LMFDB - Embedded newform 2850.2.d.k.799.2

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:	$1$
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			799.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	2850.799
Dual form			2850.2.d.k.799.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -4.00000 q^{11} +1.00000i q^{12} +2.00000i q^{13} -4.00000 q^{14} +1.00000 q^{16} -2.00000i q^{17} -1.00000i q^{18} +1.00000 q^{19} +4.00000 q^{21} -4.00000i q^{22} +8.00000i q^{23} -1.00000 q^{24} -2.00000 q^{26} +1.00000i q^{27} -4.00000i q^{28} -6.00000 q^{29} +4.00000 q^{31} +1.00000i q^{32} +4.00000i q^{33} +2.00000 q^{34} +1.00000 q^{36} -10.0000i q^{37} +1.00000i q^{38} +2.00000 q^{39} -2.00000 q^{41} +4.00000i q^{42} -12.0000i q^{43} +4.00000 q^{44} -8.00000 q^{46} -1.00000i q^{48} -9.00000 q^{49} -2.00000 q^{51} -2.00000i q^{52} -6.00000i q^{53} -1.00000 q^{54} +4.00000 q^{56} -1.00000i q^{57} -6.00000i q^{58} -10.0000 q^{61} +4.00000i q^{62} -4.00000i q^{63} -1.00000 q^{64} -4.00000 q^{66} -4.00000i q^{67} +2.00000i q^{68} +8.00000 q^{69} -8.00000 q^{71} +1.00000i q^{72} -2.00000i q^{73} +10.0000 q^{74} -1.00000 q^{76} -16.0000i q^{77} +2.00000i q^{78} +12.0000 q^{79} +1.00000 q^{81} -2.00000i q^{82} +8.00000i q^{83} -4.00000 q^{84} +12.0000 q^{86} +6.00000i q^{87} +4.00000i q^{88} -6.00000 q^{89} -8.00000 q^{91} -8.00000i q^{92} -4.00000i q^{93} +1.00000 q^{96} +18.0000i q^{97} -9.00000i q^{98} +4.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 8 q^{14} + 2 q^{16} + 2 q^{19} + 8 q^{21} - 2 q^{24} - 4 q^{26} - 12 q^{29} + 8 q^{31} + 4 q^{34} + 2 q^{36} + 4 q^{39} - 4 q^{41} + 8 q^{44} - 16 q^{46} - 18 q^{49} - 4 q^{51} - 2 q^{54} + 8 q^{56} - 20 q^{61} - 2 q^{64} - 8 q^{66} + 16 q^{69} - 16 q^{71} + 20 q^{74} - 2 q^{76} + 24 q^{79} + 2 q^{81} - 8 q^{84} + 24 q^{86} - 12 q^{89} - 16 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 - 8 * q^14 + 2 * q^16 + 2 * q^19 + 8 * q^21 - 2 * q^24 - 4 * q^26 - 12 * q^29 + 8 * q^31 + 4 * q^34 + 2 * q^36 + 4 * q^39 - 4 * q^41 + 8 * q^44 - 16 * q^46 - 18 * q^49 - 4 * q^51 - 2 * q^54 + 8 * q^56 - 20 * q^61 - 2 * q^64 - 8 * q^66 + 16 * q^69 - 16 * q^71 + 20 * q^74 - 2 * q^76 + 24 * q^79 + 2 * q^81 - 8 * q^84 + 24 * q^86 - 12 * q^89 - 16 * 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
$7$	4.00000i	1.51186i	0.654654	+	0.755929i	$0.272814\pi$
$7$	4.00000i	1.51186i	−0.654654	+	0.755929i	$0.727186\pi$
$8$	− 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	$0.0894562\pi$
$13$	2.00000i	0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$	−4.00000	−1.06904
$15$	0	0
$16$	1.00000	0.250000
$17$	− 2.00000i	− 0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$	− 2.00000i	− 0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$	− 1.00000i	− 0.235702i
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	4.00000	0.872872
$22$	− 4.00000i	− 0.852803i
$23$	8.00000i	1.66812i	0.551677	+	0.834058i	$0.313988\pi$
$23$	8.00000i	1.66812i	−0.551677	+	0.834058i	$0.686012\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	−2.00000	−0.392232
$27$	1.00000i	0.192450i
$28$	− 4.00000i	− 0.755929i
$29$	−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
$33$	4.00000i	0.696311i
$34$	2.00000	0.342997
$35$	0	0
$36$	1.00000	0.166667
$37$	− 10.0000i	− 1.64399i	−0.569495	−	0.821995i	$-0.692861\pi$
$37$	− 10.0000i	− 1.64399i	0.569495	−	0.821995i	$-0.307139\pi$
$38$	1.00000i	0.162221i
$39$	2.00000	0.320256
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	4.00000i	0.617213i
$43$	− 12.0000i	− 1.82998i	−0.403473	−	0.914991i	$-0.632197\pi$
$43$	− 12.0000i	− 1.82998i	0.403473	−	0.914991i	$-0.367803\pi$
$44$	4.00000	0.603023
$45$	0	0
$46$	−8.00000	−1.17954
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	− 1.00000i	− 0.144338i
$49$	−9.00000	−1.28571
$50$	0	0
$51$	−2.00000	−0.280056
$52$	− 2.00000i	− 0.277350i
$53$	− 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
$56$	4.00000	0.534522
$57$	− 1.00000i	− 0.132453i
$58$	− 6.00000i	− 0.787839i
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	4.00000i	0.508001i
$63$	− 4.00000i	− 0.503953i
$64$	−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	$-0.0785709\pi$
$68$	2.00000i	0.242536i
$69$	8.00000	0.963087
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	1.00000i	0.117851i
$73$	− 2.00000i	− 0.234082i	−0.993127	−	0.117041i	$-0.962659\pi$
$73$	− 2.00000i	− 0.234082i	0.993127	−	0.117041i	$-0.0373409\pi$
$74$	10.0000	1.16248
$75$	0	0
$76$	−1.00000	−0.114708
$77$	− 16.0000i	− 1.82337i
$78$	2.00000i	0.226455i
$79$	12.0000	1.35011	0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000	1.35011	0.675053	+	0.737769i	$0.264121\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 2.00000i	− 0.220863i
$83$	8.00000i	0.878114i	0.898459	+	0.439057i	$0.144687\pi$
$83$	8.00000i	0.878114i	−0.898459	+	0.439057i	$0.855313\pi$
$84$	−4.00000	−0.436436
$85$	0	0
$86$	12.0000	1.29399
$87$	6.00000i	0.643268i
$88$	4.00000i	0.426401i
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	−8.00000	−0.838628
$92$	− 8.00000i	− 0.834058i
$93$	− 4.00000i	− 0.414781i
$94$	0	0
$95$	0	0
$96$	1.00000	0.102062
$97$	18.0000i	1.82762i	0.406138	+	0.913812i	$0.366875\pi$
$97$	18.0000i	1.82762i	−0.406138	+	0.913812i	$0.633125\pi$
$98$	− 9.00000i	− 0.909137i
$99$	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	$0.665755\pi$
$102$	− 2.00000i	− 0.198030i
$103$	− 16.0000i	− 1.57653i	−0.615338	−	0.788263i	$-0.710980\pi$
$103$	− 16.0000i	− 1.57653i	0.615338	−	0.788263i	$-0.289020\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	6.00000	0.582772
$107$	− 12.0000i	− 1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$	− 12.0000i	− 1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$	− 1.00000i	− 0.0962250i
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	−10.0000	−0.949158
$112$	4.00000i	0.377964i
$113$	6.00000i	0.564433i	0.959351	+	0.282216i	$0.0910696\pi$
$113$	6.00000i	0.564433i	−0.959351	+	0.282216i	$0.908930\pi$
$114$	1.00000	0.0936586
$115$	0	0
$116$	6.00000	0.557086
$117$	− 2.00000i	− 0.184900i
$118$	0	0
$119$	8.00000	0.733359
$120$	0	0
$121$	5.00000	0.454545
$122$	− 10.0000i	− 0.905357i
$123$	2.00000i	0.180334i
$124$	−4.00000	−0.359211
$125$	0	0
$126$	4.00000	0.356348
$127$	8.00000i	0.709885i	0.934888	+	0.354943i	$0.115500\pi$
$127$	8.00000i	0.709885i	−0.934888	+	0.354943i	$0.884500\pi$
$128$	− 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
$133$	4.00000i	0.346844i
$134$	4.00000	0.345547
$135$	0	0
$136$	−2.00000	−0.171499
$137$	14.0000i	1.19610i	0.801459	+	0.598050i	$0.204058\pi$
$137$	14.0000i	1.19610i	−0.801459	+	0.598050i	$0.795942\pi$
$138$	8.00000i	0.681005i
$139$	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
$142$	− 8.00000i	− 0.671345i
$143$	− 8.00000i	− 0.668994i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	2.00000	0.165521
$147$	9.00000i	0.742307i
$148$	10.0000i	0.821995i
$149$	−22.0000	−1.80231	−0.901155	−	0.433497i	$-0.857280\pi$
$149$	−22.0000	−1.80231	−0.901155	+	0.433497i	$0.857280\pi$
$150$	0	0
$151$	4.00000	0.325515	0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000	0.325515	0.162758	+	0.986666i	$0.447961\pi$
$152$	− 1.00000i	− 0.0811107i
$153$	2.00000i	0.161690i
$154$	16.0000	1.28932
$155$	0	0
$156$	−2.00000	−0.160128
$157$	− 6.00000i	− 0.478852i	−0.970915	−	0.239426i	$-0.923041\pi$
$157$	− 6.00000i	− 0.478852i	0.970915	−	0.239426i	$-0.0769593\pi$
$158$	12.0000i	0.954669i
$159$	−6.00000	−0.475831
$160$	0	0
$161$	−32.0000	−2.52195
$162$	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	$0.949930\pi$
$164$	2.00000	0.156174
$165$	0	0
$166$	−8.00000	−0.620920
$167$	− 16.0000i	− 1.23812i	−0.785345	−	0.619059i	$-0.787514\pi$
$167$	− 16.0000i	− 1.23812i	0.785345	−	0.619059i	$-0.212486\pi$
$168$	− 4.00000i	− 0.308607i
$169$	9.00000	0.692308
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	12.0000i	0.914991i
$173$	− 14.0000i	− 1.06440i	−0.846619	−	0.532200i	$-0.821365\pi$
$173$	− 14.0000i	− 1.06440i	0.846619	−	0.532200i	$-0.178635\pi$
$174$	−6.00000	−0.454859
$175$	0	0
$176$	−4.00000	−0.301511
$177$	0	0
$178$	− 6.00000i	− 0.449719i
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	− 8.00000i	− 0.592999i
$183$	10.0000i	0.739221i
$184$	8.00000	0.589768
$185$	0	0
$186$	4.00000	0.293294
$187$	8.00000i	0.585018i
$188$	0	0
$189$	−4.00000	−0.290957
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	1.00000i	0.0721688i
$193$	− 18.0000i	− 1.29567i	−0.761781	−	0.647834i	$-0.775675\pi$
$193$	− 18.0000i	− 1.29567i	0.761781	−	0.647834i	$-0.224325\pi$
$194$	−18.0000	−1.29232
$195$	0	0
$196$	9.00000	0.642857
$197$	− 18.0000i	− 1.28245i	−0.767354	−	0.641223i	$-0.778427\pi$
$197$	− 18.0000i	− 1.28245i	0.767354	−	0.641223i	$-0.221573\pi$
$198$	4.00000i	0.284268i
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	− 10.0000i	− 0.703598i
$203$	− 24.0000i	− 1.68447i
$204$	2.00000	0.140028
$205$	0	0
$206$	16.0000	1.11477
$207$	− 8.00000i	− 0.556038i
$208$	2.00000i	0.138675i
$209$	−4.00000	−0.276686
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	6.00000i	0.412082i
$213$	8.00000i	0.548151i
$214$	12.0000	0.820303
$215$	0	0
$216$	1.00000	0.0680414
$217$	16.0000i	1.08615i
$218$	− 10.0000i	− 0.677285i
$219$	−2.00000	−0.135147
$220$	0	0
$221$	4.00000	0.269069
$222$	− 10.0000i	− 0.671156i
$223$	− 16.0000i	− 1.07144i	−0.844396	−	0.535720i	$-0.820040\pi$
$223$	− 16.0000i	− 1.07144i	0.844396	−	0.535720i	$-0.179960\pi$
$224$	−4.00000	−0.267261
$225$	0	0
$226$	−6.00000	−0.399114
$227$	− 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
$231$	−16.0000	−1.05272
$232$	6.00000i	0.393919i
$233$	− 6.00000i	− 0.393073i	−0.980497	−	0.196537i	$-0.937031\pi$
$233$	− 6.00000i	− 0.393073i	0.980497	−	0.196537i	$-0.0629694\pi$
$234$	2.00000	0.130744
$235$	0	0
$236$	0	0
$237$	− 12.0000i	− 0.779484i
$238$	8.00000i	0.518563i
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	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
$246$	−2.00000	−0.127515
$247$	2.00000i	0.127257i
$248$	− 4.00000i	− 0.254000i
$249$	8.00000	0.506979
$250$	0	0
$251$	−28.0000	−1.76734	−0.883672	−	0.468106i	$-0.844936\pi$
$251$	−28.0000	−1.76734	−0.883672	+	0.468106i	$0.844936\pi$
$252$	4.00000i	0.251976i
$253$	− 32.0000i	− 2.01182i
$254$	−8.00000	−0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	18.0000i	1.12281i	0.827541	+	0.561405i	$0.189739\pi$
$257$	18.0000i	1.12281i	−0.827541	+	0.561405i	$0.810261\pi$
$258$	− 12.0000i	− 0.747087i
$259$	40.0000	2.48548
$260$	0	0
$261$	6.00000	0.371391
$262$	− 12.0000i	− 0.741362i
$263$	− 24.0000i	− 1.47990i	−0.672660	−	0.739952i	$-0.734848\pi$
$263$	− 24.0000i	− 1.47990i	0.672660	−	0.739952i	$-0.265152\pi$
$264$	4.00000	0.246183
$265$	0	0
$266$	−4.00000	−0.245256
$267$	6.00000i	0.367194i
$268$	4.00000i	0.244339i
$269$	−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
$271$	−24.0000	−1.45790	−0.728948	−	0.684569i	$-0.759990\pi$
$271$	−24.0000	−1.45790	−0.728948	+	0.684569i	$0.759990\pi$
$272$	− 2.00000i	− 0.121268i
$273$	8.00000i	0.484182i
$274$	−14.0000	−0.845771
$275$	0	0
$276$	−8.00000	−0.481543
$277$	18.0000i	1.08152i	0.841178	+	0.540758i	$0.181862\pi$
$277$	18.0000i	1.08152i	−0.841178	+	0.540758i	$0.818138\pi$
$278$	12.0000i	0.719712i
$279$	−4.00000	−0.239474
$280$	0	0
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	0	0
$283$	4.00000i	0.237775i	0.992908	+	0.118888i	$0.0379328\pi$
$283$	4.00000i	0.237775i	−0.992908	+	0.118888i	$0.962067\pi$
$284$	8.00000	0.474713
$285$	0	0
$286$	8.00000	0.473050
$287$	− 8.00000i	− 0.472225i
$288$	− 1.00000i	− 0.0589256i
$289$	13.0000	0.764706
$290$	0	0
$291$	18.0000	1.05518
$292$	2.00000i	0.117041i
$293$	10.0000i	0.584206i	0.956387	+	0.292103i	$0.0943550\pi$
$293$	10.0000i	0.584206i	−0.956387	+	0.292103i	$0.905645\pi$
$294$	−9.00000	−0.524891
$295$	0	0
$296$	−10.0000	−0.581238
$297$	− 4.00000i	− 0.232104i
$298$	− 22.0000i	− 1.27443i
$299$	−16.0000	−0.925304
$300$	0	0
$301$	48.0000	2.76667
$302$	4.00000i	0.230174i
$303$	10.0000i	0.574485i
$304$	1.00000	0.0573539
$305$	0	0
$306$	−2.00000	−0.114332
$307$	28.0000i	1.59804i	0.601302	+	0.799022i	$0.294649\pi$
$307$	28.0000i	1.59804i	−0.601302	+	0.799022i	$0.705351\pi$
$308$	16.0000i	0.911685i
$309$	−16.0000	−0.910208
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	− 2.00000i	− 0.113228i
$313$	30.0000i	1.69570i	0.530236	+	0.847850i	$0.322103\pi$
$313$	30.0000i	1.69570i	−0.530236	+	0.847850i	$0.677897\pi$
$314$	6.00000	0.338600
$315$	0	0
$316$	−12.0000	−0.675053
$317$	30.0000i	1.68497i	0.538721	+	0.842484i	$0.318908\pi$
$317$	30.0000i	1.68497i	−0.538721	+	0.842484i	$0.681092\pi$
$318$	− 6.00000i	− 0.336463i
$319$	24.0000	1.34374
$320$	0	0
$321$	−12.0000	−0.669775
$322$	− 32.0000i	− 1.78329i
$323$	− 2.00000i	− 0.111283i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−4.00000	−0.221540
$327$	10.0000i	0.553001i
$328$	2.00000i	0.110432i
$329$	0	0
$330$	0	0
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	− 8.00000i	− 0.439057i
$333$	10.0000i	0.547997i
$334$	16.0000	0.875481
$335$	0	0
$336$	4.00000	0.218218
$337$	34.0000i	1.85210i	0.377403	+	0.926049i	$0.376817\pi$
$337$	34.0000i	1.85210i	−0.377403	+	0.926049i	$0.623183\pi$
$338$	9.00000i	0.489535i
$339$	6.00000	0.325875
$340$	0	0
$341$	−16.0000	−0.866449
$342$	− 1.00000i	− 0.0540738i
$343$	− 8.00000i	− 0.431959i
$344$	−12.0000	−0.646997
$345$	0	0
$346$	14.0000	0.752645
$347$	24.0000i	1.28839i	0.764862	+	0.644194i	$0.222807\pi$
$347$	24.0000i	1.28839i	−0.764862	+	0.644194i	$0.777193\pi$
$348$	− 6.00000i	− 0.321634i
$349$	34.0000	1.81998	0.909989	−	0.414632i	$-0.136090\pi$
$349$	34.0000	1.81998	0.909989	+	0.414632i	$0.136090\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	− 4.00000i	− 0.213201i
$353$	− 6.00000i	− 0.319348i	−0.987170	−	0.159674i	$-0.948956\pi$
$353$	− 6.00000i	− 0.319348i	0.987170	−	0.159674i	$-0.0510443\pi$
$354$	0	0
$355$	0	0
$356$	6.00000	0.317999
$357$	− 8.00000i	− 0.423405i
$358$	0	0
$359$	−32.0000	−1.68890	−0.844448	−	0.535638i	$-0.820071\pi$
$359$	−32.0000	−1.68890	−0.844448	+	0.535638i	$0.820071\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	2.00000i	0.105118i
$363$	− 5.00000i	− 0.262432i
$364$	8.00000	0.419314
$365$	0	0
$366$	−10.0000	−0.522708
$367$	− 12.0000i	− 0.626395i	−0.949688	−	0.313197i	$-0.898600\pi$
$367$	− 12.0000i	− 0.626395i	0.949688	−	0.313197i	$-0.101400\pi$
$368$	8.00000i	0.417029i
$369$	2.00000	0.104116
$370$	0	0
$371$	24.0000	1.24602
$372$	4.00000i	0.207390i
$373$	− 30.0000i	− 1.55334i	−0.629907	−	0.776671i	$-0.716907\pi$
$373$	− 30.0000i	− 1.55334i	0.629907	−	0.776671i	$-0.283093\pi$
$374$	−8.00000	−0.413670
$375$	0	0
$376$	0	0
$377$	− 12.0000i	− 0.618031i
$378$	− 4.00000i	− 0.205738i
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	0	0
$381$	8.00000	0.409852
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	18.0000	0.916176
$387$	12.0000i	0.609994i
$388$	− 18.0000i	− 0.913812i
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	16.0000	0.809155
$392$	9.00000i	0.454569i
$393$	12.0000i	0.605320i
$394$	18.0000	0.906827
$395$	0	0
$396$	−4.00000	−0.201008
$397$	2.00000i	0.100377i	0.998740	+	0.0501886i	$0.0159822\pi$
$397$	2.00000i	0.100377i	−0.998740	+	0.0501886i	$0.984018\pi$
$398$	− 16.0000i	− 0.802008i
$399$	4.00000	0.200250
$400$	0	0
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	− 4.00000i	− 0.199502i
$403$	8.00000i	0.398508i
$404$	10.0000	0.497519
$405$	0	0
$406$	24.0000	1.19110
$407$	40.0000i	1.98273i
$408$	2.00000i	0.0990148i
$409$	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
$411$	14.0000	0.690569
$412$	16.0000i	0.788263i
$413$	0	0
$414$	8.00000	0.393179
$415$	0	0
$416$	−2.00000	−0.0980581
$417$	− 12.0000i	− 0.587643i
$418$	− 4.00000i	− 0.195646i
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	−14.0000	−0.682318	−0.341159	−	0.940006i	$-0.610819\pi$
$421$	−14.0000	−0.682318	−0.341159	+	0.940006i	$0.610819\pi$
$422$	− 12.0000i	− 0.584151i
$423$	0	0
$424$	−6.00000	−0.291386
$425$	0	0
$426$	−8.00000	−0.387601
$427$	− 40.0000i	− 1.93574i
$428$	12.0000i	0.580042i
$429$	−8.00000	−0.386244
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	1.00000i	0.0481125i
$433$	− 2.00000i	− 0.0961139i	−0.998845	−	0.0480569i	$-0.984697\pi$
$433$	− 2.00000i	− 0.0961139i	0.998845	−	0.0480569i	$-0.0153029\pi$
$434$	−16.0000	−0.768025
$435$	0	0
$436$	10.0000	0.478913
$437$	8.00000i	0.382692i
$438$	− 2.00000i	− 0.0955637i
$439$	4.00000	0.190910	0.0954548	−	0.995434i	$-0.469569\pi$
$439$	4.00000	0.190910	0.0954548	+	0.995434i	$0.469569\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	4.00000i	0.190261i
$443$	24.0000i	1.14027i	0.821549	+	0.570137i	$0.193110\pi$
$443$	24.0000i	1.14027i	−0.821549	+	0.570137i	$0.806890\pi$
$444$	10.0000	0.474579
$445$	0	0
$446$	16.0000	0.757622
$447$	22.0000i	1.04056i
$448$	− 4.00000i	− 0.188982i
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	0	0
$451$	8.00000	0.376705
$452$	− 6.00000i	− 0.282216i
$453$	− 4.00000i	− 0.187936i
$454$	12.0000	0.563188
$455$	0	0
$456$	−1.00000	−0.0468293
$457$	− 6.00000i	− 0.280668i	−0.990104	−	0.140334i	$-0.955182\pi$
$457$	− 6.00000i	− 0.280668i	0.990104	−	0.140334i	$-0.0448177\pi$
$458$	− 22.0000i	− 1.02799i
$459$	2.00000	0.0933520
$460$	0	0
$461$	14.0000	0.652045	0.326023	−	0.945362i	$-0.394291\pi$
$461$	14.0000	0.652045	0.326023	+	0.945362i	$0.394291\pi$
$462$	− 16.0000i	− 0.744387i
$463$	36.0000i	1.67306i	0.547920	+	0.836531i	$0.315420\pi$
$463$	36.0000i	1.67306i	−0.547920	+	0.836531i	$0.684580\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	6.00000	0.277945
$467$	8.00000i	0.370196i	0.982720	+	0.185098i	$0.0592602\pi$
$467$	8.00000i	0.370196i	−0.982720	+	0.185098i	$0.940740\pi$
$468$	2.00000i	0.0924500i
$469$	16.0000	0.738811
$470$	0	0
$471$	−6.00000	−0.276465
$472$	0	0
$473$	48.0000i	2.20704i
$474$	12.0000	0.551178
$475$	0	0
$476$	−8.00000	−0.366679
$477$	6.00000i	0.274721i
$478$	8.00000i	0.365911i
$479$	16.0000	0.731059	0.365529	−	0.930800i	$-0.380888\pi$
$479$	16.0000	0.731059	0.365529	+	0.930800i	$0.380888\pi$
$480$	0	0
$481$	20.0000	0.911922
$482$	2.00000i	0.0910975i
$483$	32.0000i	1.45605i
$484$	−5.00000	−0.227273
$485$	0	0
$486$	1.00000	0.0453609
$487$	− 32.0000i	− 1.45006i	−0.688718	−	0.725029i	$-0.741826\pi$
$487$	− 32.0000i	− 1.45006i	0.688718	−	0.725029i	$-0.258174\pi$
$488$	10.0000i	0.452679i
$489$	4.00000	0.180886
$490$	0	0
$491$	28.0000	1.26362	0.631811	−	0.775122i	$-0.282312\pi$
$491$	28.0000	1.26362	0.631811	+	0.775122i	$0.282312\pi$
$492$	− 2.00000i	− 0.0901670i
$493$	12.0000i	0.540453i
$494$	−2.00000	−0.0899843
$495$	0	0
$496$	4.00000	0.179605
$497$	− 32.0000i	− 1.43540i
$498$	8.00000i	0.358489i
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	0	0
$501$	−16.0000	−0.714827
$502$	− 28.0000i	− 1.24970i
$503$	− 32.0000i	− 1.42681i	−0.700752	−	0.713405i	$-0.747152\pi$
$503$	− 32.0000i	− 1.42681i	0.700752	−	0.713405i	$-0.252848\pi$
$504$	−4.00000	−0.178174
$505$	0	0
$506$	32.0000	1.42257
$507$	− 9.00000i	− 0.399704i
$508$	− 8.00000i	− 0.354943i
$509$	10.0000	0.443242	0.221621	−	0.975133i	$-0.428865\pi$
$509$	10.0000	0.443242	0.221621	+	0.975133i	$0.428865\pi$
$510$	0	0
$511$	8.00000	0.353899
$512$	1.00000i	0.0441942i
$513$	1.00000i	0.0441511i
$514$	−18.0000	−0.793946
$515$	0	0
$516$	12.0000	0.528271
$517$	0	0
$518$	40.0000i	1.75750i
$519$	−14.0000	−0.614532
$520$	0	0
$521$	−10.0000	−0.438108	−0.219054	−	0.975713i	$-0.570297\pi$
$521$	−10.0000	−0.438108	−0.219054	+	0.975713i	$0.570297\pi$
$522$	6.00000i	0.262613i
$523$	− 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
$526$	24.0000	1.04645
$527$	− 8.00000i	− 0.348485i
$528$	4.00000i	0.174078i
$529$	−41.0000	−1.78261
$530$	0	0
$531$	0	0
$532$	− 4.00000i	− 0.173422i
$533$	− 4.00000i	− 0.173259i
$534$	−6.00000	−0.259645
$535$	0	0
$536$	−4.00000	−0.172774
$537$	0	0
$538$	− 6.00000i	− 0.258678i
$539$	36.0000	1.55063
$540$	0	0
$541$	−18.0000	−0.773880	−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000	−0.773880	−0.386940	+	0.922105i	$0.626468\pi$
$542$	− 24.0000i	− 1.03089i
$543$	− 2.00000i	− 0.0858282i
$544$	2.00000	0.0857493
$545$	0	0
$546$	−8.00000	−0.342368
$547$	− 20.0000i	− 0.855138i	−0.903983	−	0.427569i	$-0.859370\pi$
$547$	− 20.0000i	− 0.855138i	0.903983	−	0.427569i	$-0.140630\pi$
$548$	− 14.0000i	− 0.598050i
$549$	10.0000	0.426790
$550$	0	0
$551$	−6.00000	−0.255609
$552$	− 8.00000i	− 0.340503i
$553$	48.0000i	2.04117i
$554$	−18.0000	−0.764747
$555$	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	$0.780765\pi$
$558$	− 4.00000i	− 0.169334i
$559$	24.0000	1.01509
$560$	0	0
$561$	8.00000	0.337760
$562$	30.0000i	1.26547i
$563$	20.0000i	0.842900i	0.906852	+	0.421450i	$0.138479\pi$
$563$	20.0000i	0.842900i	−0.906852	+	0.421450i	$0.861521\pi$
$564$	0	0
$565$	0	0
$566$	−4.00000	−0.168133
$567$	4.00000i	0.167984i
$568$	8.00000i	0.335673i
$569$	18.0000	0.754599	0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000	0.754599	0.377300	+	0.926091i	$0.376853\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	8.00000i	0.334497i
$573$	0	0
$574$	8.00000	0.333914
$575$	0	0
$576$	1.00000	0.0416667
$577$	2.00000i	0.0832611i	0.999133	+	0.0416305i	$0.0132552\pi$
$577$	2.00000i	0.0832611i	−0.999133	+	0.0416305i	$0.986745\pi$
$578$	13.0000i	0.540729i
$579$	−18.0000	−0.748054
$580$	0	0
$581$	−32.0000	−1.32758
$582$	18.0000i	0.746124i
$583$	24.0000i	0.993978i
$584$	−2.00000	−0.0827606
$585$	0	0
$586$	−10.0000	−0.413096
$587$	− 40.0000i	− 1.65098i	−0.564419	−	0.825488i	$-0.690900\pi$
$587$	− 40.0000i	− 1.65098i	0.564419	−	0.825488i	$-0.309100\pi$
$588$	− 9.00000i	− 0.371154i
$589$	4.00000	0.164817
$590$	0	0
$591$	−18.0000	−0.740421
$592$	− 10.0000i	− 0.410997i
$593$	10.0000i	0.410651i	0.978694	+	0.205325i	$0.0658253\pi$
$593$	10.0000i	0.410651i	−0.978694	+	0.205325i	$0.934175\pi$
$594$	4.00000	0.164122
$595$	0	0
$596$	22.0000	0.901155
$597$	16.0000i	0.654836i
$598$	− 16.0000i	− 0.654289i
$599$	−32.0000	−1.30748	−0.653742	−	0.756717i	$-0.726802\pi$
$599$	−32.0000	−1.30748	−0.653742	+	0.756717i	$0.726802\pi$
$600$	0	0
$601$	−6.00000	−0.244745	−0.122373	−	0.992484i	$-0.539050\pi$
$601$	−6.00000	−0.244745	−0.122373	+	0.992484i	$0.539050\pi$
$602$	48.0000i	1.95633i
$603$	4.00000i	0.162893i
$604$	−4.00000	−0.162758
$605$	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	$0.838067\pi$
$608$	1.00000i	0.0405554i
$609$	−24.0000	−0.972529
$610$	0	0
$611$	0	0
$612$	− 2.00000i	− 0.0808452i
$613$	6.00000i	0.242338i	0.992632	+	0.121169i	$0.0386643\pi$
$613$	6.00000i	0.242338i	−0.992632	+	0.121169i	$0.961336\pi$
$614$	−28.0000	−1.12999
$615$	0	0
$616$	−16.0000	−0.644658
$617$	30.0000i	1.20775i	0.797077	+	0.603877i	$0.206378\pi$
$617$	30.0000i	1.20775i	−0.797077	+	0.603877i	$0.793622\pi$
$618$	− 16.0000i	− 0.643614i
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	0	0
$621$	−8.00000	−0.321029
$622$	24.0000i	0.962312i
$623$	− 24.0000i	− 0.961540i
$624$	2.00000	0.0800641
$625$	0	0
$626$	−30.0000	−1.19904
$627$	4.00000i	0.159745i
$628$	6.00000i	0.239426i
$629$	−20.0000	−0.797452
$630$	0	0
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	− 12.0000i	− 0.477334i
$633$	12.0000i	0.476957i
$634$	−30.0000	−1.19145
$635$	0	0
$636$	6.00000	0.237915
$637$	− 18.0000i	− 0.713186i
$638$	24.0000i	0.950169i
$639$	8.00000	0.316475
$640$	0	0
$641$	−10.0000	−0.394976	−0.197488	−	0.980305i	$-0.563278\pi$
$641$	−10.0000	−0.394976	−0.197488	+	0.980305i	$0.563278\pi$
$642$	− 12.0000i	− 0.473602i
$643$	28.0000i	1.10421i	0.833774	+	0.552106i	$0.186176\pi$
$643$	28.0000i	1.10421i	−0.833774	+	0.552106i	$0.813824\pi$
$644$	32.0000	1.26098
$645$	0	0
$646$	2.00000	0.0786889
$647$	− 32.0000i	− 1.25805i	−0.777385	−	0.629025i	$-0.783454\pi$
$647$	− 32.0000i	− 1.25805i	0.777385	−	0.629025i	$-0.216546\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	0	0
$650$	0	0
$651$	16.0000	0.627089
$652$	− 4.00000i	− 0.156652i
$653$	− 14.0000i	− 0.547862i	−0.961749	−	0.273931i	$-0.911676\pi$
$653$	− 14.0000i	− 0.547862i	0.961749	−	0.273931i	$-0.0883240\pi$
$654$	−10.0000	−0.391031
$655$	0	0
$656$	−2.00000	−0.0780869
$657$	2.00000i	0.0780274i
$658$	0	0
$659$	16.0000	0.623272	0.311636	−	0.950202i	$-0.399123\pi$
$659$	16.0000	0.623272	0.311636	+	0.950202i	$0.399123\pi$
$660$	0	0
$661$	18.0000	0.700119	0.350059	−	0.936727i	$-0.386161\pi$
$661$	18.0000	0.700119	0.350059	+	0.936727i	$0.386161\pi$
$662$	− 12.0000i	− 0.466393i
$663$	− 4.00000i	− 0.155347i
$664$	8.00000	0.310460
$665$	0	0
$666$	−10.0000	−0.387492
$667$	− 48.0000i	− 1.85857i
$668$	16.0000i	0.619059i
$669$	−16.0000	−0.618596
$670$	0	0
$671$	40.0000	1.54418
$672$	4.00000i	0.154303i
$673$	14.0000i	0.539660i	0.962908	+	0.269830i	$0.0869676\pi$
$673$	14.0000i	0.539660i	−0.962908	+	0.269830i	$0.913032\pi$
$674$	−34.0000	−1.30963
$675$	0	0
$676$	−9.00000	−0.346154
$677$	6.00000i	0.230599i	0.993331	+	0.115299i	$0.0367827\pi$
$677$	6.00000i	0.230599i	−0.993331	+	0.115299i	$0.963217\pi$
$678$	6.00000i	0.230429i
$679$	−72.0000	−2.76311
$680$	0	0
$681$	−12.0000	−0.459841
$682$	− 16.0000i	− 0.612672i
$683$	36.0000i	1.37750i	0.724998	+	0.688751i	$0.241841\pi$
$683$	36.0000i	1.37750i	−0.724998	+	0.688751i	$0.758159\pi$
$684$	1.00000	0.0382360
$685$	0	0
$686$	8.00000	0.305441
$687$	22.0000i	0.839352i
$688$	− 12.0000i	− 0.457496i
$689$	12.0000	0.457164
$690$	0	0
$691$	−44.0000	−1.67384	−0.836919	−	0.547326i	$-0.815646\pi$
$691$	−44.0000	−1.67384	−0.836919	+	0.547326i	$0.815646\pi$
$692$	14.0000i	0.532200i
$693$	16.0000i	0.607790i
$694$	−24.0000	−0.911028
$695$	0	0
$696$	6.00000	0.227429
$697$	4.00000i	0.151511i
$698$	34.0000i	1.28692i
$699$	−6.00000	−0.226941
$700$	0	0
$701$	−34.0000	−1.28416	−0.642081	−	0.766637i	$-0.721929\pi$
$701$	−34.0000	−1.28416	−0.642081	+	0.766637i	$0.721929\pi$
$702$	− 2.00000i	− 0.0754851i
$703$	− 10.0000i	− 0.377157i
$704$	4.00000	0.150756
$705$	0	0
$706$	6.00000	0.225813
$707$	− 40.0000i	− 1.50435i
$708$	0	0
$709$	−22.0000	−0.826227	−0.413114	−	0.910679i	$-0.635559\pi$
$709$	−22.0000	−0.826227	−0.413114	+	0.910679i	$0.635559\pi$
$710$	0	0
$711$	−12.0000	−0.450035
$712$	6.00000i	0.224860i
$713$	32.0000i	1.19841i
$714$	8.00000	0.299392
$715$	0	0
$716$	0	0
$717$	− 8.00000i	− 0.298765i
$718$	− 32.0000i	− 1.19423i
$719$	40.0000	1.49175	0.745874	−	0.666087i	$-0.232032\pi$
$719$	40.0000	1.49175	0.745874	+	0.666087i	$0.232032\pi$
$720$	0	0
$721$	64.0000	2.38348
$722$	1.00000i	0.0372161i
$723$	− 2.00000i	− 0.0743808i
$724$	−2.00000	−0.0743294
$725$	0	0
$726$	5.00000	0.185567
$727$	12.0000i	0.445055i	0.974926	+	0.222528i	$0.0714308\pi$
$727$	12.0000i	0.445055i	−0.974926	+	0.222528i	$0.928569\pi$
$728$	8.00000i	0.296500i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−24.0000	−0.887672
$732$	− 10.0000i	− 0.369611i
$733$	14.0000i	0.517102i	0.965998	+	0.258551i	$0.0832450\pi$
$733$	14.0000i	0.517102i	−0.965998	+	0.258551i	$0.916755\pi$
$734$	12.0000	0.442928
$735$	0	0
$736$	−8.00000	−0.294884
$737$	16.0000i	0.589368i
$738$	2.00000i	0.0736210i
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	2.00000	0.0734718
$742$	24.0000i	0.881068i
$743$	− 48.0000i	− 1.76095i	−0.474093	−	0.880475i	$-0.657224\pi$
$743$	− 48.0000i	− 1.76095i	0.474093	−	0.880475i	$-0.342776\pi$
$744$	−4.00000	−0.146647
$745$	0	0
$746$	30.0000	1.09838
$747$	− 8.00000i	− 0.292705i
$748$	− 8.00000i	− 0.292509i
$749$	48.0000	1.75388
$750$	0	0
$751$	52.0000	1.89751	0.948753	−	0.316017i	$-0.102346\pi$
$751$	52.0000	1.89751	0.948753	+	0.316017i	$0.102346\pi$
$752$	0	0
$753$	28.0000i	1.02038i
$754$	12.0000	0.437014
$755$	0	0
$756$	4.00000	0.145479
$757$	− 38.0000i	− 1.38113i	−0.723269	−	0.690567i	$-0.757361\pi$
$757$	− 38.0000i	− 1.38113i	0.723269	−	0.690567i	$-0.242639\pi$
$758$	20.0000i	0.726433i
$759$	−32.0000	−1.16153
$760$	0	0
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	8.00000i	0.289809i
$763$	− 40.0000i	− 1.44810i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	− 1.00000i	− 0.0360844i
$769$	−2.00000	−0.0721218	−0.0360609	−	0.999350i	$-0.511481\pi$
$769$	−2.00000	−0.0721218	−0.0360609	+	0.999350i	$0.511481\pi$
$770$	0	0
$771$	18.0000	0.648254
$772$	18.0000i	0.647834i
$773$	26.0000i	0.935155i	0.883952	+	0.467578i	$0.154873\pi$
$773$	26.0000i	0.935155i	−0.883952	+	0.467578i	$0.845127\pi$
$774$	−12.0000	−0.431331
$775$	0	0
$776$	18.0000	0.646162
$777$	− 40.0000i	− 1.43499i
$778$	− 6.00000i	− 0.215110i
$779$	−2.00000	−0.0716574
$780$	0	0
$781$	32.0000	1.14505
$782$	16.0000i	0.572159i
$783$	− 6.00000i	− 0.214423i
$784$	−9.00000	−0.321429
$785$	0	0
$786$	−12.0000	−0.428026
$787$	52.0000i	1.85360i	0.375555	+	0.926800i	$0.377452\pi$
$787$	52.0000i	1.85360i	−0.375555	+	0.926800i	$0.622548\pi$
$788$	18.0000i	0.641223i
$789$	−24.0000	−0.854423
$790$	0	0
$791$	−24.0000	−0.853342
$792$	− 4.00000i	− 0.142134i
$793$	− 20.0000i	− 0.710221i
$794$	−2.00000	−0.0709773
$795$	0	0
$796$	16.0000	0.567105
$797$	− 2.00000i	− 0.0708436i	−0.999372	−	0.0354218i	$-0.988723\pi$
$797$	− 2.00000i	− 0.0708436i	0.999372	−	0.0354218i	$-0.0112775\pi$
$798$	4.00000i	0.141598i
$799$	0	0
$800$	0	0
$801$	6.00000	0.212000
$802$	− 18.0000i	− 0.635602i
$803$	8.00000i	0.282314i
$804$	4.00000	0.141069
$805$	0	0
$806$	−8.00000	−0.281788
$807$	6.00000i	0.211210i
$808$	10.0000i	0.351799i
$809$	−18.0000	−0.632846	−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000	−0.632846	−0.316423	+	0.948618i	$0.602482\pi$
$810$	0	0
$811$	12.0000	0.421377	0.210688	−	0.977553i	$-0.432429\pi$
$811$	12.0000	0.421377	0.210688	+	0.977553i	$0.432429\pi$
$812$	24.0000i	0.842235i
$813$	24.0000i	0.841717i
$814$	−40.0000	−1.40200
$815$	0	0
$816$	−2.00000	−0.0700140
$817$	− 12.0000i	− 0.419827i
$818$	6.00000i	0.209785i
$819$	8.00000	0.279543
$820$	0	0
$821$	22.0000	0.767805	0.383903	−	0.923374i	$-0.374580\pi$
$821$	22.0000	0.767805	0.383903	+	0.923374i	$0.374580\pi$
$822$	14.0000i	0.488306i
$823$	− 52.0000i	− 1.81261i	−0.422628	−	0.906303i	$-0.638892\pi$
$823$	− 52.0000i	− 1.81261i	0.422628	−	0.906303i	$-0.361108\pi$
$824$	−16.0000	−0.557386
$825$	0	0
$826$	0	0
$827$	52.0000i	1.80822i	0.427303	+	0.904109i	$0.359464\pi$
$827$	52.0000i	1.80822i	−0.427303	+	0.904109i	$0.640536\pi$
$828$	8.00000i	0.278019i
$829$	−10.0000	−0.347314	−0.173657	−	0.984806i	$-0.555558\pi$
$829$	−10.0000	−0.347314	−0.173657	+	0.984806i	$0.555558\pi$
$830$	0	0
$831$	18.0000	0.624413
$832$	− 2.00000i	− 0.0693375i
$833$	18.0000i	0.623663i
$834$	12.0000	0.415526
$835$	0	0
$836$	4.00000	0.138343
$837$	4.00000i	0.138260i
$838$	− 12.0000i	− 0.414533i
$839$	40.0000	1.38095	0.690477	−	0.723355i	$-0.257401\pi$
$839$	40.0000	1.38095	0.690477	+	0.723355i	$0.257401\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	− 14.0000i	− 0.482472i
$843$	− 30.0000i	− 1.03325i
$844$	12.0000	0.413057
$845$	0	0
$846$	0	0
$847$	20.0000i	0.687208i
$848$	− 6.00000i	− 0.206041i
$849$	4.00000	0.137280
$850$	0	0
$851$	80.0000	2.74236
$852$	− 8.00000i	− 0.274075i
$853$	46.0000i	1.57501i	0.616308	+	0.787505i	$0.288628\pi$
$853$	46.0000i	1.57501i	−0.616308	+	0.787505i	$0.711372\pi$
$854$	40.0000	1.36877
$855$	0	0
$856$	−12.0000	−0.410152
$857$	42.0000i	1.43469i	0.696717	+	0.717346i	$0.254643\pi$
$857$	42.0000i	1.43469i	−0.696717	+	0.717346i	$0.745357\pi$
$858$	− 8.00000i	− 0.273115i
$859$	4.00000	0.136478	0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000	0.136478	0.0682391	+	0.997669i	$0.478262\pi$
$860$	0	0
$861$	−8.00000	−0.272639
$862$	0	0
$863$	− 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
$866$	2.00000	0.0679628
$867$	− 13.0000i	− 0.441503i
$868$	− 16.0000i	− 0.543075i
$869$	−48.0000	−1.62829
$870$	0	0
$871$	8.00000	0.271070
$872$	10.0000i	0.338643i
$873$	− 18.0000i	− 0.609208i
$874$	−8.00000	−0.270604
$875$	0	0
$876$	2.00000	0.0675737
$877$	6.00000i	0.202606i	0.994856	+	0.101303i	$0.0323011\pi$
$877$	6.00000i	0.202606i	−0.994856	+	0.101303i	$0.967699\pi$
$878$	4.00000i	0.134993i
$879$	10.0000	0.337292
$880$	0	0
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	9.00000i	0.303046i
$883$	− 4.00000i	− 0.134611i	−0.997732	−	0.0673054i	$-0.978560\pi$
$883$	− 4.00000i	− 0.134611i	0.997732	−	0.0673054i	$-0.0214402\pi$
$884$	−4.00000	−0.134535
$885$	0	0
$886$	−24.0000	−0.806296
$887$	48.0000i	1.61168i	0.592132	+	0.805841i	$0.298286\pi$
$887$	48.0000i	1.61168i	−0.592132	+	0.805841i	$0.701714\pi$
$888$	10.0000i	0.335578i
$889$	−32.0000	−1.07325
$890$	0	0
$891$	−4.00000	−0.134005
$892$	16.0000i	0.535720i
$893$	0	0
$894$	−22.0000	−0.735790
$895$	0	0
$896$	4.00000	0.133631
$897$	16.0000i	0.534224i
$898$	− 6.00000i	− 0.200223i
$899$	−24.0000	−0.800445
$900$	0	0
$901$	−12.0000	−0.399778
$902$	8.00000i	0.266371i
$903$	− 48.0000i	− 1.59734i
$904$	6.00000	0.199557
$905$	0	0
$906$	4.00000	0.132891
$907$	− 20.0000i	− 0.664089i	−0.943264	−	0.332045i	$-0.892262\pi$
$907$	− 20.0000i	− 0.664089i	0.943264	−	0.332045i	$-0.107738\pi$
$908$	12.0000i	0.398234i
$909$	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	$0.457691\pi$
$912$	− 1.00000i	− 0.0331133i
$913$	− 32.0000i	− 1.05905i
$914$	6.00000	0.198462
$915$	0	0
$916$	22.0000	0.726900
$917$	− 48.0000i	− 1.58510i
$918$	2.00000i	0.0660098i
$919$	24.0000	0.791687	0.395843	−	0.918318i	$-0.370452\pi$
$919$	24.0000	0.791687	0.395843	+	0.918318i	$0.370452\pi$
$920$	0	0
$921$	28.0000	0.922631
$922$	14.0000i	0.461065i
$923$	− 16.0000i	− 0.526646i
$924$	16.0000	0.526361
$925$	0	0
$926$	−36.0000	−1.18303
$927$	16.0000i	0.525509i
$928$	− 6.00000i	− 0.196960i
$929$	−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
$931$	−9.00000	−0.294963
$932$	6.00000i	0.196537i
$933$	− 24.0000i	− 0.785725i
$934$	−8.00000	−0.261768
$935$	0	0
$936$	−2.00000	−0.0653720
$937$	− 38.0000i	− 1.24141i	−0.784046	−	0.620703i	$-0.786847\pi$
$937$	− 38.0000i	− 1.24141i	0.784046	−	0.620703i	$-0.213153\pi$
$938$	16.0000i	0.522419i
$939$	30.0000	0.979013
$940$	0	0
$941$	−50.0000	−1.62995	−0.814977	−	0.579494i	$-0.803250\pi$
$941$	−50.0000	−1.62995	−0.814977	+	0.579494i	$0.803250\pi$
$942$	− 6.00000i	− 0.195491i
$943$	− 16.0000i	− 0.521032i
$944$	0	0
$945$	0	0
$946$	−48.0000	−1.56061
$947$	− 40.0000i	− 1.29983i	−0.760009	−	0.649913i	$-0.774805\pi$
$947$	− 40.0000i	− 1.29983i	0.760009	−	0.649913i	$-0.225195\pi$
$948$	12.0000i	0.389742i
$949$	4.00000	0.129845
$950$	0	0
$951$	30.0000	0.972817
$952$	− 8.00000i	− 0.259281i
$953$	− 10.0000i	− 0.323932i	−0.986796	−	0.161966i	$-0.948217\pi$
$953$	− 10.0000i	− 0.323932i	0.986796	−	0.161966i	$-0.0517835\pi$
$954$	−6.00000	−0.194257
$955$	0	0
$956$	−8.00000	−0.258738
$957$	− 24.0000i	− 0.775810i
$958$	16.0000i	0.516937i
$959$	−56.0000	−1.80833
$960$	0	0
$961$	−15.0000	−0.483871
$962$	20.0000i	0.644826i
$963$	12.0000i	0.386695i
$964$	−2.00000	−0.0644157
$965$	0	0
$966$	−32.0000	−1.02958
$967$	52.0000i	1.67221i	0.548572	+	0.836104i	$0.315172\pi$
$967$	52.0000i	1.67221i	−0.548572	+	0.836104i	$0.684828\pi$
$968$	− 5.00000i	− 0.160706i
$969$	−2.00000	−0.0642493
$970$	0	0
$971$	−24.0000	−0.770197	−0.385098	−	0.922876i	$-0.625832\pi$
$971$	−24.0000	−0.770197	−0.385098	+	0.922876i	$0.625832\pi$
$972$	1.00000i	0.0320750i
$973$	48.0000i	1.53881i
$974$	32.0000	1.02535
$975$	0	0
$976$	−10.0000	−0.320092
$977$	50.0000i	1.59964i	0.600239	+	0.799821i	$0.295072\pi$
$977$	50.0000i	1.59964i	−0.600239	+	0.799821i	$0.704928\pi$
$978$	4.00000i	0.127906i
$979$	24.0000	0.767043
$980$	0	0
$981$	10.0000	0.319275
$982$	28.0000i	0.893516i
$983$	− 48.0000i	− 1.53096i	−0.643458	−	0.765481i	$-0.722501\pi$
$983$	− 48.0000i	− 1.53096i	0.643458	−	0.765481i	$-0.277499\pi$
$984$	2.00000	0.0637577
$985$	0	0
$986$	−12.0000	−0.382158
$987$	0	0
$988$	− 2.00000i	− 0.0636285i
$989$	96.0000	3.05262
$990$	0	0
$991$	−20.0000	−0.635321	−0.317660	−	0.948205i	$-0.602897\pi$
$991$	−20.0000	−0.635321	−0.317660	+	0.948205i	$0.602897\pi$
$992$	4.00000i	0.127000i
$993$	12.0000i	0.380808i
$994$	32.0000	1.01498
$995$	0	0
$996$	−8.00000	−0.253490
$997$	− 14.0000i	− 0.443384i	−0.975117	−	0.221692i	$-0.928842\pi$
$997$	− 14.0000i	− 0.443384i	0.975117	−	0.221692i	$-0.0711580\pi$
$998$	4.00000i	0.126618i
$999$	10.0000	0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2850.2.d.k.799.2		2
5.2	odd	4		2850.2.a.a.1.1		1
5.3	odd	4		570.2.a.m.1.1	✓	1
5.4	even	2	inner	2850.2.d.k.799.1		2
15.2	even	4		8550.2.a.t.1.1		1
15.8	even	4		1710.2.a.f.1.1		1
20.3	even	4		4560.2.a.k.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
570.2.a.m.1.1	✓	1	5.3	odd	4
1710.2.a.f.1.1		1	15.8	even	4
2850.2.a.a.1.1		1	5.2	odd	4
2850.2.d.k.799.1		2	5.4	even	2	inner
2850.2.d.k.799.2		2	1.1	even	1	trivial
4560.2.a.k.1.1		1	20.3	even	4
8550.2.a.t.1.1		1	15.2	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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -4.00000 q^{11} +1.00000i q^{12} +2.00000i q^{13} -4.00000 q^{14} +1.00000 q^{16} -2.00000i q^{17} -1.00000i q^{18} +1.00000 q^{19} +4.00000 q^{21} -4.00000i q^{22} +8.00000i q^{23} -1.00000 q^{24} -2.00000 q^{26} +1.00000i q^{27} -4.00000i q^{28} -6.00000 q^{29} +4.00000 q^{31} +1.00000i q^{32} +4.00000i q^{33} +2.00000 q^{34} +1.00000 q^{36} -10.0000i q^{37} +1.00000i q^{38} +2.00000 q^{39} -2.00000 q^{41} +4.00000i q^{42} -12.0000i q^{43} +4.00000 q^{44} -8.00000 q^{46} -1.00000i q^{48} -9.00000 q^{49} -2.00000 q^{51} -2.00000i q^{52} -6.00000i q^{53} -1.00000 q^{54} +4.00000 q^{56} -1.00000i q^{57} -6.00000i q^{58} -10.0000 q^{61} +4.00000i q^{62} -4.00000i q^{63} -1.00000 q^{64} -4.00000 q^{66} -4.00000i q^{67} +2.00000i q^{68} +8.00000 q^{69} -8.00000 q^{71} +1.00000i q^{72} -2.00000i q^{73} +10.0000 q^{74} -1.00000 q^{76} -16.0000i q^{77} +2.00000i q^{78} +12.0000 q^{79} +1.00000 q^{81} -2.00000i q^{82} +8.00000i q^{83} -4.00000 q^{84} +12.0000 q^{86} +6.00000i q^{87} +4.00000i q^{88} -6.00000 q^{89} -8.00000 q^{91} -8.00000i q^{92} -4.00000i q^{93} +1.00000 q^{96} +18.0000i q^{97} -9.00000i q^{98} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 8 q^{11} - 8 q^{14} + 2 q^{16} + 2 q^{19} + 8 q^{21} - 2 q^{24} - 4 q^{26} - 12 q^{29} + 8 q^{31} + 4 q^{34} + 2 q^{36} + 4 q^{39} - 4 q^{41} + 8 q^{44} - 16 q^{46} - 18 q^{49} - 4 q^{51} - 2 q^{54} + 8 q^{56} - 20 q^{61} - 2 q^{64} - 8 q^{66} + 16 q^{69} - 16 q^{71} + 20 q^{74} - 2 q^{76} + 24 q^{79} + 2 q^{81} - 8 q^{84} + 24 q^{86} - 12 q^{89} - 16 q^{91} + 2 q^{96} + 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 - 8 * q^11 - 8 * q^14 + 2 * q^16 + 2 * q^19 + 8 * q^21 - 2 * q^24 - 4 * q^26 - 12 * q^29 + 8 * q^31 + 4 * q^34 + 2 * q^36 + 4 * q^39 - 4 * q^41 + 8 * q^44 - 16 * q^46 - 18 * q^49 - 4 * q^51 - 2 * q^54 + 8 * q^56 - 20 * q^61 - 2 * q^64 - 8 * q^66 + 16 * q^69 - 16 * q^71 + 20 * q^74 - 2 * q^76 + 24 * q^79 + 2 * q^81 - 8 * q^84 + 24 * q^86 - 12 * q^89 - 16 * q^91 + 2 * q^96 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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