LMFDB - Embedded newform 2850.2.d.d.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.d.799.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +1.00000i q^{12} -2.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} +1.00000i q^{18} -1.00000 q^{19} +2.00000 q^{21} +1.00000 q^{24} -2.00000 q^{26} +1.00000i q^{27} -2.00000i q^{28} +6.00000 q^{29} +2.00000 q^{31} -1.00000i q^{32} +1.00000 q^{36} +2.00000i q^{37} +1.00000i q^{38} -2.00000 q^{39} -2.00000i q^{42} -8.00000i q^{43} -1.00000i q^{48} +3.00000 q^{49} +2.00000i q^{52} -6.00000i q^{53} +1.00000 q^{54} -2.00000 q^{56} +1.00000i q^{57} -6.00000i q^{58} +6.00000 q^{59} +2.00000 q^{61} -2.00000i q^{62} -2.00000i q^{63} -1.00000 q^{64} -4.00000i q^{67} -1.00000i q^{72} -14.0000i q^{73} +2.00000 q^{74} +1.00000 q^{76} +2.00000i q^{78} -2.00000 q^{79} +1.00000 q^{81} -6.00000i q^{83} -2.00000 q^{84} -8.00000 q^{86} -6.00000i q^{87} +12.0000 q^{89} +4.00000 q^{91} -2.00000i q^{93} -1.00000 q^{96} -10.0000i q^{97} -3.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 4 q^{14} + 2 q^{16} - 2 q^{19} + 4 q^{21} + 2 q^{24} - 4 q^{26} + 12 q^{29} + 4 q^{31} + 2 q^{36} - 4 q^{39} + 6 q^{49} + 2 q^{54} - 4 q^{56} + 12 q^{59} + 4 q^{61} - 2 q^{64} + 4 q^{74} + 2 q^{76} - 4 q^{79} + 2 q^{81} - 4 q^{84} - 16 q^{86} + 24 q^{89} + 8 q^{91} - 2 q^{96}+O(q^{100}) $ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 + 4 * q^14 + 2 * q^16 - 2 * q^19 + 4 * q^21 + 2 * q^24 - 4 * q^26 + 12 * q^29 + 4 * q^31 + 2 * q^36 - 4 * q^39 + 6 * q^49 + 2 * q^54 - 4 * q^56 + 12 * q^59 + 4 * q^61 - 2 * q^64 + 4 * q^74 + 2 * q^76 - 4 * q^79 + 2 * q^81 - 4 * q^84 - 16 * q^86 + 24 * q^89 + 8 * q^91 - 2 * q^96

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	$0.123376\pi$
$7$	2.00000i	0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$	1.00000i	0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	1.00000i	0.288675i
$13$	− 2.00000i	− 0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$	− 2.00000i	− 0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	2.00000	0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	1.00000i	0.235702i
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	2.00000	0.436436
$22$	0	0
$23$	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
$28$	− 2.00000i	− 0.377964i
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	0	0
$35$	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
$39$	−2.00000	−0.320256
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	− 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	$-0.208828\pi$
$44$	0	0
$45$	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
$51$	0	0
$52$	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
$56$	−2.00000	−0.267261
$57$	1.00000i	0.132453i
$58$	− 6.00000i	− 0.787839i
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	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
$66$	0	0
$67$	− 4.00000i	− 0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$	− 4.00000i	− 0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$	0	0
$69$	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
$73$	− 14.0000i	− 1.63858i	−0.573382	−	0.819288i	$-0.694369\pi$
$73$	− 14.0000i	− 1.63858i	0.573382	−	0.819288i	$-0.305631\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	1.00000	0.114708
$77$	0	0
$78$	2.00000i	0.226455i
$79$	−2.00000	−0.225018	−0.112509	−	0.993651i	$-0.535889\pi$
$79$	−2.00000	−0.225018	−0.112509	+	0.993651i	$0.535889\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 6.00000i	− 0.658586i	−0.944228	−	0.329293i	$-0.893190\pi$
$83$	− 6.00000i	− 0.658586i	0.944228	−	0.329293i	$-0.106810\pi$
$84$	−2.00000	−0.218218
$85$	0	0
$86$	−8.00000	−0.862662
$87$	− 6.00000i	− 0.643268i
$88$	0	0
$89$	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
$91$	4.00000	0.419314
$92$	0	0
$93$	− 2.00000i	− 0.207390i
$94$	0	0
$95$	0	0
$96$	−1.00000	−0.102062
$97$	− 10.0000i	− 1.01535i	−0.861550	−	0.507673i	$-0.830506\pi$
$97$	− 10.0000i	− 1.01535i	0.861550	−	0.507673i	$-0.169494\pi$
$98$	− 3.00000i	− 0.303046i
$99$	0	0
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	− 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
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	− 1.00000i	− 0.0962250i
$109$	16.0000	1.53252	0.766261	−	0.642529i	$-0.222115\pi$
$109$	16.0000	1.53252	0.766261	+	0.642529i	$0.222115\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	2.00000i	0.188982i
$113$	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
$118$	− 6.00000i	− 0.552345i
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	− 2.00000i	− 0.181071i
$123$	0	0
$124$	−2.00000	−0.179605
$125$	0	0
$126$	−2.00000	−0.178174
$127$	− 16.0000i	− 1.41977i	−0.704317	−	0.709885i	$-0.748747\pi$
$127$	− 16.0000i	− 1.41977i	0.704317	−	0.709885i	$-0.251253\pi$
$128$	1.00000i	0.0883883i
$129$	−8.00000	−0.704361
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	− 2.00000i	− 0.173422i
$134$	−4.00000	−0.345547
$135$	0	0
$136$	0	0
$137$	− 12.0000i	− 1.02523i	−0.858619	−	0.512615i	$-0.828677\pi$
$137$	− 12.0000i	− 1.02523i	0.858619	−	0.512615i	$-0.171323\pi$
$138$	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
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−14.0000	−1.15865
$147$	− 3.00000i	− 0.247436i
$148$	− 2.00000i	− 0.164399i
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	−10.0000	−0.813788	−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000	−0.813788	−0.406894	+	0.913475i	$0.633388\pi$
$152$	− 1.00000i	− 0.0811107i
$153$	0	0
$154$	0	0
$155$	0	0
$156$	2.00000	0.160128
$157$	− 4.00000i	− 0.319235i	−0.987179	−	0.159617i	$-0.948974\pi$
$157$	− 4.00000i	− 0.319235i	0.987179	−	0.159617i	$-0.0510260\pi$
$158$	2.00000i	0.159111i
$159$	−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	$0.949930\pi$
$164$	0	0
$165$	0	0
$166$	−6.00000	−0.465690
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	2.00000i	0.154303i
$169$	9.00000	0.692308
$170$	0	0
$171$	1.00000	0.0764719
$172$	8.00000i	0.609994i
$173$	18.0000i	1.36851i	0.729241	+	0.684257i	$0.239873\pi$
$173$	18.0000i	1.36851i	−0.729241	+	0.684257i	$0.760127\pi$
$174$	−6.00000	−0.454859
$175$	0	0
$176$	0	0
$177$	− 6.00000i	− 0.450988i
$178$	− 12.0000i	− 0.899438i
$179$	−6.00000	−0.448461	−0.224231	−	0.974536i	$-0.571987\pi$
$179$	−6.00000	−0.448461	−0.224231	+	0.974536i	$0.571987\pi$
$180$	0	0
$181$	−16.0000	−1.18927	−0.594635	−	0.803996i	$-0.702704\pi$
$181$	−16.0000	−1.18927	−0.594635	+	0.803996i	$0.702704\pi$
$182$	− 4.00000i	− 0.296500i
$183$	− 2.00000i	− 0.147844i
$184$	0	0
$185$	0	0
$186$	−2.00000	−0.146647
$187$	0	0
$188$	0	0
$189$	−2.00000	−0.145479
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	1.00000i	0.0721688i
$193$	− 2.00000i	− 0.143963i	−0.997406	−	0.0719816i	$-0.977068\pi$
$193$	− 2.00000i	− 0.143963i	0.997406	−	0.0719816i	$-0.0229323\pi$
$194$	−10.0000	−0.717958
$195$	0	0
$196$	−3.00000	−0.214286
$197$	18.0000i	1.28245i	0.767354	+	0.641223i	$0.221573\pi$
$197$	18.0000i	1.28245i	−0.767354	+	0.641223i	$0.778427\pi$
$198$	0	0
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	6.00000i	0.422159i
$203$	12.0000i	0.842235i
$204$	0	0
$205$	0	0
$206$	−8.00000	−0.557386
$207$	0	0
$208$	− 2.00000i	− 0.138675i
$209$	0	0
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	6.00000i	0.412082i
$213$	0	0
$214$	0	0
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	4.00000i	0.271538i
$218$	− 16.0000i	− 1.08366i
$219$	−14.0000	−0.946032
$220$	0	0
$221$	0	0
$222$	− 2.00000i	− 0.134231i
$223$	4.00000i	0.267860i	0.990991	+	0.133930i	$0.0427597\pi$
$223$	4.00000i	0.267860i	−0.990991	+	0.133930i	$0.957240\pi$
$224$	2.00000	0.133631
$225$	0	0
$226$	6.00000	0.399114
$227$	12.0000i	0.796468i	0.917284	+	0.398234i	$0.130377\pi$
$227$	12.0000i	0.796468i	−0.917284	+	0.398234i	$0.869623\pi$
$228$	− 1.00000i	− 0.0662266i
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	0	0
$231$	0	0
$232$	6.00000i	0.393919i
$233$	− 24.0000i	− 1.57229i	−0.618041	−	0.786146i	$-0.712073\pi$
$233$	− 24.0000i	− 1.57229i	0.618041	−	0.786146i	$-0.287927\pi$
$234$	2.00000	0.130744
$235$	0	0
$236$	−6.00000	−0.390567
$237$	2.00000i	0.129914i
$238$	0	0
$239$	−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
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	11.0000i	0.707107i
$243$	− 1.00000i	− 0.0641500i
$244$	−2.00000	−0.128037
$245$	0	0
$246$	0	0
$247$	2.00000i	0.127257i
$248$	2.00000i	0.127000i
$249$	−6.00000	−0.380235
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	2.00000i	0.125988i
$253$	0	0
$254$	−16.0000	−1.00393
$255$	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
$261$	−6.00000	−0.371391
$262$	0	0
$263$	24.0000i	1.47990i	0.672660	+	0.739952i	$0.265152\pi$
$263$	24.0000i	1.47990i	−0.672660	+	0.739952i	$0.734848\pi$
$264$	0	0
$265$	0	0
$266$	−2.00000	−0.122628
$267$	− 12.0000i	− 0.734388i
$268$	4.00000i	0.244339i
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	0	0
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	0	0
$273$	− 4.00000i	− 0.242091i
$274$	−12.0000	−0.724947
$275$	0	0
$276$	0	0
$277$	− 28.0000i	− 1.68236i	−0.540758	−	0.841178i	$-0.681862\pi$
$277$	− 28.0000i	− 1.68236i	0.540758	−	0.841178i	$-0.318138\pi$
$278$	− 4.00000i	− 0.239904i
$279$	−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
$283$	− 32.0000i	− 1.90220i	−0.308879	−	0.951101i	$-0.599954\pi$
$283$	− 32.0000i	− 1.90220i	0.308879	−	0.951101i	$-0.400046\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	1.00000i	0.0589256i
$289$	17.0000	1.00000
$290$	0	0
$291$	−10.0000	−0.586210
$292$	14.0000i	0.819288i
$293$	6.00000i	0.350524i	0.984522	+	0.175262i	$0.0560772\pi$
$293$	6.00000i	0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$	−3.00000	−0.174964
$295$	0	0
$296$	−2.00000	−0.116248
$297$	0	0
$298$	− 6.00000i	− 0.347571i
$299$	0	0
$300$	0	0
$301$	16.0000	0.922225
$302$	10.0000i	0.575435i
$303$	6.00000i	0.344691i
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	0	0
$307$	20.0000i	1.14146i	0.821138	+	0.570730i	$0.193340\pi$
$307$	20.0000i	1.14146i	−0.821138	+	0.570730i	$0.806660\pi$
$308$	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	$0.908798\pi$
$314$	−4.00000	−0.225733
$315$	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
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	4.00000	0.221540
$327$	− 16.0000i	− 0.884802i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	6.00000i	0.329293i
$333$	− 2.00000i	− 0.109599i
$334$	0	0
$335$	0	0
$336$	2.00000	0.109109
$337$	2.00000i	0.108947i	0.998515	+	0.0544735i	$0.0173480\pi$
$337$	2.00000i	0.108947i	−0.998515	+	0.0544735i	$0.982652\pi$
$338$	− 9.00000i	− 0.489535i
$339$	6.00000	0.325875
$340$	0	0
$341$	0	0
$342$	− 1.00000i	− 0.0540738i
$343$	20.0000i	1.07990i
$344$	8.00000	0.431331
$345$	0	0
$346$	18.0000	0.967686
$347$	18.0000i	0.966291i	0.875540	+	0.483145i	$0.160506\pi$
$347$	18.0000i	0.966291i	−0.875540	+	0.483145i	$0.839494\pi$
$348$	6.00000i	0.321634i
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	0	0
$353$	− 12.0000i	− 0.638696i	−0.947638	−	0.319348i	$-0.896536\pi$
$353$	− 12.0000i	− 0.638696i	0.947638	−	0.319348i	$-0.103464\pi$
$354$	−6.00000	−0.318896
$355$	0	0
$356$	−12.0000	−0.635999
$357$	0	0
$358$	6.00000i	0.317110i
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	16.0000i	0.840941i
$363$	11.0000i	0.577350i
$364$	−4.00000	−0.209657
$365$	0	0
$366$	−2.00000	−0.104542
$367$	− 10.0000i	− 0.521996i	−0.965339	−	0.260998i	$-0.915948\pi$
$367$	− 10.0000i	− 0.521996i	0.965339	−	0.260998i	$-0.0840516\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	12.0000	0.623009
$372$	2.00000i	0.103695i
$373$	− 14.0000i	− 0.724893i	−0.932005	−	0.362446i	$-0.881942\pi$
$373$	− 14.0000i	− 0.724893i	0.932005	−	0.362446i	$-0.118058\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 12.0000i	− 0.618031i
$378$	2.00000i	0.102869i
$379$	−8.00000	−0.410932	−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000	−0.410932	−0.205466	+	0.978664i	$0.565871\pi$
$380$	0	0
$381$	−16.0000	−0.819705
$382$	0	0
$383$	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
$386$	−2.00000	−0.101797
$387$	8.00000i	0.406663i
$388$	10.0000i	0.507673i
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	0	0
$391$	0	0
$392$	3.00000i	0.151523i
$393$	0	0
$394$	18.0000	0.906827
$395$	0	0
$396$	0	0
$397$	8.00000i	0.401508i	0.979642	+	0.200754i	$0.0643393\pi$
$397$	8.00000i	0.401508i	−0.979642	+	0.200754i	$0.935661\pi$
$398$	− 4.00000i	− 0.200502i
$399$	−2.00000	−0.100125
$400$	0	0
$401$	12.0000	0.599251	0.299626	−	0.954057i	$-0.403138\pi$
$401$	12.0000	0.599251	0.299626	+	0.954057i	$0.403138\pi$
$402$	4.00000i	0.199502i
$403$	− 4.00000i	− 0.199254i
$404$	6.00000	0.298511
$405$	0	0
$406$	12.0000	0.595550
$407$	0	0
$408$	0	0
$409$	22.0000	1.08783	0.543915	−	0.839140i	$-0.316941\pi$
$409$	22.0000	1.08783	0.543915	+	0.839140i	$0.316941\pi$
$410$	0	0
$411$	−12.0000	−0.591916
$412$	8.00000i	0.394132i
$413$	12.0000i	0.590481i
$414$	0	0
$415$	0	0
$416$	−2.00000	−0.0980581
$417$	− 4.00000i	− 0.195881i
$418$	0	0
$419$	−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
$421$	8.00000	0.389896	0.194948	−	0.980814i	$-0.437546\pi$
$421$	8.00000	0.389896	0.194948	+	0.980814i	$0.437546\pi$
$422$	− 8.00000i	− 0.389434i
$423$	0	0
$424$	6.00000	0.291386
$425$	0	0
$426$	0	0
$427$	4.00000i	0.193574i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	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	$-0.0153029\pi$
$434$	4.00000	0.192006
$435$	0	0
$436$	−16.0000	−0.766261
$437$	0	0
$438$	14.0000i	0.668946i
$439$	22.0000	1.05000	0.525001	−	0.851101i	$-0.324065\pi$
$439$	22.0000	1.05000	0.525001	+	0.851101i	$0.324065\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	0	0
$443$	30.0000i	1.42534i	0.701498	+	0.712672i	$0.252515\pi$
$443$	30.0000i	1.42534i	−0.701498	+	0.712672i	$0.747485\pi$
$444$	−2.00000	−0.0949158
$445$	0	0
$446$	4.00000	0.189405
$447$	− 6.00000i	− 0.283790i
$448$	− 2.00000i	− 0.0944911i
$449$	−24.0000	−1.13263	−0.566315	−	0.824189i	$-0.691631\pi$
$449$	−24.0000	−1.13263	−0.566315	+	0.824189i	$0.691631\pi$
$450$	0	0
$451$	0	0
$452$	− 6.00000i	− 0.282216i
$453$	10.0000i	0.469841i
$454$	12.0000	0.563188
$455$	0	0
$456$	−1.00000	−0.0468293
$457$	− 22.0000i	− 1.02912i	−0.857455	−	0.514558i	$-0.827956\pi$
$457$	− 22.0000i	− 1.02912i	0.857455	−	0.514558i	$-0.172044\pi$
$458$	− 10.0000i	− 0.467269i
$459$	0	0
$460$	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
$463$	− 26.0000i	− 1.20832i	−0.796862	−	0.604161i	$-0.793508\pi$
$463$	− 26.0000i	− 1.20832i	0.796862	−	0.604161i	$-0.206492\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	−24.0000	−1.11178
$467$	6.00000i	0.277647i	0.990317	+	0.138823i	$0.0443321\pi$
$467$	6.00000i	0.277647i	−0.990317	+	0.138823i	$0.955668\pi$
$468$	− 2.00000i	− 0.0924500i
$469$	8.00000	0.369406
$470$	0	0
$471$	−4.00000	−0.184310
$472$	6.00000i	0.276172i
$473$	0	0
$474$	2.00000	0.0918630
$475$	0	0
$476$	0	0
$477$	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
$481$	4.00000	0.182384
$482$	22.0000i	1.00207i
$483$	0	0
$484$	11.0000	0.500000
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	− 16.0000i	− 0.725029i	−0.931978	−	0.362515i	$-0.881918\pi$
$487$	− 16.0000i	− 0.725029i	0.931978	−	0.362515i	$-0.118082\pi$
$488$	2.00000i	0.0905357i
$489$	4.00000	0.180886
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	0	0
$494$	2.00000	0.0899843
$495$	0	0
$496$	2.00000	0.0898027
$497$	0	0
$498$	6.00000i	0.268866i
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	0	0
$501$	0	0
$502$	− 12.0000i	− 0.535586i
$503$	12.0000i	0.535054i	0.963550	+	0.267527i	$0.0862064\pi$
$503$	12.0000i	0.535054i	−0.963550	+	0.267527i	$0.913794\pi$
$504$	2.00000	0.0890871
$505$	0	0
$506$	0	0
$507$	− 9.00000i	− 0.399704i
$508$	16.0000i	0.709885i
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	0	0
$511$	28.0000	1.23865
$512$	− 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
$519$	18.0000	0.790112
$520$	0	0
$521$	12.0000	0.525730	0.262865	−	0.964833i	$-0.415333\pi$
$521$	12.0000	0.525730	0.262865	+	0.964833i	$0.415333\pi$
$522$	6.00000i	0.262613i
$523$	4.00000i	0.174908i	0.996169	+	0.0874539i	$0.0278730\pi$
$523$	4.00000i	0.174908i	−0.996169	+	0.0874539i	$0.972127\pi$
$524$	0	0
$525$	0	0
$526$	24.0000	1.04645
$527$	0	0
$528$	0	0
$529$	23.0000	1.00000
$530$	0	0
$531$	−6.00000	−0.260378
$532$	2.00000i	0.0867110i
$533$	0	0
$534$	−12.0000	−0.519291
$535$	0	0
$536$	4.00000	0.172774
$537$	6.00000i	0.258919i
$538$	18.0000i	0.776035i
$539$	0	0
$540$	0	0
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	− 20.0000i	− 0.859074i
$543$	16.0000i	0.686626i
$544$	0	0
$545$	0	0
$546$	−4.00000	−0.171184
$547$	− 4.00000i	− 0.171028i	−0.996337	−	0.0855138i	$-0.972747\pi$
$547$	− 4.00000i	− 0.171028i	0.996337	−	0.0855138i	$-0.0272532\pi$
$548$	12.0000i	0.512615i
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	−6.00000	−0.255609
$552$	0	0
$553$	− 4.00000i	− 0.170097i
$554$	−28.0000	−1.18961
$555$	0	0
$556$	−4.00000	−0.169638
$557$	6.00000i	0.254228i	0.991888	+	0.127114i	$0.0405714\pi$
$557$	6.00000i	0.254228i	−0.991888	+	0.127114i	$0.959429\pi$
$558$	2.00000i	0.0846668i
$559$	−16.0000	−0.676728
$560$	0	0
$561$	0	0
$562$	− 24.0000i	− 1.01238i
$563$	24.0000i	1.01148i	0.862686	+	0.505740i	$0.168780\pi$
$563$	24.0000i	1.01148i	−0.862686	+	0.505740i	$0.831220\pi$
$564$	0	0
$565$	0	0
$566$	−32.0000	−1.34506
$567$	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
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	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	$0.986745\pi$
$578$	− 17.0000i	− 0.707107i
$579$	−2.00000	−0.0831172
$580$	0	0
$581$	12.0000	0.497844
$582$	10.0000i	0.414513i
$583$	0	0
$584$	14.0000	0.579324
$585$	0	0
$586$	6.00000	0.247858
$587$	18.0000i	0.742940i	0.928445	+	0.371470i	$0.121146\pi$
$587$	18.0000i	0.742940i	−0.928445	+	0.371470i	$0.878854\pi$
$588$	3.00000i	0.123718i
$589$	−2.00000	−0.0824086
$590$	0	0
$591$	18.0000	0.740421
$592$	2.00000i	0.0821995i
$593$	− 36.0000i	− 1.47834i	−0.673517	−	0.739171i	$-0.735217\pi$
$593$	− 36.0000i	− 1.47834i	0.673517	−	0.739171i	$-0.264783\pi$
$594$	0	0
$595$	0	0
$596$	−6.00000	−0.245770
$597$	− 4.00000i	− 0.163709i
$598$	0	0
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	0	0
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	− 16.0000i	− 0.652111i
$603$	4.00000i	0.162893i
$604$	10.0000	0.406894
$605$	0	0
$606$	6.00000	0.243733
$607$	32.0000i	1.29884i	0.760430	+	0.649420i	$0.224988\pi$
$607$	32.0000i	1.29884i	−0.760430	+	0.649420i	$0.775012\pi$
$608$	1.00000i	0.0405554i
$609$	12.0000	0.486265
$610$	0	0
$611$	0	0
$612$	0	0
$613$	16.0000i	0.646234i	0.946359	+	0.323117i	$0.104731\pi$
$613$	16.0000i	0.646234i	−0.946359	+	0.323117i	$0.895269\pi$
$614$	20.0000	0.807134
$615$	0	0
$616$	0	0
$617$	12.0000i	0.483102i	0.970388	+	0.241551i	$0.0776561\pi$
$617$	12.0000i	0.483102i	−0.970388	+	0.241551i	$0.922344\pi$
$618$	8.00000i	0.321807i
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	24.0000i	0.961540i
$624$	−2.00000	−0.0800641
$625$	0	0
$626$	10.0000	0.399680
$627$	0	0
$628$	4.00000i	0.159617i
$629$	0	0
$630$	0	0
$631$	32.0000	1.27390	0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000	1.27390	0.636950	+	0.770905i	$0.280196\pi$
$632$	− 2.00000i	− 0.0795557i
$633$	− 8.00000i	− 0.317971i
$634$	−30.0000	−1.19145
$635$	0	0
$636$	6.00000	0.237915
$637$	− 6.00000i	− 0.237729i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−24.0000	−0.947943	−0.473972	−	0.880540i	$-0.657180\pi$
$641$	−24.0000	−0.947943	−0.473972	+	0.880540i	$0.657180\pi$
$642$	0	0
$643$	− 32.0000i	− 1.26196i	−0.775800	−	0.630978i	$-0.782654\pi$
$643$	− 32.0000i	− 1.26196i	0.775800	−	0.630978i	$-0.217346\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	12.0000i	0.471769i	0.971781	+	0.235884i	$0.0757987\pi$
$647$	12.0000i	0.471769i	−0.971781	+	0.235884i	$0.924201\pi$
$648$	1.00000i	0.0392837i
$649$	0	0
$650$	0	0
$651$	4.00000	0.156772
$652$	− 4.00000i	− 0.156652i
$653$	18.0000i	0.704394i	0.935926	+	0.352197i	$0.114565\pi$
$653$	18.0000i	0.704394i	−0.935926	+	0.352197i	$0.885435\pi$
$654$	−16.0000	−0.625650
$655$	0	0
$656$	0	0
$657$	14.0000i	0.546192i
$658$	0	0
$659$	−6.00000	−0.233727	−0.116863	−	0.993148i	$-0.537284\pi$
$659$	−6.00000	−0.233727	−0.116863	+	0.993148i	$0.537284\pi$
$660$	0	0
$661$	32.0000	1.24466	0.622328	−	0.782757i	$-0.286187\pi$
$661$	32.0000	1.24466	0.622328	+	0.782757i	$0.286187\pi$
$662$	− 20.0000i	− 0.777322i
$663$	0	0
$664$	6.00000	0.232845
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	0	0
$668$	0	0
$669$	4.00000	0.154649
$670$	0	0
$671$	0	0
$672$	− 2.00000i	− 0.0771517i
$673$	− 14.0000i	− 0.539660i	−0.962908	−	0.269830i	$-0.913032\pi$
$673$	− 14.0000i	− 0.539660i	0.962908	−	0.269830i	$-0.0869676\pi$
$674$	2.00000	0.0770371
$675$	0	0
$676$	−9.00000	−0.346154
$677$	− 30.0000i	− 1.15299i	−0.817099	−	0.576497i	$-0.804419\pi$
$677$	− 30.0000i	− 1.15299i	0.817099	−	0.576497i	$-0.195581\pi$
$678$	− 6.00000i	− 0.230429i
$679$	20.0000	0.767530
$680$	0	0
$681$	12.0000	0.459841
$682$	0	0
$683$	12.0000i	0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$683$	12.0000i	0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$684$	−1.00000	−0.0382360
$685$	0	0
$686$	20.0000	0.763604
$687$	− 10.0000i	− 0.381524i
$688$	− 8.00000i	− 0.304997i
$689$	−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	$0.678779\pi$
$692$	− 18.0000i	− 0.684257i
$693$	0	0
$694$	18.0000	0.683271
$695$	0	0
$696$	6.00000	0.227429
$697$	0	0
$698$	− 10.0000i	− 0.378506i
$699$	−24.0000	−0.907763
$700$	0	0
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	− 2.00000i	− 0.0754851i
$703$	− 2.00000i	− 0.0754314i
$704$	0	0
$705$	0	0
$706$	−12.0000	−0.451626
$707$	− 12.0000i	− 0.451306i
$708$	6.00000i	0.225494i
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	0	0
$711$	2.00000	0.0750059
$712$	12.0000i	0.449719i
$713$	0	0
$714$	0	0
$715$	0	0
$716$	6.00000	0.224231
$717$	24.0000i	0.896296i
$718$	24.0000i	0.895672i
$719$	48.0000	1.79010	0.895049	−	0.445968i	$-0.147140\pi$
$719$	48.0000	1.79010	0.895049	+	0.445968i	$0.147140\pi$
$720$	0	0
$721$	16.0000	0.595871
$722$	− 1.00000i	− 0.0372161i
$723$	22.0000i	0.818189i
$724$	16.0000	0.594635
$725$	0	0
$726$	11.0000	0.408248
$727$	− 34.0000i	− 1.26099i	−0.776193	−	0.630495i	$-0.782852\pi$
$727$	− 34.0000i	− 1.26099i	0.776193	−	0.630495i	$-0.217148\pi$
$728$	4.00000i	0.148250i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	0	0
$732$	2.00000i	0.0739221i
$733$	− 32.0000i	− 1.18195i	−0.806691	−	0.590973i	$-0.798744\pi$
$733$	− 32.0000i	− 1.18195i	0.806691	−	0.590973i	$-0.201256\pi$
$734$	−10.0000	−0.369107
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−44.0000	−1.61857	−0.809283	−	0.587419i	$-0.800144\pi$
$739$	−44.0000	−1.61857	−0.809283	+	0.587419i	$0.800144\pi$
$740$	0	0
$741$	2.00000	0.0734718
$742$	− 12.0000i	− 0.440534i
$743$	48.0000i	1.76095i	0.474093	+	0.880475i	$0.342776\pi$
$743$	48.0000i	1.76095i	−0.474093	+	0.880475i	$0.657224\pi$
$744$	2.00000	0.0733236
$745$	0	0
$746$	−14.0000	−0.512576
$747$	6.00000i	0.219529i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−22.0000	−0.802791	−0.401396	−	0.915905i	$-0.631475\pi$
$751$	−22.0000	−0.802791	−0.401396	+	0.915905i	$0.631475\pi$
$752$	0	0
$753$	− 12.0000i	− 0.437304i
$754$	−12.0000	−0.437014
$755$	0	0
$756$	2.00000	0.0727393
$757$	32.0000i	1.16306i	0.813525	+	0.581530i	$0.197546\pi$
$757$	32.0000i	1.16306i	−0.813525	+	0.581530i	$0.802454\pi$
$758$	8.00000i	0.290573i
$759$	0	0
$760$	0	0
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	16.0000i	0.579619i
$763$	32.0000i	1.15848i
$764$	0	0
$765$	0	0
$766$	24.0000	0.867155
$767$	− 12.0000i	− 0.433295i
$768$	− 1.00000i	− 0.0360844i
$769$	34.0000	1.22607	0.613036	−	0.790055i	$-0.289948\pi$
$769$	34.0000	1.22607	0.613036	+	0.790055i	$0.289948\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	2.00000i	0.0719816i
$773$	18.0000i	0.647415i	0.946157	+	0.323708i	$0.104929\pi$
$773$	18.0000i	0.647415i	−0.946157	+	0.323708i	$0.895071\pi$
$774$	8.00000	0.287554
$775$	0	0
$776$	10.0000	0.358979
$777$	4.00000i	0.143499i
$778$	− 30.0000i	− 1.07555i
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	6.00000i	0.214423i
$784$	3.00000	0.107143
$785$	0	0
$786$	0	0
$787$	44.0000i	1.56843i	0.620489	+	0.784215i	$0.286934\pi$
$787$	44.0000i	1.56843i	−0.620489	+	0.784215i	$0.713066\pi$
$788$	− 18.0000i	− 0.641223i
$789$	24.0000	0.854423
$790$	0	0
$791$	−12.0000	−0.426671
$792$	0	0
$793$	− 4.00000i	− 0.142044i
$794$	8.00000	0.283909
$795$	0	0
$796$	−4.00000	−0.141776
$797$	42.0000i	1.48772i	0.668338	+	0.743858i	$0.267006\pi$
$797$	42.0000i	1.48772i	−0.668338	+	0.743858i	$0.732994\pi$
$798$	2.00000i	0.0707992i
$799$	0	0
$800$	0	0
$801$	−12.0000	−0.423999
$802$	− 12.0000i	− 0.423735i
$803$	0	0
$804$	4.00000	0.141069
$805$	0	0
$806$	−4.00000	−0.140894
$807$	18.0000i	0.633630i
$808$	− 6.00000i	− 0.211079i
$809$	−54.0000	−1.89854	−0.949269	−	0.314464i	$-0.898175\pi$
$809$	−54.0000	−1.89854	−0.949269	+	0.314464i	$0.898175\pi$
$810$	0	0
$811$	−16.0000	−0.561836	−0.280918	−	0.959732i	$-0.590639\pi$
$811$	−16.0000	−0.561836	−0.280918	+	0.959732i	$0.590639\pi$
$812$	− 12.0000i	− 0.421117i
$813$	− 20.0000i	− 0.701431i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	8.00000i	0.279885i
$818$	− 22.0000i	− 0.769212i
$819$	−4.00000	−0.139771
$820$	0	0
$821$	42.0000	1.46581	0.732905	−	0.680331i	$-0.238164\pi$
$821$	42.0000	1.46581	0.732905	+	0.680331i	$0.238164\pi$
$822$	12.0000i	0.418548i
$823$	22.0000i	0.766872i	0.923567	+	0.383436i	$0.125259\pi$
$823$	22.0000i	0.766872i	−0.923567	+	0.383436i	$0.874741\pi$
$824$	8.00000	0.278693
$825$	0	0
$826$	12.0000	0.417533
$827$	48.0000i	1.66912i	0.550914	+	0.834562i	$0.314279\pi$
$827$	48.0000i	1.66912i	−0.550914	+	0.834562i	$0.685721\pi$
$828$	0	0
$829$	−8.00000	−0.277851	−0.138926	−	0.990303i	$-0.544365\pi$
$829$	−8.00000	−0.277851	−0.138926	+	0.990303i	$0.544365\pi$
$830$	0	0
$831$	−28.0000	−0.971309
$832$	2.00000i	0.0693375i
$833$	0	0
$834$	−4.00000	−0.138509
$835$	0	0
$836$	0	0
$837$	2.00000i	0.0691301i
$838$	12.0000i	0.414533i
$839$	24.0000	0.828572	0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000	0.828572	0.414286	+	0.910147i	$0.364031\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	− 8.00000i	− 0.275698i
$843$	− 24.0000i	− 0.826604i
$844$	−8.00000	−0.275371
$845$	0	0
$846$	0	0
$847$	− 22.0000i	− 0.755929i
$848$	− 6.00000i	− 0.206041i
$849$	−32.0000	−1.09824
$850$	0	0
$851$	0	0
$852$	0	0
$853$	− 44.0000i	− 1.50653i	−0.657716	−	0.753266i	$-0.728477\pi$
$853$	− 44.0000i	− 1.50653i	0.657716	−	0.753266i	$-0.271523\pi$
$854$	4.00000	0.136877
$855$	0	0
$856$	0	0
$857$	− 18.0000i	− 0.614868i	−0.951569	−	0.307434i	$-0.900530\pi$
$857$	− 18.0000i	− 0.614868i	0.951569	−	0.307434i	$-0.0994704\pi$
$858$	0	0
$859$	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
$861$	0	0
$862$	24.0000i	0.817443i
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	−2.00000	−0.0679628
$867$	− 17.0000i	− 0.577350i
$868$	− 4.00000i	− 0.135769i
$869$	0	0
$870$	0	0
$871$	−8.00000	−0.271070
$872$	16.0000i	0.541828i
$873$	10.0000i	0.338449i
$874$	0	0
$875$	0	0
$876$	14.0000	0.473016
$877$	38.0000i	1.28317i	0.767052	+	0.641584i	$0.221723\pi$
$877$	38.0000i	1.28317i	−0.767052	+	0.641584i	$0.778277\pi$
$878$	− 22.0000i	− 0.742464i
$879$	6.00000	0.202375
$880$	0	0
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	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	$0.913234\pi$
$884$	0	0
$885$	0	0
$886$	30.0000	1.00787
$887$	− 48.0000i	− 1.61168i	−0.592132	−	0.805841i	$-0.701714\pi$
$887$	− 48.0000i	− 1.61168i	0.592132	−	0.805841i	$-0.298286\pi$
$888$	2.00000i	0.0671156i
$889$	32.0000	1.07325
$890$	0	0
$891$	0	0
$892$	− 4.00000i	− 0.133930i
$893$	0	0
$894$	−6.00000	−0.200670
$895$	0	0
$896$	−2.00000	−0.0668153
$897$	0	0
$898$	24.0000i	0.800890i
$899$	12.0000	0.400222
$900$	0	0
$901$	0	0
$902$	0	0
$903$	− 16.0000i	− 0.532447i
$904$	−6.00000	−0.199557
$905$	0	0
$906$	10.0000	0.332228
$907$	44.0000i	1.46100i	0.682915	+	0.730498i	$0.260712\pi$
$907$	44.0000i	1.46100i	−0.682915	+	0.730498i	$0.739288\pi$
$908$	− 12.0000i	− 0.398234i
$909$	6.00000	0.199007
$910$	0	0
$911$	60.0000	1.98789	0.993944	−	0.109885i	$-0.0350482\pi$
$911$	60.0000	1.98789	0.993944	+	0.109885i	$0.0350482\pi$
$912$	1.00000i	0.0331133i
$913$	0	0
$914$	−22.0000	−0.727695
$915$	0	0
$916$	−10.0000	−0.330409
$917$	0	0
$918$	0	0
$919$	40.0000	1.31948	0.659739	−	0.751495i	$-0.270667\pi$
$919$	40.0000	1.31948	0.659739	+	0.751495i	$0.270667\pi$
$920$	0	0
$921$	20.0000	0.659022
$922$	− 30.0000i	− 0.987997i
$923$	0	0
$924$	0	0
$925$	0	0
$926$	−26.0000	−0.854413
$927$	8.00000i	0.262754i
$928$	− 6.00000i	− 0.196960i
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	0	0
$931$	−3.00000	−0.0983210
$932$	24.0000i	0.786146i
$933$	0	0
$934$	6.00000	0.196326
$935$	0	0
$936$	−2.00000	−0.0653720
$937$	26.0000i	0.849383i	0.905338	+	0.424691i	$0.139617\pi$
$937$	26.0000i	0.849383i	−0.905338	+	0.424691i	$0.860383\pi$
$938$	− 8.00000i	− 0.261209i
$939$	10.0000	0.326338
$940$	0	0
$941$	6.00000	0.195594	0.0977972	−	0.995206i	$-0.468820\pi$
$941$	6.00000	0.195594	0.0977972	+	0.995206i	$0.468820\pi$
$942$	4.00000i	0.130327i
$943$	0	0
$944$	6.00000	0.195283
$945$	0	0
$946$	0	0
$947$	− 18.0000i	− 0.584921i	−0.956278	−	0.292461i	$-0.905526\pi$
$947$	− 18.0000i	− 0.584921i	0.956278	−	0.292461i	$-0.0944741\pi$
$948$	− 2.00000i	− 0.0649570i
$949$	−28.0000	−0.908918
$950$	0	0
$951$	−30.0000	−0.972817
$952$	0	0
$953$	− 30.0000i	− 0.971795i	−0.874016	−	0.485898i	$-0.838493\pi$
$953$	− 30.0000i	− 0.971795i	0.874016	−	0.485898i	$-0.161507\pi$
$954$	6.00000	0.194257
$955$	0	0
$956$	24.0000	0.776215
$957$	0	0
$958$	0	0
$959$	24.0000	0.775000
$960$	0	0
$961$	−27.0000	−0.870968
$962$	− 4.00000i	− 0.128965i
$963$	0	0
$964$	22.0000	0.708572
$965$	0	0
$966$	0	0
$967$	26.0000i	0.836104i	0.908423	+	0.418052i	$0.137287\pi$
$967$	26.0000i	0.836104i	−0.908423	+	0.418052i	$0.862713\pi$
$968$	− 11.0000i	− 0.353553i
$969$	0	0
$970$	0	0
$971$	−42.0000	−1.34784	−0.673922	−	0.738802i	$-0.735392\pi$
$971$	−42.0000	−1.34784	−0.673922	+	0.738802i	$0.735392\pi$
$972$	1.00000i	0.0320750i
$973$	8.00000i	0.256468i
$974$	−16.0000	−0.512673
$975$	0	0
$976$	2.00000	0.0640184
$977$	− 42.0000i	− 1.34370i	−0.740688	−	0.671850i	$-0.765500\pi$
$977$	− 42.0000i	− 1.34370i	0.740688	−	0.671850i	$-0.234500\pi$
$978$	− 4.00000i	− 0.127906i
$979$	0	0
$980$	0	0
$981$	−16.0000	−0.510841
$982$	0	0
$983$	− 24.0000i	− 0.765481i	−0.923856	−	0.382741i	$-0.874980\pi$
$983$	− 24.0000i	− 0.765481i	0.923856	−	0.382741i	$-0.125020\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	− 2.00000i	− 0.0636285i
$989$	0	0
$990$	0	0
$991$	−58.0000	−1.84243	−0.921215	−	0.389053i	$-0.872802\pi$
$991$	−58.0000	−1.84243	−0.921215	+	0.389053i	$0.872802\pi$
$992$	− 2.00000i	− 0.0635001i
$993$	− 20.0000i	− 0.634681i
$994$	0	0
$995$	0	0
$996$	6.00000	0.190117
$997$	44.0000i	1.39349i	0.717317	+	0.696747i	$0.245370\pi$
$997$	44.0000i	1.39349i	−0.717317	+	0.696747i	$0.754630\pi$
$998$	20.0000i	0.633089i
$999$	−2.00000	−0.0632772

Twists

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

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

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +1.00000i q^{12} -2.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} +1.00000i q^{18} -1.00000 q^{19} +2.00000 q^{21} +1.00000 q^{24} -2.00000 q^{26} +1.00000i q^{27} -2.00000i q^{28} +6.00000 q^{29} +2.00000 q^{31} -1.00000i q^{32} +1.00000 q^{36} +2.00000i q^{37} +1.00000i q^{38} -2.00000 q^{39} -2.00000i q^{42} -8.00000i q^{43} -1.00000i q^{48} +3.00000 q^{49} +2.00000i q^{52} -6.00000i q^{53} +1.00000 q^{54} -2.00000 q^{56} +1.00000i q^{57} -6.00000i q^{58} +6.00000 q^{59} +2.00000 q^{61} -2.00000i q^{62} -2.00000i q^{63} -1.00000 q^{64} -4.00000i q^{67} -1.00000i q^{72} -14.0000i q^{73} +2.00000 q^{74} +1.00000 q^{76} +2.00000i q^{78} -2.00000 q^{79} +1.00000 q^{81} -6.00000i q^{83} -2.00000 q^{84} -8.00000 q^{86} -6.00000i q^{87} +12.0000 q^{89} +4.00000 q^{91} -2.00000i q^{93} -1.00000 q^{96} -10.0000i q^{97} -3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + 4 q^{14} + 2 q^{16} - 2 q^{19} + 4 q^{21} + 2 q^{24} - 4 q^{26} + 12 q^{29} + 4 q^{31} + 2 q^{36} - 4 q^{39} + 6 q^{49} + 2 q^{54} - 4 q^{56} + 12 q^{59} + 4 q^{61} - 2 q^{64} + 4 q^{74} + 2 q^{76} - 4 q^{79} + 2 q^{81} - 4 q^{84} - 16 q^{86} + 24 q^{89} + 8 q^{91} - 2 q^{96}+O(q^{100}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 + 4 * q^14 + 2 * q^16 - 2 * q^19 + 4 * q^21 + 2 * q^24 - 4 * q^26 + 12 * q^29 + 4 * q^31 + 2 * q^36 - 4 * q^39 + 6 * q^49 + 2 * q^54 - 4 * q^56 + 12 * q^59 + 4 * q^61 - 2 * q^64 + 4 * q^74 + 2 * q^76 - 4 * q^79 + 2 * q^81 - 4 * q^84 - 16 * q^86 + 24 * q^89 + 8 * q^91 - 2 * q^96

Introduction

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