LMFDB - Embedded newform 2550.2.a.bl.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2550,2,Mod(1,2550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2550, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2550.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2550.a (trivial)

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:	yes
Analytic conductor:	$20.3618525154$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 510)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.44949$ of defining polynomial
Character	$\chi$	$=$	2550.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -4.89898 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -4.89898 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{12} +2.89898 q^{13} -4.89898 q^{14} +1.00000 q^{16} +1.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} -4.89898 q^{21} +4.00000 q^{23} +1.00000 q^{24} +2.89898 q^{26} +1.00000 q^{27} -4.89898 q^{28} +6.00000 q^{29} +4.00000 q^{31} +1.00000 q^{32} +1.00000 q^{34} +1.00000 q^{36} -6.00000 q^{37} +4.00000 q^{38} +2.89898 q^{39} +6.89898 q^{41} -4.89898 q^{42} +0.898979 q^{43} +4.00000 q^{46} +9.79796 q^{47} +1.00000 q^{48} +17.0000 q^{49} +1.00000 q^{51} +2.89898 q^{52} -11.7980 q^{53} +1.00000 q^{54} -4.89898 q^{56} +4.00000 q^{57} +6.00000 q^{58} -4.89898 q^{59} -7.79796 q^{61} +4.00000 q^{62} -4.89898 q^{63} +1.00000 q^{64} +8.89898 q^{67} +1.00000 q^{68} +4.00000 q^{69} +0.898979 q^{71} +1.00000 q^{72} +1.10102 q^{73} -6.00000 q^{74} +4.00000 q^{76} +2.89898 q^{78} +13.7980 q^{79} +1.00000 q^{81} +6.89898 q^{82} +5.79796 q^{83} -4.89898 q^{84} +0.898979 q^{86} +6.00000 q^{87} +11.7980 q^{89} -14.2020 q^{91} +4.00000 q^{92} +4.00000 q^{93} +9.79796 q^{94} +1.00000 q^{96} -16.6969 q^{97} +17.0000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 + 2 * q^8 + 2 * q^9	$ 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 2 q^{8} + 2 q^{9} + 2 q^{12} - 4 q^{13} + 2 q^{16} + 2 q^{17} + 2 q^{18} + 8 q^{19} + 8 q^{23} + 2 q^{24} - 4 q^{26} + 2 q^{27} + 12 q^{29} + 8 q^{31} + 2 q^{32} + 2 q^{34} + 2 q^{36} - 12 q^{37} + 8 q^{38} - 4 q^{39} + 4 q^{41} - 8 q^{43} + 8 q^{46} + 2 q^{48} + 34 q^{49} + 2 q^{51} - 4 q^{52} - 4 q^{53} + 2 q^{54} + 8 q^{57} + 12 q^{58} + 4 q^{61} + 8 q^{62} + 2 q^{64} + 8 q^{67} + 2 q^{68} + 8 q^{69} - 8 q^{71} + 2 q^{72} + 12 q^{73} - 12 q^{74} + 8 q^{76} - 4 q^{78} + 8 q^{79} + 2 q^{81} + 4 q^{82} - 8 q^{83} - 8 q^{86} + 12 q^{87} + 4 q^{89} - 48 q^{91} + 8 q^{92} + 8 q^{93} + 2 q^{96} - 4 q^{97} + 34 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 + 2 * q^8 + 2 * q^9 + 2 * q^12 - 4 * q^13 + 2 * q^16 + 2 * q^17 + 2 * q^18 + 8 * q^19 + 8 * q^23 + 2 * q^24 - 4 * q^26 + 2 * q^27 + 12 * q^29 + 8 * q^31 + 2 * q^32 + 2 * q^34 + 2 * q^36 - 12 * q^37 + 8 * q^38 - 4 * q^39 + 4 * q^41 - 8 * q^43 + 8 * q^46 + 2 * q^48 + 34 * q^49 + 2 * q^51 - 4 * q^52 - 4 * q^53 + 2 * q^54 + 8 * q^57 + 12 * q^58 + 4 * q^61 + 8 * q^62 + 2 * q^64 + 8 * q^67 + 2 * q^68 + 8 * q^69 - 8 * q^71 + 2 * q^72 + 12 * q^73 - 12 * q^74 + 8 * q^76 - 4 * q^78 + 8 * q^79 + 2 * q^81 + 4 * q^82 - 8 * q^83 - 8 * q^86 + 12 * q^87 + 4 * q^89 - 48 * q^91 + 8 * q^92 + 8 * q^93 + 2 * q^96 - 4 * q^97 + 34 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	−4.89898	−1.85164	−0.925820	−	0.377964i	$-0.876624\pi$
$7$	−4.89898	−1.85164	−0.925820	+	0.377964i	$0.876624\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	1.00000	0.288675
$13$	2.89898	0.804032	0.402016	−	0.915633i	$-0.368310\pi$
$13$	2.89898	0.804032	0.402016	+	0.915633i	$0.368310\pi$
$14$	−4.89898	−1.30931
$15$	0	0
$16$	1.00000	0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	1.00000	0.235702
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	−4.89898	−1.06904
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	2.89898	0.568537
$27$	1.00000	0.192450
$28$	−4.89898	−0.925820
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	1.00000	0.171499
$35$	0	0
$36$	1.00000	0.166667
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	4.00000	0.648886
$39$	2.89898	0.464208
$40$	0	0
$41$	6.89898	1.07744	0.538720	−	0.842485i	$-0.318908\pi$
$41$	6.89898	1.07744	0.538720	+	0.842485i	$0.318908\pi$
$42$	−4.89898	−0.755929
$43$	0.898979	0.137093	0.0685465	−	0.997648i	$-0.478164\pi$
$43$	0.898979	0.137093	0.0685465	+	0.997648i	$0.478164\pi$
$44$	0	0
$45$	0	0
$46$	4.00000	0.589768
$47$	9.79796	1.42918	0.714590	−	0.699544i	$-0.246613\pi$
$47$	9.79796	1.42918	0.714590	+	0.699544i	$0.246613\pi$
$48$	1.00000	0.144338
$49$	17.0000	2.42857
$50$	0	0
$51$	1.00000	0.140028
$52$	2.89898	0.402016
$53$	−11.7980	−1.62057	−0.810287	−	0.586033i	$-0.800689\pi$
$53$	−11.7980	−1.62057	−0.810287	+	0.586033i	$0.800689\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	−4.89898	−0.654654
$57$	4.00000	0.529813
$58$	6.00000	0.787839
$59$	−4.89898	−0.637793	−0.318896	−	0.947790i	$-0.603312\pi$
$59$	−4.89898	−0.637793	−0.318896	+	0.947790i	$0.603312\pi$
$60$	0	0
$61$	−7.79796	−0.998426	−0.499213	−	0.866479i	$-0.666378\pi$
$61$	−7.79796	−0.998426	−0.499213	+	0.866479i	$0.666378\pi$
$62$	4.00000	0.508001
$63$	−4.89898	−0.617213
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	8.89898	1.08718	0.543592	−	0.839350i	$-0.317064\pi$
$67$	8.89898	1.08718	0.543592	+	0.839350i	$0.317064\pi$
$68$	1.00000	0.121268
$69$	4.00000	0.481543
$70$	0	0
$71$	0.898979	0.106689	0.0533446	−	0.998576i	$-0.483012\pi$
$71$	0.898979	0.106689	0.0533446	+	0.998576i	$0.483012\pi$
$72$	1.00000	0.117851
$73$	1.10102	0.128865	0.0644324	−	0.997922i	$-0.479476\pi$
$73$	1.10102	0.128865	0.0644324	+	0.997922i	$0.479476\pi$
$74$	−6.00000	−0.697486
$75$	0	0
$76$	4.00000	0.458831
$77$	0	0
$78$	2.89898	0.328245
$79$	13.7980	1.55239	0.776196	−	0.630492i	$-0.217147\pi$
$79$	13.7980	1.55239	0.776196	+	0.630492i	$0.217147\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	6.89898	0.761865
$83$	5.79796	0.636409	0.318204	−	0.948022i	$-0.396920\pi$
$83$	5.79796	0.636409	0.318204	+	0.948022i	$0.396920\pi$
$84$	−4.89898	−0.534522
$85$	0	0
$86$	0.898979	0.0969395
$87$	6.00000	0.643268
$88$	0	0
$89$	11.7980	1.25058	0.625291	−	0.780392i	$-0.284980\pi$
$89$	11.7980	1.25058	0.625291	+	0.780392i	$0.284980\pi$
$90$	0	0
$91$	−14.2020	−1.48878
$92$	4.00000	0.417029
$93$	4.00000	0.414781
$94$	9.79796	1.01058
$95$	0	0
$96$	1.00000	0.102062
$97$	−16.6969	−1.69532	−0.847659	−	0.530542i	$-0.821988\pi$
$97$	−16.6969	−1.69532	−0.847659	+	0.530542i	$0.821988\pi$
$98$	17.0000	1.71726
$99$	0	0
$100$	0	0
$101$	−9.10102	−0.905585	−0.452793	−	0.891616i	$-0.649572\pi$
$101$	−9.10102	−0.905585	−0.452793	+	0.891616i	$0.649572\pi$
$102$	1.00000	0.0990148
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	2.89898	0.284268
$105$	0	0
$106$	−11.7980	−1.14592
$107$	13.7980	1.33390	0.666950	−	0.745103i	$-0.267600\pi$
$107$	13.7980	1.33390	0.666950	+	0.745103i	$0.267600\pi$
$108$	1.00000	0.0962250
$109$	−7.79796	−0.746909	−0.373455	−	0.927648i	$-0.621827\pi$
$109$	−7.79796	−0.746909	−0.373455	+	0.927648i	$0.621827\pi$
$110$	0	0
$111$	−6.00000	−0.569495
$112$	−4.89898	−0.462910
$113$	11.7980	1.10986	0.554929	−	0.831898i	$-0.312745\pi$
$113$	11.7980	1.10986	0.554929	+	0.831898i	$0.312745\pi$
$114$	4.00000	0.374634
$115$	0	0
$116$	6.00000	0.557086
$117$	2.89898	0.268011
$118$	−4.89898	−0.450988
$119$	−4.89898	−0.449089
$120$	0	0
$121$	−11.0000	−1.00000
$122$	−7.79796	−0.705994
$123$	6.89898	0.622060
$124$	4.00000	0.359211
$125$	0	0
$126$	−4.89898	−0.436436
$127$	12.0000	1.06483	0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000	1.06483	0.532414	+	0.846484i	$0.321285\pi$
$128$	1.00000	0.0883883
$129$	0.898979	0.0791507
$130$	0	0
$131$	−9.79796	−0.856052	−0.428026	−	0.903767i	$-0.640791\pi$
$131$	−9.79796	−0.856052	−0.428026	+	0.903767i	$0.640791\pi$
$132$	0	0
$133$	−19.5959	−1.69918
$134$	8.89898	0.768755
$135$	0	0
$136$	1.00000	0.0857493
$137$	−14.0000	−1.19610	−0.598050	−	0.801459i	$-0.704058\pi$
$137$	−14.0000	−1.19610	−0.598050	+	0.801459i	$0.704058\pi$
$138$	4.00000	0.340503
$139$	−5.79796	−0.491776	−0.245888	−	0.969298i	$-0.579080\pi$
$139$	−5.79796	−0.491776	−0.245888	+	0.969298i	$0.579080\pi$
$140$	0	0
$141$	9.79796	0.825137
$142$	0.898979	0.0754407
$143$	0	0
$144$	1.00000	0.0833333
$145$	0	0
$146$	1.10102	0.0911211
$147$	17.0000	1.40214
$148$	−6.00000	−0.493197
$149$	−9.10102	−0.745585	−0.372792	−	0.927915i	$-0.621600\pi$
$149$	−9.10102	−0.745585	−0.372792	+	0.927915i	$0.621600\pi$
$150$	0	0
$151$	9.79796	0.797347	0.398673	−	0.917093i	$-0.369471\pi$
$151$	9.79796	0.797347	0.398673	+	0.917093i	$0.369471\pi$
$152$	4.00000	0.324443
$153$	1.00000	0.0808452
$154$	0	0
$155$	0	0
$156$	2.89898	0.232104
$157$	10.8990	0.869833	0.434917	−	0.900471i	$-0.356778\pi$
$157$	10.8990	0.869833	0.434917	+	0.900471i	$0.356778\pi$
$158$	13.7980	1.09771
$159$	−11.7980	−0.935639
$160$	0	0
$161$	−19.5959	−1.54437
$162$	1.00000	0.0785674
$163$	21.7980	1.70735	0.853674	−	0.520808i	$-0.174369\pi$
$163$	21.7980	1.70735	0.853674	+	0.520808i	$0.174369\pi$
$164$	6.89898	0.538720
$165$	0	0
$166$	5.79796	0.450009
$167$	−21.7980	−1.68678	−0.843388	−	0.537304i	$-0.819443\pi$
$167$	−21.7980	−1.68678	−0.843388	+	0.537304i	$0.819443\pi$
$168$	−4.89898	−0.377964
$169$	−4.59592	−0.353532
$170$	0	0
$171$	4.00000	0.305888
$172$	0.898979	0.0685465
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	6.00000	0.454859
$175$	0	0
$176$	0	0
$177$	−4.89898	−0.368230
$178$	11.7980	0.884294
$179$	4.89898	0.366167	0.183083	−	0.983097i	$-0.441392\pi$
$179$	4.89898	0.366167	0.183083	+	0.983097i	$0.441392\pi$
$180$	0	0
$181$	−23.7980	−1.76889	−0.884444	−	0.466646i	$-0.845462\pi$
$181$	−23.7980	−1.76889	−0.884444	+	0.466646i	$0.845462\pi$
$182$	−14.2020	−1.05273
$183$	−7.79796	−0.576442
$184$	4.00000	0.294884
$185$	0	0
$186$	4.00000	0.293294
$187$	0	0
$188$	9.79796	0.714590
$189$	−4.89898	−0.356348
$190$	0	0
$191$	9.79796	0.708955	0.354478	−	0.935064i	$-0.384659\pi$
$191$	9.79796	0.708955	0.354478	+	0.935064i	$0.384659\pi$
$192$	1.00000	0.0721688
$193$	10.8990	0.784526	0.392263	−	0.919853i	$-0.371692\pi$
$193$	10.8990	0.784526	0.392263	+	0.919853i	$0.371692\pi$
$194$	−16.6969	−1.19877
$195$	0	0
$196$	17.0000	1.21429
$197$	−25.5959	−1.82363	−0.911817	−	0.410597i	$-0.865320\pi$
$197$	−25.5959	−1.82363	−0.911817	+	0.410597i	$0.865320\pi$
$198$	0	0
$199$	−23.5959	−1.67267	−0.836335	−	0.548219i	$-0.815306\pi$
$199$	−23.5959	−1.67267	−0.836335	+	0.548219i	$0.815306\pi$
$200$	0	0
$201$	8.89898	0.627686
$202$	−9.10102	−0.640346
$203$	−29.3939	−2.06305
$204$	1.00000	0.0700140
$205$	0	0
$206$	−4.00000	−0.278693
$207$	4.00000	0.278019
$208$	2.89898	0.201008
$209$	0	0
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	−11.7980	−0.810287
$213$	0.898979	0.0615971
$214$	13.7980	0.943209
$215$	0	0
$216$	1.00000	0.0680414
$217$	−19.5959	−1.33026
$218$	−7.79796	−0.528144
$219$	1.10102	0.0744001
$220$	0	0
$221$	2.89898	0.195006
$222$	−6.00000	−0.402694
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	−4.89898	−0.327327
$225$	0	0
$226$	11.7980	0.784789
$227$	2.20204	0.146155	0.0730773	−	0.997326i	$-0.476718\pi$
$227$	2.20204	0.146155	0.0730773	+	0.997326i	$0.476718\pi$
$228$	4.00000	0.264906
$229$	25.5959	1.69143	0.845713	−	0.533638i	$-0.179176\pi$
$229$	25.5959	1.69143	0.845713	+	0.533638i	$0.179176\pi$
$230$	0	0
$231$	0	0
$232$	6.00000	0.393919
$233$	−14.0000	−0.917170	−0.458585	−	0.888650i	$-0.651644\pi$
$233$	−14.0000	−0.917170	−0.458585	+	0.888650i	$0.651644\pi$
$234$	2.89898	0.189512
$235$	0	0
$236$	−4.89898	−0.318896
$237$	13.7980	0.896274
$238$	−4.89898	−0.317554
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−25.5959	−1.64878	−0.824389	−	0.566024i	$-0.808481\pi$
$241$	−25.5959	−1.64878	−0.824389	+	0.566024i	$0.808481\pi$
$242$	−11.0000	−0.707107
$243$	1.00000	0.0641500
$244$	−7.79796	−0.499213
$245$	0	0
$246$	6.89898	0.439863
$247$	11.5959	0.737831
$248$	4.00000	0.254000
$249$	5.79796	0.367431
$250$	0	0
$251$	−4.89898	−0.309221	−0.154610	−	0.987976i	$-0.549412\pi$
$251$	−4.89898	−0.309221	−0.154610	+	0.987976i	$0.549412\pi$
$252$	−4.89898	−0.308607
$253$	0	0
$254$	12.0000	0.752947
$255$	0	0
$256$	1.00000	0.0625000
$257$	2.00000	0.124757	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000	0.124757	0.0623783	+	0.998053i	$0.480131\pi$
$258$	0.898979	0.0559680
$259$	29.3939	1.82645
$260$	0	0
$261$	6.00000	0.371391
$262$	−9.79796	−0.605320
$263$	−8.00000	−0.493301	−0.246651	−	0.969104i	$-0.579330\pi$
$263$	−8.00000	−0.493301	−0.246651	+	0.969104i	$0.579330\pi$
$264$	0	0
$265$	0	0
$266$	−19.5959	−1.20150
$267$	11.7980	0.722023
$268$	8.89898	0.543592
$269$	−29.5959	−1.80449	−0.902247	−	0.431219i	$-0.858084\pi$
$269$	−29.5959	−1.80449	−0.902247	+	0.431219i	$0.858084\pi$
$270$	0	0
$271$	17.7980	1.08115	0.540575	−	0.841296i	$-0.318207\pi$
$271$	17.7980	1.08115	0.540575	+	0.841296i	$0.318207\pi$
$272$	1.00000	0.0606339
$273$	−14.2020	−0.859547
$274$	−14.0000	−0.845771
$275$	0	0
$276$	4.00000	0.240772
$277$	11.7980	0.708871	0.354435	−	0.935081i	$-0.384673\pi$
$277$	11.7980	0.708871	0.354435	+	0.935081i	$0.384673\pi$
$278$	−5.79796	−0.347738
$279$	4.00000	0.239474
$280$	0	0
$281$	8.20204	0.489293	0.244646	−	0.969612i	$-0.421328\pi$
$281$	8.20204	0.489293	0.244646	+	0.969612i	$0.421328\pi$
$282$	9.79796	0.583460
$283$	15.5959	0.927081	0.463541	−	0.886076i	$-0.346579\pi$
$283$	15.5959	0.927081	0.463541	+	0.886076i	$0.346579\pi$
$284$	0.898979	0.0533446
$285$	0	0
$286$	0	0
$287$	−33.7980	−1.99503
$288$	1.00000	0.0589256
$289$	1.00000	0.0588235
$290$	0	0
$291$	−16.6969	−0.978792
$292$	1.10102	0.0644324
$293$	25.5959	1.49533	0.747665	−	0.664076i	$-0.231175\pi$
$293$	25.5959	1.49533	0.747665	+	0.664076i	$0.231175\pi$
$294$	17.0000	0.991460
$295$	0	0
$296$	−6.00000	−0.348743
$297$	0	0
$298$	−9.10102	−0.527208
$299$	11.5959	0.670609
$300$	0	0
$301$	−4.40408	−0.253847
$302$	9.79796	0.563809
$303$	−9.10102	−0.522840
$304$	4.00000	0.229416
$305$	0	0
$306$	1.00000	0.0571662
$307$	8.89898	0.507892	0.253946	−	0.967218i	$-0.418272\pi$
$307$	8.89898	0.507892	0.253946	+	0.967218i	$0.418272\pi$
$308$	0	0
$309$	−4.00000	−0.227552
$310$	0	0
$311$	−16.8990	−0.958253	−0.479127	−	0.877746i	$-0.659047\pi$
$311$	−16.8990	−0.958253	−0.479127	+	0.877746i	$0.659047\pi$
$312$	2.89898	0.164122
$313$	2.89898	0.163860	0.0819300	−	0.996638i	$-0.473892\pi$
$313$	2.89898	0.163860	0.0819300	+	0.996638i	$0.473892\pi$
$314$	10.8990	0.615065
$315$	0	0
$316$	13.7980	0.776196
$317$	−14.0000	−0.786318	−0.393159	−	0.919470i	$-0.628618\pi$
$317$	−14.0000	−0.786318	−0.393159	+	0.919470i	$0.628618\pi$
$318$	−11.7980	−0.661597
$319$	0	0
$320$	0	0
$321$	13.7980	0.770127
$322$	−19.5959	−1.09204
$323$	4.00000	0.222566
$324$	1.00000	0.0555556
$325$	0	0
$326$	21.7980	1.20728
$327$	−7.79796	−0.431228
$328$	6.89898	0.380932
$329$	−48.0000	−2.64633
$330$	0	0
$331$	13.7980	0.758404	0.379202	−	0.925314i	$-0.376198\pi$
$331$	13.7980	0.758404	0.379202	+	0.925314i	$0.376198\pi$
$332$	5.79796	0.318204
$333$	−6.00000	−0.328798
$334$	−21.7980	−1.19273
$335$	0	0
$336$	−4.89898	−0.267261
$337$	−3.30306	−0.179929	−0.0899646	−	0.995945i	$-0.528675\pi$
$337$	−3.30306	−0.179929	−0.0899646	+	0.995945i	$0.528675\pi$
$338$	−4.59592	−0.249985
$339$	11.7980	0.640777
$340$	0	0
$341$	0	0
$342$	4.00000	0.216295
$343$	−48.9898	−2.64520
$344$	0.898979	0.0484697
$345$	0	0
$346$	−6.00000	−0.322562
$347$	−2.20204	−0.118212	−0.0591059	−	0.998252i	$-0.518825\pi$
$347$	−2.20204	−0.118212	−0.0591059	+	0.998252i	$0.518825\pi$
$348$	6.00000	0.321634
$349$	23.7980	1.27388	0.636938	−	0.770915i	$-0.280201\pi$
$349$	23.7980	1.27388	0.636938	+	0.770915i	$0.280201\pi$
$350$	0	0
$351$	2.89898	0.154736
$352$	0	0
$353$	26.0000	1.38384	0.691920	−	0.721974i	$-0.256765\pi$
$353$	26.0000	1.38384	0.691920	+	0.721974i	$0.256765\pi$
$354$	−4.89898	−0.260378
$355$	0	0
$356$	11.7980	0.625291
$357$	−4.89898	−0.259281
$358$	4.89898	0.258919
$359$	−21.3939	−1.12913	−0.564563	−	0.825390i	$-0.690955\pi$
$359$	−21.3939	−1.12913	−0.564563	+	0.825390i	$0.690955\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	−23.7980	−1.25079
$363$	−11.0000	−0.577350
$364$	−14.2020	−0.744389
$365$	0	0
$366$	−7.79796	−0.407606
$367$	36.8990	1.92611	0.963056	−	0.269303i	$-0.0867933\pi$
$367$	36.8990	1.92611	0.963056	+	0.269303i	$0.0867933\pi$
$368$	4.00000	0.208514
$369$	6.89898	0.359147
$370$	0	0
$371$	57.7980	3.00072
$372$	4.00000	0.207390
$373$	4.69694	0.243198	0.121599	−	0.992579i	$-0.461198\pi$
$373$	4.69694	0.243198	0.121599	+	0.992579i	$0.461198\pi$
$374$	0	0
$375$	0	0
$376$	9.79796	0.505291
$377$	17.3939	0.895830
$378$	−4.89898	−0.251976
$379$	−10.2020	−0.524044	−0.262022	−	0.965062i	$-0.584389\pi$
$379$	−10.2020	−0.524044	−0.262022	+	0.965062i	$0.584389\pi$
$380$	0	0
$381$	12.0000	0.614779
$382$	9.79796	0.501307
$383$	−8.00000	−0.408781	−0.204390	−	0.978889i	$-0.565521\pi$
$383$	−8.00000	−0.408781	−0.204390	+	0.978889i	$0.565521\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	10.8990	0.554743
$387$	0.898979	0.0456977
$388$	−16.6969	−0.847659
$389$	10.4949	0.532112	0.266056	−	0.963958i	$-0.414279\pi$
$389$	10.4949	0.532112	0.266056	+	0.963958i	$0.414279\pi$
$390$	0	0
$391$	4.00000	0.202289
$392$	17.0000	0.858630
$393$	−9.79796	−0.494242
$394$	−25.5959	−1.28950
$395$	0	0
$396$	0	0
$397$	37.5959	1.88689	0.943443	−	0.331536i	$-0.107567\pi$
$397$	37.5959	1.88689	0.943443	+	0.331536i	$0.107567\pi$
$398$	−23.5959	−1.18276
$399$	−19.5959	−0.981023
$400$	0	0
$401$	13.1010	0.654234	0.327117	−	0.944984i	$-0.393923\pi$
$401$	13.1010	0.654234	0.327117	+	0.944984i	$0.393923\pi$
$402$	8.89898	0.443841
$403$	11.5959	0.577634
$404$	−9.10102	−0.452793
$405$	0	0
$406$	−29.3939	−1.45879
$407$	0	0
$408$	1.00000	0.0495074
$409$	21.5959	1.06785	0.533925	−	0.845532i	$-0.320717\pi$
$409$	21.5959	1.06785	0.533925	+	0.845532i	$0.320717\pi$
$410$	0	0
$411$	−14.0000	−0.690569
$412$	−4.00000	−0.197066
$413$	24.0000	1.18096
$414$	4.00000	0.196589
$415$	0	0
$416$	2.89898	0.142134
$417$	−5.79796	−0.283927
$418$	0	0
$419$	1.79796	0.0878360	0.0439180	−	0.999035i	$-0.486016\pi$
$419$	1.79796	0.0878360	0.0439180	+	0.999035i	$0.486016\pi$
$420$	0	0
$421$	14.0000	0.682318	0.341159	−	0.940006i	$-0.389181\pi$
$421$	14.0000	0.682318	0.341159	+	0.940006i	$0.389181\pi$
$422$	−12.0000	−0.584151
$423$	9.79796	0.476393
$424$	−11.7980	−0.572960
$425$	0	0
$426$	0.898979	0.0435557
$427$	38.2020	1.84873
$428$	13.7980	0.666950
$429$	0	0
$430$	0	0
$431$	−32.8990	−1.58469	−0.792344	−	0.610075i	$-0.791139\pi$
$431$	−32.8990	−1.58469	−0.792344	+	0.610075i	$0.791139\pi$
$432$	1.00000	0.0481125
$433$	−23.3939	−1.12424	−0.562119	−	0.827056i	$-0.690014\pi$
$433$	−23.3939	−1.12424	−0.562119	+	0.827056i	$0.690014\pi$
$434$	−19.5959	−0.940634
$435$	0	0
$436$	−7.79796	−0.373455
$437$	16.0000	0.765384
$438$	1.10102	0.0526088
$439$	−13.7980	−0.658541	−0.329270	−	0.944236i	$-0.606803\pi$
$439$	−13.7980	−0.658541	−0.329270	+	0.944236i	$0.606803\pi$
$440$	0	0
$441$	17.0000	0.809524
$442$	2.89898	0.137890
$443$	−18.2020	−0.864805	−0.432403	−	0.901681i	$-0.642334\pi$
$443$	−18.2020	−0.864805	−0.432403	+	0.901681i	$0.642334\pi$
$444$	−6.00000	−0.284747
$445$	0	0
$446$	4.00000	0.189405
$447$	−9.10102	−0.430463
$448$	−4.89898	−0.231455
$449$	−2.89898	−0.136811	−0.0684057	−	0.997658i	$-0.521791\pi$
$449$	−2.89898	−0.136811	−0.0684057	+	0.997658i	$0.521791\pi$
$450$	0	0
$451$	0	0
$452$	11.7980	0.554929
$453$	9.79796	0.460348
$454$	2.20204	0.103347
$455$	0	0
$456$	4.00000	0.187317
$457$	−35.7980	−1.67456	−0.837279	−	0.546776i	$-0.815855\pi$
$457$	−35.7980	−1.67456	−0.837279	+	0.546776i	$0.815855\pi$
$458$	25.5959	1.19602
$459$	1.00000	0.0466760
$460$	0	0
$461$	0.696938	0.0324597	0.0162298	−	0.999868i	$-0.494834\pi$
$461$	0.696938	0.0324597	0.0162298	+	0.999868i	$0.494834\pi$
$462$	0	0
$463$	31.5959	1.46839	0.734193	−	0.678940i	$-0.237561\pi$
$463$	31.5959	1.46839	0.734193	+	0.678940i	$0.237561\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	−14.0000	−0.648537
$467$	−31.5959	−1.46208	−0.731042	−	0.682332i	$-0.760966\pi$
$467$	−31.5959	−1.46208	−0.731042	+	0.682332i	$0.760966\pi$
$468$	2.89898	0.134005
$469$	−43.5959	−2.01307
$470$	0	0
$471$	10.8990	0.502198
$472$	−4.89898	−0.225494
$473$	0	0
$474$	13.7980	0.633761
$475$	0	0
$476$	−4.89898	−0.224544
$477$	−11.7980	−0.540191
$478$	0	0
$479$	−23.1010	−1.05551	−0.527756	−	0.849396i	$-0.676967\pi$
$479$	−23.1010	−1.05551	−0.527756	+	0.849396i	$0.676967\pi$
$480$	0	0
$481$	−17.3939	−0.793093
$482$	−25.5959	−1.16586
$483$	−19.5959	−0.891645
$484$	−11.0000	−0.500000
$485$	0	0
$486$	1.00000	0.0453609
$487$	11.1010	0.503035	0.251518	−	0.967853i	$-0.419070\pi$
$487$	11.1010	0.503035	0.251518	+	0.967853i	$0.419070\pi$
$488$	−7.79796	−0.352997
$489$	21.7980	0.985738
$490$	0	0
$491$	−30.6969	−1.38533	−0.692667	−	0.721258i	$-0.743564\pi$
$491$	−30.6969	−1.38533	−0.692667	+	0.721258i	$0.743564\pi$
$492$	6.89898	0.311030
$493$	6.00000	0.270226
$494$	11.5959	0.521725
$495$	0	0
$496$	4.00000	0.179605
$497$	−4.40408	−0.197550
$498$	5.79796	0.259813
$499$	−0.404082	−0.0180892	−0.00904460	−	0.999959i	$-0.502879\pi$
$499$	−0.404082	−0.0180892	−0.00904460	+	0.999959i	$0.502879\pi$
$500$	0	0
$501$	−21.7980	−0.973861
$502$	−4.89898	−0.218652
$503$	−4.00000	−0.178351	−0.0891756	−	0.996016i	$-0.528423\pi$
$503$	−4.00000	−0.178351	−0.0891756	+	0.996016i	$0.528423\pi$
$504$	−4.89898	−0.218218
$505$	0	0
$506$	0	0
$507$	−4.59592	−0.204112
$508$	12.0000	0.532414
$509$	−44.6969	−1.98116	−0.990578	−	0.136946i	$-0.956271\pi$
$509$	−44.6969	−1.98116	−0.990578	+	0.136946i	$0.956271\pi$
$510$	0	0
$511$	−5.39388	−0.238611
$512$	1.00000	0.0441942
$513$	4.00000	0.176604
$514$	2.00000	0.0882162
$515$	0	0
$516$	0.898979	0.0395754
$517$	0	0
$518$	29.3939	1.29149
$519$	−6.00000	−0.263371
$520$	0	0
$521$	28.2929	1.23953	0.619766	−	0.784786i	$-0.287227\pi$
$521$	28.2929	1.23953	0.619766	+	0.784786i	$0.287227\pi$
$522$	6.00000	0.262613
$523$	10.6969	0.467744	0.233872	−	0.972267i	$-0.424860\pi$
$523$	10.6969	0.467744	0.233872	+	0.972267i	$0.424860\pi$
$524$	−9.79796	−0.428026
$525$	0	0
$526$	−8.00000	−0.348817
$527$	4.00000	0.174243
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	−4.89898	−0.212598
$532$	−19.5959	−0.849591
$533$	20.0000	0.866296
$534$	11.7980	0.510548
$535$	0	0
$536$	8.89898	0.384377
$537$	4.89898	0.211407
$538$	−29.5959	−1.27597
$539$	0	0
$540$	0	0
$541$	11.7980	0.507234	0.253617	−	0.967305i	$-0.418380\pi$
$541$	11.7980	0.507234	0.253617	+	0.967305i	$0.418380\pi$
$542$	17.7980	0.764488
$543$	−23.7980	−1.02127
$544$	1.00000	0.0428746
$545$	0	0
$546$	−14.2020	−0.607791
$547$	−39.5959	−1.69300	−0.846500	−	0.532389i	$-0.821294\pi$
$547$	−39.5959	−1.69300	−0.846500	+	0.532389i	$0.821294\pi$
$548$	−14.0000	−0.598050
$549$	−7.79796	−0.332809
$550$	0	0
$551$	24.0000	1.02243
$552$	4.00000	0.170251
$553$	−67.5959	−2.87447
$554$	11.7980	0.501247
$555$	0	0
$556$	−5.79796	−0.245888
$557$	−0.202041	−0.00856075	−0.00428038	−	0.999991i	$-0.501362\pi$
$557$	−0.202041	−0.00856075	−0.00428038	+	0.999991i	$0.501362\pi$
$558$	4.00000	0.169334
$559$	2.60612	0.110227
$560$	0	0
$561$	0	0
$562$	8.20204	0.345982
$563$	−2.20204	−0.0928050	−0.0464025	−	0.998923i	$-0.514776\pi$
$563$	−2.20204	−0.0928050	−0.0464025	+	0.998923i	$0.514776\pi$
$564$	9.79796	0.412568
$565$	0	0
$566$	15.5959	0.655545
$567$	−4.89898	−0.205738
$568$	0.898979	0.0377203
$569$	34.0000	1.42535	0.712677	−	0.701492i	$-0.247483\pi$
$569$	34.0000	1.42535	0.712677	+	0.701492i	$0.247483\pi$
$570$	0	0
$571$	10.2020	0.426942	0.213471	−	0.976949i	$-0.431523\pi$
$571$	10.2020	0.426942	0.213471	+	0.976949i	$0.431523\pi$
$572$	0	0
$573$	9.79796	0.409316
$574$	−33.7980	−1.41070
$575$	0	0
$576$	1.00000	0.0416667
$577$	−31.3939	−1.30694	−0.653472	−	0.756951i	$-0.726688\pi$
$577$	−31.3939	−1.30694	−0.653472	+	0.756951i	$0.726688\pi$
$578$	1.00000	0.0415945
$579$	10.8990	0.452946
$580$	0	0
$581$	−28.4041	−1.17840
$582$	−16.6969	−0.692110
$583$	0	0
$584$	1.10102	0.0455606
$585$	0	0
$586$	25.5959	1.05736
$587$	−17.3939	−0.717922	−0.358961	−	0.933353i	$-0.616869\pi$
$587$	−17.3939	−0.717922	−0.358961	+	0.933353i	$0.616869\pi$
$588$	17.0000	0.701068
$589$	16.0000	0.659269
$590$	0	0
$591$	−25.5959	−1.05288
$592$	−6.00000	−0.246598
$593$	37.5959	1.54388	0.771940	−	0.635696i	$-0.219287\pi$
$593$	37.5959	1.54388	0.771940	+	0.635696i	$0.219287\pi$
$594$	0	0
$595$	0	0
$596$	−9.10102	−0.372792
$597$	−23.5959	−0.965717
$598$	11.5959	0.474192
$599$	−37.3939	−1.52787	−0.763936	−	0.645292i	$-0.776736\pi$
$599$	−37.3939	−1.52787	−0.763936	+	0.645292i	$0.776736\pi$
$600$	0	0
$601$	35.7980	1.46023	0.730115	−	0.683325i	$-0.239467\pi$
$601$	35.7980	1.46023	0.730115	+	0.683325i	$0.239467\pi$
$602$	−4.40408	−0.179497
$603$	8.89898	0.362394
$604$	9.79796	0.398673
$605$	0	0
$606$	−9.10102	−0.369704
$607$	−16.4949	−0.669507	−0.334754	−	0.942306i	$-0.608653\pi$
$607$	−16.4949	−0.669507	−0.334754	+	0.942306i	$0.608653\pi$
$608$	4.00000	0.162221
$609$	−29.3939	−1.19110
$610$	0	0
$611$	28.4041	1.14911
$612$	1.00000	0.0404226
$613$	−34.4949	−1.39324	−0.696618	−	0.717442i	$-0.745313\pi$
$613$	−34.4949	−1.39324	−0.696618	+	0.717442i	$0.745313\pi$
$614$	8.89898	0.359134
$615$	0	0
$616$	0	0
$617$	−1.59592	−0.0642492	−0.0321246	−	0.999484i	$-0.510227\pi$
$617$	−1.59592	−0.0642492	−0.0321246	+	0.999484i	$0.510227\pi$
$618$	−4.00000	−0.160904
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	0	0
$621$	4.00000	0.160514
$622$	−16.8990	−0.677587
$623$	−57.7980	−2.31563
$624$	2.89898	0.116052
$625$	0	0
$626$	2.89898	0.115867
$627$	0	0
$628$	10.8990	0.434917
$629$	−6.00000	−0.239236
$630$	0	0
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	13.7980	0.548853
$633$	−12.0000	−0.476957
$634$	−14.0000	−0.556011
$635$	0	0
$636$	−11.7980	−0.467820
$637$	49.2827	1.95265
$638$	0	0
$639$	0.898979	0.0355631
$640$	0	0
$641$	8.69694	0.343508	0.171754	−	0.985140i	$-0.445057\pi$
$641$	8.69694	0.343508	0.171754	+	0.985140i	$0.445057\pi$
$642$	13.7980	0.544562
$643$	−10.2020	−0.402329	−0.201165	−	0.979557i	$-0.564473\pi$
$643$	−10.2020	−0.402329	−0.201165	+	0.979557i	$0.564473\pi$
$644$	−19.5959	−0.772187
$645$	0	0
$646$	4.00000	0.157378
$647$	32.0000	1.25805	0.629025	−	0.777385i	$-0.283454\pi$
$647$	32.0000	1.25805	0.629025	+	0.777385i	$0.283454\pi$
$648$	1.00000	0.0392837
$649$	0	0
$650$	0	0
$651$	−19.5959	−0.768025
$652$	21.7980	0.853674
$653$	−30.0000	−1.17399	−0.586995	−	0.809590i	$-0.699689\pi$
$653$	−30.0000	−1.17399	−0.586995	+	0.809590i	$0.699689\pi$
$654$	−7.79796	−0.304924
$655$	0	0
$656$	6.89898	0.269360
$657$	1.10102	0.0429549
$658$	−48.0000	−1.87123
$659$	1.30306	0.0507601	0.0253800	−	0.999678i	$-0.491920\pi$
$659$	1.30306	0.0507601	0.0253800	+	0.999678i	$0.491920\pi$
$660$	0	0
$661$	−0.202041	−0.00785849	−0.00392924	−	0.999992i	$-0.501251\pi$
$661$	−0.202041	−0.00785849	−0.00392924	+	0.999992i	$0.501251\pi$
$662$	13.7980	0.536273
$663$	2.89898	0.112587
$664$	5.79796	0.225004
$665$	0	0
$666$	−6.00000	−0.232495
$667$	24.0000	0.929284
$668$	−21.7980	−0.843388
$669$	4.00000	0.154649
$670$	0	0
$671$	0	0
$672$	−4.89898	−0.188982
$673$	42.8990	1.65363	0.826817	−	0.562471i	$-0.190149\pi$
$673$	42.8990	1.65363	0.826817	+	0.562471i	$0.190149\pi$
$674$	−3.30306	−0.127229
$675$	0	0
$676$	−4.59592	−0.176766
$677$	−25.5959	−0.983731	−0.491866	−	0.870671i	$-0.663685\pi$
$677$	−25.5959	−0.983731	−0.491866	+	0.870671i	$0.663685\pi$
$678$	11.7980	0.453098
$679$	81.7980	3.13912
$680$	0	0
$681$	2.20204	0.0843824
$682$	0	0
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	4.00000	0.152944
$685$	0	0
$686$	−48.9898	−1.87044
$687$	25.5959	0.976545
$688$	0.898979	0.0342733
$689$	−34.2020	−1.30299
$690$	0	0
$691$	51.1918	1.94743	0.973715	−	0.227772i	$-0.0731439\pi$
$691$	51.1918	1.94743	0.973715	+	0.227772i	$0.0731439\pi$
$692$	−6.00000	−0.228086
$693$	0	0
$694$	−2.20204	−0.0835883
$695$	0	0
$696$	6.00000	0.227429
$697$	6.89898	0.261317
$698$	23.7980	0.900766
$699$	−14.0000	−0.529529
$700$	0	0
$701$	5.10102	0.192663	0.0963314	−	0.995349i	$-0.469289\pi$
$701$	5.10102	0.192663	0.0963314	+	0.995349i	$0.469289\pi$
$702$	2.89898	0.109415
$703$	−24.0000	−0.905177
$704$	0	0
$705$	0	0
$706$	26.0000	0.978523
$707$	44.5857	1.67682
$708$	−4.89898	−0.184115
$709$	−12.2020	−0.458257	−0.229129	−	0.973396i	$-0.573588\pi$
$709$	−12.2020	−0.458257	−0.229129	+	0.973396i	$0.573588\pi$
$710$	0	0
$711$	13.7980	0.517464
$712$	11.7980	0.442147
$713$	16.0000	0.599205
$714$	−4.89898	−0.183340
$715$	0	0
$716$	4.89898	0.183083
$717$	0	0
$718$	−21.3939	−0.798412
$719$	36.4949	1.36103	0.680515	−	0.732734i	$-0.261756\pi$
$719$	36.4949	1.36103	0.680515	+	0.732734i	$0.261756\pi$
$720$	0	0
$721$	19.5959	0.729790
$722$	−3.00000	−0.111648
$723$	−25.5959	−0.951922
$724$	−23.7980	−0.884444
$725$	0	0
$726$	−11.0000	−0.408248
$727$	−39.5959	−1.46853	−0.734266	−	0.678862i	$-0.762473\pi$
$727$	−39.5959	−1.46853	−0.734266	+	0.678862i	$0.762473\pi$
$728$	−14.2020	−0.526363
$729$	1.00000	0.0370370
$730$	0	0
$731$	0.898979	0.0332500
$732$	−7.79796	−0.288221
$733$	−2.49490	−0.0921511	−0.0460756	−	0.998938i	$-0.514672\pi$
$733$	−2.49490	−0.0921511	−0.0460756	+	0.998938i	$0.514672\pi$
$734$	36.8990	1.36197
$735$	0	0
$736$	4.00000	0.147442
$737$	0	0
$738$	6.89898	0.253955
$739$	49.3939	1.81698	0.908492	−	0.417903i	$-0.137235\pi$
$739$	49.3939	1.81698	0.908492	+	0.417903i	$0.137235\pi$
$740$	0	0
$741$	11.5959	0.425987
$742$	57.7980	2.12183
$743$	−49.3939	−1.81209	−0.906043	−	0.423186i	$-0.860912\pi$
$743$	−49.3939	−1.81209	−0.906043	+	0.423186i	$0.860912\pi$
$744$	4.00000	0.146647
$745$	0	0
$746$	4.69694	0.171967
$747$	5.79796	0.212136
$748$	0	0
$749$	−67.5959	−2.46990
$750$	0	0
$751$	2.20204	0.0803536	0.0401768	−	0.999193i	$-0.487208\pi$
$751$	2.20204	0.0803536	0.0401768	+	0.999193i	$0.487208\pi$
$752$	9.79796	0.357295
$753$	−4.89898	−0.178529
$754$	17.3939	0.633448
$755$	0	0
$756$	−4.89898	−0.178174
$757$	−24.6969	−0.897625	−0.448813	−	0.893626i	$-0.648153\pi$
$757$	−24.6969	−0.897625	−0.448813	+	0.893626i	$0.648153\pi$
$758$	−10.2020	−0.370555
$759$	0	0
$760$	0	0
$761$	37.5959	1.36285	0.681425	−	0.731888i	$-0.261360\pi$
$761$	37.5959	1.36285	0.681425	+	0.731888i	$0.261360\pi$
$762$	12.0000	0.434714
$763$	38.2020	1.38301
$764$	9.79796	0.354478
$765$	0	0
$766$	−8.00000	−0.289052
$767$	−14.2020	−0.512806
$768$	1.00000	0.0360844
$769$	26.0000	0.937584	0.468792	−	0.883309i	$-0.344689\pi$
$769$	26.0000	0.937584	0.468792	+	0.883309i	$0.344689\pi$
$770$	0	0
$771$	2.00000	0.0720282
$772$	10.8990	0.392263
$773$	−23.3939	−0.841419	−0.420710	−	0.907195i	$-0.638219\pi$
$773$	−23.3939	−0.841419	−0.420710	+	0.907195i	$0.638219\pi$
$774$	0.898979	0.0323132
$775$	0	0
$776$	−16.6969	−0.599385
$777$	29.3939	1.05450
$778$	10.4949	0.376260
$779$	27.5959	0.988726
$780$	0	0
$781$	0	0
$782$	4.00000	0.143040
$783$	6.00000	0.214423
$784$	17.0000	0.607143
$785$	0	0
$786$	−9.79796	−0.349482
$787$	26.2020	0.934002	0.467001	−	0.884257i	$-0.345334\pi$
$787$	26.2020	0.934002	0.467001	+	0.884257i	$0.345334\pi$
$788$	−25.5959	−0.911817
$789$	−8.00000	−0.284808
$790$	0	0
$791$	−57.7980	−2.05506
$792$	0	0
$793$	−22.6061	−0.802767
$794$	37.5959	1.33423
$795$	0	0
$796$	−23.5959	−0.836335
$797$	−26.0000	−0.920967	−0.460484	−	0.887668i	$-0.652324\pi$
$797$	−26.0000	−0.920967	−0.460484	+	0.887668i	$0.652324\pi$
$798$	−19.5959	−0.693688
$799$	9.79796	0.346627
$800$	0	0
$801$	11.7980	0.416860
$802$	13.1010	0.462613
$803$	0	0
$804$	8.89898	0.313843
$805$	0	0
$806$	11.5959	0.408449
$807$	−29.5959	−1.04183
$808$	−9.10102	−0.320173
$809$	3.30306	0.116129	0.0580647	−	0.998313i	$-0.481507\pi$
$809$	3.30306	0.116129	0.0580647	+	0.998313i	$0.481507\pi$
$810$	0	0
$811$	−33.3939	−1.17262	−0.586309	−	0.810088i	$-0.699419\pi$
$811$	−33.3939	−1.17262	−0.586309	+	0.810088i	$0.699419\pi$
$812$	−29.3939	−1.03152
$813$	17.7980	0.624202
$814$	0	0
$815$	0	0
$816$	1.00000	0.0350070
$817$	3.59592	0.125805
$818$	21.5959	0.755084
$819$	−14.2020	−0.496259
$820$	0	0
$821$	−3.79796	−0.132550	−0.0662748	−	0.997801i	$-0.521111\pi$
$821$	−3.79796	−0.132550	−0.0662748	+	0.997801i	$0.521111\pi$
$822$	−14.0000	−0.488306
$823$	−11.1010	−0.386957	−0.193479	−	0.981104i	$-0.561977\pi$
$823$	−11.1010	−0.386957	−0.193479	+	0.981104i	$0.561977\pi$
$824$	−4.00000	−0.139347
$825$	0	0
$826$	24.0000	0.835067
$827$	4.00000	0.139094	0.0695468	−	0.997579i	$-0.477845\pi$
$827$	4.00000	0.139094	0.0695468	+	0.997579i	$0.477845\pi$
$828$	4.00000	0.139010
$829$	−55.3939	−1.92391	−0.961954	−	0.273210i	$-0.911915\pi$
$829$	−55.3939	−1.92391	−0.961954	+	0.273210i	$0.911915\pi$
$830$	0	0
$831$	11.7980	0.409267
$832$	2.89898	0.100504
$833$	17.0000	0.589015
$834$	−5.79796	−0.200767
$835$	0	0
$836$	0	0
$837$	4.00000	0.138260
$838$	1.79796	0.0621095
$839$	−16.8990	−0.583418	−0.291709	−	0.956507i	$-0.594224\pi$
$839$	−16.8990	−0.583418	−0.291709	+	0.956507i	$0.594224\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	14.0000	0.482472
$843$	8.20204	0.282493
$844$	−12.0000	−0.413057
$845$	0	0
$846$	9.79796	0.336861
$847$	53.8888	1.85164
$848$	−11.7980	−0.405144
$849$	15.5959	0.535251
$850$	0	0
$851$	−24.0000	−0.822709
$852$	0.898979	0.0307985
$853$	47.3939	1.62274	0.811368	−	0.584536i	$-0.198723\pi$
$853$	47.3939	1.62274	0.811368	+	0.584536i	$0.198723\pi$
$854$	38.2020	1.30725
$855$	0	0
$856$	13.7980	0.471605
$857$	10.0000	0.341593	0.170797	−	0.985306i	$-0.445366\pi$
$857$	10.0000	0.341593	0.170797	+	0.985306i	$0.445366\pi$
$858$	0	0
$859$	−13.7980	−0.470780	−0.235390	−	0.971901i	$-0.575637\pi$
$859$	−13.7980	−0.470780	−0.235390	+	0.971901i	$0.575637\pi$
$860$	0	0
$861$	−33.7980	−1.15183
$862$	−32.8990	−1.12054
$863$	−13.3939	−0.455933	−0.227966	−	0.973669i	$-0.573208\pi$
$863$	−13.3939	−0.455933	−0.227966	+	0.973669i	$0.573208\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	−23.3939	−0.794956
$867$	1.00000	0.0339618
$868$	−19.5959	−0.665129
$869$	0	0
$870$	0	0
$871$	25.7980	0.874130
$872$	−7.79796	−0.264072
$873$	−16.6969	−0.565106
$874$	16.0000	0.541208
$875$	0	0
$876$	1.10102	0.0372000
$877$	−27.3939	−0.925025	−0.462513	−	0.886613i	$-0.653052\pi$
$877$	−27.3939	−0.925025	−0.462513	+	0.886613i	$0.653052\pi$
$878$	−13.7980	−0.465659
$879$	25.5959	0.863329
$880$	0	0
$881$	−14.4949	−0.488346	−0.244173	−	0.969732i	$-0.578516\pi$
$881$	−14.4949	−0.488346	−0.244173	+	0.969732i	$0.578516\pi$
$882$	17.0000	0.572420
$883$	2.69694	0.0907592	0.0453796	−	0.998970i	$-0.485550\pi$
$883$	2.69694	0.0907592	0.0453796	+	0.998970i	$0.485550\pi$
$884$	2.89898	0.0975032
$885$	0	0
$886$	−18.2020	−0.611510
$887$	12.0000	0.402921	0.201460	−	0.979497i	$-0.435431\pi$
$887$	12.0000	0.402921	0.201460	+	0.979497i	$0.435431\pi$
$888$	−6.00000	−0.201347
$889$	−58.7878	−1.97168
$890$	0	0
$891$	0	0
$892$	4.00000	0.133930
$893$	39.1918	1.31150
$894$	−9.10102	−0.304384
$895$	0	0
$896$	−4.89898	−0.163663
$897$	11.5959	0.387176
$898$	−2.89898	−0.0967402
$899$	24.0000	0.800445
$900$	0	0
$901$	−11.7980	−0.393047
$902$	0	0
$903$	−4.40408	−0.146559
$904$	11.7980	0.392394
$905$	0	0
$906$	9.79796	0.325515
$907$	−25.3939	−0.843190	−0.421595	−	0.906784i	$-0.638530\pi$
$907$	−25.3939	−0.843190	−0.421595	+	0.906784i	$0.638530\pi$
$908$	2.20204	0.0730773
$909$	−9.10102	−0.301862
$910$	0	0
$911$	32.8990	1.08999	0.544996	−	0.838439i	$-0.316531\pi$
$911$	32.8990	1.08999	0.544996	+	0.838439i	$0.316531\pi$
$912$	4.00000	0.132453
$913$	0	0
$914$	−35.7980	−1.18409
$915$	0	0
$916$	25.5959	0.845713
$917$	48.0000	1.58510
$918$	1.00000	0.0330049
$919$	−17.7980	−0.587100	−0.293550	−	0.955944i	$-0.594837\pi$
$919$	−17.7980	−0.587100	−0.293550	+	0.955944i	$0.594837\pi$
$920$	0	0
$921$	8.89898	0.293231
$922$	0.696938	0.0229524
$923$	2.60612	0.0857816
$924$	0	0
$925$	0	0
$926$	31.5959	1.03831
$927$	−4.00000	−0.131377
$928$	6.00000	0.196960
$929$	−32.2929	−1.05949	−0.529747	−	0.848156i	$-0.677713\pi$
$929$	−32.2929	−1.05949	−0.529747	+	0.848156i	$0.677713\pi$
$930$	0	0
$931$	68.0000	2.22861
$932$	−14.0000	−0.458585
$933$	−16.8990	−0.553248
$934$	−31.5959	−1.03385
$935$	0	0
$936$	2.89898	0.0947561
$937$	19.3939	0.633570	0.316785	−	0.948497i	$-0.397397\pi$
$937$	19.3939	0.633570	0.316785	+	0.948497i	$0.397397\pi$
$938$	−43.5959	−1.42346
$939$	2.89898	0.0946046
$940$	0	0
$941$	−35.7980	−1.16698	−0.583490	−	0.812120i	$-0.698313\pi$
$941$	−35.7980	−1.16698	−0.583490	+	0.812120i	$0.698313\pi$
$942$	10.8990	0.355108
$943$	27.5959	0.898647
$944$	−4.89898	−0.159448
$945$	0	0
$946$	0	0
$947$	−2.20204	−0.0715567	−0.0357784	−	0.999360i	$-0.511391\pi$
$947$	−2.20204	−0.0715567	−0.0357784	+	0.999360i	$0.511391\pi$
$948$	13.7980	0.448137
$949$	3.19184	0.103611
$950$	0	0
$951$	−14.0000	−0.453981
$952$	−4.89898	−0.158777
$953$	−37.1918	−1.20476	−0.602381	−	0.798209i	$-0.705781\pi$
$953$	−37.1918	−1.20476	−0.602381	+	0.798209i	$0.705781\pi$
$954$	−11.7980	−0.381973
$955$	0	0
$956$	0	0
$957$	0	0
$958$	−23.1010	−0.746360
$959$	68.5857	2.21475
$960$	0	0
$961$	−15.0000	−0.483871
$962$	−17.3939	−0.560801
$963$	13.7980	0.444633
$964$	−25.5959	−0.824389
$965$	0	0
$966$	−19.5959	−0.630488
$967$	−17.3939	−0.559349	−0.279675	−	0.960095i	$-0.590227\pi$
$967$	−17.3939	−0.559349	−0.279675	+	0.960095i	$0.590227\pi$
$968$	−11.0000	−0.353553
$969$	4.00000	0.128499
$970$	0	0
$971$	9.30306	0.298549	0.149275	−	0.988796i	$-0.452306\pi$
$971$	9.30306	0.298549	0.149275	+	0.988796i	$0.452306\pi$
$972$	1.00000	0.0320750
$973$	28.4041	0.910593
$974$	11.1010	0.355700
$975$	0	0
$976$	−7.79796	−0.249607
$977$	−9.59592	−0.307001	−0.153500	−	0.988149i	$-0.549055\pi$
$977$	−9.59592	−0.307001	−0.153500	+	0.988149i	$0.549055\pi$
$978$	21.7980	0.697022
$979$	0	0
$980$	0	0
$981$	−7.79796	−0.248970
$982$	−30.6969	−0.979579
$983$	49.3939	1.57542	0.787710	−	0.616046i	$-0.211267\pi$
$983$	49.3939	1.57542	0.787710	+	0.616046i	$0.211267\pi$
$984$	6.89898	0.219931
$985$	0	0
$986$	6.00000	0.191079
$987$	−48.0000	−1.52786
$988$	11.5959	0.368915
$989$	3.59592	0.114344
$990$	0	0
$991$	23.5959	0.749549	0.374775	−	0.927116i	$-0.377720\pi$
$991$	23.5959	0.749549	0.374775	+	0.927116i	$0.377720\pi$
$992$	4.00000	0.127000
$993$	13.7980	0.437865
$994$	−4.40408	−0.139689
$995$	0	0
$996$	5.79796	0.183715
$997$	−17.5959	−0.557268	−0.278634	−	0.960397i	$-0.589882\pi$
$997$	−17.5959	−0.557268	−0.278634	+	0.960397i	$0.589882\pi$
$998$	−0.404082	−0.0127910
$999$	−6.00000	−0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2550.2.a.bl.1.1		2
3.2	odd	2		7650.2.a.cu.1.1		2
5.2	odd	4		2550.2.d.u.2449.3		4
5.3	odd	4		2550.2.d.u.2449.2		4
5.4	even	2		510.2.a.h.1.2	✓	2
15.14	odd	2		1530.2.a.s.1.2		2
20.19	odd	2		4080.2.a.bq.1.1		2
85.84	even	2		8670.2.a.be.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
510.2.a.h.1.2	✓	2	5.4	even	2
1530.2.a.s.1.2		2	15.14	odd	2
2550.2.a.bl.1.1		2	1.1	even	1	trivial
2550.2.d.u.2449.2		4	5.3	odd	4
2550.2.d.u.2449.3		4	5.2	odd	4
4080.2.a.bq.1.1		2	20.19	odd	2
7650.2.a.cu.1.1		2	3.2	odd	2
8670.2.a.be.1.1		2	85.84	even	2

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:	\( N \)	\(=\)	\( 2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2550.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -4.89898 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -4.89898 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{12} +2.89898 q^{13} -4.89898 q^{14} +1.00000 q^{16} +1.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} -4.89898 q^{21} +4.00000 q^{23} +1.00000 q^{24} +2.89898 q^{26} +1.00000 q^{27} -4.89898 q^{28} +6.00000 q^{29} +4.00000 q^{31} +1.00000 q^{32} +1.00000 q^{34} +1.00000 q^{36} -6.00000 q^{37} +4.00000 q^{38} +2.89898 q^{39} +6.89898 q^{41} -4.89898 q^{42} +0.898979 q^{43} +4.00000 q^{46} +9.79796 q^{47} +1.00000 q^{48} +17.0000 q^{49} +1.00000 q^{51} +2.89898 q^{52} -11.7980 q^{53} +1.00000 q^{54} -4.89898 q^{56} +4.00000 q^{57} +6.00000 q^{58} -4.89898 q^{59} -7.79796 q^{61} +4.00000 q^{62} -4.89898 q^{63} +1.00000 q^{64} +8.89898 q^{67} +1.00000 q^{68} +4.00000 q^{69} +0.898979 q^{71} +1.00000 q^{72} +1.10102 q^{73} -6.00000 q^{74} +4.00000 q^{76} +2.89898 q^{78} +13.7980 q^{79} +1.00000 q^{81} +6.89898 q^{82} +5.79796 q^{83} -4.89898 q^{84} +0.898979 q^{86} +6.00000 q^{87} +11.7980 q^{89} -14.2020 q^{91} +4.00000 q^{92} +4.00000 q^{93} +9.79796 q^{94} +1.00000 q^{96} -16.6969 q^{97} +17.0000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 + 2 * q^8 + 2 * q^9	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 2 q^{8} + 2 q^{9} + 2 q^{12} - 4 q^{13} + 2 q^{16} + 2 q^{17} + 2 q^{18} + 8 q^{19} + 8 q^{23} + 2 q^{24} - 4 q^{26} + 2 q^{27} + 12 q^{29} + 8 q^{31} + 2 q^{32} + 2 q^{34} + 2 q^{36} - 12 q^{37} + 8 q^{38} - 4 q^{39} + 4 q^{41} - 8 q^{43} + 8 q^{46} + 2 q^{48} + 34 q^{49} + 2 q^{51} - 4 q^{52} - 4 q^{53} + 2 q^{54} + 8 q^{57} + 12 q^{58} + 4 q^{61} + 8 q^{62} + 2 q^{64} + 8 q^{67} + 2 q^{68} + 8 q^{69} - 8 q^{71} + 2 q^{72} + 12 q^{73} - 12 q^{74} + 8 q^{76} - 4 q^{78} + 8 q^{79} + 2 q^{81} + 4 q^{82} - 8 q^{83} - 8 q^{86} + 12 q^{87} + 4 q^{89} - 48 q^{91} + 8 q^{92} + 8 q^{93} + 2 q^{96} - 4 q^{97} + 34 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 + 2 * q^8 + 2 * q^9 + 2 * q^12 - 4 * q^13 + 2 * q^16 + 2 * q^17 + 2 * q^18 + 8 * q^19 + 8 * q^23 + 2 * q^24 - 4 * q^26 + 2 * q^27 + 12 * q^29 + 8 * q^31 + 2 * q^32 + 2 * q^34 + 2 * q^36 - 12 * q^37 + 8 * q^38 - 4 * q^39 + 4 * q^41 - 8 * q^43 + 8 * q^46 + 2 * q^48 + 34 * q^49 + 2 * q^51 - 4 * q^52 - 4 * q^53 + 2 * q^54 + 8 * q^57 + 12 * q^58 + 4 * q^61 + 8 * q^62 + 2 * q^64 + 8 * q^67 + 2 * q^68 + 8 * q^69 - 8 * q^71 + 2 * q^72 + 12 * q^73 - 12 * q^74 + 8 * q^76 - 4 * q^78 + 8 * q^79 + 2 * q^81 + 4 * q^82 - 8 * q^83 - 8 * q^86 + 12 * q^87 + 4 * q^89 - 48 * q^91 + 8 * q^92 + 8 * q^93 + 2 * q^96 - 4 * q^97 + 34 * q^98

Introduction

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