LMFDB - Embedded newform 2550.2.d.m.2449.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2550,2,Mod(2449,2550)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [2,0,0,-2,0,2,0,0,-2,0,-8,0,0,0,0,2,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$20.3618525154$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 102)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2449.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	2550.2449
Dual form			2550.2.d.m.2449.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} -4.00000 q^{11} -1.00000i q^{12} -2.00000i q^{13} +1.00000 q^{16} -1.00000i q^{17} +1.00000i q^{18} -4.00000 q^{19} +4.00000i q^{22} -1.00000 q^{24} -2.00000 q^{26} -1.00000i q^{27} +10.0000 q^{29} +8.00000 q^{31} -1.00000i q^{32} -4.00000i q^{33} -1.00000 q^{34} +1.00000 q^{36} +2.00000i q^{37} +4.00000i q^{38} +2.00000 q^{39} +10.0000 q^{41} +12.0000i q^{43} +4.00000 q^{44} +1.00000i q^{48} +7.00000 q^{49} +1.00000 q^{51} +2.00000i q^{52} +6.00000i q^{53} -1.00000 q^{54} -4.00000i q^{57} -10.0000i q^{58} -12.0000 q^{59} -10.0000 q^{61} -8.00000i q^{62} -1.00000 q^{64} -4.00000 q^{66} +12.0000i q^{67} +1.00000i q^{68} -1.00000i q^{72} +10.0000i q^{73} +2.00000 q^{74} +4.00000 q^{76} -2.00000i q^{78} +8.00000 q^{79} +1.00000 q^{81} -10.0000i q^{82} +4.00000i q^{83} +12.0000 q^{86} +10.0000i q^{87} -4.00000i q^{88} +6.00000 q^{89} +8.00000i q^{93} +1.00000 q^{96} +14.0000i q^{97} -7.00000i q^{98} +4.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 8 q^{11} + 2 q^{16} - 8 q^{19} - 2 q^{24} - 4 q^{26} + 20 q^{29} + 16 q^{31} - 2 q^{34} + 2 q^{36} + 4 q^{39} + 20 q^{41} + 8 q^{44} + 14 q^{49} + 2 q^{51} - 2 q^{54}+ \cdots + 8 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 - 8 * q^11 + 2 * q^16 - 8 * q^19 - 2 * q^24 - 4 * q^26 + 20 * q^29 + 16 * q^31 - 2 * q^34 + 2 * q^36 + 4 * q^39 + 20 * q^41 + 8 * q^44 + 14 * q^49 + 2 * q^51 - 2 * q^54 - 24 * q^59 - 20 * q^61 - 2 * q^64 - 8 * q^66 + 4 * q^74 + 8 * q^76 + 16 * q^79 + 2 * q^81 + 24 * q^86 + 12 * q^89 + 2 * q^96 + 8 * q^99

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
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	1.00000i	0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	− 1.00000i	− 0.288675i
$13$	− 2.00000i	− 0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$	− 2.00000i	− 0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	0	0
$15$	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
$21$	0	0
$22$	4.00000i	0.852803i
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	−2.00000	−0.392232
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	10.0000	1.85695	0.928477	−	0.371391i	$-0.121119\pi$
$29$	10.0000	1.85695	0.928477	+	0.371391i	$0.121119\pi$
$30$	0	0
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	− 1.00000i	− 0.176777i
$33$	− 4.00000i	− 0.696311i
$34$	−1.00000	−0.171499
$35$	0	0
$36$	1.00000	0.166667
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	4.00000i	0.648886i
$39$	2.00000	0.320256
$40$	0	0
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	0	0
$43$	12.0000i	1.82998i	0.403473	+	0.914991i	$0.367803\pi$
$43$	12.0000i	1.82998i	−0.403473	+	0.914991i	$0.632197\pi$
$44$	4.00000	0.603023
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	1.00000i	0.144338i
$49$	7.00000	1.00000
$50$	0	0
$51$	1.00000	0.140028
$52$	2.00000i	0.277350i
$53$	6.00000i	0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$	6.00000i	0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	0	0
$57$	− 4.00000i	− 0.529813i
$58$	− 10.0000i	− 1.31306i
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	− 8.00000i	− 1.01600i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−4.00000	−0.492366
$67$	12.0000i	1.46603i	0.680211	+	0.733017i	$0.261888\pi$
$67$	12.0000i	1.46603i	−0.680211	+	0.733017i	$0.738112\pi$
$68$	1.00000i	0.121268i
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	− 1.00000i	− 0.117851i
$73$	10.0000i	1.17041i	0.810885	+	0.585206i	$0.198986\pi$
$73$	10.0000i	1.17041i	−0.810885	+	0.585206i	$0.801014\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	4.00000	0.458831
$77$	0	0
$78$	− 2.00000i	− 0.226455i
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 10.0000i	− 1.10432i
$83$	4.00000i	0.439057i	0.975606	+	0.219529i	$0.0704519\pi$
$83$	4.00000i	0.439057i	−0.975606	+	0.219529i	$0.929548\pi$
$84$	0	0
$85$	0	0
$86$	12.0000	1.29399
$87$	10.0000i	1.07211i
$88$	− 4.00000i	− 0.426401i
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	8.00000i	0.829561i
$94$	0	0
$95$	0	0
$96$	1.00000	0.102062
$97$	14.0000i	1.42148i	0.703452	+	0.710742i	$0.251641\pi$
$97$	14.0000i	1.42148i	−0.703452	+	0.710742i	$0.748359\pi$
$98$	− 7.00000i	− 0.707107i
$99$	4.00000	0.402015
$100$	0	0
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	− 1.00000i	− 0.0990148i
$103$	− 8.00000i	− 0.788263i	−0.919054	−	0.394132i	$-0.871045\pi$
$103$	− 8.00000i	− 0.788263i	0.919054	−	0.394132i	$-0.128955\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	6.00000	0.582772
$107$	4.00000i	0.386695i	0.981130	+	0.193347i	$0.0619344\pi$
$107$	4.00000i	0.386695i	−0.981130	+	0.193347i	$0.938066\pi$
$108$	1.00000i	0.0962250i
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	0	0
$113$	2.00000i	0.188144i	0.995565	+	0.0940721i	$0.0299884\pi$
$113$	2.00000i	0.188144i	−0.995565	+	0.0940721i	$0.970012\pi$
$114$	−4.00000	−0.374634
$115$	0	0
$116$	−10.0000	−0.928477
$117$	2.00000i	0.184900i
$118$	12.0000i	1.10469i
$119$	0	0
$120$	0	0
$121$	5.00000	0.454545
$122$	10.0000i	0.905357i
$123$	10.0000i	0.901670i
$124$	−8.00000	−0.718421
$125$	0	0
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	1.00000i	0.0883883i
$129$	−12.0000	−1.05654
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	4.00000i	0.348155i
$133$	0	0
$134$	12.0000	1.03664
$135$	0	0
$136$	1.00000	0.0857493
$137$	− 10.0000i	− 0.854358i	−0.904167	−	0.427179i	$-0.859507\pi$
$137$	− 10.0000i	− 0.854358i	0.904167	−	0.427179i	$-0.140493\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	8.00000i	0.668994i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	10.0000	0.827606
$147$	7.00000i	0.577350i
$148$	− 2.00000i	− 0.164399i
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	0	0
$151$	24.0000	1.95309	0.976546	−	0.215308i	$-0.0690756\pi$
$151$	24.0000	1.95309	0.976546	+	0.215308i	$0.0690756\pi$
$152$	− 4.00000i	− 0.324443i
$153$	1.00000i	0.0808452i
$154$	0	0
$155$	0	0
$156$	−2.00000	−0.160128
$157$	2.00000i	0.159617i	0.996810	+	0.0798087i	$0.0254309\pi$
$157$	2.00000i	0.159617i	−0.996810	+	0.0798087i	$0.974569\pi$
$158$	− 8.00000i	− 0.636446i
$159$	−6.00000	−0.475831
$160$	0	0
$161$	0	0
$162$	− 1.00000i	− 0.0785674i
$163$	4.00000i	0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$	4.00000i	0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$	−10.0000	−0.780869
$165$	0	0
$166$	4.00000	0.310460
$167$	− 16.0000i	− 1.23812i	−0.785345	−	0.619059i	$-0.787514\pi$
$167$	− 16.0000i	− 1.23812i	0.785345	−	0.619059i	$-0.212486\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	4.00000	0.305888
$172$	− 12.0000i	− 0.914991i
$173$	6.00000i	0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$	6.00000i	0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$	10.0000	0.758098
$175$	0	0
$176$	−4.00000	−0.301511
$177$	− 12.0000i	− 0.901975i
$178$	− 6.00000i	− 0.449719i
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	0	0
$183$	− 10.0000i	− 0.739221i
$184$	0	0
$185$	0	0
$186$	8.00000	0.586588
$187$	4.00000i	0.292509i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−16.0000	−1.15772	−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000	−1.15772	−0.578860	+	0.815427i	$0.696502\pi$
$192$	− 1.00000i	− 0.0721688i
$193$	18.0000i	1.29567i	0.761781	+	0.647834i	$0.224325\pi$
$193$	18.0000i	1.29567i	−0.761781	+	0.647834i	$0.775675\pi$
$194$	14.0000	1.00514
$195$	0	0
$196$	−7.00000	−0.500000
$197$	− 14.0000i	− 0.997459i	−0.866758	−	0.498729i	$-0.833800\pi$
$197$	− 14.0000i	− 0.997459i	0.866758	−	0.498729i	$-0.166200\pi$
$198$	− 4.00000i	− 0.284268i
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	−12.0000	−0.846415
$202$	10.0000i	0.703598i
$203$	0	0
$204$	−1.00000	−0.0700140
$205$	0	0
$206$	−8.00000	−0.557386
$207$	0	0
$208$	− 2.00000i	− 0.138675i
$209$	16.0000	1.10674
$210$	0	0
$211$	−28.0000	−1.92760	−0.963800	−	0.266627i	$-0.914091\pi$
$211$	−28.0000	−1.92760	−0.963800	+	0.266627i	$0.914091\pi$
$212$	− 6.00000i	− 0.412082i
$213$	0	0
$214$	4.00000	0.273434
$215$	0	0
$216$	1.00000	0.0680414
$217$	0	0
$218$	− 10.0000i	− 0.677285i
$219$	−10.0000	−0.675737
$220$	0	0
$221$	−2.00000	−0.134535
$222$	2.00000i	0.134231i
$223$	16.0000i	1.07144i	0.844396	+	0.535720i	$0.179960\pi$
$223$	16.0000i	1.07144i	−0.844396	+	0.535720i	$0.820040\pi$
$224$	0	0
$225$	0	0
$226$	2.00000	0.133038
$227$	− 4.00000i	− 0.265489i	−0.991150	−	0.132745i	$-0.957621\pi$
$227$	− 4.00000i	− 0.265489i	0.991150	−	0.132745i	$-0.0423790\pi$
$228$	4.00000i	0.264906i
$229$	26.0000	1.71813	0.859064	−	0.511868i	$-0.171046\pi$
$229$	26.0000	1.71813	0.859064	+	0.511868i	$0.171046\pi$
$230$	0	0
$231$	0	0
$232$	10.0000i	0.656532i
$233$	26.0000i	1.70332i	0.524097	+	0.851658i	$0.324403\pi$
$233$	26.0000i	1.70332i	−0.524097	+	0.851658i	$0.675597\pi$
$234$	2.00000	0.130744
$235$	0	0
$236$	12.0000	0.781133
$237$	8.00000i	0.519656i
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	− 5.00000i	− 0.321412i
$243$	1.00000i	0.0641500i
$244$	10.0000	0.640184
$245$	0	0
$246$	10.0000	0.637577
$247$	8.00000i	0.509028i
$248$	8.00000i	0.508001i
$249$	−4.00000	−0.253490
$250$	0	0
$251$	28.0000	1.76734	0.883672	−	0.468106i	$-0.155064\pi$
$251$	28.0000	1.76734	0.883672	+	0.468106i	$0.155064\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 2.00000i	− 0.124757i	−0.998053	−	0.0623783i	$-0.980131\pi$
$257$	− 2.00000i	− 0.124757i	0.998053	−	0.0623783i	$-0.0198685\pi$
$258$	12.0000i	0.747087i
$259$	0	0
$260$	0	0
$261$	−10.0000	−0.618984
$262$	12.0000i	0.741362i
$263$	− 8.00000i	− 0.493301i	−0.969104	−	0.246651i	$-0.920670\pi$
$263$	− 8.00000i	− 0.493301i	0.969104	−	0.246651i	$-0.0793300\pi$
$264$	4.00000	0.246183
$265$	0	0
$266$	0	0
$267$	6.00000i	0.367194i
$268$	− 12.0000i	− 0.733017i
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	0	0
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	− 1.00000i	− 0.0606339i
$273$	0	0
$274$	−10.0000	−0.604122
$275$	0	0
$276$	0	0
$277$	− 30.0000i	− 1.80253i	−0.433273	−	0.901263i	$-0.642641\pi$
$277$	− 30.0000i	− 1.80253i	0.433273	−	0.901263i	$-0.357359\pi$
$278$	− 4.00000i	− 0.239904i
$279$	−8.00000	−0.478947
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	12.0000i	0.713326i	0.934233	+	0.356663i	$0.116086\pi$
$283$	12.0000i	0.713326i	−0.934233	+	0.356663i	$0.883914\pi$
$284$	0	0
$285$	0	0
$286$	8.00000	0.473050
$287$	0	0
$288$	1.00000i	0.0589256i
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−14.0000	−0.820695
$292$	− 10.0000i	− 0.585206i
$293$	− 26.0000i	− 1.51894i	−0.650545	−	0.759468i	$-0.725459\pi$
$293$	− 26.0000i	− 1.51894i	0.650545	−	0.759468i	$-0.274541\pi$
$294$	7.00000	0.408248
$295$	0	0
$296$	−2.00000	−0.116248
$297$	4.00000i	0.232104i
$298$	− 10.0000i	− 0.579284i
$299$	0	0
$300$	0	0
$301$	0	0
$302$	− 24.0000i	− 1.38104i
$303$	− 10.0000i	− 0.574485i
$304$	−4.00000	−0.229416
$305$	0	0
$306$	1.00000	0.0571662
$307$	12.0000i	0.684876i	0.939540	+	0.342438i	$0.111253\pi$
$307$	12.0000i	0.684876i	−0.939540	+	0.342438i	$0.888747\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	2.00000i	0.113228i
$313$	10.0000i	0.565233i	0.959233	+	0.282617i	$0.0912024\pi$
$313$	10.0000i	0.565233i	−0.959233	+	0.282617i	$0.908798\pi$
$314$	2.00000	0.112867
$315$	0	0
$316$	−8.00000	−0.450035
$317$	− 6.00000i	− 0.336994i	−0.985702	−	0.168497i	$-0.946109\pi$
$317$	− 6.00000i	− 0.336994i	0.985702	−	0.168497i	$-0.0538913\pi$
$318$	6.00000i	0.336463i
$319$	−40.0000	−2.23957
$320$	0	0
$321$	−4.00000	−0.223258
$322$	0	0
$323$	4.00000i	0.222566i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	4.00000	0.221540
$327$	10.0000i	0.553001i
$328$	10.0000i	0.552158i
$329$	0	0
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	− 4.00000i	− 0.219529i
$333$	− 2.00000i	− 0.109599i
$334$	−16.0000	−0.875481
$335$	0	0
$336$	0	0
$337$	14.0000i	0.762629i	0.924445	+	0.381314i	$0.124528\pi$
$337$	14.0000i	0.762629i	−0.924445	+	0.381314i	$0.875472\pi$
$338$	− 9.00000i	− 0.489535i
$339$	−2.00000	−0.108625
$340$	0	0
$341$	−32.0000	−1.73290
$342$	− 4.00000i	− 0.216295i
$343$	0	0
$344$	−12.0000	−0.646997
$345$	0	0
$346$	6.00000	0.322562
$347$	− 28.0000i	− 1.50312i	−0.659665	−	0.751559i	$-0.729302\pi$
$347$	− 28.0000i	− 1.50312i	0.659665	−	0.751559i	$-0.270698\pi$
$348$	− 10.0000i	− 0.536056i
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	4.00000i	0.213201i
$353$	− 30.0000i	− 1.59674i	−0.602168	−	0.798369i	$-0.705696\pi$
$353$	− 30.0000i	− 1.59674i	0.602168	−	0.798369i	$-0.294304\pi$
$354$	−12.0000	−0.637793
$355$	0	0
$356$	−6.00000	−0.317999
$357$	0	0
$358$	− 12.0000i	− 0.634220i
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	− 14.0000i	− 0.735824i
$363$	5.00000i	0.262432i
$364$	0	0
$365$	0	0
$366$	−10.0000	−0.522708
$367$	− 24.0000i	− 1.25279i	−0.779506	−	0.626395i	$-0.784530\pi$
$367$	− 24.0000i	− 1.25279i	0.779506	−	0.626395i	$-0.215470\pi$
$368$	0	0
$369$	−10.0000	−0.520579
$370$	0	0
$371$	0	0
$372$	− 8.00000i	− 0.414781i
$373$	6.00000i	0.310668i	0.987862	+	0.155334i	$0.0496454\pi$
$373$	6.00000i	0.310668i	−0.987862	+	0.155334i	$0.950355\pi$
$374$	4.00000	0.206835
$375$	0	0
$376$	0	0
$377$	− 20.0000i	− 1.03005i
$378$	0	0
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	0	0
$381$	0	0
$382$	16.0000i	0.818631i
$383$	− 16.0000i	− 0.817562i	−0.912633	−	0.408781i	$-0.865954\pi$
$383$	− 16.0000i	− 0.817562i	0.912633	−	0.408781i	$-0.134046\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	18.0000	0.916176
$387$	− 12.0000i	− 0.609994i
$388$	− 14.0000i	− 0.710742i
$389$	26.0000	1.31825	0.659126	−	0.752032i	$-0.270926\pi$
$389$	26.0000	1.31825	0.659126	+	0.752032i	$0.270926\pi$
$390$	0	0
$391$	0	0
$392$	7.00000i	0.353553i
$393$	− 12.0000i	− 0.605320i
$394$	−14.0000	−0.705310
$395$	0	0
$396$	−4.00000	−0.201008
$397$	26.0000i	1.30490i	0.757831	+	0.652451i	$0.226259\pi$
$397$	26.0000i	1.30490i	−0.757831	+	0.652451i	$0.773741\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−14.0000	−0.699127	−0.349563	−	0.936913i	$-0.613670\pi$
$401$	−14.0000	−0.699127	−0.349563	+	0.936913i	$0.613670\pi$
$402$	12.0000i	0.598506i
$403$	− 16.0000i	− 0.797017i
$404$	10.0000	0.497519
$405$	0	0
$406$	0	0
$407$	− 8.00000i	− 0.396545i
$408$	1.00000i	0.0495074i
$409$	−26.0000	−1.28562	−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000	−1.28562	−0.642809	+	0.766027i	$0.722231\pi$
$410$	0	0
$411$	10.0000	0.493264
$412$	8.00000i	0.394132i
$413$	0	0
$414$	0	0
$415$	0	0
$416$	−2.00000	−0.0980581
$417$	4.00000i	0.195881i
$418$	− 16.0000i	− 0.782586i
$419$	−4.00000	−0.195413	−0.0977064	−	0.995215i	$-0.531151\pi$
$419$	−4.00000	−0.195413	−0.0977064	+	0.995215i	$0.531151\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	28.0000i	1.36302i
$423$	0	0
$424$	−6.00000	−0.291386
$425$	0	0
$426$	0	0
$427$	0	0
$428$	− 4.00000i	− 0.193347i
$429$	−8.00000	−0.386244
$430$	0	0
$431$	−8.00000	−0.385346	−0.192673	−	0.981263i	$-0.561716\pi$
$431$	−8.00000	−0.385346	−0.192673	+	0.981263i	$0.561716\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	− 14.0000i	− 0.672797i	−0.941720	−	0.336399i	$-0.890791\pi$
$433$	− 14.0000i	− 0.672797i	0.941720	−	0.336399i	$-0.109209\pi$
$434$	0	0
$435$	0	0
$436$	−10.0000	−0.478913
$437$	0	0
$438$	10.0000i	0.477818i
$439$	−16.0000	−0.763638	−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000	−0.763638	−0.381819	+	0.924237i	$0.624702\pi$
$440$	0	0
$441$	−7.00000	−0.333333
$442$	2.00000i	0.0951303i
$443$	− 4.00000i	− 0.190046i	−0.995475	−	0.0950229i	$-0.969708\pi$
$443$	− 4.00000i	− 0.190046i	0.995475	−	0.0950229i	$-0.0302924\pi$
$444$	2.00000	0.0949158
$445$	0	0
$446$	16.0000	0.757622
$447$	10.0000i	0.472984i
$448$	0	0
$449$	−2.00000	−0.0943858	−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−2.00000	−0.0943858	−0.0471929	+	0.998886i	$0.515028\pi$
$450$	0	0
$451$	−40.0000	−1.88353
$452$	− 2.00000i	− 0.0940721i
$453$	24.0000i	1.12762i
$454$	−4.00000	−0.187729
$455$	0	0
$456$	4.00000	0.187317
$457$	− 10.0000i	− 0.467780i	−0.972263	−	0.233890i	$-0.924854\pi$
$457$	− 10.0000i	− 0.467780i	0.972263	−	0.233890i	$-0.0751456\pi$
$458$	− 26.0000i	− 1.21490i
$459$	−1.00000	−0.0466760
$460$	0	0
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	0	0
$463$	32.0000i	1.48717i	0.668644	+	0.743583i	$0.266875\pi$
$463$	32.0000i	1.48717i	−0.668644	+	0.743583i	$0.733125\pi$
$464$	10.0000	0.464238
$465$	0	0
$466$	26.0000	1.20443
$467$	12.0000i	0.555294i	0.960683	+	0.277647i	$0.0895545\pi$
$467$	12.0000i	0.555294i	−0.960683	+	0.277647i	$0.910445\pi$
$468$	− 2.00000i	− 0.0924500i
$469$	0	0
$470$	0	0
$471$	−2.00000	−0.0921551
$472$	− 12.0000i	− 0.552345i
$473$	− 48.0000i	− 2.20704i
$474$	8.00000	0.367452
$475$	0	0
$476$	0	0
$477$	− 6.00000i	− 0.274721i
$478$	0	0
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	− 2.00000i	− 0.0910975i
$483$	0	0
$484$	−5.00000	−0.227273
$485$	0	0
$486$	1.00000	0.0453609
$487$	16.0000i	0.725029i	0.931978	+	0.362515i	$0.118082\pi$
$487$	16.0000i	0.725029i	−0.931978	+	0.362515i	$0.881918\pi$
$488$	− 10.0000i	− 0.452679i
$489$	−4.00000	−0.180886
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	− 10.0000i	− 0.450835i
$493$	− 10.0000i	− 0.450377i
$494$	8.00000	0.359937
$495$	0	0
$496$	8.00000	0.359211
$497$	0	0
$498$	4.00000i	0.179244i
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	0	0
$501$	16.0000	0.714827
$502$	− 28.0000i	− 1.24970i
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	9.00000i	0.399704i
$508$	0	0
$509$	−30.0000	−1.32973	−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000	−1.32973	−0.664863	+	0.746965i	$0.731510\pi$
$510$	0	0
$511$	0	0
$512$	− 1.00000i	− 0.0441942i
$513$	4.00000i	0.176604i
$514$	−2.00000	−0.0882162
$515$	0	0
$516$	12.0000	0.528271
$517$	0	0
$518$	0	0
$519$	−6.00000	−0.263371
$520$	0	0
$521$	−22.0000	−0.963837	−0.481919	−	0.876216i	$-0.660060\pi$
$521$	−22.0000	−0.963837	−0.481919	+	0.876216i	$0.660060\pi$
$522$	10.0000i	0.437688i
$523$	− 4.00000i	− 0.174908i	−0.996169	−	0.0874539i	$-0.972127\pi$
$523$	− 4.00000i	− 0.174908i	0.996169	−	0.0874539i	$-0.0278730\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	−8.00000	−0.348817
$527$	− 8.00000i	− 0.348485i
$528$	− 4.00000i	− 0.174078i
$529$	23.0000	1.00000
$530$	0	0
$531$	12.0000	0.520756
$532$	0	0
$533$	− 20.0000i	− 0.866296i
$534$	6.00000	0.259645
$535$	0	0
$536$	−12.0000	−0.518321
$537$	12.0000i	0.517838i
$538$	6.00000i	0.258678i
$539$	−28.0000	−1.20605
$540$	0	0
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	− 16.0000i	− 0.687259i
$543$	14.0000i	0.600798i
$544$	−1.00000	−0.0428746
$545$	0	0
$546$	0	0
$547$	12.0000i	0.513083i	0.966533	+	0.256541i	$0.0825830\pi$
$547$	12.0000i	0.513083i	−0.966533	+	0.256541i	$0.917417\pi$
$548$	10.0000i	0.427179i
$549$	10.0000	0.426790
$550$	0	0
$551$	−40.0000	−1.70406
$552$	0	0
$553$	0	0
$554$	−30.0000	−1.27458
$555$	0	0
$556$	−4.00000	−0.169638
$557$	34.0000i	1.44063i	0.693649	+	0.720313i	$0.256002\pi$
$557$	34.0000i	1.44063i	−0.693649	+	0.720313i	$0.743998\pi$
$558$	8.00000i	0.338667i
$559$	24.0000	1.01509
$560$	0	0
$561$	−4.00000	−0.168880
$562$	6.00000i	0.253095i
$563$	36.0000i	1.51722i	0.651546	+	0.758610i	$0.274121\pi$
$563$	36.0000i	1.51722i	−0.651546	+	0.758610i	$0.725879\pi$
$564$	0	0
$565$	0	0
$566$	12.0000	0.504398
$567$	0	0
$568$	0	0
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	28.0000	1.17176	0.585882	−	0.810397i	$-0.300748\pi$
$571$	28.0000	1.17176	0.585882	+	0.810397i	$0.300748\pi$
$572$	− 8.00000i	− 0.334497i
$573$	− 16.0000i	− 0.668410i
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 2.00000i	− 0.0832611i	−0.999133	−	0.0416305i	$-0.986745\pi$
$577$	− 2.00000i	− 0.0832611i	0.999133	−	0.0416305i	$-0.0132552\pi$
$578$	1.00000i	0.0415945i
$579$	−18.0000	−0.748054
$580$	0	0
$581$	0	0
$582$	14.0000i	0.580319i
$583$	− 24.0000i	− 0.993978i
$584$	−10.0000	−0.413803
$585$	0	0
$586$	−26.0000	−1.07405
$587$	20.0000i	0.825488i	0.910847	+	0.412744i	$0.135430\pi$
$587$	20.0000i	0.825488i	−0.910847	+	0.412744i	$0.864570\pi$
$588$	− 7.00000i	− 0.288675i
$589$	−32.0000	−1.31854
$590$	0	0
$591$	14.0000	0.575883
$592$	2.00000i	0.0821995i
$593$	− 14.0000i	− 0.574911i	−0.957794	−	0.287456i	$-0.907191\pi$
$593$	− 14.0000i	− 0.574911i	0.957794	−	0.287456i	$-0.0928094\pi$
$594$	4.00000	0.164122
$595$	0	0
$596$	−10.0000	−0.409616
$597$	0	0
$598$	0	0
$599$	8.00000	0.326871	0.163436	−	0.986554i	$-0.447742\pi$
$599$	8.00000	0.326871	0.163436	+	0.986554i	$0.447742\pi$
$600$	0	0
$601$	−6.00000	−0.244745	−0.122373	−	0.992484i	$-0.539050\pi$
$601$	−6.00000	−0.244745	−0.122373	+	0.992484i	$0.539050\pi$
$602$	0	0
$603$	− 12.0000i	− 0.488678i
$604$	−24.0000	−0.976546
$605$	0	0
$606$	−10.0000	−0.406222
$607$	24.0000i	0.974130i	0.873366	+	0.487065i	$0.161933\pi$
$607$	24.0000i	0.974130i	−0.873366	+	0.487065i	$0.838067\pi$
$608$	4.00000i	0.162221i
$609$	0	0
$610$	0	0
$611$	0	0
$612$	− 1.00000i	− 0.0404226i
$613$	38.0000i	1.53481i	0.641165	+	0.767403i	$0.278451\pi$
$613$	38.0000i	1.53481i	−0.641165	+	0.767403i	$0.721549\pi$
$614$	12.0000	0.484281
$615$	0	0
$616$	0	0
$617$	6.00000i	0.241551i	0.992680	+	0.120775i	$0.0385381\pi$
$617$	6.00000i	0.241551i	−0.992680	+	0.120775i	$0.961462\pi$
$618$	− 8.00000i	− 0.321807i
$619$	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
$621$	0	0
$622$	0	0
$623$	0	0
$624$	2.00000	0.0800641
$625$	0	0
$626$	10.0000	0.399680
$627$	16.0000i	0.638978i
$628$	− 2.00000i	− 0.0798087i
$629$	2.00000	0.0797452
$630$	0	0
$631$	−24.0000	−0.955425	−0.477712	−	0.878516i	$-0.658534\pi$
$631$	−24.0000	−0.955425	−0.477712	+	0.878516i	$0.658534\pi$
$632$	8.00000i	0.318223i
$633$	− 28.0000i	− 1.11290i
$634$	−6.00000	−0.238290
$635$	0	0
$636$	6.00000	0.237915
$637$	− 14.0000i	− 0.554700i
$638$	40.0000i	1.58362i
$639$	0	0
$640$	0	0
$641$	2.00000	0.0789953	0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000	0.0789953	0.0394976	+	0.999220i	$0.487424\pi$
$642$	4.00000i	0.157867i
$643$	− 28.0000i	− 1.10421i	−0.833774	−	0.552106i	$-0.813824\pi$
$643$	− 28.0000i	− 1.10421i	0.833774	−	0.552106i	$-0.186176\pi$
$644$	0	0
$645$	0	0
$646$	4.00000	0.157378
$647$	40.0000i	1.57256i	0.617869	+	0.786281i	$0.287996\pi$
$647$	40.0000i	1.57256i	−0.617869	+	0.786281i	$0.712004\pi$
$648$	1.00000i	0.0392837i
$649$	48.0000	1.88416
$650$	0	0
$651$	0	0
$652$	− 4.00000i	− 0.156652i
$653$	6.00000i	0.234798i	0.993085	+	0.117399i	$0.0374557\pi$
$653$	6.00000i	0.234798i	−0.993085	+	0.117399i	$0.962544\pi$
$654$	10.0000	0.391031
$655$	0	0
$656$	10.0000	0.390434
$657$	− 10.0000i	− 0.390137i
$658$	0	0
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	0	0
$661$	6.00000	0.233373	0.116686	−	0.993169i	$-0.462773\pi$
$661$	6.00000	0.233373	0.116686	+	0.993169i	$0.462773\pi$
$662$	20.0000i	0.777322i
$663$	− 2.00000i	− 0.0776736i
$664$	−4.00000	−0.155230
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	0	0
$668$	16.0000i	0.619059i
$669$	−16.0000	−0.618596
$670$	0	0
$671$	40.0000	1.54418
$672$	0	0
$673$	− 46.0000i	− 1.77317i	−0.462566	−	0.886585i	$-0.653071\pi$
$673$	− 46.0000i	− 1.77317i	0.462566	−	0.886585i	$-0.346929\pi$
$674$	14.0000	0.539260
$675$	0	0
$676$	−9.00000	−0.346154
$677$	− 46.0000i	− 1.76792i	−0.467559	−	0.883962i	$-0.654866\pi$
$677$	− 46.0000i	− 1.76792i	0.467559	−	0.883962i	$-0.345134\pi$
$678$	2.00000i	0.0768095i
$679$	0	0
$680$	0	0
$681$	4.00000	0.153280
$682$	32.0000i	1.22534i
$683$	− 20.0000i	− 0.765279i	−0.923898	−	0.382639i	$-0.875015\pi$
$683$	− 20.0000i	− 0.765279i	0.923898	−	0.382639i	$-0.124985\pi$
$684$	−4.00000	−0.152944
$685$	0	0
$686$	0	0
$687$	26.0000i	0.991962i
$688$	12.0000i	0.457496i
$689$	12.0000	0.457164
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	− 6.00000i	− 0.228086i
$693$	0	0
$694$	−28.0000	−1.06287
$695$	0	0
$696$	−10.0000	−0.379049
$697$	− 10.0000i	− 0.378777i
$698$	14.0000i	0.529908i
$699$	−26.0000	−0.983410
$700$	0	0
$701$	46.0000	1.73740	0.868698	−	0.495342i	$-0.164957\pi$
$701$	46.0000	1.73740	0.868698	+	0.495342i	$0.164957\pi$
$702$	2.00000i	0.0754851i
$703$	− 8.00000i	− 0.301726i
$704$	4.00000	0.150756
$705$	0	0
$706$	−30.0000	−1.12906
$707$	0	0
$708$	12.0000i	0.450988i
$709$	−46.0000	−1.72757	−0.863783	−	0.503864i	$-0.831911\pi$
$709$	−46.0000	−1.72757	−0.863783	+	0.503864i	$0.831911\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	6.00000i	0.224860i
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−12.0000	−0.448461
$717$	0	0
$718$	− 24.0000i	− 0.895672i
$719$	−40.0000	−1.49175	−0.745874	−	0.666087i	$-0.767968\pi$
$719$	−40.0000	−1.49175	−0.745874	+	0.666087i	$0.767968\pi$
$720$	0	0
$721$	0	0
$722$	3.00000i	0.111648i
$723$	2.00000i	0.0743808i
$724$	−14.0000	−0.520306
$725$	0	0
$726$	5.00000	0.185567
$727$	8.00000i	0.296704i	0.988935	+	0.148352i	$0.0473968\pi$
$727$	8.00000i	0.296704i	−0.988935	+	0.148352i	$0.952603\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	12.0000	0.443836
$732$	10.0000i	0.369611i
$733$	46.0000i	1.69905i	0.527549	+	0.849524i	$0.323111\pi$
$733$	46.0000i	1.69905i	−0.527549	+	0.849524i	$0.676889\pi$
$734$	−24.0000	−0.885856
$735$	0	0
$736$	0	0
$737$	− 48.0000i	− 1.76810i
$738$	10.0000i	0.368105i
$739$	−52.0000	−1.91285	−0.956425	−	0.291977i	$-0.905687\pi$
$739$	−52.0000	−1.91285	−0.956425	+	0.291977i	$0.905687\pi$
$740$	0	0
$741$	−8.00000	−0.293887
$742$	0	0
$743$	− 16.0000i	− 0.586983i	−0.955962	−	0.293492i	$-0.905183\pi$
$743$	− 16.0000i	− 0.586983i	0.955962	−	0.293492i	$-0.0948173\pi$
$744$	−8.00000	−0.293294
$745$	0	0
$746$	6.00000	0.219676
$747$	− 4.00000i	− 0.146352i
$748$	− 4.00000i	− 0.146254i
$749$	0	0
$750$	0	0
$751$	8.00000	0.291924	0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	0	0
$753$	28.0000i	1.02038i
$754$	−20.0000	−0.728357
$755$	0	0
$756$	0	0
$757$	10.0000i	0.363456i	0.983349	+	0.181728i	$0.0581691\pi$
$757$	10.0000i	0.363456i	−0.983349	+	0.181728i	$0.941831\pi$
$758$	− 4.00000i	− 0.145287i
$759$	0	0
$760$	0	0
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	0	0
$763$	0	0
$764$	16.0000	0.578860
$765$	0	0
$766$	−16.0000	−0.578103
$767$	24.0000i	0.866590i
$768$	1.00000i	0.0360844i
$769$	30.0000	1.08183	0.540914	−	0.841078i	$-0.318079\pi$
$769$	30.0000	1.08183	0.540914	+	0.841078i	$0.318079\pi$
$770$	0	0
$771$	2.00000	0.0720282
$772$	− 18.0000i	− 0.647834i
$773$	38.0000i	1.36677i	0.730061	+	0.683383i	$0.239492\pi$
$773$	38.0000i	1.36677i	−0.730061	+	0.683383i	$0.760508\pi$
$774$	−12.0000	−0.431331
$775$	0	0
$776$	−14.0000	−0.502571
$777$	0	0
$778$	− 26.0000i	− 0.932145i
$779$	−40.0000	−1.43315
$780$	0	0
$781$	0	0
$782$	0	0
$783$	− 10.0000i	− 0.357371i
$784$	7.00000	0.250000
$785$	0	0
$786$	−12.0000	−0.428026
$787$	− 4.00000i	− 0.142585i	−0.997455	−	0.0712923i	$-0.977288\pi$
$787$	− 4.00000i	− 0.142585i	0.997455	−	0.0712923i	$-0.0227123\pi$
$788$	14.0000i	0.498729i
$789$	8.00000	0.284808
$790$	0	0
$791$	0	0
$792$	4.00000i	0.142134i
$793$	20.0000i	0.710221i
$794$	26.0000	0.922705
$795$	0	0
$796$	0	0
$797$	− 14.0000i	− 0.495905i	−0.968772	−	0.247953i	$-0.920242\pi$
$797$	− 14.0000i	− 0.495905i	0.968772	−	0.247953i	$-0.0797578\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−6.00000	−0.212000
$802$	14.0000i	0.494357i
$803$	− 40.0000i	− 1.41157i
$804$	12.0000	0.423207
$805$	0	0
$806$	−16.0000	−0.563576
$807$	− 6.00000i	− 0.211210i
$808$	− 10.0000i	− 0.351799i
$809$	22.0000	0.773479	0.386739	−	0.922189i	$-0.373601\pi$
$809$	22.0000	0.773479	0.386739	+	0.922189i	$0.373601\pi$
$810$	0	0
$811$	−20.0000	−0.702295	−0.351147	−	0.936320i	$-0.614208\pi$
$811$	−20.0000	−0.702295	−0.351147	+	0.936320i	$0.614208\pi$
$812$	0	0
$813$	16.0000i	0.561144i
$814$	−8.00000	−0.280400
$815$	0	0
$816$	1.00000	0.0350070
$817$	− 48.0000i	− 1.67931i
$818$	26.0000i	0.909069i
$819$	0	0
$820$	0	0
$821$	−34.0000	−1.18661	−0.593304	−	0.804978i	$-0.702177\pi$
$821$	−34.0000	−1.18661	−0.593304	+	0.804978i	$0.702177\pi$
$822$	− 10.0000i	− 0.348790i
$823$	32.0000i	1.11545i	0.830026	+	0.557725i	$0.188326\pi$
$823$	32.0000i	1.11545i	−0.830026	+	0.557725i	$0.811674\pi$
$824$	8.00000	0.278693
$825$	0	0
$826$	0	0
$827$	20.0000i	0.695468i	0.937593	+	0.347734i	$0.113049\pi$
$827$	20.0000i	0.695468i	−0.937593	+	0.347734i	$0.886951\pi$
$828$	0	0
$829$	34.0000	1.18087	0.590434	−	0.807086i	$-0.298956\pi$
$829$	34.0000	1.18087	0.590434	+	0.807086i	$0.298956\pi$
$830$	0	0
$831$	30.0000	1.04069
$832$	2.00000i	0.0693375i
$833$	− 7.00000i	− 0.242536i
$834$	4.00000	0.138509
$835$	0	0
$836$	−16.0000	−0.553372
$837$	− 8.00000i	− 0.276520i
$838$	4.00000i	0.138178i
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	71.0000	2.44828
$842$	− 22.0000i	− 0.758170i
$843$	− 6.00000i	− 0.206651i
$844$	28.0000	0.963800
$845$	0	0
$846$	0	0
$847$	0	0
$848$	6.00000i	0.206041i
$849$	−12.0000	−0.411839
$850$	0	0
$851$	0	0
$852$	0	0
$853$	− 2.00000i	− 0.0684787i	−0.999414	−	0.0342393i	$-0.989099\pi$
$853$	− 2.00000i	− 0.0684787i	0.999414	−	0.0342393i	$-0.0109009\pi$
$854$	0	0
$855$	0	0
$856$	−4.00000	−0.136717
$857$	6.00000i	0.204956i	0.994735	+	0.102478i	$0.0326771\pi$
$857$	6.00000i	0.204956i	−0.994735	+	0.102478i	$0.967323\pi$
$858$	8.00000i	0.273115i
$859$	36.0000	1.22830	0.614152	−	0.789188i	$-0.289498\pi$
$859$	36.0000	1.22830	0.614152	+	0.789188i	$0.289498\pi$
$860$	0	0
$861$	0	0
$862$	8.00000i	0.272481i
$863$	− 32.0000i	− 1.08929i	−0.838666	−	0.544646i	$-0.816664\pi$
$863$	− 32.0000i	− 1.08929i	0.838666	−	0.544646i	$-0.183336\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	−14.0000	−0.475739
$867$	− 1.00000i	− 0.0339618i
$868$	0	0
$869$	−32.0000	−1.08553
$870$	0	0
$871$	24.0000	0.813209
$872$	10.0000i	0.338643i
$873$	− 14.0000i	− 0.473828i
$874$	0	0
$875$	0	0
$876$	10.0000	0.337869
$877$	− 22.0000i	− 0.742887i	−0.928456	−	0.371444i	$-0.878863\pi$
$877$	− 22.0000i	− 0.742887i	0.928456	−	0.371444i	$-0.121137\pi$
$878$	16.0000i	0.539974i
$879$	26.0000	0.876958
$880$	0	0
$881$	2.00000	0.0673817	0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000	0.0673817	0.0336909	+	0.999432i	$0.489274\pi$
$882$	7.00000i	0.235702i
$883$	36.0000i	1.21150i	0.795656	+	0.605748i	$0.207126\pi$
$883$	36.0000i	1.21150i	−0.795656	+	0.605748i	$0.792874\pi$
$884$	2.00000	0.0672673
$885$	0	0
$886$	−4.00000	−0.134383
$887$	48.0000i	1.61168i	0.592132	+	0.805841i	$0.298286\pi$
$887$	48.0000i	1.61168i	−0.592132	+	0.805841i	$0.701714\pi$
$888$	− 2.00000i	− 0.0671156i
$889$	0	0
$890$	0	0
$891$	−4.00000	−0.134005
$892$	− 16.0000i	− 0.535720i
$893$	0	0
$894$	10.0000	0.334450
$895$	0	0
$896$	0	0
$897$	0	0
$898$	2.00000i	0.0667409i
$899$	80.0000	2.66815
$900$	0	0
$901$	6.00000	0.199889
$902$	40.0000i	1.33185i
$903$	0	0
$904$	−2.00000	−0.0665190
$905$	0	0
$906$	24.0000	0.797347
$907$	− 28.0000i	− 0.929725i	−0.885383	−	0.464862i	$-0.846104\pi$
$907$	− 28.0000i	− 0.929725i	0.885383	−	0.464862i	$-0.153896\pi$
$908$	4.00000i	0.132745i
$909$	10.0000	0.331679
$910$	0	0
$911$	8.00000	0.265052	0.132526	−	0.991180i	$-0.457691\pi$
$911$	8.00000	0.265052	0.132526	+	0.991180i	$0.457691\pi$
$912$	− 4.00000i	− 0.132453i
$913$	− 16.0000i	− 0.529523i
$914$	−10.0000	−0.330771
$915$	0	0
$916$	−26.0000	−0.859064
$917$	0	0
$918$	1.00000i	0.0330049i
$919$	−24.0000	−0.791687	−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000	−0.791687	−0.395843	+	0.918318i	$0.629548\pi$
$920$	0	0
$921$	−12.0000	−0.395413
$922$	− 30.0000i	− 0.987997i
$923$	0	0
$924$	0	0
$925$	0	0
$926$	32.0000	1.05159
$927$	8.00000i	0.262754i
$928$	− 10.0000i	− 0.328266i
$929$	−2.00000	−0.0656179	−0.0328089	−	0.999462i	$-0.510445\pi$
$929$	−2.00000	−0.0656179	−0.0328089	+	0.999462i	$0.510445\pi$
$930$	0	0
$931$	−28.0000	−0.917663
$932$	− 26.0000i	− 0.851658i
$933$	0	0
$934$	12.0000	0.392652
$935$	0	0
$936$	−2.00000	−0.0653720
$937$	22.0000i	0.718709i	0.933201	+	0.359354i	$0.117003\pi$
$937$	22.0000i	0.718709i	−0.933201	+	0.359354i	$0.882997\pi$
$938$	0	0
$939$	−10.0000	−0.326338
$940$	0	0
$941$	−10.0000	−0.325991	−0.162995	−	0.986627i	$-0.552116\pi$
$941$	−10.0000	−0.325991	−0.162995	+	0.986627i	$0.552116\pi$
$942$	2.00000i	0.0651635i
$943$	0	0
$944$	−12.0000	−0.390567
$945$	0	0
$946$	−48.0000	−1.56061
$947$	28.0000i	0.909878i	0.890523	+	0.454939i	$0.150339\pi$
$947$	28.0000i	0.909878i	−0.890523	+	0.454939i	$0.849661\pi$
$948$	− 8.00000i	− 0.259828i
$949$	20.0000	0.649227
$950$	0	0
$951$	6.00000	0.194563
$952$	0	0
$953$	− 6.00000i	− 0.194359i	−0.995267	−	0.0971795i	$-0.969018\pi$
$953$	− 6.00000i	− 0.194359i	0.995267	−	0.0971795i	$-0.0309821\pi$
$954$	−6.00000	−0.194257
$955$	0	0
$956$	0	0
$957$	− 40.0000i	− 1.29302i
$958$	24.0000i	0.775405i
$959$	0	0
$960$	0	0
$961$	33.0000	1.06452
$962$	− 4.00000i	− 0.128965i
$963$	− 4.00000i	− 0.128898i
$964$	−2.00000	−0.0644157
$965$	0	0
$966$	0	0
$967$	8.00000i	0.257263i	0.991692	+	0.128631i	$0.0410584\pi$
$967$	8.00000i	0.257263i	−0.991692	+	0.128631i	$0.958942\pi$
$968$	5.00000i	0.160706i
$969$	−4.00000	−0.128499
$970$	0	0
$971$	−4.00000	−0.128366	−0.0641831	−	0.997938i	$-0.520444\pi$
$971$	−4.00000	−0.128366	−0.0641831	+	0.997938i	$0.520444\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	0	0
$974$	16.0000	0.512673
$975$	0	0
$976$	−10.0000	−0.320092
$977$	46.0000i	1.47167i	0.677161	+	0.735835i	$0.263210\pi$
$977$	46.0000i	1.47167i	−0.677161	+	0.735835i	$0.736790\pi$
$978$	4.00000i	0.127906i
$979$	−24.0000	−0.767043
$980$	0	0
$981$	−10.0000	−0.319275
$982$	− 12.0000i	− 0.382935i
$983$	48.0000i	1.53096i	0.643458	+	0.765481i	$0.277499\pi$
$983$	48.0000i	1.53096i	−0.643458	+	0.765481i	$0.722501\pi$
$984$	−10.0000	−0.318788
$985$	0	0
$986$	−10.0000	−0.318465
$987$	0	0
$988$	− 8.00000i	− 0.254514i
$989$	0	0
$990$	0	0
$991$	−40.0000	−1.27064	−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000	−1.27064	−0.635321	+	0.772248i	$0.719132\pi$
$992$	− 8.00000i	− 0.254000i
$993$	− 20.0000i	− 0.634681i
$994$	0	0
$995$	0	0
$996$	4.00000	0.126745
$997$	2.00000i	0.0633406i	0.999498	+	0.0316703i	$0.0100827\pi$
$997$	2.00000i	0.0633406i	−0.999498	+	0.0316703i	$0.989917\pi$
$998$	4.00000i	0.126618i
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2550.2.d.m.2449.1		2
5.2	odd	4		102.2.a.c.1.1	✓	1
5.3	odd	4		2550.2.a.c.1.1		1
5.4	even	2	inner	2550.2.d.m.2449.2		2
15.2	even	4		306.2.a.b.1.1		1
15.8	even	4		7650.2.a.ca.1.1		1
20.7	even	4		816.2.a.b.1.1		1
35.27	even	4		4998.2.a.be.1.1		1
40.27	even	4		3264.2.a.bc.1.1		1
40.37	odd	4		3264.2.a.m.1.1		1
60.47	odd	4		2448.2.a.p.1.1		1
85.2	odd	8		1734.2.f.e.1483.2		4
85.32	odd	8		1734.2.f.e.1483.1		4
85.42	odd	8		1734.2.f.e.829.1		4
85.47	odd	4		1734.2.b.b.577.2		2
85.67	odd	4		1734.2.a.j.1.1		1
85.72	odd	4		1734.2.b.b.577.1		2
85.77	odd	8		1734.2.f.e.829.2		4
120.77	even	4		9792.2.a.k.1.1		1
120.107	odd	4		9792.2.a.l.1.1		1
255.152	even	4		5202.2.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
102.2.a.c.1.1	✓	1	5.2	odd	4
306.2.a.b.1.1		1	15.2	even	4
816.2.a.b.1.1		1	20.7	even	4
1734.2.a.j.1.1		1	85.67	odd	4
1734.2.b.b.577.1		2	85.72	odd	4
1734.2.b.b.577.2		2	85.47	odd	4
1734.2.f.e.829.1		4	85.42	odd	8
1734.2.f.e.829.2		4	85.77	odd	8
1734.2.f.e.1483.1		4	85.32	odd	8
1734.2.f.e.1483.2		4	85.2	odd	8
2448.2.a.p.1.1		1	60.47	odd	4
2550.2.a.c.1.1		1	5.3	odd	4
2550.2.d.m.2449.1		2	1.1	even	1	trivial
2550.2.d.m.2449.2		2	5.4	even	2	inner
3264.2.a.m.1.1		1	40.37	odd	4
3264.2.a.bc.1.1		1	40.27	even	4
4998.2.a.be.1.1		1	35.27	even	4
5202.2.a.c.1.1		1	255.152	even	4
7650.2.a.ca.1.1		1	15.8	even	4
9792.2.a.k.1.1		1	120.77	even	4
9792.2.a.l.1.1		1	120.107	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 $\F_{q}$
Belyi maps

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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} -4.00000 q^{11} -1.00000i q^{12} -2.00000i q^{13} +1.00000 q^{16} -1.00000i q^{17} +1.00000i q^{18} -4.00000 q^{19} +4.00000i q^{22} -1.00000 q^{24} -2.00000 q^{26} -1.00000i q^{27} +10.0000 q^{29} +8.00000 q^{31} -1.00000i q^{32} -4.00000i q^{33} -1.00000 q^{34} +1.00000 q^{36} +2.00000i q^{37} +4.00000i q^{38} +2.00000 q^{39} +10.0000 q^{41} +12.0000i q^{43} +4.00000 q^{44} +1.00000i q^{48} +7.00000 q^{49} +1.00000 q^{51} +2.00000i q^{52} +6.00000i q^{53} -1.00000 q^{54} -4.00000i q^{57} -10.0000i q^{58} -12.0000 q^{59} -10.0000 q^{61} -8.00000i q^{62} -1.00000 q^{64} -4.00000 q^{66} +12.0000i q^{67} +1.00000i q^{68} -1.00000i q^{72} +10.0000i q^{73} +2.00000 q^{74} +4.00000 q^{76} -2.00000i q^{78} +8.00000 q^{79} +1.00000 q^{81} -10.0000i q^{82} +4.00000i q^{83} +12.0000 q^{86} +10.0000i q^{87} -4.00000i q^{88} +6.00000 q^{89} +8.00000i q^{93} +1.00000 q^{96} +14.0000i q^{97} -7.00000i q^{98} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 8 q^{11} + 2 q^{16} - 8 q^{19} - 2 q^{24} - 4 q^{26} + 20 q^{29} + 16 q^{31} - 2 q^{34} + 2 q^{36} + 4 q^{39} + 20 q^{41} + 8 q^{44} + 14 q^{49} + 2 q^{51} - 2 q^{54}+ \cdots + 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 - 8 * q^11 + 2 * q^16 - 8 * q^19 - 2 * q^24 - 4 * q^26 + 20 * q^29 + 16 * q^31 - 2 * q^34 + 2 * q^36 + 4 * q^39 + 20 * q^41 + 8 * q^44 + 14 * q^49 + 2 * q^51 - 2 * q^54 - 24 * q^59 - 20 * q^61 - 2 * q^64 - 8 * q^66 + 4 * q^74 + 8 * q^76 + 16 * q^79 + 2 * q^81 + 24 * q^86 + 12 * q^89 + 2 * q^96 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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