LMFDB - Embedded newform 2550.2.d.u.2449.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2550,2,Mod(2449,2550)]

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, 1, 0]))

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

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

chi := DirichletCharacter("2550.2449");

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.d (of order $2$, degree $1$, 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:	$20.3618525154$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 510)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2449.3
Root			$1.22474 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	2550.2449
Dual form			2550.2.d.u.2449.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2550\mathbb{Z}\right)^\times$.

$n$	$751$	$851$	$1327$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	− 4.89898i	− 1.85164i	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	− 4.89898i	− 1.85164i	0.377964	−	0.925820i	$-0.376624\pi$
$8$	− 1.00000i	− 0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	1.00000i	0.288675i
$13$	− 2.89898i	− 0.804032i	−0.915633	−	0.402016i	$-0.868310\pi$
$13$	− 2.89898i	− 0.804032i	0.915633	−	0.402016i	$-0.131690\pi$
$14$	4.89898	1.30931
$15$	0	0
$16$	1.00000	0.250000
$17$	1.00000i	0.242536i
$17$	1.00000i	0.242536i
$18$	− 1.00000i	− 0.235702i
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	−4.89898	−1.06904
$22$	0	0
$23$	− 4.00000i	− 0.834058i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$	− 4.00000i	− 0.834058i	0.908893	−	0.417029i	$-0.136929\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	2.89898	0.568537
$27$	1.00000i	0.192450i
$28$	4.89898i	0.925820i
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	−1.00000	−0.171499
$35$	0	0
$36$	1.00000	0.166667
$37$	− 6.00000i	− 0.986394i	−0.869918	−	0.493197i	$-0.835828\pi$
$37$	− 6.00000i	− 0.986394i	0.869918	−	0.493197i	$-0.164172\pi$
$38$	− 4.00000i	− 0.648886i
$39$	−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	$0.318908\pi$
$42$	− 4.89898i	− 0.755929i
$43$	− 0.898979i	− 0.137093i	−0.997648	−	0.0685465i	$-0.978164\pi$
$43$	− 0.898979i	− 0.137093i	0.997648	−	0.0685465i	$-0.0218362\pi$
$44$	0	0
$45$	0	0
$46$	4.00000	0.589768
$47$	9.79796i	1.42918i	0.699544	+	0.714590i	$0.253387\pi$
$47$	9.79796i	1.42918i	−0.699544	+	0.714590i	$0.746613\pi$
$48$	− 1.00000i	− 0.144338i
$49$	−17.0000	−2.42857
$50$	0	0
$51$	1.00000	0.140028
$52$	2.89898i	0.402016i
$53$	11.7980i	1.62057i	0.586033	+	0.810287i	$0.300689\pi$
$53$	11.7980i	1.62057i	−0.586033	+	0.810287i	$0.699311\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	−4.89898	−0.654654
$57$	4.00000i	0.529813i
$58$	− 6.00000i	− 0.787839i
$59$	4.89898	0.637793	0.318896	−	0.947790i	$-0.396688\pi$
$59$	4.89898	0.637793	0.318896	+	0.947790i	$0.396688\pi$
$60$	0	0
$61$	−7.79796	−0.998426	−0.499213	−	0.866479i	$-0.666378\pi$
$61$	−7.79796	−0.998426	−0.499213	+	0.866479i	$0.666378\pi$
$62$	4.00000i	0.508001i
$63$	4.89898i	0.617213i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	8.89898i	1.08718i	0.839350	+	0.543592i	$0.182936\pi$
$67$	8.89898i	1.08718i	−0.839350	+	0.543592i	$0.817064\pi$
$68$	− 1.00000i	− 0.121268i
$69$	−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	$0.483012\pi$
$72$	1.00000i	0.117851i
$73$	− 1.10102i	− 0.128865i	−0.997922	−	0.0644324i	$-0.979476\pi$
$73$	− 1.10102i	− 0.128865i	0.997922	−	0.0644324i	$-0.0205237\pi$
$74$	6.00000	0.697486
$75$	0	0
$76$	4.00000	0.458831
$77$	0	0
$78$	− 2.89898i	− 0.328245i
$79$	−13.7980	−1.55239	−0.776196	−	0.630492i	$-0.782853\pi$
$79$	−13.7980	−1.55239	−0.776196	+	0.630492i	$0.782853\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	6.89898i	0.761865i
$83$	− 5.79796i	− 0.636409i	−0.948022	−	0.318204i	$-0.896920\pi$
$83$	− 5.79796i	− 0.636409i	0.948022	−	0.318204i	$-0.103080\pi$
$84$	4.89898	0.534522
$85$	0	0
$86$	0.898979	0.0969395
$87$	6.00000i	0.643268i
$88$	0	0
$89$	−11.7980	−1.25058	−0.625291	−	0.780392i	$-0.715020\pi$
$89$	−11.7980	−1.25058	−0.625291	+	0.780392i	$0.715020\pi$
$90$	0	0
$91$	−14.2020	−1.48878
$92$	4.00000i	0.417029i
$93$	− 4.00000i	− 0.414781i
$94$	−9.79796	−1.01058
$95$	0	0
$96$	1.00000	0.102062
$97$	− 16.6969i	− 1.69532i	−0.530542	−	0.847659i	$-0.678012\pi$
$97$	− 16.6969i	− 1.69532i	0.530542	−	0.847659i	$-0.321988\pi$
$98$	− 17.0000i	− 1.71726i
$99$	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	$0.649572\pi$
$102$	1.00000i	0.0990148i
$103$	4.00000i	0.394132i	0.980390	+	0.197066i	$0.0631413\pi$
$103$	4.00000i	0.394132i	−0.980390	+	0.197066i	$0.936859\pi$
$104$	−2.89898	−0.284268
$105$	0	0
$106$	−11.7980	−1.14592
$107$	13.7980i	1.33390i	0.745103	+	0.666950i	$0.232400\pi$
$107$	13.7980i	1.33390i	−0.745103	+	0.666950i	$0.767600\pi$
$108$	− 1.00000i	− 0.0962250i
$109$	7.79796	0.746909	0.373455	−	0.927648i	$-0.378173\pi$
$109$	7.79796	0.746909	0.373455	+	0.927648i	$0.378173\pi$
$110$	0	0
$111$	−6.00000	−0.569495
$112$	− 4.89898i	− 0.462910i
$113$	− 11.7980i	− 1.10986i	−0.831898	−	0.554929i	$-0.812745\pi$
$113$	− 11.7980i	− 1.10986i	0.831898	−	0.554929i	$-0.187255\pi$
$114$	−4.00000	−0.374634
$115$	0	0
$116$	6.00000	0.557086
$117$	2.89898i	0.268011i
$118$	4.89898i	0.450988i
$119$	4.89898	0.449089
$120$	0	0
$121$	−11.0000	−1.00000
$122$	− 7.79796i	− 0.705994i
$123$	− 6.89898i	− 0.622060i
$124$	−4.00000	−0.359211
$125$	0	0
$126$	−4.89898	−0.436436
$127$	12.0000i	1.06483i	0.846484	+	0.532414i	$0.178715\pi$
$127$	12.0000i	1.06483i	−0.846484	+	0.532414i	$0.821285\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	−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
$133$	19.5959i	1.69918i
$134$	−8.89898	−0.768755
$135$	0	0
$136$	1.00000	0.0857493
$137$	− 14.0000i	− 1.19610i	−0.801459	−	0.598050i	$-0.795942\pi$
$137$	− 14.0000i	− 1.19610i	0.801459	−	0.598050i	$-0.204058\pi$
$138$	− 4.00000i	− 0.340503i
$139$	5.79796	0.491776	0.245888	−	0.969298i	$-0.420920\pi$
$139$	5.79796	0.491776	0.245888	+	0.969298i	$0.420920\pi$
$140$	0	0
$141$	9.79796	0.825137
$142$	0.898979i	0.0754407i
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	1.10102	0.0911211
$147$	17.0000i	1.40214i
$148$	6.00000i	0.493197i
$149$	9.10102	0.745585	0.372792	−	0.927915i	$-0.378400\pi$
$149$	9.10102	0.745585	0.372792	+	0.927915i	$0.378400\pi$
$150$	0	0
$151$	9.79796	0.797347	0.398673	−	0.917093i	$-0.369471\pi$
$151$	9.79796	0.797347	0.398673	+	0.917093i	$0.369471\pi$
$152$	4.00000i	0.324443i
$153$	− 1.00000i	− 0.0808452i
$154$	0	0
$155$	0	0
$156$	2.89898	0.232104
$157$	10.8990i	0.869833i	0.900471	+	0.434917i	$0.143222\pi$
$157$	10.8990i	0.869833i	−0.900471	+	0.434917i	$0.856778\pi$
$158$	− 13.7980i	− 1.09771i
$159$	11.7980	0.935639
$160$	0	0
$161$	−19.5959	−1.54437
$162$	1.00000i	0.0785674i
$163$	− 21.7980i	− 1.70735i	−0.520808	−	0.853674i	$-0.674369\pi$
$163$	− 21.7980i	− 1.70735i	0.520808	−	0.853674i	$-0.325631\pi$
$164$	−6.89898	−0.538720
$165$	0	0
$166$	5.79796	0.450009
$167$	− 21.7980i	− 1.68678i	−0.537304	−	0.843388i	$-0.680557\pi$
$167$	− 21.7980i	− 1.68678i	0.537304	−	0.843388i	$-0.319443\pi$
$168$	4.89898i	0.377964i
$169$	4.59592	0.353532
$170$	0	0
$171$	4.00000	0.305888
$172$	0.898979i	0.0685465i
$173$	6.00000i	0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$	6.00000i	0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$	−6.00000	−0.454859
$175$	0	0
$176$	0	0
$177$	− 4.89898i	− 0.368230i
$178$	− 11.7980i	− 0.884294i
$179$	−4.89898	−0.366167	−0.183083	−	0.983097i	$-0.558608\pi$
$179$	−4.89898	−0.366167	−0.183083	+	0.983097i	$0.558608\pi$
$180$	0	0
$181$	−23.7980	−1.76889	−0.884444	−	0.466646i	$-0.845462\pi$
$181$	−23.7980	−1.76889	−0.884444	+	0.466646i	$0.845462\pi$
$182$	− 14.2020i	− 1.05273i
$183$	7.79796i	0.576442i
$184$	−4.00000	−0.294884
$185$	0	0
$186$	4.00000	0.293294
$187$	0	0
$188$	− 9.79796i	− 0.714590i
$189$	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	$0.384659\pi$
$192$	1.00000i	0.0721688i
$193$	− 10.8990i	− 0.784526i	−0.919853	−	0.392263i	$-0.871692\pi$
$193$	− 10.8990i	− 0.784526i	0.919853	−	0.392263i	$-0.128308\pi$
$194$	16.6969	1.19877
$195$	0	0
$196$	17.0000	1.21429
$197$	− 25.5959i	− 1.82363i	−0.410597	−	0.911817i	$-0.634680\pi$
$197$	− 25.5959i	− 1.82363i	0.410597	−	0.911817i	$-0.365320\pi$
$198$	0	0
$199$	23.5959	1.67267	0.836335	−	0.548219i	$-0.184694\pi$
$199$	23.5959	1.67267	0.836335	+	0.548219i	$0.184694\pi$
$200$	0	0
$201$	8.89898	0.627686
$202$	− 9.10102i	− 0.640346i
$203$	29.3939i	2.06305i
$204$	−1.00000	−0.0700140
$205$	0	0
$206$	−4.00000	−0.278693
$207$	4.00000i	0.278019i
$208$	− 2.89898i	− 0.201008i
$209$	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	$0.635539\pi$
$212$	− 11.7980i	− 0.810287i
$213$	− 0.898979i	− 0.0615971i
$214$	−13.7980	−0.943209
$215$	0	0
$216$	1.00000	0.0680414
$217$	− 19.5959i	− 1.33026i
$218$	7.79796i	0.528144i
$219$	−1.10102	−0.0744001
$220$	0	0
$221$	2.89898	0.195006
$222$	− 6.00000i	− 0.402694i
$223$	− 4.00000i	− 0.267860i	−0.990991	−	0.133930i	$-0.957240\pi$
$223$	− 4.00000i	− 0.267860i	0.990991	−	0.133930i	$-0.0427597\pi$
$224$	4.89898	0.327327
$225$	0	0
$226$	11.7980	0.784789
$227$	2.20204i	0.146155i	0.997326	+	0.0730773i	$0.0232820\pi$
$227$	2.20204i	0.146155i	−0.997326	+	0.0730773i	$0.976718\pi$
$228$	− 4.00000i	− 0.264906i
$229$	−25.5959	−1.69143	−0.845713	−	0.533638i	$-0.820824\pi$
$229$	−25.5959	−1.69143	−0.845713	+	0.533638i	$0.820824\pi$
$230$	0	0
$231$	0	0
$232$	6.00000i	0.393919i
$233$	14.0000i	0.917170i	0.888650	+	0.458585i	$0.151644\pi$
$233$	14.0000i	0.917170i	−0.888650	+	0.458585i	$0.848356\pi$
$234$	−2.89898	−0.189512
$235$	0	0
$236$	−4.89898	−0.318896
$237$	13.7980i	0.896274i
$238$	4.89898i	0.317554i
$239$	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	$0.808481\pi$
$242$	− 11.0000i	− 0.707107i
$243$	− 1.00000i	− 0.0641500i
$244$	7.79796	0.499213
$245$	0	0
$246$	6.89898	0.439863
$247$	11.5959i	0.737831i
$248$	− 4.00000i	− 0.254000i
$249$	−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	$0.549412\pi$
$252$	− 4.89898i	− 0.308607i
$253$	0	0
$254$	−12.0000	−0.752947
$255$	0	0
$256$	1.00000	0.0625000
$257$	2.00000i	0.124757i	0.998053	+	0.0623783i	$0.0198685\pi$
$257$	2.00000i	0.124757i	−0.998053	+	0.0623783i	$0.980131\pi$
$258$	− 0.898979i	− 0.0559680i
$259$	−29.3939	−1.82645
$260$	0	0
$261$	6.00000	0.371391
$262$	− 9.79796i	− 0.605320i
$263$	8.00000i	0.493301i	0.969104	+	0.246651i	$0.0793300\pi$
$263$	8.00000i	0.493301i	−0.969104	+	0.246651i	$0.920670\pi$
$264$	0	0
$265$	0	0
$266$	−19.5959	−1.20150
$267$	11.7980i	0.722023i
$268$	− 8.89898i	− 0.543592i
$269$	29.5959	1.80449	0.902247	−	0.431219i	$-0.141916\pi$
$269$	29.5959	1.80449	0.902247	+	0.431219i	$0.141916\pi$
$270$	0	0
$271$	17.7980	1.08115	0.540575	−	0.841296i	$-0.318207\pi$
$271$	17.7980	1.08115	0.540575	+	0.841296i	$0.318207\pi$
$272$	1.00000i	0.0606339i
$273$	14.2020i	0.859547i
$274$	14.0000	0.845771
$275$	0	0
$276$	4.00000	0.240772
$277$	11.7980i	0.708871i	0.935081	+	0.354435i	$0.115327\pi$
$277$	11.7980i	0.708871i	−0.935081	+	0.354435i	$0.884673\pi$
$278$	5.79796i	0.347738i
$279$	−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	$0.421328\pi$
$282$	9.79796i	0.583460i
$283$	− 15.5959i	− 0.927081i	−0.886076	−	0.463541i	$-0.846579\pi$
$283$	− 15.5959i	− 0.927081i	0.886076	−	0.463541i	$-0.153421\pi$
$284$	−0.898979	−0.0533446
$285$	0	0
$286$	0	0
$287$	− 33.7980i	− 1.99503i
$288$	− 1.00000i	− 0.0589256i
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−16.6969	−0.978792
$292$	1.10102i	0.0644324i
$293$	− 25.5959i	− 1.49533i	−0.664076	−	0.747665i	$-0.731175\pi$
$293$	− 25.5959i	− 1.49533i	0.664076	−	0.747665i	$-0.268825\pi$
$294$	−17.0000	−0.991460
$295$	0	0
$296$	−6.00000	−0.348743
$297$	0	0
$298$	9.10102i	0.527208i
$299$	−11.5959	−0.670609
$300$	0	0
$301$	−4.40408	−0.253847
$302$	9.79796i	0.563809i
$303$	9.10102i	0.522840i
$304$	−4.00000	−0.229416
$305$	0	0
$306$	1.00000	0.0571662
$307$	8.89898i	0.507892i	0.967218	+	0.253946i	$0.0817285\pi$
$307$	8.89898i	0.507892i	−0.967218	+	0.253946i	$0.918272\pi$
$308$	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	$0.659047\pi$
$312$	2.89898i	0.164122i
$313$	− 2.89898i	− 0.163860i	−0.996638	−	0.0819300i	$-0.973892\pi$
$313$	− 2.89898i	− 0.163860i	0.996638	−	0.0819300i	$-0.0261084\pi$
$314$	−10.8990	−0.615065
$315$	0	0
$316$	13.7980	0.776196
$317$	− 14.0000i	− 0.786318i	−0.919470	−	0.393159i	$-0.871382\pi$
$317$	− 14.0000i	− 0.786318i	0.919470	−	0.393159i	$-0.128618\pi$
$318$	11.7980i	0.661597i
$319$	0	0
$320$	0	0
$321$	13.7980	0.770127
$322$	− 19.5959i	− 1.09204i
$323$	− 4.00000i	− 0.222566i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	21.7980	1.20728
$327$	− 7.79796i	− 0.431228i
$328$	− 6.89898i	− 0.380932i
$329$	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	$0.376198\pi$
$332$	5.79796i	0.318204i
$333$	6.00000i	0.328798i
$334$	21.7980	1.19273
$335$	0	0
$336$	−4.89898	−0.267261
$337$	− 3.30306i	− 0.179929i	−0.995945	−	0.0899646i	$-0.971325\pi$
$337$	− 3.30306i	− 0.179929i	0.995945	−	0.0899646i	$-0.0286754\pi$
$338$	4.59592i	0.249985i
$339$	−11.7980	−0.640777
$340$	0	0
$341$	0	0
$342$	4.00000i	0.216295i
$343$	48.9898i	2.64520i
$344$	−0.898979	−0.0484697
$345$	0	0
$346$	−6.00000	−0.322562
$347$	− 2.20204i	− 0.118212i	−0.998252	−	0.0591059i	$-0.981175\pi$
$347$	− 2.20204i	− 0.118212i	0.998252	−	0.0591059i	$-0.0188250\pi$
$348$	− 6.00000i	− 0.321634i
$349$	−23.7980	−1.27388	−0.636938	−	0.770915i	$-0.719799\pi$
$349$	−23.7980	−1.27388	−0.636938	+	0.770915i	$0.719799\pi$
$350$	0	0
$351$	2.89898	0.154736
$352$	0	0
$353$	− 26.0000i	− 1.38384i	−0.721974	−	0.691920i	$-0.756765\pi$
$353$	− 26.0000i	− 1.38384i	0.721974	−	0.691920i	$-0.243235\pi$
$354$	4.89898	0.260378
$355$	0	0
$356$	11.7980	0.625291
$357$	− 4.89898i	− 0.259281i
$358$	− 4.89898i	− 0.258919i
$359$	21.3939	1.12913	0.564563	−	0.825390i	$-0.309045\pi$
$359$	21.3939	1.12913	0.564563	+	0.825390i	$0.309045\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	− 23.7980i	− 1.25079i
$363$	11.0000i	0.577350i
$364$	14.2020	0.744389
$365$	0	0
$366$	−7.79796	−0.407606
$367$	36.8990i	1.92611i	0.269303	+	0.963056i	$0.413207\pi$
$367$	36.8990i	1.92611i	−0.269303	+	0.963056i	$0.586793\pi$
$368$	− 4.00000i	− 0.208514i
$369$	−6.89898	−0.359147
$370$	0	0
$371$	57.7980	3.00072
$372$	4.00000i	0.207390i
$373$	− 4.69694i	− 0.243198i	−0.992579	−	0.121599i	$-0.961198\pi$
$373$	− 4.69694i	− 0.243198i	0.992579	−	0.121599i	$-0.0388022\pi$
$374$	0	0
$375$	0	0
$376$	9.79796	0.505291
$377$	17.3939i	0.895830i
$378$	4.89898i	0.251976i
$379$	10.2020	0.524044	0.262022	−	0.965062i	$-0.415611\pi$
$379$	10.2020	0.524044	0.262022	+	0.965062i	$0.415611\pi$
$380$	0	0
$381$	12.0000	0.614779
$382$	9.79796i	0.501307i
$383$	8.00000i	0.408781i	0.978889	+	0.204390i	$0.0655212\pi$
$383$	8.00000i	0.408781i	−0.978889	+	0.204390i	$0.934479\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	10.8990	0.554743
$387$	0.898979i	0.0456977i
$388$	16.6969i	0.847659i
$389$	−10.4949	−0.532112	−0.266056	−	0.963958i	$-0.585721\pi$
$389$	−10.4949	−0.532112	−0.266056	+	0.963958i	$0.585721\pi$
$390$	0	0
$391$	4.00000	0.202289
$392$	17.0000i	0.858630i
$393$	9.79796i	0.494242i
$394$	25.5959	1.28950
$395$	0	0
$396$	0	0
$397$	37.5959i	1.88689i	0.331536	+	0.943443i	$0.392433\pi$
$397$	37.5959i	1.88689i	−0.331536	+	0.943443i	$0.607567\pi$
$398$	23.5959i	1.18276i
$399$	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	$0.393923\pi$
$402$	8.89898i	0.443841i
$403$	− 11.5959i	− 0.577634i
$404$	9.10102	0.452793
$405$	0	0
$406$	−29.3939	−1.45879
$407$	0	0
$408$	− 1.00000i	− 0.0495074i
$409$	−21.5959	−1.06785	−0.533925	−	0.845532i	$-0.679283\pi$
$409$	−21.5959	−1.06785	−0.533925	+	0.845532i	$0.679283\pi$
$410$	0	0
$411$	−14.0000	−0.690569
$412$	− 4.00000i	− 0.197066i
$413$	− 24.0000i	− 1.18096i
$414$	−4.00000	−0.196589
$415$	0	0
$416$	2.89898	0.142134
$417$	− 5.79796i	− 0.283927i
$418$	0	0
$419$	−1.79796	−0.0878360	−0.0439180	−	0.999035i	$-0.513984\pi$
$419$	−1.79796	−0.0878360	−0.0439180	+	0.999035i	$0.513984\pi$
$420$	0	0
$421$	14.0000	0.682318	0.341159	−	0.940006i	$-0.389181\pi$
$421$	14.0000	0.682318	0.341159	+	0.940006i	$0.389181\pi$
$422$	− 12.0000i	− 0.584151i
$423$	− 9.79796i	− 0.476393i
$424$	11.7980	0.572960
$425$	0	0
$426$	0.898979	0.0435557
$427$	38.2020i	1.84873i
$428$	− 13.7980i	− 0.666950i
$429$	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	$0.791139\pi$
$432$	1.00000i	0.0481125i
$433$	23.3939i	1.12424i	0.827056	+	0.562119i	$0.190014\pi$
$433$	23.3939i	1.12424i	−0.827056	+	0.562119i	$0.809986\pi$
$434$	19.5959	0.940634
$435$	0	0
$436$	−7.79796	−0.373455
$437$	16.0000i	0.765384i
$438$	− 1.10102i	− 0.0526088i
$439$	13.7980	0.658541	0.329270	−	0.944236i	$-0.393197\pi$
$439$	13.7980	0.658541	0.329270	+	0.944236i	$0.393197\pi$
$440$	0	0
$441$	17.0000	0.809524
$442$	2.89898i	0.137890i
$443$	18.2020i	0.864805i	0.901681	+	0.432403i	$0.142334\pi$
$443$	18.2020i	0.864805i	−0.901681	+	0.432403i	$0.857666\pi$
$444$	6.00000	0.284747
$445$	0	0
$446$	4.00000	0.189405
$447$	− 9.10102i	− 0.430463i
$448$	4.89898i	0.231455i
$449$	2.89898	0.136811	0.0684057	−	0.997658i	$-0.478209\pi$
$449$	2.89898	0.136811	0.0684057	+	0.997658i	$0.478209\pi$
$450$	0	0
$451$	0	0
$452$	11.7980i	0.554929i
$453$	− 9.79796i	− 0.460348i
$454$	−2.20204	−0.103347
$455$	0	0
$456$	4.00000	0.187317
$457$	− 35.7980i	− 1.67456i	−0.546776	−	0.837279i	$-0.684145\pi$
$457$	− 35.7980i	− 1.67456i	0.546776	−	0.837279i	$-0.315855\pi$
$458$	− 25.5959i	− 1.19602i
$459$	−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
$463$	− 31.5959i	− 1.46839i	−0.678940	−	0.734193i	$-0.737561\pi$
$463$	− 31.5959i	− 1.46839i	0.678940	−	0.734193i	$-0.262439\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	−14.0000	−0.648537
$467$	− 31.5959i	− 1.46208i	−0.682332	−	0.731042i	$-0.739034\pi$
$467$	− 31.5959i	− 1.46208i	0.682332	−	0.731042i	$-0.260966\pi$
$468$	− 2.89898i	− 0.134005i
$469$	43.5959	2.01307
$470$	0	0
$471$	10.8990	0.502198
$472$	− 4.89898i	− 0.225494i
$473$	0	0
$474$	−13.7980	−0.633761
$475$	0	0
$476$	−4.89898	−0.224544
$477$	− 11.7980i	− 0.540191i
$478$	0	0
$479$	23.1010	1.05551	0.527756	−	0.849396i	$-0.323033\pi$
$479$	23.1010	1.05551	0.527756	+	0.849396i	$0.323033\pi$
$480$	0	0
$481$	−17.3939	−0.793093
$482$	− 25.5959i	− 1.16586i
$483$	19.5959i	0.891645i
$484$	11.0000	0.500000
$485$	0	0
$486$	1.00000	0.0453609
$487$	11.1010i	0.503035i	0.967853	+	0.251518i	$0.0809296\pi$
$487$	11.1010i	0.503035i	−0.967853	+	0.251518i	$0.919070\pi$
$488$	7.79796i	0.352997i
$489$	−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	$0.743564\pi$
$492$	6.89898i	0.311030i
$493$	− 6.00000i	− 0.270226i
$494$	−11.5959	−0.521725
$495$	0	0
$496$	4.00000	0.179605
$497$	− 4.40408i	− 0.197550i
$498$	− 5.79796i	− 0.259813i
$499$	0.404082	0.0180892	0.00904460	−	0.999959i	$-0.497121\pi$
$499$	0.404082	0.0180892	0.00904460	+	0.999959i	$0.497121\pi$
$500$	0	0
$501$	−21.7980	−0.973861
$502$	− 4.89898i	− 0.218652i
$503$	4.00000i	0.178351i	0.996016	+	0.0891756i	$0.0284232\pi$
$503$	4.00000i	0.178351i	−0.996016	+	0.0891756i	$0.971577\pi$
$504$	4.89898	0.218218
$505$	0	0
$506$	0	0
$507$	− 4.59592i	− 0.204112i
$508$	− 12.0000i	− 0.532414i
$509$	44.6969	1.98116	0.990578	−	0.136946i	$-0.0437288\pi$
$509$	44.6969	1.98116	0.990578	+	0.136946i	$0.0437288\pi$
$510$	0	0
$511$	−5.39388	−0.238611
$512$	1.00000i	0.0441942i
$513$	− 4.00000i	− 0.176604i
$514$	−2.00000	−0.0882162
$515$	0	0
$516$	0.898979	0.0395754
$517$	0	0
$518$	− 29.3939i	− 1.29149i
$519$	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	$0.287227\pi$
$522$	6.00000i	0.262613i
$523$	− 10.6969i	− 0.467744i	−0.972267	−	0.233872i	$-0.924860\pi$
$523$	− 10.6969i	− 0.467744i	0.972267	−	0.233872i	$-0.0751397\pi$
$524$	9.79796	0.428026
$525$	0	0
$526$	−8.00000	−0.348817
$527$	4.00000i	0.174243i
$528$	0	0
$529$	7.00000	0.304348
$530$	0	0
$531$	−4.89898	−0.212598
$532$	− 19.5959i	− 0.849591i
$533$	− 20.0000i	− 0.866296i
$534$	−11.7980	−0.510548
$535$	0	0
$536$	8.89898	0.384377
$537$	4.89898i	0.211407i
$538$	29.5959i	1.27597i
$539$	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	$0.418380\pi$
$542$	17.7980i	0.764488i
$543$	23.7980i	1.02127i
$544$	−1.00000	−0.0428746
$545$	0	0
$546$	−14.2020	−0.607791
$547$	− 39.5959i	− 1.69300i	−0.532389	−	0.846500i	$-0.678706\pi$
$547$	− 39.5959i	− 1.69300i	0.532389	−	0.846500i	$-0.321294\pi$
$548$	14.0000i	0.598050i
$549$	7.79796	0.332809
$550$	0	0
$551$	24.0000	1.02243
$552$	4.00000i	0.170251i
$553$	67.5959i	2.87447i
$554$	−11.7980	−0.501247
$555$	0	0
$556$	−5.79796	−0.245888
$557$	− 0.202041i	− 0.00856075i	−0.999991	−	0.00428038i	$-0.998638\pi$
$557$	− 0.202041i	− 0.00856075i	0.999991	−	0.00428038i	$-0.00136249\pi$
$558$	− 4.00000i	− 0.169334i
$559$	−2.60612	−0.110227
$560$	0	0
$561$	0	0
$562$	8.20204i	0.345982i
$563$	2.20204i	0.0928050i	0.998923	+	0.0464025i	$0.0147757\pi$
$563$	2.20204i	0.0928050i	−0.998923	+	0.0464025i	$0.985224\pi$
$564$	−9.79796	−0.412568
$565$	0	0
$566$	15.5959	0.655545
$567$	− 4.89898i	− 0.205738i
$568$	− 0.898979i	− 0.0377203i
$569$	−34.0000	−1.42535	−0.712677	−	0.701492i	$-0.752517\pi$
$569$	−34.0000	−1.42535	−0.712677	+	0.701492i	$0.752517\pi$
$570$	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
$573$	− 9.79796i	− 0.409316i
$574$	33.7980	1.41070
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 31.3939i	− 1.30694i	−0.756951	−	0.653472i	$-0.773312\pi$
$577$	− 31.3939i	− 1.30694i	0.756951	−	0.653472i	$-0.226688\pi$
$578$	− 1.00000i	− 0.0415945i
$579$	−10.8990	−0.452946
$580$	0	0
$581$	−28.4041	−1.17840
$582$	− 16.6969i	− 0.692110i
$583$	0	0
$584$	−1.10102	−0.0455606
$585$	0	0
$586$	25.5959	1.05736
$587$	− 17.3939i	− 0.717922i	−0.933353	−	0.358961i	$-0.883131\pi$
$587$	− 17.3939i	− 0.717922i	0.933353	−	0.358961i	$-0.116869\pi$
$588$	− 17.0000i	− 0.701068i
$589$	−16.0000	−0.659269
$590$	0	0
$591$	−25.5959	−1.05288
$592$	− 6.00000i	− 0.246598i
$593$	− 37.5959i	− 1.54388i	−0.635696	−	0.771940i	$-0.719287\pi$
$593$	− 37.5959i	− 1.54388i	0.635696	−	0.771940i	$-0.280713\pi$
$594$	0	0
$595$	0	0
$596$	−9.10102	−0.372792
$597$	− 23.5959i	− 0.965717i
$598$	− 11.5959i	− 0.474192i
$599$	37.3939	1.52787	0.763936	−	0.645292i	$-0.223264\pi$
$599$	37.3939	1.52787	0.763936	+	0.645292i	$0.223264\pi$
$600$	0	0
$601$	35.7980	1.46023	0.730115	−	0.683325i	$-0.239467\pi$
$601$	35.7980	1.46023	0.730115	+	0.683325i	$0.239467\pi$
$602$	− 4.40408i	− 0.179497i
$603$	− 8.89898i	− 0.362394i
$604$	−9.79796	−0.398673
$605$	0	0
$606$	−9.10102	−0.369704
$607$	− 16.4949i	− 0.669507i	−0.942306	−	0.334754i	$-0.891347\pi$
$607$	− 16.4949i	− 0.669507i	0.942306	−	0.334754i	$-0.108653\pi$
$608$	− 4.00000i	− 0.162221i
$609$	29.3939	1.19110
$610$	0	0
$611$	28.4041	1.14911
$612$	1.00000i	0.0404226i
$613$	34.4949i	1.39324i	0.717442	+	0.696618i	$0.245313\pi$
$613$	34.4949i	1.39324i	−0.717442	+	0.696618i	$0.754687\pi$
$614$	−8.89898	−0.359134
$615$	0	0
$616$	0	0
$617$	− 1.59592i	− 0.0642492i	−0.999484	−	0.0321246i	$-0.989773\pi$
$617$	− 1.59592i	− 0.0642492i	0.999484	−	0.0321246i	$-0.0102273\pi$
$618$	4.00000i	0.160904i
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	0	0
$621$	4.00000	0.160514
$622$	− 16.8990i	− 0.677587i
$623$	57.7980i	2.31563i
$624$	−2.89898	−0.116052
$625$	0	0
$626$	2.89898	0.115867
$627$	0	0
$628$	− 10.8990i	− 0.434917i
$629$	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	$0.793153\pi$
$632$	13.7980i	0.548853i
$633$	12.0000i	0.476957i
$634$	14.0000	0.556011
$635$	0	0
$636$	−11.7980	−0.467820
$637$	49.2827i	1.95265i
$638$	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	$0.445057\pi$
$642$	13.7980i	0.544562i
$643$	10.2020i	0.402329i	0.979557	+	0.201165i	$0.0644726\pi$
$643$	10.2020i	0.402329i	−0.979557	+	0.201165i	$0.935527\pi$
$644$	19.5959	0.772187
$645$	0	0
$646$	4.00000	0.157378
$647$	32.0000i	1.25805i	0.777385	+	0.629025i	$0.216546\pi$
$647$	32.0000i	1.25805i	−0.777385	+	0.629025i	$0.783454\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	0	0
$650$	0	0
$651$	−19.5959	−0.768025
$652$	21.7980i	0.853674i
$653$	30.0000i	1.17399i	0.809590	+	0.586995i	$0.199689\pi$
$653$	30.0000i	1.17399i	−0.809590	+	0.586995i	$0.800311\pi$
$654$	7.79796	0.304924
$655$	0	0
$656$	6.89898	0.269360
$657$	1.10102i	0.0429549i
$658$	48.0000i	1.87123i
$659$	−1.30306	−0.0507601	−0.0253800	−	0.999678i	$-0.508080\pi$
$659$	−1.30306	−0.0507601	−0.0253800	+	0.999678i	$0.508080\pi$
$660$	0	0
$661$	−0.202041	−0.00785849	−0.00392924	−	0.999992i	$-0.501251\pi$
$661$	−0.202041	−0.00785849	−0.00392924	+	0.999992i	$0.501251\pi$
$662$	13.7980i	0.536273i
$663$	− 2.89898i	− 0.112587i
$664$	−5.79796	−0.225004
$665$	0	0
$666$	−6.00000	−0.232495
$667$	24.0000i	0.929284i
$668$	21.7980i	0.843388i
$669$	−4.00000	−0.154649
$670$	0	0
$671$	0	0
$672$	− 4.89898i	− 0.188982i
$673$	− 42.8990i	− 1.65363i	−0.562471	−	0.826817i	$-0.690149\pi$
$673$	− 42.8990i	− 1.65363i	0.562471	−	0.826817i	$-0.309851\pi$
$674$	3.30306	0.127229
$675$	0	0
$676$	−4.59592	−0.176766
$677$	− 25.5959i	− 0.983731i	−0.870671	−	0.491866i	$-0.836315\pi$
$677$	− 25.5959i	− 0.983731i	0.870671	−	0.491866i	$-0.163685\pi$
$678$	− 11.7980i	− 0.453098i
$679$	−81.7980	−3.13912
$680$	0	0
$681$	2.20204	0.0843824
$682$	0	0
$683$	12.0000i	0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$683$	12.0000i	0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$684$	−4.00000	−0.152944
$685$	0	0
$686$	−48.9898	−1.87044
$687$	25.5959i	0.976545i
$688$	− 0.898979i	− 0.0342733i
$689$	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	$0.0731439\pi$
$692$	− 6.00000i	− 0.228086i
$693$	0	0
$694$	2.20204	0.0835883
$695$	0	0
$696$	6.00000	0.227429
$697$	6.89898i	0.261317i
$698$	− 23.7980i	− 0.900766i
$699$	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	$0.469289\pi$
$702$	2.89898i	0.109415i
$703$	24.0000i	0.905177i
$704$	0	0
$705$	0	0
$706$	26.0000	0.978523
$707$	44.5857i	1.67682i
$708$	4.89898i	0.184115i
$709$	12.2020	0.458257	0.229129	−	0.973396i	$-0.426412\pi$
$709$	12.2020	0.458257	0.229129	+	0.973396i	$0.426412\pi$
$710$	0	0
$711$	13.7980	0.517464
$712$	11.7980i	0.442147i
$713$	− 16.0000i	− 0.599205i
$714$	4.89898	0.183340
$715$	0	0
$716$	4.89898	0.183083
$717$	0	0
$718$	21.3939i	0.798412i
$719$	−36.4949	−1.36103	−0.680515	−	0.732734i	$-0.738244\pi$
$719$	−36.4949	−1.36103	−0.680515	+	0.732734i	$0.738244\pi$
$720$	0	0
$721$	19.5959	0.729790
$722$	− 3.00000i	− 0.111648i
$723$	25.5959i	0.951922i
$724$	23.7980	0.884444
$725$	0	0
$726$	−11.0000	−0.408248
$727$	− 39.5959i	− 1.46853i	−0.678862	−	0.734266i	$-0.737527\pi$
$727$	− 39.5959i	− 1.46853i	0.678862	−	0.734266i	$-0.262473\pi$
$728$	14.2020i	0.526363i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	0.898979	0.0332500
$732$	− 7.79796i	− 0.288221i
$733$	2.49490i	0.0921511i	0.998938	+	0.0460756i	$0.0146715\pi$
$733$	2.49490i	0.0921511i	−0.998938	+	0.0460756i	$0.985328\pi$
$734$	−36.8990	−1.36197
$735$	0	0
$736$	4.00000	0.147442
$737$	0	0
$738$	− 6.89898i	− 0.253955i
$739$	−49.3939	−1.81698	−0.908492	−	0.417903i	$-0.862765\pi$
$739$	−49.3939	−1.81698	−0.908492	+	0.417903i	$0.862765\pi$
$740$	0	0
$741$	11.5959	0.425987
$742$	57.7980i	2.12183i
$743$	49.3939i	1.81209i	0.423186	+	0.906043i	$0.360912\pi$
$743$	49.3939i	1.81209i	−0.423186	+	0.906043i	$0.639088\pi$
$744$	−4.00000	−0.146647
$745$	0	0
$746$	4.69694	0.171967
$747$	5.79796i	0.212136i
$748$	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	$0.487208\pi$
$752$	9.79796i	0.357295i
$753$	4.89898i	0.178529i
$754$	−17.3939	−0.633448
$755$	0	0
$756$	−4.89898	−0.178174
$757$	− 24.6969i	− 0.897625i	−0.893626	−	0.448813i	$-0.851847\pi$
$757$	− 24.6969i	− 0.897625i	0.893626	−	0.448813i	$-0.148153\pi$
$758$	10.2020i	0.370555i
$759$	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	$0.261360\pi$
$762$	12.0000i	0.434714i
$763$	− 38.2020i	− 1.38301i
$764$	−9.79796	−0.354478
$765$	0	0
$766$	−8.00000	−0.289052
$767$	− 14.2020i	− 0.512806i
$768$	− 1.00000i	− 0.0360844i
$769$	−26.0000	−0.937584	−0.468792	−	0.883309i	$-0.655311\pi$
$769$	−26.0000	−0.937584	−0.468792	+	0.883309i	$0.655311\pi$
$770$	0	0
$771$	2.00000	0.0720282
$772$	10.8990i	0.392263i
$773$	23.3939i	0.841419i	0.907195	+	0.420710i	$0.138219\pi$
$773$	23.3939i	0.841419i	−0.907195	+	0.420710i	$0.861781\pi$
$774$	−0.898979	−0.0323132
$775$	0	0
$776$	−16.6969	−0.599385
$777$	29.3939i	1.05450i
$778$	− 10.4949i	− 0.376260i
$779$	−27.5959	−0.988726
$780$	0	0
$781$	0	0
$782$	4.00000i	0.143040i
$783$	− 6.00000i	− 0.214423i
$784$	−17.0000	−0.607143
$785$	0	0
$786$	−9.79796	−0.349482
$787$	26.2020i	0.934002i	0.884257	+	0.467001i	$0.154666\pi$
$787$	26.2020i	0.934002i	−0.884257	+	0.467001i	$0.845334\pi$
$788$	25.5959i	0.911817i
$789$	8.00000	0.284808
$790$	0	0
$791$	−57.7980	−2.05506
$792$	0	0
$793$	22.6061i	0.802767i
$794$	−37.5959	−1.33423
$795$	0	0
$796$	−23.5959	−0.836335
$797$	− 26.0000i	− 0.920967i	−0.887668	−	0.460484i	$-0.847676\pi$
$797$	− 26.0000i	− 0.920967i	0.887668	−	0.460484i	$-0.152324\pi$
$798$	19.5959i	0.693688i
$799$	−9.79796	−0.346627
$800$	0	0
$801$	11.7980	0.416860
$802$	13.1010i	0.462613i
$803$	0	0
$804$	−8.89898	−0.313843
$805$	0	0
$806$	11.5959	0.408449
$807$	− 29.5959i	− 1.04183i
$808$	9.10102i	0.320173i
$809$	−3.30306	−0.116129	−0.0580647	−	0.998313i	$-0.518493\pi$
$809$	−3.30306	−0.116129	−0.0580647	+	0.998313i	$0.518493\pi$
$810$	0	0
$811$	−33.3939	−1.17262	−0.586309	−	0.810088i	$-0.699419\pi$
$811$	−33.3939	−1.17262	−0.586309	+	0.810088i	$0.699419\pi$
$812$	− 29.3939i	− 1.03152i
$813$	− 17.7980i	− 0.624202i
$814$	0	0
$815$	0	0
$816$	1.00000	0.0350070
$817$	3.59592i	0.125805i
$818$	− 21.5959i	− 0.755084i
$819$	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	$0.521111\pi$
$822$	− 14.0000i	− 0.488306i
$823$	11.1010i	0.386957i	0.981104	+	0.193479i	$0.0619770\pi$
$823$	11.1010i	0.386957i	−0.981104	+	0.193479i	$0.938023\pi$
$824$	4.00000	0.139347
$825$	0	0
$826$	24.0000	0.835067
$827$	4.00000i	0.139094i	0.997579	+	0.0695468i	$0.0221553\pi$
$827$	4.00000i	0.139094i	−0.997579	+	0.0695468i	$0.977845\pi$
$828$	− 4.00000i	− 0.139010i
$829$	55.3939	1.92391	0.961954	−	0.273210i	$-0.0880854\pi$
$829$	55.3939	1.92391	0.961954	+	0.273210i	$0.0880854\pi$
$830$	0	0
$831$	11.7980	0.409267
$832$	2.89898i	0.100504i
$833$	− 17.0000i	− 0.589015i
$834$	5.79796	0.200767
$835$	0	0
$836$	0	0
$837$	4.00000i	0.138260i
$838$	− 1.79796i	− 0.0621095i
$839$	16.8990	0.583418	0.291709	−	0.956507i	$-0.405776\pi$
$839$	16.8990	0.583418	0.291709	+	0.956507i	$0.405776\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	14.0000i	0.482472i
$843$	− 8.20204i	− 0.282493i
$844$	12.0000	0.413057
$845$	0	0
$846$	9.79796	0.336861
$847$	53.8888i	1.85164i
$848$	11.7980i	0.405144i
$849$	−15.5959	−0.535251
$850$	0	0
$851$	−24.0000	−0.822709
$852$	0.898979i	0.0307985i
$853$	− 47.3939i	− 1.62274i	−0.584536	−	0.811368i	$-0.698723\pi$
$853$	− 47.3939i	− 1.62274i	0.584536	−	0.811368i	$-0.301277\pi$
$854$	−38.2020	−1.30725
$855$	0	0
$856$	13.7980	0.471605
$857$	10.0000i	0.341593i	0.985306	+	0.170797i	$0.0546341\pi$
$857$	10.0000i	0.341593i	−0.985306	+	0.170797i	$0.945366\pi$
$858$	0	0
$859$	13.7980	0.470780	0.235390	−	0.971901i	$-0.424363\pi$
$859$	13.7980	0.470780	0.235390	+	0.971901i	$0.424363\pi$
$860$	0	0
$861$	−33.7980	−1.15183
$862$	− 32.8990i	− 1.12054i
$863$	13.3939i	0.455933i	0.973669	+	0.227966i	$0.0732076\pi$
$863$	13.3939i	0.455933i	−0.973669	+	0.227966i	$0.926792\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	−23.3939	−0.794956
$867$	1.00000i	0.0339618i
$868$	19.5959i	0.665129i
$869$	0	0
$870$	0	0
$871$	25.7980	0.874130
$872$	− 7.79796i	− 0.264072i
$873$	16.6969i	0.565106i
$874$	−16.0000	−0.541208
$875$	0	0
$876$	1.10102	0.0372000
$877$	− 27.3939i	− 0.925025i	−0.886613	−	0.462513i	$-0.846948\pi$
$877$	− 27.3939i	− 0.925025i	0.886613	−	0.462513i	$-0.153052\pi$
$878$	13.7980i	0.465659i
$879$	−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	$0.578516\pi$
$882$	17.0000i	0.572420i
$883$	− 2.69694i	− 0.0907592i	−0.998970	−	0.0453796i	$-0.985550\pi$
$883$	− 2.69694i	− 0.0907592i	0.998970	−	0.0453796i	$-0.0144497\pi$
$884$	−2.89898	−0.0975032
$885$	0	0
$886$	−18.2020	−0.611510
$887$	12.0000i	0.402921i	0.979497	+	0.201460i	$0.0645687\pi$
$887$	12.0000i	0.402921i	−0.979497	+	0.201460i	$0.935431\pi$
$888$	6.00000i	0.201347i
$889$	58.7878	1.97168
$890$	0	0
$891$	0	0
$892$	4.00000i	0.133930i
$893$	− 39.1918i	− 1.31150i
$894$	9.10102	0.304384
$895$	0	0
$896$	−4.89898	−0.163663
$897$	11.5959i	0.387176i
$898$	2.89898i	0.0967402i
$899$	−24.0000	−0.800445
$900$	0	0
$901$	−11.7980	−0.393047
$902$	0	0
$903$	4.40408i	0.146559i
$904$	−11.7980	−0.392394
$905$	0	0
$906$	9.79796	0.325515
$907$	− 25.3939i	− 0.843190i	−0.906784	−	0.421595i	$-0.861470\pi$
$907$	− 25.3939i	− 0.843190i	0.906784	−	0.421595i	$-0.138530\pi$
$908$	− 2.20204i	− 0.0730773i
$909$	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	$0.316531\pi$
$912$	4.00000i	0.132453i
$913$	0	0
$914$	35.7980	1.18409
$915$	0	0
$916$	25.5959	0.845713
$917$	48.0000i	1.58510i
$918$	− 1.00000i	− 0.0330049i
$919$	17.7980	0.587100	0.293550	−	0.955944i	$-0.405163\pi$
$919$	17.7980	0.587100	0.293550	+	0.955944i	$0.405163\pi$
$920$	0	0
$921$	8.89898	0.293231
$922$	0.696938i	0.0229524i
$923$	− 2.60612i	− 0.0857816i
$924$	0	0
$925$	0	0
$926$	31.5959	1.03831
$927$	− 4.00000i	− 0.131377i
$928$	− 6.00000i	− 0.196960i
$929$	32.2929	1.05949	0.529747	−	0.848156i	$-0.322287\pi$
$929$	32.2929	1.05949	0.529747	+	0.848156i	$0.322287\pi$
$930$	0	0
$931$	68.0000	2.22861
$932$	− 14.0000i	− 0.458585i
$933$	16.8990i	0.553248i
$934$	31.5959	1.03385
$935$	0	0
$936$	2.89898	0.0947561
$937$	19.3939i	0.633570i	0.948497	+	0.316785i	$0.102603\pi$
$937$	19.3939i	0.633570i	−0.948497	+	0.316785i	$0.897397\pi$
$938$	43.5959i	1.42346i
$939$	−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	$0.698313\pi$
$942$	10.8990i	0.355108i
$943$	− 27.5959i	− 0.898647i
$944$	4.89898	0.159448
$945$	0	0
$946$	0	0
$947$	− 2.20204i	− 0.0715567i	−0.999360	−	0.0357784i	$-0.988609\pi$
$947$	− 2.20204i	− 0.0715567i	0.999360	−	0.0357784i	$-0.0113910\pi$
$948$	− 13.7980i	− 0.448137i
$949$	−3.19184	−0.103611
$950$	0	0
$951$	−14.0000	−0.453981
$952$	− 4.89898i	− 0.158777i
$953$	37.1918i	1.20476i	0.798209	+	0.602381i	$0.205781\pi$
$953$	37.1918i	1.20476i	−0.798209	+	0.602381i	$0.794219\pi$
$954$	11.7980	0.381973
$955$	0	0
$956$	0	0
$957$	0	0
$958$	23.1010i	0.746360i
$959$	−68.5857	−2.21475
$960$	0	0
$961$	−15.0000	−0.483871
$962$	− 17.3939i	− 0.560801i
$963$	− 13.7980i	− 0.444633i
$964$	25.5959	0.824389
$965$	0	0
$966$	−19.5959	−0.630488
$967$	− 17.3939i	− 0.559349i	−0.960095	−	0.279675i	$-0.909773\pi$
$967$	− 17.3939i	− 0.559349i	0.960095	−	0.279675i	$-0.0902266\pi$
$968$	11.0000i	0.353553i
$969$	−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	$0.452306\pi$
$972$	1.00000i	0.0320750i
$973$	− 28.4041i	− 0.910593i
$974$	−11.1010	−0.355700
$975$	0	0
$976$	−7.79796	−0.249607
$977$	− 9.59592i	− 0.307001i	−0.988149	−	0.153500i	$-0.950945\pi$
$977$	− 9.59592i	− 0.307001i	0.988149	−	0.153500i	$-0.0490546\pi$
$978$	− 21.7980i	− 0.697022i
$979$	0	0
$980$	0	0
$981$	−7.79796	−0.248970
$982$	− 30.6969i	− 0.979579i
$983$	− 49.3939i	− 1.57542i	−0.616046	−	0.787710i	$-0.711267\pi$
$983$	− 49.3939i	− 1.57542i	0.616046	−	0.787710i	$-0.288733\pi$
$984$	−6.89898	−0.219931
$985$	0	0
$986$	6.00000	0.191079
$987$	− 48.0000i	− 1.52786i
$988$	− 11.5959i	− 0.368915i
$989$	−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	$0.377720\pi$
$992$	4.00000i	0.127000i
$993$	− 13.7980i	− 0.437865i
$994$	4.40408	0.139689
$995$	0	0
$996$	5.79796	0.183715
$997$	− 17.5959i	− 0.557268i	−0.960397	−	0.278634i	$-0.910118\pi$
$997$	− 17.5959i	− 0.557268i	0.960397	−	0.278634i	$-0.0898817\pi$
$998$	0.404082i	0.0127910i
$999$	6.00000	0.189832

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
510.2.a.h.1.2	✓	2	5.2	odd	4
1530.2.a.s.1.2		2	15.2	even	4
2550.2.a.bl.1.1		2	5.3	odd	4
2550.2.d.u.2449.2		4	5.4	even	2	inner
2550.2.d.u.2449.3		4	1.1	even	1	trivial
4080.2.a.bq.1.1		2	20.7	even	4
7650.2.a.cu.1.1		2	15.8	even	4
8670.2.a.be.1.1		2	85.67	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:	\( N \)	\(=\)	\( 2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2550.d (of order \(2\), degree \(1\), not minimal)

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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