LMFDB - Embedded newform 2850.2.d.l.799.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2850,2,Mod(799,2850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2850, 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("2850.799");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2850.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:	$22.7573645761$
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 570)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			799.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	2850.799
Dual form			2850.2.d.l.799.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -4.00000 q^{11} -1.00000i q^{12} +6.00000i q^{13} -2.00000 q^{14} +1.00000 q^{16} -4.00000i q^{17} +1.00000i q^{18} -1.00000 q^{19} +2.00000 q^{21} +4.00000i q^{22} -1.00000 q^{24} +6.00000 q^{26} -1.00000i q^{27} +2.00000i q^{28} +10.0000 q^{29} -2.00000 q^{31} -1.00000i q^{32} -4.00000i q^{33} -4.00000 q^{34} +1.00000 q^{36} +2.00000i q^{37} +1.00000i q^{38} -6.00000 q^{39} +8.00000 q^{41} -2.00000i q^{42} -8.00000i q^{43} +4.00000 q^{44} +1.00000i q^{48} +3.00000 q^{49} +4.00000 q^{51} -6.00000i q^{52} -6.00000i q^{53} -1.00000 q^{54} +2.00000 q^{56} -1.00000i q^{57} -10.0000i q^{58} +2.00000 q^{59} +2.00000 q^{61} +2.00000i q^{62} +2.00000i q^{63} -1.00000 q^{64} -4.00000 q^{66} -4.00000i q^{67} +4.00000i q^{68} -1.00000i q^{72} -10.0000i q^{73} +2.00000 q^{74} +1.00000 q^{76} +8.00000i q^{77} +6.00000i q^{78} +2.00000 q^{79} +1.00000 q^{81} -8.00000i q^{82} -10.0000i q^{83} -2.00000 q^{84} -8.00000 q^{86} +10.0000i q^{87} -4.00000i q^{88} +12.0000 q^{89} +12.0000 q^{91} -2.00000i q^{93} +1.00000 q^{96} +2.00000i q^{97} -3.00000i q^{98} +4.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9	$ 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 8 q^{11} - 4 q^{14} + 2 q^{16} - 2 q^{19} + 4 q^{21} - 2 q^{24} + 12 q^{26} + 20 q^{29} - 4 q^{31} - 8 q^{34} + 2 q^{36} - 12 q^{39} + 16 q^{41} + 8 q^{44} + 6 q^{49} + 8 q^{51} - 2 q^{54} + 4 q^{56} + 4 q^{59} + 4 q^{61} - 2 q^{64} - 8 q^{66} + 4 q^{74} + 2 q^{76} + 4 q^{79} + 2 q^{81} - 4 q^{84} - 16 q^{86} + 24 q^{89} + 24 q^{91} + 2 q^{96} + 8 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 - 8 * q^11 - 4 * q^14 + 2 * q^16 - 2 * q^19 + 4 * q^21 - 2 * q^24 + 12 * q^26 + 20 * q^29 - 4 * q^31 - 8 * q^34 + 2 * q^36 - 12 * q^39 + 16 * q^41 + 8 * q^44 + 6 * q^49 + 8 * q^51 - 2 * q^54 + 4 * q^56 + 4 * q^59 + 4 * q^61 - 2 * q^64 - 8 * q^66 + 4 * q^74 + 2 * q^76 + 4 * q^79 + 2 * q^81 - 4 * q^84 - 16 * q^86 + 24 * q^89 + 24 * q^91 + 2 * q^96 + 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2850\mathbb{Z}\right)^\times$.

$n$	$1027$	$1351$	$1901$
$\chi(n)$	$-1$	$1$	$1$

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
$7$	− 2.00000i	− 0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$	− 2.00000i	− 0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	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
$13$	6.00000i	1.66410i	0.554700	+	0.832050i	$0.312833\pi$
$13$	6.00000i	1.66410i	−0.554700	+	0.832050i	$0.687167\pi$
$14$	−2.00000	−0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	− 4.00000i	− 0.970143i	−0.874475	−	0.485071i	$-0.838794\pi$
$17$	− 4.00000i	− 0.970143i	0.874475	−	0.485071i	$-0.161206\pi$
$18$	1.00000i	0.235702i
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	2.00000	0.436436
$22$	4.00000i	0.852803i
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	6.00000	1.17670
$27$	− 1.00000i	− 0.192450i
$28$	2.00000i	0.377964i
$29$	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
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	− 1.00000i	− 0.176777i
$33$	− 4.00000i	− 0.696311i
$34$	−4.00000	−0.685994
$35$	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	$0.947432\pi$
$38$	1.00000i	0.162221i
$39$	−6.00000	−0.960769
$40$	0	0
$41$	8.00000	1.24939	0.624695	−	0.780869i	$-0.285223\pi$
$41$	8.00000	1.24939	0.624695	+	0.780869i	$0.285223\pi$
$42$	− 2.00000i	− 0.308607i
$43$	− 8.00000i	− 1.21999i	−0.792406	−	0.609994i	$-0.791172\pi$
$43$	− 8.00000i	− 1.21999i	0.792406	−	0.609994i	$-0.208828\pi$
$44$	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
$49$	3.00000	0.428571
$50$	0	0
$51$	4.00000	0.560112
$52$	− 6.00000i	− 0.832050i
$53$	− 6.00000i	− 0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$	− 6.00000i	− 0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	2.00000	0.267261
$57$	− 1.00000i	− 0.132453i
$58$	− 10.0000i	− 1.31306i
$59$	2.00000	0.260378	0.130189	−	0.991489i	$-0.458442\pi$
$59$	2.00000	0.260378	0.130189	+	0.991489i	$0.458442\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	2.00000i	0.254000i
$63$	2.00000i	0.251976i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−4.00000	−0.492366
$67$	− 4.00000i	− 0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$	− 4.00000i	− 0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$	4.00000i	0.485071i
$69$	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	$-0.801014\pi$
$73$	− 10.0000i	− 1.17041i	0.810885	−	0.585206i	$-0.198986\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	1.00000	0.114708
$77$	8.00000i	0.911685i
$78$	6.00000i	0.679366i
$79$	2.00000	0.225018	0.112509	−	0.993651i	$-0.464111\pi$
$79$	2.00000	0.225018	0.112509	+	0.993651i	$0.464111\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 8.00000i	− 0.883452i
$83$	− 10.0000i	− 1.09764i	−0.835940	−	0.548821i	$-0.815077\pi$
$83$	− 10.0000i	− 1.09764i	0.835940	−	0.548821i	$-0.184923\pi$
$84$	−2.00000	−0.218218
$85$	0	0
$86$	−8.00000	−0.862662
$87$	10.0000i	1.07211i
$88$	− 4.00000i	− 0.426401i
$89$	12.0000	1.27200	0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000	1.27200	0.635999	+	0.771690i	$0.280588\pi$
$90$	0	0
$91$	12.0000	1.25794
$92$	0	0
$93$	− 2.00000i	− 0.207390i
$94$	0	0
$95$	0	0
$96$	1.00000	0.102062
$97$	2.00000i	0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$	2.00000i	0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$	− 3.00000i	− 0.303046i
$99$	4.00000	0.402015
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	− 4.00000i	− 0.396059i
$103$	4.00000i	0.394132i	0.980390	+	0.197066i	$0.0631413\pi$
$103$	4.00000i	0.394132i	−0.980390	+	0.197066i	$0.936859\pi$
$104$	−6.00000	−0.588348
$105$	0	0
$106$	−6.00000	−0.582772
$107$	− 8.00000i	− 0.773389i	−0.922208	−	0.386695i	$-0.873617\pi$
$107$	− 8.00000i	− 0.773389i	0.922208	−	0.386695i	$-0.126383\pi$
$108$	1.00000i	0.0962250i
$109$	−12.0000	−1.14939	−0.574696	−	0.818367i	$-0.694880\pi$
$109$	−12.0000	−1.14939	−0.574696	+	0.818367i	$0.694880\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	− 2.00000i	− 0.188982i
$113$	14.0000i	1.31701i	0.752577	+	0.658505i	$0.228811\pi$
$113$	14.0000i	1.31701i	−0.752577	+	0.658505i	$0.771189\pi$
$114$	−1.00000	−0.0936586
$115$	0	0
$116$	−10.0000	−0.928477
$117$	− 6.00000i	− 0.554700i
$118$	− 2.00000i	− 0.184115i
$119$	−8.00000	−0.733359
$120$	0	0
$121$	5.00000	0.454545
$122$	− 2.00000i	− 0.181071i
$123$	8.00000i	0.721336i
$124$	2.00000	0.179605
$125$	0	0
$126$	2.00000	0.178174
$127$	20.0000i	1.77471i	0.461084	+	0.887357i	$0.347461\pi$
$127$	20.0000i	1.77471i	−0.461084	+	0.887357i	$0.652539\pi$
$128$	1.00000i	0.0883883i
$129$	8.00000	0.704361
$130$	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
$133$	2.00000i	0.173422i
$134$	−4.00000	−0.345547
$135$	0	0
$136$	4.00000	0.342997
$137$	− 16.0000i	− 1.36697i	−0.729964	−	0.683486i	$-0.760463\pi$
$137$	− 16.0000i	− 1.36697i	0.729964	−	0.683486i	$-0.239537\pi$
$138$	0	0
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 24.0000i	− 2.00698i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−10.0000	−0.827606
$147$	3.00000i	0.247436i
$148$	− 2.00000i	− 0.164399i
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	0	0
$151$	18.0000	1.46482	0.732410	−	0.680864i	$-0.238396\pi$
$151$	18.0000	1.46482	0.732410	+	0.680864i	$0.238396\pi$
$152$	− 1.00000i	− 0.0811107i
$153$	4.00000i	0.323381i
$154$	8.00000	0.644658
$155$	0	0
$156$	6.00000	0.480384
$157$	− 20.0000i	− 1.59617i	−0.602542	−	0.798087i	$-0.705846\pi$
$157$	− 20.0000i	− 1.59617i	0.602542	−	0.798087i	$-0.294154\pi$
$158$	− 2.00000i	− 0.159111i
$159$	6.00000	0.475831
$160$	0	0
$161$	0	0
$162$	− 1.00000i	− 0.0785674i
$163$	− 20.0000i	− 1.56652i	−0.621694	−	0.783260i	$-0.713555\pi$
$163$	− 20.0000i	− 1.56652i	0.621694	−	0.783260i	$-0.286445\pi$
$164$	−8.00000	−0.624695
$165$	0	0
$166$	−10.0000	−0.776151
$167$	16.0000i	1.23812i	0.785345	+	0.619059i	$0.212486\pi$
$167$	16.0000i	1.23812i	−0.785345	+	0.619059i	$0.787514\pi$
$168$	2.00000i	0.154303i
$169$	−23.0000	−1.76923
$170$	0	0
$171$	1.00000	0.0764719
$172$	8.00000i	0.609994i
$173$	− 14.0000i	− 1.06440i	−0.846619	−	0.532200i	$-0.821365\pi$
$173$	− 14.0000i	− 1.06440i	0.846619	−	0.532200i	$-0.178635\pi$
$174$	10.0000	0.758098
$175$	0	0
$176$	−4.00000	−0.301511
$177$	2.00000i	0.150329i
$178$	− 12.0000i	− 0.899438i
$179$	22.0000	1.64436	0.822179	−	0.569230i	$-0.192758\pi$
$179$	22.0000	1.64436	0.822179	+	0.569230i	$0.192758\pi$
$180$	0	0
$181$	12.0000	0.891953	0.445976	−	0.895045i	$-0.352856\pi$
$181$	12.0000	0.891953	0.445976	+	0.895045i	$0.352856\pi$
$182$	− 12.0000i	− 0.889499i
$183$	2.00000i	0.147844i
$184$	0	0
$185$	0	0
$186$	−2.00000	−0.146647
$187$	16.0000i	1.17004i
$188$	0	0
$189$	−2.00000	−0.145479
$190$	0	0
$191$	16.0000	1.15772	0.578860	−	0.815427i	$-0.303498\pi$
$191$	16.0000	1.15772	0.578860	+	0.815427i	$0.303498\pi$
$192$	− 1.00000i	− 0.0721688i
$193$	10.0000i	0.719816i	0.932988	+	0.359908i	$0.117192\pi$
$193$	10.0000i	0.719816i	−0.932988	+	0.359908i	$0.882808\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	−3.00000	−0.214286
$197$	− 6.00000i	− 0.427482i	−0.976890	−	0.213741i	$-0.931435\pi$
$197$	− 6.00000i	− 0.427482i	0.976890	−	0.213741i	$-0.0685649\pi$
$198$	− 4.00000i	− 0.284268i
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	− 6.00000i	− 0.422159i
$203$	− 20.0000i	− 1.40372i
$204$	−4.00000	−0.280056
$205$	0	0
$206$	4.00000	0.278693
$207$	0	0
$208$	6.00000i	0.416025i
$209$	4.00000	0.276686
$210$	0	0
$211$	16.0000	1.10149	0.550743	−	0.834675i	$-0.314345\pi$
$211$	16.0000	1.10149	0.550743	+	0.834675i	$0.314345\pi$
$212$	6.00000i	0.412082i
$213$	0	0
$214$	−8.00000	−0.546869
$215$	0	0
$216$	1.00000	0.0680414
$217$	4.00000i	0.271538i
$218$	12.0000i	0.812743i
$219$	10.0000	0.675737
$220$	0	0
$221$	24.0000	1.61441
$222$	2.00000i	0.134231i
$223$	− 16.0000i	− 1.07144i	−0.844396	−	0.535720i	$-0.820040\pi$
$223$	− 16.0000i	− 1.07144i	0.844396	−	0.535720i	$-0.179960\pi$
$224$	−2.00000	−0.133631
$225$	0	0
$226$	14.0000	0.931266
$227$	− 12.0000i	− 0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$	− 12.0000i	− 0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$	1.00000i	0.0662266i
$229$	−22.0000	−1.45380	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000	−1.45380	−0.726900	+	0.686743i	$0.759040\pi$
$230$	0	0
$231$	−8.00000	−0.526361
$232$	10.0000i	0.656532i
$233$	− 4.00000i	− 0.262049i	−0.991379	−	0.131024i	$-0.958173\pi$
$233$	− 4.00000i	− 0.262049i	0.991379	−	0.131024i	$-0.0418266\pi$
$234$	−6.00000	−0.392232
$235$	0	0
$236$	−2.00000	−0.130189
$237$	2.00000i	0.129914i
$238$	8.00000i	0.518563i
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	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
$244$	−2.00000	−0.128037
$245$	0	0
$246$	8.00000	0.510061
$247$	− 6.00000i	− 0.381771i
$248$	− 2.00000i	− 0.127000i
$249$	10.0000	0.633724
$250$	0	0
$251$	16.0000	1.00991	0.504956	−	0.863145i	$-0.331509\pi$
$251$	16.0000	1.00991	0.504956	+	0.863145i	$0.331509\pi$
$252$	− 2.00000i	− 0.125988i
$253$	0	0
$254$	20.0000	1.25491
$255$	0	0
$256$	1.00000	0.0625000
$257$	6.00000i	0.374270i	0.982334	+	0.187135i	$0.0599201\pi$
$257$	6.00000i	0.374270i	−0.982334	+	0.187135i	$0.940080\pi$
$258$	− 8.00000i	− 0.498058i
$259$	4.00000	0.248548
$260$	0	0
$261$	−10.0000	−0.618984
$262$	12.0000i	0.741362i
$263$	8.00000i	0.493301i	0.969104	+	0.246651i	$0.0793300\pi$
$263$	8.00000i	0.493301i	−0.969104	+	0.246651i	$0.920670\pi$
$264$	4.00000	0.246183
$265$	0	0
$266$	2.00000	0.122628
$267$	12.0000i	0.734388i
$268$	4.00000i	0.244339i
$269$	26.0000	1.58525	0.792624	−	0.609711i	$-0.208714\pi$
$269$	26.0000	1.58525	0.792624	+	0.609711i	$0.208714\pi$
$270$	0	0
$271$	−20.0000	−1.21491	−0.607457	−	0.794353i	$-0.707810\pi$
$271$	−20.0000	−1.21491	−0.607457	+	0.794353i	$0.707810\pi$
$272$	− 4.00000i	− 0.242536i
$273$	12.0000i	0.726273i
$274$	−16.0000	−0.966595
$275$	0	0
$276$	0	0
$277$	− 20.0000i	− 1.20168i	−0.799368	−	0.600842i	$-0.794832\pi$
$277$	− 20.0000i	− 1.20168i	0.799368	−	0.600842i	$-0.205168\pi$
$278$	− 12.0000i	− 0.719712i
$279$	2.00000	0.119737
$280$	0	0
$281$	24.0000	1.43172	0.715860	−	0.698244i	$-0.246035\pi$
$281$	24.0000	1.43172	0.715860	+	0.698244i	$0.246035\pi$
$282$	0	0
$283$	− 16.0000i	− 0.951101i	−0.879688	−	0.475551i	$-0.842249\pi$
$283$	− 16.0000i	− 0.951101i	0.879688	−	0.475551i	$-0.157751\pi$
$284$	0	0
$285$	0	0
$286$	−24.0000	−1.41915
$287$	− 16.0000i	− 0.944450i
$288$	1.00000i	0.0589256i
$289$	1.00000	0.0588235
$290$	0	0
$291$	−2.00000	−0.117242
$292$	10.0000i	0.585206i
$293$	22.0000i	1.28525i	0.766179	+	0.642627i	$0.222155\pi$
$293$	22.0000i	1.28525i	−0.766179	+	0.642627i	$0.777845\pi$
$294$	3.00000	0.174964
$295$	0	0
$296$	−2.00000	−0.116248
$297$	4.00000i	0.232104i
$298$	− 18.0000i	− 1.04271i
$299$	0	0
$300$	0	0
$301$	−16.0000	−0.922225
$302$	− 18.0000i	− 1.03578i
$303$	6.00000i	0.344691i
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	4.00000	0.228665
$307$	− 4.00000i	− 0.228292i	−0.993464	−	0.114146i	$-0.963587\pi$
$307$	− 4.00000i	− 0.228292i	0.993464	−	0.114146i	$-0.0364132\pi$
$308$	− 8.00000i	− 0.455842i
$309$	−4.00000	−0.227552
$310$	0	0
$311$	32.0000	1.81455	0.907277	−	0.420534i	$-0.138157\pi$
$311$	32.0000	1.81455	0.907277	+	0.420534i	$0.138157\pi$
$312$	− 6.00000i	− 0.339683i
$313$	6.00000i	0.339140i	0.985518	+	0.169570i	$0.0542379\pi$
$313$	6.00000i	0.339140i	−0.985518	+	0.169570i	$0.945762\pi$
$314$	−20.0000	−1.12867
$315$	0	0
$316$	−2.00000	−0.112509
$317$	− 30.0000i	− 1.68497i	−0.538721	−	0.842484i	$-0.681092\pi$
$317$	− 30.0000i	− 1.68497i	0.538721	−	0.842484i	$-0.318908\pi$
$318$	− 6.00000i	− 0.336463i
$319$	−40.0000	−2.23957
$320$	0	0
$321$	8.00000	0.446516
$322$	0	0
$323$	4.00000i	0.222566i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−20.0000	−1.10770
$327$	− 12.0000i	− 0.663602i
$328$	8.00000i	0.441726i
$329$	0	0
$330$	0	0
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	10.0000i	0.548821i
$333$	− 2.00000i	− 0.109599i
$334$	16.0000	0.875481
$335$	0	0
$336$	2.00000	0.109109
$337$	22.0000i	1.19842i	0.800593	+	0.599208i	$0.204518\pi$
$337$	22.0000i	1.19842i	−0.800593	+	0.599208i	$0.795482\pi$
$338$	23.0000i	1.25104i
$339$	−14.0000	−0.760376
$340$	0	0
$341$	8.00000	0.433224
$342$	− 1.00000i	− 0.0540738i
$343$	− 20.0000i	− 1.07990i
$344$	8.00000	0.431331
$345$	0	0
$346$	−14.0000	−0.752645
$347$	6.00000i	0.322097i	0.986947	+	0.161048i	$0.0514875\pi$
$347$	6.00000i	0.322097i	−0.986947	+	0.161048i	$0.948512\pi$
$348$	− 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
$351$	6.00000	0.320256
$352$	4.00000i	0.213201i
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	2.00000	0.106299
$355$	0	0
$356$	−12.0000	−0.635999
$357$	− 8.00000i	− 0.423405i
$358$	− 22.0000i	− 1.16274i
$359$	−16.0000	−0.844448	−0.422224	−	0.906492i	$-0.638750\pi$
$359$	−16.0000	−0.844448	−0.422224	+	0.906492i	$0.638750\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	− 12.0000i	− 0.630706i
$363$	5.00000i	0.262432i
$364$	−12.0000	−0.628971
$365$	0	0
$366$	2.00000	0.104542
$367$	34.0000i	1.77479i	0.461014	+	0.887393i	$0.347486\pi$
$367$	34.0000i	1.77479i	−0.461014	+	0.887393i	$0.652514\pi$
$368$	0	0
$369$	−8.00000	−0.416463
$370$	0	0
$371$	−12.0000	−0.623009
$372$	2.00000i	0.103695i
$373$	26.0000i	1.34623i	0.739538	+	0.673114i	$0.235044\pi$
$373$	26.0000i	1.34623i	−0.739538	+	0.673114i	$0.764956\pi$
$374$	16.0000	0.827340
$375$	0	0
$376$	0	0
$377$	60.0000i	3.09016i
$378$	2.00000i	0.102869i
$379$	−16.0000	−0.821865	−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000	−0.821865	−0.410932	+	0.911666i	$0.634797\pi$
$380$	0	0
$381$	−20.0000	−1.02463
$382$	− 16.0000i	− 0.818631i
$383$	24.0000i	1.22634i	0.789950	+	0.613171i	$0.210106\pi$
$383$	24.0000i	1.22634i	−0.789950	+	0.613171i	$0.789894\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	10.0000	0.508987
$387$	8.00000i	0.406663i
$388$	− 2.00000i	− 0.101535i
$389$	−38.0000	−1.92668	−0.963338	−	0.268290i	$-0.913542\pi$
$389$	−38.0000	−1.92668	−0.963338	+	0.268290i	$0.913542\pi$
$390$	0	0
$391$	0	0
$392$	3.00000i	0.151523i
$393$	− 12.0000i	− 0.605320i
$394$	−6.00000	−0.302276
$395$	0	0
$396$	−4.00000	−0.201008
$397$	16.0000i	0.803017i	0.915855	+	0.401508i	$0.131514\pi$
$397$	16.0000i	0.803017i	−0.915855	+	0.401508i	$0.868486\pi$
$398$	20.0000i	1.00251i
$399$	−2.00000	−0.100125
$400$	0	0
$401$	−20.0000	−0.998752	−0.499376	−	0.866385i	$-0.666437\pi$
$401$	−20.0000	−0.998752	−0.499376	+	0.866385i	$0.666437\pi$
$402$	− 4.00000i	− 0.199502i
$403$	− 12.0000i	− 0.597763i
$404$	−6.00000	−0.298511
$405$	0	0
$406$	−20.0000	−0.992583
$407$	− 8.00000i	− 0.396545i
$408$	4.00000i	0.198030i
$409$	6.00000	0.296681	0.148340	−	0.988936i	$-0.452607\pi$
$409$	6.00000	0.296681	0.148340	+	0.988936i	$0.452607\pi$
$410$	0	0
$411$	16.0000	0.789222
$412$	− 4.00000i	− 0.197066i
$413$	− 4.00000i	− 0.196827i
$414$	0	0
$415$	0	0
$416$	6.00000	0.294174
$417$	12.0000i	0.587643i
$418$	− 4.00000i	− 0.195646i
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	−20.0000	−0.974740	−0.487370	−	0.873195i	$-0.662044\pi$
$421$	−20.0000	−0.974740	−0.487370	+	0.873195i	$0.662044\pi$
$422$	− 16.0000i	− 0.778868i
$423$	0	0
$424$	6.00000	0.291386
$425$	0	0
$426$	0	0
$427$	− 4.00000i	− 0.193574i
$428$	8.00000i	0.386695i
$429$	24.0000	1.15873
$430$	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
$433$	26.0000i	1.24948i	0.780833	+	0.624740i	$0.214795\pi$
$433$	26.0000i	1.24948i	−0.780833	+	0.624740i	$0.785205\pi$
$434$	4.00000	0.192006
$435$	0	0
$436$	12.0000	0.574696
$437$	0	0
$438$	− 10.0000i	− 0.477818i
$439$	2.00000	0.0954548	0.0477274	−	0.998860i	$-0.484802\pi$
$439$	2.00000	0.0954548	0.0477274	+	0.998860i	$0.484802\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	− 24.0000i	− 1.14156i
$443$	− 6.00000i	− 0.285069i	−0.989790	−	0.142534i	$-0.954475\pi$
$443$	− 6.00000i	− 0.285069i	0.989790	−	0.142534i	$-0.0455251\pi$
$444$	2.00000	0.0949158
$445$	0	0
$446$	−16.0000	−0.757622
$447$	18.0000i	0.851371i
$448$	2.00000i	0.0944911i
$449$	8.00000	0.377543	0.188772	−	0.982021i	$-0.439549\pi$
$449$	8.00000	0.377543	0.188772	+	0.982021i	$0.439549\pi$
$450$	0	0
$451$	−32.0000	−1.50682
$452$	− 14.0000i	− 0.658505i
$453$	18.0000i	0.845714i
$454$	−12.0000	−0.563188
$455$	0	0
$456$	1.00000	0.0468293
$457$	22.0000i	1.02912i	0.857455	+	0.514558i	$0.172044\pi$
$457$	22.0000i	1.02912i	−0.857455	+	0.514558i	$0.827956\pi$
$458$	22.0000i	1.02799i
$459$	−4.00000	−0.186704
$460$	0	0
$461$	18.0000	0.838344	0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000	0.838344	0.419172	+	0.907907i	$0.362320\pi$
$462$	8.00000i	0.372194i
$463$	34.0000i	1.58011i	0.613033	+	0.790057i	$0.289949\pi$
$463$	34.0000i	1.58011i	−0.613033	+	0.790057i	$0.710051\pi$
$464$	10.0000	0.464238
$465$	0	0
$466$	−4.00000	−0.185296
$467$	− 6.00000i	− 0.277647i	−0.990317	−	0.138823i	$-0.955668\pi$
$467$	− 6.00000i	− 0.277647i	0.990317	−	0.138823i	$-0.0443321\pi$
$468$	6.00000i	0.277350i
$469$	−8.00000	−0.369406
$470$	0	0
$471$	20.0000	0.921551
$472$	2.00000i	0.0920575i
$473$	32.0000i	1.47136i
$474$	2.00000	0.0918630
$475$	0	0
$476$	8.00000	0.366679
$477$	6.00000i	0.274721i
$478$	24.0000i	1.09773i
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	−12.0000	−0.547153
$482$	− 2.00000i	− 0.0910975i
$483$	0	0
$484$	−5.00000	−0.227273
$485$	0	0
$486$	1.00000	0.0453609
$487$	− 4.00000i	− 0.181257i	−0.995885	−	0.0906287i	$-0.971112\pi$
$487$	− 4.00000i	− 0.181257i	0.995885	−	0.0906287i	$-0.0288876\pi$
$488$	2.00000i	0.0905357i
$489$	20.0000	0.904431
$490$	0	0
$491$	4.00000	0.180517	0.0902587	−	0.995918i	$-0.471231\pi$
$491$	4.00000	0.180517	0.0902587	+	0.995918i	$0.471231\pi$
$492$	− 8.00000i	− 0.360668i
$493$	− 40.0000i	− 1.80151i
$494$	−6.00000	−0.269953
$495$	0	0
$496$	−2.00000	−0.0898027
$497$	0	0
$498$	− 10.0000i	− 0.448111i
$499$	28.0000	1.25345	0.626726	−	0.779240i	$-0.284395\pi$
$499$	28.0000	1.25345	0.626726	+	0.779240i	$0.284395\pi$
$500$	0	0
$501$	−16.0000	−0.714827
$502$	− 16.0000i	− 0.714115i
$503$	− 28.0000i	− 1.24846i	−0.781241	−	0.624229i	$-0.785413\pi$
$503$	− 28.0000i	− 1.24846i	0.781241	−	0.624229i	$-0.214587\pi$
$504$	−2.00000	−0.0890871
$505$	0	0
$506$	0	0
$507$	− 23.0000i	− 1.02147i
$508$	− 20.0000i	− 0.887357i
$509$	26.0000	1.15243	0.576215	−	0.817298i	$-0.304529\pi$
$509$	26.0000	1.15243	0.576215	+	0.817298i	$0.304529\pi$
$510$	0	0
$511$	−20.0000	−0.884748
$512$	− 1.00000i	− 0.0441942i
$513$	1.00000i	0.0441511i
$514$	6.00000	0.264649
$515$	0	0
$516$	−8.00000	−0.352180
$517$	0	0
$518$	− 4.00000i	− 0.175750i
$519$	14.0000	0.614532
$520$	0	0
$521$	−4.00000	−0.175243	−0.0876216	−	0.996154i	$-0.527927\pi$
$521$	−4.00000	−0.175243	−0.0876216	+	0.996154i	$0.527927\pi$
$522$	10.0000i	0.437688i
$523$	− 44.0000i	− 1.92399i	−0.273075	−	0.961993i	$-0.588041\pi$
$523$	− 44.0000i	− 1.92399i	0.273075	−	0.961993i	$-0.411959\pi$
$524$	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
$531$	−2.00000	−0.0867926
$532$	− 2.00000i	− 0.0867110i
$533$	48.0000i	2.07911i
$534$	12.0000	0.519291
$535$	0	0
$536$	4.00000	0.172774
$537$	22.0000i	0.949370i
$538$	− 26.0000i	− 1.12094i
$539$	−12.0000	−0.516877
$540$	0	0
$541$	30.0000	1.28980	0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000	1.28980	0.644900	+	0.764267i	$0.276899\pi$
$542$	20.0000i	0.859074i
$543$	12.0000i	0.514969i
$544$	−4.00000	−0.171499
$545$	0	0
$546$	12.0000	0.513553
$547$	36.0000i	1.53925i	0.638497	+	0.769624i	$0.279557\pi$
$547$	36.0000i	1.53925i	−0.638497	+	0.769624i	$0.720443\pi$
$548$	16.0000i	0.683486i
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	−10.0000	−0.426014
$552$	0	0
$553$	− 4.00000i	− 0.170097i
$554$	−20.0000	−0.849719
$555$	0	0
$556$	−12.0000	−0.508913
$557$	30.0000i	1.27114i	0.772043	+	0.635570i	$0.219235\pi$
$557$	30.0000i	1.27114i	−0.772043	+	0.635570i	$0.780765\pi$
$558$	− 2.00000i	− 0.0846668i
$559$	48.0000	2.03018
$560$	0	0
$561$	−16.0000	−0.675521
$562$	− 24.0000i	− 1.01238i
$563$	− 8.00000i	− 0.337160i	−0.985688	−	0.168580i	$-0.946082\pi$
$563$	− 8.00000i	− 0.337160i	0.985688	−	0.168580i	$-0.0539181\pi$
$564$	0	0
$565$	0	0
$566$	−16.0000	−0.672530
$567$	− 2.00000i	− 0.0839921i
$568$	0	0
$569$	−24.0000	−1.00613	−0.503066	−	0.864248i	$-0.667795\pi$
$569$	−24.0000	−1.00613	−0.503066	+	0.864248i	$0.667795\pi$
$570$	0	0
$571$	−36.0000	−1.50655	−0.753277	−	0.657704i	$-0.771528\pi$
$571$	−36.0000	−1.50655	−0.753277	+	0.657704i	$0.771528\pi$
$572$	24.0000i	1.00349i
$573$	16.0000i	0.668410i
$574$	−16.0000	−0.667827
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 42.0000i	− 1.74848i	−0.485491	−	0.874241i	$-0.661359\pi$
$577$	− 42.0000i	− 1.74848i	0.485491	−	0.874241i	$-0.338641\pi$
$578$	− 1.00000i	− 0.0415945i
$579$	−10.0000	−0.415586
$580$	0	0
$581$	−20.0000	−0.829740
$582$	2.00000i	0.0829027i
$583$	24.0000i	0.993978i
$584$	10.0000	0.413803
$585$	0	0
$586$	22.0000	0.908812
$587$	− 18.0000i	− 0.742940i	−0.928445	−	0.371470i	$-0.878854\pi$
$587$	− 18.0000i	− 0.742940i	0.928445	−	0.371470i	$-0.121146\pi$
$588$	− 3.00000i	− 0.123718i
$589$	2.00000	0.0824086
$590$	0	0
$591$	6.00000	0.246807
$592$	2.00000i	0.0821995i
$593$	16.0000i	0.657041i	0.944497	+	0.328521i	$0.106550\pi$
$593$	16.0000i	0.657041i	−0.944497	+	0.328521i	$0.893450\pi$
$594$	4.00000	0.164122
$595$	0	0
$596$	−18.0000	−0.737309
$597$	− 20.0000i	− 0.818546i
$598$	0	0
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	0	0
$601$	−14.0000	−0.571072	−0.285536	−	0.958368i	$-0.592172\pi$
$601$	−14.0000	−0.571072	−0.285536	+	0.958368i	$0.592172\pi$
$602$	16.0000i	0.652111i
$603$	4.00000i	0.162893i
$604$	−18.0000	−0.732410
$605$	0	0
$606$	6.00000	0.243733
$607$	12.0000i	0.487065i	0.969893	+	0.243532i	$0.0783062\pi$
$607$	12.0000i	0.487065i	−0.969893	+	0.243532i	$0.921694\pi$
$608$	1.00000i	0.0405554i
$609$	20.0000	0.810441
$610$	0	0
$611$	0	0
$612$	− 4.00000i	− 0.161690i
$613$	8.00000i	0.323117i	0.986863	+	0.161558i	$0.0516520\pi$
$613$	8.00000i	0.323117i	−0.986863	+	0.161558i	$0.948348\pi$
$614$	−4.00000	−0.161427
$615$	0	0
$616$	−8.00000	−0.322329
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	4.00000i	0.160904i
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	0	0
$621$	0	0
$622$	− 32.0000i	− 1.28308i
$623$	− 24.0000i	− 0.961540i
$624$	−6.00000	−0.240192
$625$	0	0
$626$	6.00000	0.239808
$627$	4.00000i	0.159745i
$628$	20.0000i	0.798087i
$629$	8.00000	0.318981
$630$	0	0
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	2.00000i	0.0795557i
$633$	16.0000i	0.635943i
$634$	−30.0000	−1.19145
$635$	0	0
$636$	−6.00000	−0.237915
$637$	18.0000i	0.713186i
$638$	40.0000i	1.58362i
$639$	0	0
$640$	0	0
$641$	−48.0000	−1.89589	−0.947943	−	0.318440i	$-0.896841\pi$
$641$	−48.0000	−1.89589	−0.947943	+	0.318440i	$0.896841\pi$
$642$	− 8.00000i	− 0.315735i
$643$	− 16.0000i	− 0.630978i	−0.948929	−	0.315489i	$-0.897831\pi$
$643$	− 16.0000i	− 0.630978i	0.948929	−	0.315489i	$-0.102169\pi$
$644$	0	0
$645$	0	0
$646$	4.00000	0.157378
$647$	− 28.0000i	− 1.10079i	−0.834903	−	0.550397i	$-0.814476\pi$
$647$	− 28.0000i	− 1.10079i	0.834903	−	0.550397i	$-0.185524\pi$
$648$	1.00000i	0.0392837i
$649$	−8.00000	−0.314027
$650$	0	0
$651$	−4.00000	−0.156772
$652$	20.0000i	0.783260i
$653$	− 30.0000i	− 1.17399i	−0.809590	−	0.586995i	$-0.800311\pi$
$653$	− 30.0000i	− 1.17399i	0.809590	−	0.586995i	$-0.199689\pi$
$654$	−12.0000	−0.469237
$655$	0	0
$656$	8.00000	0.312348
$657$	10.0000i	0.390137i
$658$	0	0
$659$	30.0000	1.16863	0.584317	−	0.811525i	$-0.301362\pi$
$659$	30.0000	1.16863	0.584317	+	0.811525i	$0.301362\pi$
$660$	0	0
$661$	−20.0000	−0.777910	−0.388955	−	0.921257i	$-0.627164\pi$
$661$	−20.0000	−0.777910	−0.388955	+	0.921257i	$0.627164\pi$
$662$	28.0000i	1.08825i
$663$	24.0000i	0.932083i
$664$	10.0000	0.388075
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	0	0
$668$	− 16.0000i	− 0.619059i
$669$	16.0000	0.618596
$670$	0	0
$671$	−8.00000	−0.308837
$672$	− 2.00000i	− 0.0771517i
$673$	− 2.00000i	− 0.0770943i	−0.999257	−	0.0385472i	$-0.987727\pi$
$673$	− 2.00000i	− 0.0770943i	0.999257	−	0.0385472i	$-0.0122730\pi$
$674$	22.0000	0.847408
$675$	0	0
$676$	23.0000	0.884615
$677$	18.0000i	0.691796i	0.938272	+	0.345898i	$0.112426\pi$
$677$	18.0000i	0.691796i	−0.938272	+	0.345898i	$0.887574\pi$
$678$	14.0000i	0.537667i
$679$	4.00000	0.153506
$680$	0	0
$681$	12.0000	0.459841
$682$	− 8.00000i	− 0.306336i
$683$	12.0000i	0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$683$	12.0000i	0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$684$	−1.00000	−0.0382360
$685$	0	0
$686$	−20.0000	−0.763604
$687$	− 22.0000i	− 0.839352i
$688$	− 8.00000i	− 0.304997i
$689$	36.0000	1.37149
$690$	0	0
$691$	28.0000	1.06517	0.532585	−	0.846376i	$-0.321221\pi$
$691$	28.0000	1.06517	0.532585	+	0.846376i	$0.321221\pi$
$692$	14.0000i	0.532200i
$693$	− 8.00000i	− 0.303895i
$694$	6.00000	0.227757
$695$	0	0
$696$	−10.0000	−0.379049
$697$	− 32.0000i	− 1.21209i
$698$	14.0000i	0.529908i
$699$	4.00000	0.151294
$700$	0	0
$701$	10.0000	0.377695	0.188847	−	0.982006i	$-0.439525\pi$
$701$	10.0000	0.377695	0.188847	+	0.982006i	$0.439525\pi$
$702$	− 6.00000i	− 0.226455i
$703$	− 2.00000i	− 0.0754314i
$704$	4.00000	0.150756
$705$	0	0
$706$	0	0
$707$	− 12.0000i	− 0.451306i
$708$	− 2.00000i	− 0.0751646i
$709$	26.0000	0.976450	0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000	0.976450	0.488225	+	0.872718i	$0.337644\pi$
$710$	0	0
$711$	−2.00000	−0.0750059
$712$	12.0000i	0.449719i
$713$	0	0
$714$	−8.00000	−0.299392
$715$	0	0
$716$	−22.0000	−0.822179
$717$	− 24.0000i	− 0.896296i
$718$	16.0000i	0.597115i
$719$	−8.00000	−0.298350	−0.149175	−	0.988811i	$-0.547662\pi$
$719$	−8.00000	−0.298350	−0.149175	+	0.988811i	$0.547662\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	− 1.00000i	− 0.0372161i
$723$	2.00000i	0.0743808i
$724$	−12.0000	−0.445976
$725$	0	0
$726$	5.00000	0.185567
$727$	− 30.0000i	− 1.11264i	−0.830969	−	0.556319i	$-0.812213\pi$
$727$	− 30.0000i	− 1.11264i	0.830969	−	0.556319i	$-0.187787\pi$
$728$	12.0000i	0.444750i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−32.0000	−1.18356
$732$	− 2.00000i	− 0.0739221i
$733$	8.00000i	0.295487i	0.989026	+	0.147743i	$0.0472010\pi$
$733$	8.00000i	0.295487i	−0.989026	+	0.147743i	$0.952799\pi$
$734$	34.0000	1.25496
$735$	0	0
$736$	0	0
$737$	16.0000i	0.589368i
$738$	8.00000i	0.294484i
$739$	−4.00000	−0.147142	−0.0735712	−	0.997290i	$-0.523440\pi$
$739$	−4.00000	−0.147142	−0.0735712	+	0.997290i	$0.523440\pi$
$740$	0	0
$741$	6.00000	0.220416
$742$	12.0000i	0.440534i
$743$	24.0000i	0.880475i	0.897881	+	0.440237i	$0.145106\pi$
$743$	24.0000i	0.880475i	−0.897881	+	0.440237i	$0.854894\pi$
$744$	2.00000	0.0733236
$745$	0	0
$746$	26.0000	0.951928
$747$	10.0000i	0.365881i
$748$	− 16.0000i	− 0.585018i
$749$	−16.0000	−0.584627
$750$	0	0
$751$	−26.0000	−0.948753	−0.474377	−	0.880322i	$-0.657327\pi$
$751$	−26.0000	−0.948753	−0.474377	+	0.880322i	$0.657327\pi$
$752$	0	0
$753$	16.0000i	0.583072i
$754$	60.0000	2.18507
$755$	0	0
$756$	2.00000	0.0727393
$757$	− 8.00000i	− 0.290765i	−0.989376	−	0.145382i	$-0.953559\pi$
$757$	− 8.00000i	− 0.290765i	0.989376	−	0.145382i	$-0.0464413\pi$
$758$	16.0000i	0.581146i
$759$	0	0
$760$	0	0
$761$	46.0000	1.66750	0.833749	−	0.552143i	$-0.186190\pi$
$761$	46.0000	1.66750	0.833749	+	0.552143i	$0.186190\pi$
$762$	20.0000i	0.724524i
$763$	24.0000i	0.868858i
$764$	−16.0000	−0.578860
$765$	0	0
$766$	24.0000	0.867155
$767$	12.0000i	0.433295i
$768$	1.00000i	0.0360844i
$769$	−30.0000	−1.08183	−0.540914	−	0.841078i	$-0.681921\pi$
$769$	−30.0000	−1.08183	−0.540914	+	0.841078i	$0.681921\pi$
$770$	0	0
$771$	−6.00000	−0.216085
$772$	− 10.0000i	− 0.359908i
$773$	− 30.0000i	− 1.07903i	−0.841978	−	0.539513i	$-0.818609\pi$
$773$	− 30.0000i	− 1.07903i	0.841978	−	0.539513i	$-0.181391\pi$
$774$	8.00000	0.287554
$775$	0	0
$776$	−2.00000	−0.0717958
$777$	4.00000i	0.143499i
$778$	38.0000i	1.36237i
$779$	−8.00000	−0.286630
$780$	0	0
$781$	0	0
$782$	0	0
$783$	− 10.0000i	− 0.357371i
$784$	3.00000	0.107143
$785$	0	0
$786$	−12.0000	−0.428026
$787$	− 52.0000i	− 1.85360i	−0.375555	−	0.926800i	$-0.622548\pi$
$787$	− 52.0000i	− 1.85360i	0.375555	−	0.926800i	$-0.377452\pi$
$788$	6.00000i	0.213741i
$789$	−8.00000	−0.284808
$790$	0	0
$791$	28.0000	0.995565
$792$	4.00000i	0.142134i
$793$	12.0000i	0.426132i
$794$	16.0000	0.567819
$795$	0	0
$796$	20.0000	0.708881
$797$	− 38.0000i	− 1.34603i	−0.739629	−	0.673015i	$-0.764999\pi$
$797$	− 38.0000i	− 1.34603i	0.739629	−	0.673015i	$-0.235001\pi$
$798$	2.00000i	0.0707992i
$799$	0	0
$800$	0	0
$801$	−12.0000	−0.423999
$802$	20.0000i	0.706225i
$803$	40.0000i	1.41157i
$804$	−4.00000	−0.141069
$805$	0	0
$806$	−12.0000	−0.422682
$807$	26.0000i	0.915243i
$808$	6.00000i	0.211079i
$809$	10.0000	0.351581	0.175791	−	0.984428i	$-0.443752\pi$
$809$	10.0000	0.351581	0.175791	+	0.984428i	$0.443752\pi$
$810$	0	0
$811$	−40.0000	−1.40459	−0.702295	−	0.711886i	$-0.747841\pi$
$811$	−40.0000	−1.40459	−0.702295	+	0.711886i	$0.747841\pi$
$812$	20.0000i	0.701862i
$813$	− 20.0000i	− 0.701431i
$814$	−8.00000	−0.280400
$815$	0	0
$816$	4.00000	0.140028
$817$	8.00000i	0.279885i
$818$	− 6.00000i	− 0.209785i
$819$	−12.0000	−0.419314
$820$	0	0
$821$	−10.0000	−0.349002	−0.174501	−	0.984657i	$-0.555831\pi$
$821$	−10.0000	−0.349002	−0.174501	+	0.984657i	$0.555831\pi$
$822$	− 16.0000i	− 0.558064i
$823$	34.0000i	1.18517i	0.805510	+	0.592583i	$0.201892\pi$
$823$	34.0000i	1.18517i	−0.805510	+	0.592583i	$0.798108\pi$
$824$	−4.00000	−0.139347
$825$	0	0
$826$	−4.00000	−0.139178
$827$	− 32.0000i	− 1.11275i	−0.830932	−	0.556375i	$-0.812192\pi$
$827$	− 32.0000i	− 1.11275i	0.830932	−	0.556375i	$-0.187808\pi$
$828$	0	0
$829$	28.0000	0.972480	0.486240	−	0.873825i	$-0.338368\pi$
$829$	28.0000	0.972480	0.486240	+	0.873825i	$0.338368\pi$
$830$	0	0
$831$	20.0000	0.693792
$832$	− 6.00000i	− 0.208013i
$833$	− 12.0000i	− 0.415775i
$834$	12.0000	0.415526
$835$	0	0
$836$	−4.00000	−0.138343
$837$	2.00000i	0.0691301i
$838$	0	0
$839$	−24.0000	−0.828572	−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000	−0.828572	−0.414286	+	0.910147i	$0.635969\pi$
$840$	0	0
$841$	71.0000	2.44828
$842$	20.0000i	0.689246i
$843$	24.0000i	0.826604i
$844$	−16.0000	−0.550743
$845$	0	0
$846$	0	0
$847$	− 10.0000i	− 0.343604i
$848$	− 6.00000i	− 0.206041i
$849$	16.0000	0.549119
$850$	0	0
$851$	0	0
$852$	0	0
$853$	28.0000i	0.958702i	0.877623	+	0.479351i	$0.159128\pi$
$853$	28.0000i	0.958702i	−0.877623	+	0.479351i	$0.840872\pi$
$854$	−4.00000	−0.136877
$855$	0	0
$856$	8.00000	0.273434
$857$	54.0000i	1.84460i	0.386469	+	0.922302i	$0.373695\pi$
$857$	54.0000i	1.84460i	−0.386469	+	0.922302i	$0.626305\pi$
$858$	− 24.0000i	− 0.819346i
$859$	−28.0000	−0.955348	−0.477674	−	0.878537i	$-0.658520\pi$
$859$	−28.0000	−0.955348	−0.477674	+	0.878537i	$0.658520\pi$
$860$	0	0
$861$	16.0000	0.545279
$862$	8.00000i	0.272481i
$863$	− 56.0000i	− 1.90626i	−0.302558	−	0.953131i	$-0.597840\pi$
$863$	− 56.0000i	− 1.90626i	0.302558	−	0.953131i	$-0.402160\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	26.0000	0.883516
$867$	1.00000i	0.0339618i
$868$	− 4.00000i	− 0.135769i
$869$	−8.00000	−0.271381
$870$	0	0
$871$	24.0000	0.813209
$872$	− 12.0000i	− 0.406371i
$873$	− 2.00000i	− 0.0676897i
$874$	0	0
$875$	0	0
$876$	−10.0000	−0.337869
$877$	30.0000i	1.01303i	0.862232	+	0.506514i	$0.169066\pi$
$877$	30.0000i	1.01303i	−0.862232	+	0.506514i	$0.830934\pi$
$878$	− 2.00000i	− 0.0674967i
$879$	−22.0000	−0.742042
$880$	0	0
$881$	10.0000	0.336909	0.168454	−	0.985709i	$-0.446122\pi$
$881$	10.0000	0.336909	0.168454	+	0.985709i	$0.446122\pi$
$882$	3.00000i	0.101015i
$883$	16.0000i	0.538443i	0.963078	+	0.269221i	$0.0867663\pi$
$883$	16.0000i	0.538443i	−0.963078	+	0.269221i	$0.913234\pi$
$884$	−24.0000	−0.807207
$885$	0	0
$886$	−6.00000	−0.201574
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	− 2.00000i	− 0.0671156i
$889$	40.0000	1.34156
$890$	0	0
$891$	−4.00000	−0.134005
$892$	16.0000i	0.535720i
$893$	0	0
$894$	18.0000	0.602010
$895$	0	0
$896$	2.00000	0.0668153
$897$	0	0
$898$	− 8.00000i	− 0.266963i
$899$	−20.0000	−0.667037
$900$	0	0
$901$	−24.0000	−0.799556
$902$	32.0000i	1.06548i
$903$	− 16.0000i	− 0.532447i
$904$	−14.0000	−0.465633
$905$	0	0
$906$	18.0000	0.598010
$907$	36.0000i	1.19536i	0.801735	+	0.597680i	$0.203911\pi$
$907$	36.0000i	1.19536i	−0.801735	+	0.597680i	$0.796089\pi$
$908$	12.0000i	0.398234i
$909$	−6.00000	−0.199007
$910$	0	0
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	− 1.00000i	− 0.0331133i
$913$	40.0000i	1.32381i
$914$	22.0000	0.727695
$915$	0	0
$916$	22.0000	0.726900
$917$	24.0000i	0.792550i
$918$	4.00000i	0.132020i
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	4.00000	0.131804
$922$	− 18.0000i	− 0.592798i
$923$	0	0
$924$	8.00000	0.263181
$925$	0	0
$926$	34.0000	1.11731
$927$	− 4.00000i	− 0.131377i
$928$	− 10.0000i	− 0.328266i
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	0	0
$931$	−3.00000	−0.0983210
$932$	4.00000i	0.131024i
$933$	32.0000i	1.04763i
$934$	−6.00000	−0.196326
$935$	0	0
$936$	6.00000	0.196116
$937$	30.0000i	0.980057i	0.871706	+	0.490029i	$0.163014\pi$
$937$	30.0000i	0.980057i	−0.871706	+	0.490029i	$0.836986\pi$
$938$	8.00000i	0.261209i
$939$	−6.00000	−0.195803
$940$	0	0
$941$	26.0000	0.847576	0.423788	−	0.905761i	$-0.360700\pi$
$941$	26.0000	0.847576	0.423788	+	0.905761i	$0.360700\pi$
$942$	− 20.0000i	− 0.651635i
$943$	0	0
$944$	2.00000	0.0650945
$945$	0	0
$946$	32.0000	1.04041
$947$	10.0000i	0.324956i	0.986712	+	0.162478i	$0.0519487\pi$
$947$	10.0000i	0.324956i	−0.986712	+	0.162478i	$0.948051\pi$
$948$	− 2.00000i	− 0.0649570i
$949$	60.0000	1.94768
$950$	0	0
$951$	30.0000	0.972817
$952$	− 8.00000i	− 0.259281i
$953$	18.0000i	0.583077i	0.956559	+	0.291539i	$0.0941672\pi$
$953$	18.0000i	0.583077i	−0.956559	+	0.291539i	$0.905833\pi$
$954$	6.00000	0.194257
$955$	0	0
$956$	24.0000	0.776215
$957$	− 40.0000i	− 1.29302i
$958$	0	0
$959$	−32.0000	−1.03333
$960$	0	0
$961$	−27.0000	−0.870968
$962$	12.0000i	0.386896i
$963$	8.00000i	0.257796i
$964$	−2.00000	−0.0644157
$965$	0	0
$966$	0	0
$967$	− 26.0000i	− 0.836104i	−0.908423	−	0.418052i	$-0.862713\pi$
$967$	− 26.0000i	− 0.836104i	0.908423	−	0.418052i	$-0.137287\pi$
$968$	5.00000i	0.160706i
$969$	−4.00000	−0.128499
$970$	0	0
$971$	−14.0000	−0.449281	−0.224641	−	0.974442i	$-0.572121\pi$
$971$	−14.0000	−0.449281	−0.224641	+	0.974442i	$0.572121\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	− 24.0000i	− 0.769405i
$974$	−4.00000	−0.128168
$975$	0	0
$976$	2.00000	0.0640184
$977$	− 50.0000i	− 1.59964i	−0.600239	−	0.799821i	$-0.704928\pi$
$977$	− 50.0000i	− 1.59964i	0.600239	−	0.799821i	$-0.295072\pi$
$978$	− 20.0000i	− 0.639529i
$979$	−48.0000	−1.53409
$980$	0	0
$981$	12.0000	0.383131
$982$	− 4.00000i	− 0.127645i
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	−8.00000	−0.255031
$985$	0	0
$986$	−40.0000	−1.27386
$987$	0	0
$988$	6.00000i	0.190885i
$989$	0	0
$990$	0	0
$991$	2.00000	0.0635321	0.0317660	−	0.999495i	$-0.489887\pi$
$991$	2.00000	0.0635321	0.0317660	+	0.999495i	$0.489887\pi$
$992$	2.00000i	0.0635001i
$993$	− 28.0000i	− 0.888553i
$994$	0	0
$995$	0	0
$996$	−10.0000	−0.316862
$997$	− 20.0000i	− 0.633406i	−0.948525	−	0.316703i	$-0.897424\pi$
$997$	− 20.0000i	− 0.633406i	0.948525	−	0.316703i	$-0.102576\pi$
$998$	− 28.0000i	− 0.886325i
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2850.2.d.l.799.1		2
5.2	odd	4		570.2.a.j.1.1	✓	1
5.3	odd	4		2850.2.a.b.1.1		1
5.4	even	2	inner	2850.2.d.l.799.2		2
15.2	even	4		1710.2.a.k.1.1		1
15.8	even	4		8550.2.a.w.1.1		1
20.7	even	4		4560.2.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
570.2.a.j.1.1	✓	1	5.2	odd	4
1710.2.a.k.1.1		1	15.2	even	4
2850.2.a.b.1.1		1	5.3	odd	4
2850.2.d.l.799.1		2	1.1	even	1	trivial
2850.2.d.l.799.2		2	5.4	even	2	inner
4560.2.a.d.1.1		1	20.7	even	4
8550.2.a.w.1.1		1	15.8	even	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:	\( N \)	\(=\)	\( 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2850.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -4.00000 q^{11} -1.00000i q^{12} +6.00000i q^{13} -2.00000 q^{14} +1.00000 q^{16} -4.00000i q^{17} +1.00000i q^{18} -1.00000 q^{19} +2.00000 q^{21} +4.00000i q^{22} -1.00000 q^{24} +6.00000 q^{26} -1.00000i q^{27} +2.00000i q^{28} +10.0000 q^{29} -2.00000 q^{31} -1.00000i q^{32} -4.00000i q^{33} -4.00000 q^{34} +1.00000 q^{36} +2.00000i q^{37} +1.00000i q^{38} -6.00000 q^{39} +8.00000 q^{41} -2.00000i q^{42} -8.00000i q^{43} +4.00000 q^{44} +1.00000i q^{48} +3.00000 q^{49} +4.00000 q^{51} -6.00000i q^{52} -6.00000i q^{53} -1.00000 q^{54} +2.00000 q^{56} -1.00000i q^{57} -10.0000i q^{58} +2.00000 q^{59} +2.00000 q^{61} +2.00000i q^{62} +2.00000i q^{63} -1.00000 q^{64} -4.00000 q^{66} -4.00000i q^{67} +4.00000i q^{68} -1.00000i q^{72} -10.0000i q^{73} +2.00000 q^{74} +1.00000 q^{76} +8.00000i q^{77} +6.00000i q^{78} +2.00000 q^{79} +1.00000 q^{81} -8.00000i q^{82} -10.0000i q^{83} -2.00000 q^{84} -8.00000 q^{86} +10.0000i q^{87} -4.00000i q^{88} +12.0000 q^{89} +12.0000 q^{91} -2.00000i q^{93} +1.00000 q^{96} +2.00000i q^{97} -3.00000i q^{98} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 8 q^{11} - 4 q^{14} + 2 q^{16} - 2 q^{19} + 4 q^{21} - 2 q^{24} + 12 q^{26} + 20 q^{29} - 4 q^{31} - 8 q^{34} + 2 q^{36} - 12 q^{39} + 16 q^{41} + 8 q^{44} + 6 q^{49} + 8 q^{51} - 2 q^{54} + 4 q^{56} + 4 q^{59} + 4 q^{61} - 2 q^{64} - 8 q^{66} + 4 q^{74} + 2 q^{76} + 4 q^{79} + 2 q^{81} - 4 q^{84} - 16 q^{86} + 24 q^{89} + 24 q^{91} + 2 q^{96} + 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 - 8 * q^11 - 4 * q^14 + 2 * q^16 - 2 * q^19 + 4 * q^21 - 2 * q^24 + 12 * q^26 + 20 * q^29 - 4 * q^31 - 8 * q^34 + 2 * q^36 - 12 * q^39 + 16 * q^41 + 8 * q^44 + 6 * q^49 + 8 * q^51 - 2 * q^54 + 4 * q^56 + 4 * q^59 + 4 * q^61 - 2 * q^64 - 8 * q^66 + 4 * q^74 + 2 * q^76 + 4 * q^79 + 2 * q^81 - 4 * q^84 - 16 * q^86 + 24 * q^89 + 24 * q^91 + 2 * q^96 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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