	−	0.948607i	$-0.397507\pi$
$79$	50.0000	0.632911	0.316456	+	0.948607i	$0.397507\pi$
$80$	− 209.122i	− 2.61402i
$81$	81.0000	1.00000
$82$	−286.000	−3.48780
$83$	165.855i	1.99826i	0.0417362	+	0.999129i	$0.486711\pi$
$83$	165.855i	1.99826i	−0.0417362	+	0.999129i	$0.513289\pi$
$84$	0	0
$85$	0	0
$86$	252.389i	2.93475i
$87$	0	0
$88$	130.000	1.47727
$89$	− 79.3221i	− 0.891260i	−0.895217	−	0.445630i	$-0.852980\pi$
$89$	− 79.3221i	− 0.891260i	0.895217	−	0.445630i	$-0.147020\pi$
$90$	234.000	2.60000
$91$	0	0
$92$	0	0
$93$	0	0
$94$	−338.000	−3.59574
$95$	0	0
$96$	− 97.3499i	− 1.01406i
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	− 176.672i	− 1.80278i
$99$	64.8999i	0.655555i
$100$	243.000	2.43000
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	−50.0000	−0.485437	−0.242718	−	0.970097i	$-0.578039\pi$
$103$	−50.0000	−0.485437	−0.242718	+	0.970097i	$0.578039\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	243.000	2.25000
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	187.489i	1.70444i
$111$	0	0
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	−26.0000	−0.220339
$119$	0	0
$120$	390.000	3.25000
$121$	69.0000	0.570248
$122$	− 252.389i	− 2.06876i
$123$	− 237.966i	− 1.93469i
$124$	0	0
$125$	14.4222i	0.115378i
$126$	0	0
$127$	46.0000	0.362205	0.181102	−	0.983464i	$-0.442033\pi$
$127$	46.0000	0.362205	0.181102	+	0.983464i	$0.442033\pi$
$128$	126.194i	0.985893i
$129$	−210.000	−1.62791
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	194.700i	1.47500i
$133$	0	0
$134$	0	0
$135$	194.700i	1.44222i
$136$	0	0
$137$	266.811i	1.94752i	0.227569	+	0.973762i	$0.426922\pi$
$137$	266.811i	1.94752i	−0.227569	+	0.973762i	$0.573078\pi$
$138$	0	0
$139$	122.000	0.877698	0.438849	−	0.898561i	$-0.355386\pi$
$139$	122.000	0.877698	0.438849	+	0.898561i	$0.355386\pi$
$140$	0	0
$141$	− 281.233i	− 1.99456i
$142$	−286.000	−2.01408
$143$	0	0
$144$	261.000	1.81250
$145$	0	0
$146$	0	0
$147$	147.000	1.00000
$148$	0	0
$149$	− 7.21110i	− 0.0483967i	−0.999707	−	0.0241983i	$-0.992297\pi$
$149$	− 7.21110i	− 0.0483967i	0.999707	−	0.0241983i	$-0.00770332\pi$
$150$	292.050i	1.94700i
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−310.000	−1.97452	−0.987261	−	0.159108i	$-0.949138\pi$
$157$	−310.000	−1.97452	−0.987261	+	0.159108i	$0.949138\pi$
$158$	180.278i	1.14100i
$159$	0	0
$160$	234.000	1.46250
$161$	0	0
$162$	292.050i	1.80278i
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	− 713.899i	− 4.35304i
$165$	−156.000	−0.945455
$166$	−598.000	−3.60241
$167$	266.811i	1.59767i	0.601551	+	0.798835i	$0.294550\pi$
$167$	266.811i	1.59767i	−0.601551	+	0.798835i	$0.705450\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	0	0
$172$	−630.000	−3.66279
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0	0
$176$	209.122i	1.18819i
$177$	− 21.6333i	− 0.122222i
$178$	286.000	1.60674
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	584.099i	3.24500i
$181$	262.000	1.44751	0.723757	−	0.690055i	$-0.242414\pi$
$181$	262.000	1.44751	0.723757	+	0.690055i	$0.242414\pi$
$182$	0	0
$183$	210.000	1.14754
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	− 843.699i	− 4.48776i
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	3.00000	0.0156250
$193$	0	0		−	1.00000i	$-0.5\pi$
$193$	0	0			1.00000i	$0.5\pi$
$194$	0	0
$195$	0	0
$196$	441.000	2.25000
$197$	338.922i	1.72042i	0.509944	+	0.860208i	$0.329666\pi$
$197$	338.922i	1.72042i	−0.509944	+	0.860208i	$0.670334\pi$
$198$	−234.000	−1.18182
$199$	190.000	0.954774	0.477387	−	0.878693i	$-0.341584\pi$
$199$	190.000	0.954774	0.477387	+	0.878693i	$0.341584\pi$
$200$	486.749i	2.43375i
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	572.000	2.79024
$206$	− 180.278i	− 0.875134i
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	410.000	1.94313	0.971564	−	0.236777i	$-0.0760911\pi$
$211$	410.000	1.94313	0.971564	+	0.236777i	$0.0760911\pi$
$212$	0	0
$213$	− 237.966i	− 1.11721i
$214$	0	0
$215$	− 504.777i	− 2.34780i
$216$	486.749i	2.25347i
$217$	0	0
$218$	0	0
$219$	0	0
$220$	−468.000	−2.12727
$221$	0	0
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	−243.000	−1.08000
$226$	0	0
$227$	165.855i	0.730640i	0.930882	+	0.365320i	$0.119040\pi$
$227$	165.855i	0.730640i	−0.930882	+	0.365320i	$0.880960\pi$
$228$	0	0
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	676.000	2.87660
$236$	− 64.8999i	− 0.275000i
$237$	−150.000	−0.632911
$238$	0	0
$239$	− 439.877i	− 1.84049i	−0.391342	−	0.920245i	$-0.627989\pi$
$239$	− 439.877i	− 1.84049i	0.391342	−	0.920245i	$-0.372011\pi$
$240$	627.366i	2.61402i
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	248.783i	1.02803i
$243$	−243.000	−1.00000
$244$	630.000	2.58197
$245$	353.344i	1.44222i
$246$	858.000	3.48780
$247$	0	0
$248$	0	0
$249$	− 497.566i	− 1.99826i
$250$	−52.0000	−0.208000
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	0	0
$254$	165.855i	0.652974i
$255$	0	0
$256$	−459.000	−1.79297
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	− 757.166i	− 2.93475i
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	−390.000	−1.47727
$265$	0	0
$266$	0	0
$267$	237.966i	0.891260i
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	−702.000	−2.60000
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	0	0
$274$	−962.000	−3.51095
$275$	− 194.700i	− 0.707999i
$276$	0	0
$277$	70.0000	0.252708	0.126354	−	0.991985i	$-0.459673\pi$
$277$	70.0000	0.252708	0.126354	+	0.991985i	$0.459673\pi$
$278$	439.877i	1.58229i
$279$	0	0
$280$	0	0
$281$	− 425.455i	− 1.51407i	−0.653371	−	0.757037i	$-0.726646\pi$
$281$	− 425.455i	− 1.51407i	0.653371	−	0.757037i	$-0.273354\pi$
$282$	1014.00	3.59574
$283$	−266.000	−0.939929	−0.469965	−	0.882685i	$-0.655733\pi$
$283$	−266.000	−0.939929	−0.469965	+	0.882685i	$0.655733\pi$
$284$	− 713.899i	− 2.51373i
$285$	0	0
$286$	0	0
$287$	0	0
$288$	292.050i	1.01406i
$289$	289.000	1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 338.922i	− 1.15673i	−0.815778	−	0.578365i	$-0.803691\pi$
$293$	− 338.922i	− 1.15673i	0.815778	−	0.578365i	$-0.196309\pi$
$294$	530.016i	1.80278i
$295$	52.0000	0.176271
$296$	0	0
$297$	− 194.700i	− 0.655555i
$298$	26.0000	0.0872483
$299$	0	0
$300$	−729.000	−2.43000
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	504.777i	1.65501i
$306$	0	0
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	0	0
$309$	150.000	0.485437
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	−574.000	−1.83387	−0.916933	−	0.399041i	$-0.869343\pi$
$313$	−574.000	−1.83387	−0.916933	+	0.399041i	$0.869343\pi$
$314$	− 1117.72i	− 3.55962i
$315$	0	0
$316$	−450.000	−1.42405
$317$	− 526.410i	− 1.66060i	−0.557316	−	0.830300i	$-0.688169\pi$
$317$	− 526.410i	− 1.66060i	0.557316	−	0.830300i	$-0.311831\pi$
$318$	0	0
$319$	0	0
$320$	7.21110i	0.0225347i
$321$	0	0
$322$	0	0
$323$	0	0
$324$	−729.000	−2.25000
$325$	0	0
$326$	0	0
$327$	0	0
$328$	1430.00	4.35976
$329$	0	0
$330$	− 562.466i	− 1.70444i
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	− 1492.70i	− 4.49608i
$333$	0	0
$334$	−962.000	−2.88024
$335$	0	0
$336$	0	0
$337$	−626.000	−1.85757	−0.928783	−	0.370623i	$-0.879144\pi$
$337$	−626.000	−1.85757	−0.928783	+	0.370623i	$0.879144\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	− 1261.94i	− 3.66844i
$345$	0	0
$346$	0	0
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	0	0
$351$	0	0
$352$	−234.000	−0.664773
$353$	− 93.7443i	− 0.265565i	−0.991145	−	0.132782i	$-0.957609\pi$
$353$	− 93.7443i	− 0.265565i	0.991145	−	0.132782i	$-0.0423911\pi$
$354$	78.0000	0.220339
$355$	572.000	1.61127
$356$	713.899i	2.00533i
$357$	0	0
$358$	0	0
$359$	− 79.3221i	− 0.220953i	−0.993879	−	0.110477i	$-0.964762\pi$
$359$	− 79.3221i	− 0.220953i	0.993879	−	0.110477i	$-0.0352377\pi$
$360$	−1170.00	−3.25000
$361$	−361.000	−1.00000
$362$	944.654i	2.60954i
$363$	−207.000	−0.570248
$364$	0	0
$365$	0	0
$366$	757.166i	2.06876i
$367$	−670.000	−1.82561	−0.912807	−	0.408392i	$-0.866090\pi$
$367$	−670.000	−1.82561	−0.912807	+	0.408392i	$0.866090\pi$
$368$	0	0
$369$	713.899i	1.93469i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	554.000	1.48525	0.742627	−	0.669705i	$-0.233579\pi$
$373$	554.000	1.48525	0.742627	+	0.669705i	$0.233579\pi$
$374$	0	0
$375$	− 43.2666i	− 0.115378i
$376$	1690.00	4.49468
$377$	0	0
$378$	0	0
$379$	0	0		−	1.00000i	$-0.5\pi$
$379$	0	0			1.00000i	$0.5\pi$
$380$	0	0
$381$	−138.000	−0.362205
$382$	0	0
$383$	598.522i	1.56272i	0.624081	+	0.781360i	$0.285474\pi$
$383$	598.522i	1.56272i	−0.624081	+	0.781360i	$0.714526\pi$
$384$	− 378.583i	− 0.985893i
$385$	0	0
$386$	0	0
$387$	630.000	1.62791
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	883.360i	2.25347i
$393$	0	0
$394$	−1222.00	−3.10152
$395$	− 360.555i	− 0.912798i
$396$	− 584.099i	− 1.47500i
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	685.055i	1.72124i
$399$	0	0
$400$	−783.000	−1.95750
$401$	786.010i	1.96013i	0.198689	+	0.980063i	$0.436332\pi$
$401$	786.010i	1.96013i	−0.198689	+	0.980063i	$0.563668\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	− 584.099i	− 1.44222i
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0	0		−	1.00000i	$-0.5\pi$
$409$	0	0			1.00000i	$0.5\pi$
$410$	2062.38i	5.03018i
$411$	− 800.432i	− 1.94752i
$412$	450.000	1.09223
$413$	0	0
$414$	0	0
$415$	1196.00	2.88193
$416$	0	0
$417$	−366.000	−0.877698
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	0	0		−	1.00000i	$-0.5\pi$
$421$	0	0			1.00000i	$0.5\pi$
$422$	1478.28i	3.50302i
$423$	843.699i	1.99456i
$424$	0	0
$425$	0	0
$426$	858.000	2.01408
$427$	0	0
$428$	0	0
$429$	0	0
$430$	1820.00	4.23256
$431$	− 439.877i	− 1.02060i	−0.859997	−	0.510298i	$-0.829535\pi$
$431$	− 439.877i	− 1.02060i	0.859997	−	0.510298i	$-0.170465\pi$
$432$	−783.000	−1.81250
$433$	−434.000	−1.00231	−0.501155	−	0.865358i	$-0.667091\pi$
$433$	−434.000	−1.00231	−0.501155	+	0.865358i	$0.667091\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−722.000	−1.64465	−0.822323	−	0.569020i	$-0.807323\pi$
$439$	−722.000	−1.64465	−0.822323	+	0.569020i	$0.807323\pi$
$440$	− 937.443i	− 2.13055i
$441$	−441.000	−1.00000
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	−572.000	−1.28539
$446$	0	0
$447$	21.6333i	0.0483967i
$448$	0	0
$449$	439.877i	0.979682i	0.871812	+	0.489841i	$0.162945\pi$
$449$	439.877i	0.979682i	−0.871812	+	0.489841i	$0.837055\pi$
$450$	− 876.149i	− 1.94700i
$451$	−572.000	−1.26829
$452$	0	0
$453$	0	0
$454$	−598.000	−1.31718
$455$	0	0
$456$	0	0
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 872.543i	− 1.89272i	−0.323116	−	0.946359i	$-0.604730\pi$
$461$	− 872.543i	− 1.89272i	0.323116	−	0.946359i	$-0.395270\pi$
$462$	0	0
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	0	0
$469$	0	0
$470$	2437.35i	5.18586i
$471$	930.000	1.97452
$472$	130.000	0.275424
$473$	504.777i	1.06718i
$474$	− 540.833i	− 1.14100i
$475$	0	0
$476$	0	0
$477$	0	0
$478$	1586.00	3.31799
$479$	− 944.654i	− 1.97214i	−0.166335	−	0.986069i	$-0.553193\pi$
$479$	− 944.654i	− 1.97214i	0.166335	−	0.986069i	$-0.446807\pi$
$480$	−702.000	−1.46250
$481$	0	0
$482$	0	0
$483$	0	0
$484$	−621.000	−1.28306
$485$	0	0
$486$	− 876.149i	− 1.80278i
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	1261.94i	2.58595i
$489$	0	0
$490$	−1274.00	−2.60000
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	2141.70i	4.35304i
$493$	0	0
$494$	0	0
$495$	468.000	0.945455
$496$	0	0
$497$	0	0
$498$	1794.00	3.60241
$499$	0	0		−	1.00000i	$-0.5\pi$
$499$	0	0			1.00000i	$0.5\pi$
$500$	− 129.800i	− 0.259600i
$501$	− 800.432i	− 1.59767i
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	−414.000	−0.814961
$509$	511.988i	1.00587i	0.864324	+	0.502935i	$0.167747\pi$
$509$	511.988i	1.00587i	−0.864324	+	0.502935i	$0.832253\pi$
$510$	0	0
$511$	0	0
$512$	− 1150.17i	− 2.24643i
$513$	0	0
$514$	0	0
$515$	360.555i	0.700107i
$516$	1890.00	3.66279
$517$	−676.000	−1.30754
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	890.000	1.70172	0.850860	−	0.525392i	$-0.176081\pi$
$523$	890.000	1.70172	0.850860	+	0.525392i	$0.176081\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	− 627.366i	− 1.18819i
$529$	529.000	1.00000
$530$	0	0
$531$	64.8999i	0.122222i
$532$	0	0
$533$	0	0
$534$	−858.000	−1.60674
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 353.344i	− 0.655555i
$540$	− 1752.30i	− 3.24500i
$541$	0	0		−	1.00000i	$-0.5\pi$
$541$	0	0			1.00000i	$0.5\pi$
$542$	0	0
$543$	−786.000	−1.44751
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−310.000	−0.566728	−0.283364	−	0.959012i	$-0.591450\pi$
$547$	−310.000	−0.566728	−0.283364	+	0.959012i	$0.591450\pi$
$548$	− 2401.30i	− 4.38193i
$549$	−630.000	−1.14754
$550$	702.000	1.27636
$551$	0	0
$552$	0	0
$553$	0	0
$554$	252.389i	0.455575i
$555$	0	0
$556$	−1098.00	−1.97482
$557$	− 165.855i	− 0.297765i	−0.988855	−	0.148883i	$-0.952432\pi$
$557$	− 165.855i	− 0.297765i	0.988855	−	0.148883i	$-0.0475677\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	1534.00	2.72954
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	2531.10i	4.48776i
$565$	0	0
$566$	− 959.077i	− 1.69448i
$567$	0	0
$568$	1430.00	2.51761
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	310.000	0.542907	0.271454	−	0.962452i	$-0.412496\pi$
$571$	310.000	0.542907	0.271454	+	0.962452i	$0.412496\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	−9.00000	−0.0156250
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	1042.00i	1.80278i
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	1222.00	2.08532
$587$	1031.19i	1.75671i	0.478011	+	0.878354i	$0.341358\pi$
$587$	1031.19i	1.75671i	−0.478011	+	0.878354i	$0.658642\pi$
$588$	−1323.00	−2.25000
$589$	0	0
$590$	187.489i	0.317777i
$591$	− 1016.77i	− 1.72042i
$592$	0	0
$593$	1132.14i	1.90918i	0.297924	+	0.954589i	$0.403706\pi$
$593$	1132.14i	1.90918i	−0.297924	+	0.954589i	$0.596294\pi$
$594$	702.000	1.18182
$595$	0	0
$596$	64.8999i	0.108892i
$597$	−570.000	−0.954774
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	− 1460.25i	− 2.43375i
$601$	−1150.00	−1.91348	−0.956739	−	0.290948i	$-0.906029\pi$
$601$	−1150.00	−1.91348	−0.956739	+	0.290948i	$0.906029\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 497.566i	− 0.822423i
$606$	0	0
$607$	−190.000	−0.313015	−0.156507	−	0.987677i	$-0.550024\pi$
$607$	−190.000	−0.313015	−0.156507	+	0.987677i	$0.550024\pi$
$608$	0	0
$609$	0	0
$610$	−1820.00	−2.98361
$611$	0	0
$612$	0	0
$613$	0	0		−	1.00000i	$-0.5\pi$
$613$	0	0			1.00000i	$0.5\pi$
$614$	0	0
$615$	−1716.00	−2.79024
$616$	0	0
$617$	771.588i	1.25055i	0.780405	+	0.625274i	$0.215013\pi$
$617$	771.588i	1.25055i	−0.780405	+	0.625274i	$0.784987\pi$
$618$	540.833i	0.875134i
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−571.000	−0.913600
$626$	− 2069.59i	− 3.30605i
$627$	0	0
$628$	2790.00	4.44268
$629$	0	0
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	− 901.388i	− 1.42625i
$633$	−1230.00	−1.94313
$634$	1898.00	2.99369
$635$	− 331.711i	− 0.522379i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	713.899i	1.11721i
$640$	910.000	1.42188
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	0	0		−	1.00000i	$-0.5\pi$
$643$	0	0			1.00000i	$0.5\pi$
$644$	0	0
$645$	1514.33i	2.34780i
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	− 1460.25i	− 2.25347i
$649$	−52.0000	−0.0801233
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	2300.34i	3.50662i
$657$	0	0
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	1404.00	2.12727
$661$	0	0		−	1.00000i	$-0.5\pi$
$661$	0	0			1.00000i	$0.5\pi$
$662$	0	0
$663$	0	0
$664$	2990.00	4.50301
$665$	0	0
$666$	0	0
$667$	0	0
$668$	− 2401.30i	− 3.59476i
$669$	0	0
$670$	0	0
$671$	− 504.777i	− 0.752276i
$672$	0	0
$673$	1150.00	1.70877	0.854383	−	0.519643i	$-0.173935\pi$
$673$	1150.00	1.70877	0.854383	+	0.519643i	$0.173935\pi$
$674$	− 2257.08i	− 3.34878i
$675$	729.000	1.08000
$676$	0	0
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	− 497.566i	− 0.730640i
$682$	0	0
$683$	− 338.922i	− 0.496225i	−0.968731	−	0.248113i	$-0.920190\pi$
$683$	− 338.922i	− 0.496225i	0.968731	−	0.248113i	$-0.0798103\pi$
$684$	0	0
$685$	1924.00	2.80876
$686$	0	0
$687$	0	0
$688$	2030.00	2.95058
$689$	0	0
$690$	0	0
$691$	0	0		−	1.00000i	$-0.5\pi$
$691$	0	0			1.00000i	$0.5\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 879.755i	− 1.26583i
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	− 7.21110i	− 0.0102430i
$705$	−2028.00	−2.87660
$706$	338.000	0.478754
$707$	0	0
$708$	194.700i	0.275000i
$709$	0	0		−	1.00000i	$-0.5\pi$
$709$	0	0			1.00000i	$0.5\pi$
$710$	2062.38i	2.90475i
$711$	450.000	0.632911
$712$	−1430.00	−2.00843
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	1319.63i	1.84049i
$718$	286.000	0.398329
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	− 1882.10i	− 2.61402i
$721$	0	0
$722$	− 1301.60i	− 1.80278i
$723$	0	0
$724$	−2358.00	−3.25691
$725$	0	0
$726$	− 746.349i	− 1.02803i
$727$	1246.00	1.71389	0.856946	−	0.515406i	$-0.172359\pi$
$727$	1246.00	1.71389	0.856946	+	0.515406i	$0.172359\pi$
$728$	0	0
$729$	729.000	1.00000
$730$	0	0
$731$	0	0
$732$	−1890.00	−2.58197
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	− 2415.72i	− 3.29117i
$735$	− 1060.03i	− 1.44222i
$736$	0	0
$737$	0	0
$738$	−2574.00	−3.48780
$739$	0	0		−	1.00000i	$-0.5\pi$
$739$	0	0			1.00000i	$0.5\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 266.811i	− 0.359099i	−0.983749	−	0.179550i	$-0.942536\pi$
$743$	− 266.811i	− 0.359099i	0.983749	−	0.179550i	$-0.0574641\pi$
$744$	0	0
$745$	−52.0000	−0.0697987
$746$	1997.48i	2.67758i
$747$	1492.70i	1.99826i
$748$	0	0
$749$	0	0
$750$	156.000	0.208000
$751$	−98.0000	−0.130493	−0.0652463	−	0.997869i	$-0.520783\pi$
$751$	−98.0000	−0.130493	−0.0652463	+	0.997869i	$0.520783\pi$
$752$	2718.59i	3.61514i
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−214.000	−0.282695	−0.141347	−	0.989960i	$-0.545143\pi$
$757$	−214.000	−0.282695	−0.141347	+	0.989960i	$0.545143\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1305.21i	1.71512i	0.514380	+	0.857562i	$0.328022\pi$
$761$	1305.21i	1.71512i	−0.514380	+	0.857562i	$0.671978\pi$
$762$	− 497.566i	− 0.652974i
$763$	0	0
$764$	0	0
$765$	0	0
$766$	−2158.00	−2.81723
$767$	0	0
$768$	1377.00	1.79297
$769$	0	0		−	1.00000i	$-0.5\pi$
$769$	0	0			1.00000i	$0.5\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 699.477i	− 0.904886i	−0.891793	−	0.452443i	$-0.850553\pi$
$773$	− 699.477i	− 0.904886i	0.891793	−	0.452443i	$-0.149447\pi$
$774$	2271.50i	2.93475i
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−572.000	−0.732394
$782$	0	0
$783$	0	0
$784$	−1421.00	−1.81250
$785$	2235.44i	2.84770i
$786$	0	0
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	− 3050.30i	− 3.87093i
$789$	0	0
$790$	1300.00	1.64557
$791$	0	0
$792$	1170.00	1.47727
$793$	0	0
$794$	0	0
$795$	0	0
$796$	−1710.00	−2.14824
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	− 876.149i	− 1.09519i
$801$	− 713.899i	− 0.891260i
$802$	−2834.00	−3.53367
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	2106.00	2.60000
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	−5148.00	−6.27805
$821$	− 7.21110i	− 0.00878332i	−0.999990	−	0.00439166i	$-0.998602\pi$
$821$	− 7.21110i	− 0.00878332i	0.999990	−	0.00439166i	$-0.00139791\pi$
$822$	2886.00	3.51095
$823$	−1490.00	−1.81045	−0.905225	−	0.424933i	$-0.860298\pi$
$823$	−1490.00	−1.81045	−0.905225	+	0.424933i	$0.860298\pi$
$824$	901.388i	1.09392i
$825$	584.099i	0.707999i
$826$	0	0
$827$	1391.74i	1.68288i	0.540350	+	0.841441i	$0.318292\pi$
$827$	1391.74i	1.68288i	−0.540350	+	0.841441i	$0.681708\pi$
$828$	0	0
$829$	−890.000	−1.07358	−0.536791	−	0.843715i	$-0.680364\pi$
$829$	−890.000	−1.07358	−0.536791	+	0.843715i	$0.680364\pi$
$830$	4312.24i	5.19547i
$831$	−210.000	−0.252708
$832$	0	0
$833$	0	0
$834$	− 1319.63i	− 1.58229i
$835$	1924.00	2.30419
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1651.34i	1.96823i	0.177540	+	0.984114i	$0.443186\pi$
$839$	1651.34i	1.96823i	−0.177540	+	0.984114i	$0.556814\pi$
$840$	0	0
$841$	841.000	1.00000
$842$	0	0
$843$	1276.37i	1.51407i
$844$	−3690.00	−4.37204
$845$	0	0
$846$	−3042.00	−3.59574
$847$	0	0
$848$	0	0
$849$	798.000	0.939929
$850$	0	0
$851$	0	0
$852$	2141.70i	2.51373i
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	1610.00	1.87427	0.937136	−	0.348964i	$-0.113466\pi$
$859$	1610.00	1.87427	0.937136	+	0.348964i	$0.113466\pi$
$860$	4542.99i	5.28255i
$861$	0	0
$862$	1586.00	1.83991
$863$	1636.92i	1.89678i	0.317108	+	0.948390i	$0.397288\pi$
$863$	1636.92i	1.89678i	−0.317108	+	0.948390i	$0.602712\pi$
$864$	− 876.149i	− 1.01406i
$865$	0	0
$866$	− 1564.81i	− 1.80694i
$867$	−867.000	−1.00000
$868$	0	0
$869$	360.555i	0.414908i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0	0		−	1.00000i	$-0.5\pi$
$877$	0	0			1.00000i	$0.5\pi$
$878$	− 2603.21i	− 2.96493i
$879$	1016.77i	1.15673i
$880$	1508.00	1.71364
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	− 1590.05i	− 1.80278i
$883$	934.000	1.05776	0.528879	−	0.848697i	$-0.322613\pi$
$883$	934.000	1.05776	0.528879	+	0.848697i	$0.322613\pi$
$884$	0	0
$885$	−156.000	−0.176271
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	− 2062.38i	− 2.31728i
$891$	584.099i	0.655555i
$892$	0	0
$893$	0	0
$894$	−78.0000	−0.0872483
$895$	0	0
$896$	0	0
$897$	0	0
$898$	−1586.00	−1.76615
$899$	0	0
$900$	2187.00	2.43000
$901$	0	0
$902$	− 2062.38i	− 2.28645i
$903$	0	0
$904$	0	0
$905$	− 1889.31i	− 2.08763i
$906$	0	0
$907$	−1514.00	−1.66924	−0.834620	−	0.550827i	$-0.814313\pi$
$907$	−1514.00	−1.66924	−0.834620	+	0.550827i	$0.814313\pi$
$908$	− 1492.70i	− 1.64394i
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	−1196.00	−1.30997
$914$	0	0
$915$	− 1514.33i	− 1.65501i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−1630.00	−1.77367	−0.886834	−	0.462089i	$-0.847100\pi$
$919$	−1630.00	−1.77367	−0.886834	+	0.462089i	$0.847100\pi$
$920$	0	0
$921$	0	0
$922$	3146.00	3.41215
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−450.000	−0.485437
$928$	0	0
$929$	− 786.010i	− 0.846082i	−0.906111	−	0.423041i	$-0.860963\pi$
$929$	− 786.010i	− 0.846082i	0.906111	−	0.423041i	$-0.139037\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	674.000	0.719317	0.359658	−	0.933084i	$-0.382893\pi$
$937$	674.000	0.719317	0.359658	+	0.933084i	$0.382893\pi$
$938$	0	0
$939$	1722.00	1.83387
$940$	−6084.00	−6.47234
$941$	− 1218.68i	− 1.29509i	−0.762029	−	0.647543i	$-0.775797\pi$
$941$	− 1218.68i	− 1.29509i	0.762029	−	0.647543i	$-0.224203\pi$
$942$	3353.16i	3.55962i
$943$	0	0
$944$	209.122i	0.221528i
$945$	0	0
$946$	−1820.00	−1.92389
$947$	− 1204.25i	− 1.27165i	−0.771833	−	0.635826i	$-0.780660\pi$
$947$	− 1204.25i	− 1.27165i	0.771833	−	0.635826i	$-0.219340\pi$
$948$	1350.00	1.42405
$949$	0	0
$950$	0	0
$951$	1579.23i	1.66060i
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0	0
$956$	3958.90i	4.14110i
$957$	0	0
$958$	3406.00	3.55532
$959$	0	0
$960$	− 21.6333i	− 0.0225347i
$961$	−961.000	−1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	− 1243.92i	− 1.28504i
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	2187.00	2.25000
$973$	0	0
$974$	0	0
$975$	0	0
$976$	−2030.00	−2.07992
$977$	− 1824.41i	− 1.86736i	−0.358111	−	0.933679i	$-0.616579\pi$
$977$	− 1824.41i	− 1.86736i	0.358111	−	0.933679i	$-0.383421\pi$
$978$	0	0
$979$	572.000	0.584270
$980$	− 3180.10i	− 3.24500i
$981$	0	0
$982$	0	0
$983$	− 771.588i	− 0.784932i	−0.919767	−	0.392466i	$-0.871622\pi$
$983$	− 771.588i	− 0.784932i	0.919767	−	0.392466i	$-0.128378\pi$
$984$	−4290.00	−4.35976
$985$	2444.00	2.48122
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	1687.40i	1.70444i
$991$	−1918.00	−1.93542	−0.967709	−	0.252069i	$-0.918889\pi$
$991$	−1918.00	−1.93542	−0.967709	+	0.252069i	$0.918889\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 1370.11i	− 1.37699i
$996$	4478.09i	4.49608i
$997$	1370.00	1.37412	0.687061	−	0.726600i	$-0.258900\pi$
$997$	1370.00	1.37412	0.687061	+	0.726600i	$0.258900\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.3.c.c.170.2		2
3.2	odd	2	inner	507.3.c.c.170.1		2
13.5	odd	4		39.3.d.a.38.2	yes	2
13.8	odd	4		39.3.d.a.38.1	✓	2
13.12	even	2	inner	507.3.c.c.170.1		2
39.5	even	4		39.3.d.a.38.1	✓	2
39.8	even	4		39.3.d.a.38.2	yes	2
39.38	odd	2	CM	507.3.c.c.170.2		2
52.31	even	4		624.3.l.c.545.1		2
52.47	even	4		624.3.l.c.545.2		2
156.47	odd	4		624.3.l.c.545.1		2
156.83	odd	4		624.3.l.c.545.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.3.d.a.38.1	✓	2	13.8	odd	4
39.3.d.a.38.1	✓	2	39.5	even	4
39.3.d.a.38.2	yes	2	13.5	odd	4
39.3.d.a.38.2	yes	2	39.8	even	4
507.3.c.c.170.1		2	3.2	odd	2	inner
507.3.c.c.170.1		2	13.12	even	2	inner
507.3.c.c.170.2		2	1.1	even	1	trivial
507.3.c.c.170.2		2	39.38	odd	2	CM
624.3.l.c.545.1		2	52.31	even	4
624.3.l.c.545.1		2	156.47	odd	4
624.3.l.c.545.2		2	52.47	even	4
624.3.l.c.545.2		2	156.83	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:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	507.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.60555i q^{2} -3.00000 q^{3} -9.00000 q^{4} -7.21110i q^{5} -10.8167i q^{6} -18.0278i q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+3.60555i q^{2} -3.00000 q^{3} -9.00000 q^{4} -7.21110i q^{5} -10.8167i q^{6} -18.0278i q^{8} +9.00000 q^{9} +26.0000 q^{10} +7.21110i q^{11} +27.0000 q^{12} +21.6333i q^{15} +29.0000 q^{16} +32.4500i q^{18} +64.8999i q^{20} -26.0000 q^{22} +54.0833i q^{24} -27.0000 q^{25} -27.0000 q^{27} -78.0000 q^{30} +32.4500i q^{32} -21.6333i q^{33} -81.0000 q^{36} -130.000 q^{40} +79.3221i q^{41} +70.0000 q^{43} -64.8999i q^{44} -64.8999i q^{45} +93.7443i q^{47} -87.0000 q^{48} -49.0000 q^{49} -97.3499i q^{50} -97.3499i q^{54} +52.0000 q^{55} +7.21110i q^{59} -194.700i q^{60} -70.0000 q^{61} -1.00000 q^{64} +78.0000 q^{66} +79.3221i q^{71} -162.250i q^{72} +81.0000 q^{75} +50.0000 q^{79} -209.122i q^{80} +81.0000 q^{81} -286.000 q^{82} +165.855i q^{83} +252.389i q^{86} +130.000 q^{88} -79.3221i q^{89} +234.000 q^{90} -338.000 q^{94} -97.3499i q^{96} -176.672i q^{98} +64.8999i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} - 18 q^{4} + 18 q^{9}+O(q^{10}) \) 2 * q - 6 * q^3 - 18 * q^4 + 18 * q^9	\( 2 q - 6 q^{3} - 18 q^{4} + 18 q^{9} + 52 q^{10} + 54 q^{12} + 58 q^{16} - 52 q^{22} - 54 q^{25} - 54 q^{27} - 156 q^{30} - 162 q^{36} - 260 q^{40} + 140 q^{43} - 174 q^{48} - 98 q^{49} + 104 q^{55} - 140 q^{61} - 2 q^{64} + 156 q^{66} + 162 q^{75} + 100 q^{79} + 162 q^{81} - 572 q^{82} + 260 q^{88} + 468 q^{90} - 676 q^{94}+O(q^{100}) \) 2 * q - 6 * q^3 - 18 * q^4 + 18 * q^9 + 52 * q^10 + 54 * q^12 + 58 * q^16 - 52 * q^22 - 54 * q^25 - 54 * q^27 - 156 * q^30 - 162 * q^36 - 260 * q^40 + 140 * q^43 - 174 * q^48 - 98 * q^49 + 104 * q^55 - 140 * q^61 - 2 * q^64 + 156 * q^66 + 162 * q^75 + 100 * q^79 + 162 * q^81 - 572 * q^82 + 260 * q^88 + 468 * q^90 - 676 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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