LMFDB - Embedded newform 1050.2.g.c.799.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1050,2,Mod(799,1050)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1050, 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("1050.799");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1050.g (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:	$8.38429221223$
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 210)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			799.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1050.799
Dual form			1050.2.g.c.799.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +1.00000i q^{12} -2.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} -6.00000i q^{17} +1.00000i q^{18} -8.00000 q^{19} +1.00000 q^{21} +1.00000 q^{24} -2.00000 q^{26} +1.00000i q^{27} -1.00000i q^{28} -6.00000 q^{29} -4.00000 q^{31} -1.00000i q^{32} -6.00000 q^{34} +1.00000 q^{36} -10.0000i q^{37} +8.00000i q^{38} -2.00000 q^{39} -6.00000 q^{41} -1.00000i q^{42} +4.00000i q^{43} -1.00000i q^{48} -1.00000 q^{49} -6.00000 q^{51} +2.00000i q^{52} +6.00000i q^{53} +1.00000 q^{54} -1.00000 q^{56} +8.00000i q^{57} +6.00000i q^{58} +12.0000 q^{59} -10.0000 q^{61} +4.00000i q^{62} -1.00000i q^{63} -1.00000 q^{64} -4.00000i q^{67} +6.00000i q^{68} +12.0000 q^{71} -1.00000i q^{72} +10.0000i q^{73} -10.0000 q^{74} +8.00000 q^{76} +2.00000i q^{78} -8.00000 q^{79} +1.00000 q^{81} +6.00000i q^{82} -12.0000i q^{83} -1.00000 q^{84} +4.00000 q^{86} +6.00000i q^{87} +6.00000 q^{89} +2.00000 q^{91} +4.00000i q^{93} -1.00000 q^{96} -10.0000i q^{97} +1.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 2 q^{14} + 2 q^{16} - 16 q^{19} + 2 q^{21} + 2 q^{24} - 4 q^{26} - 12 q^{29} - 8 q^{31} - 12 q^{34} + 2 q^{36} - 4 q^{39} - 12 q^{41} - 2 q^{49} - 12 q^{51} + 2 q^{54} - 2 q^{56} + 24 q^{59} - 20 q^{61} - 2 q^{64} + 24 q^{71} - 20 q^{74} + 16 q^{76} - 16 q^{79} + 2 q^{81} - 2 q^{84} + 8 q^{86} + 12 q^{89} + 4 q^{91} - 2 q^{96}+O(q^{100}) $ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 + 2 * q^14 + 2 * q^16 - 16 * q^19 + 2 * q^21 + 2 * q^24 - 4 * q^26 - 12 * q^29 - 8 * q^31 - 12 * q^34 + 2 * q^36 - 4 * q^39 - 12 * q^41 - 2 * q^49 - 12 * q^51 + 2 * q^54 - 2 * q^56 + 24 * q^59 - 20 * q^61 - 2 * q^64 + 24 * q^71 - 20 * q^74 + 16 * q^76 - 16 * q^79 + 2 * q^81 - 2 * q^84 + 8 * q^86 + 12 * q^89 + 4 * q^91 - 2 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$701$
$\chi(n)$	$-1$	$1$	$1$

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
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	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	$-0.0894562\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	− 6.00000i	− 1.45521i	−0.685994	−	0.727607i	$-0.740633\pi$
$17$	− 6.00000i	− 1.45521i	0.685994	−	0.727607i	$-0.259367\pi$
$18$	1.00000i	0.235702i
$19$	−8.00000	−1.83533	−0.917663	−	0.397360i	$-0.869927\pi$
$19$	−8.00000	−1.83533	−0.917663	+	0.397360i	$0.869927\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	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
$28$	− 1.00000i	− 0.188982i
$29$	−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	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	−6.00000	−1.02899
$35$	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	$-0.307139\pi$
$38$	8.00000i	1.29777i
$39$	−2.00000	−0.320256
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	− 1.00000i	− 0.154303i
$43$	4.00000i	0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$	4.00000i	0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$	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
$49$	−1.00000	−0.142857
$50$	0	0
$51$	−6.00000	−0.840168
$52$	2.00000i	0.277350i
$53$	6.00000i	0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$	6.00000i	0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	−1.00000	−0.133631
$57$	8.00000i	1.05963i
$58$	6.00000i	0.787839i
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	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
$63$	− 1.00000i	− 0.125988i
$64$	−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	$-0.0785709\pi$
$68$	6.00000i	0.727607i
$69$	0	0
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	− 1.00000i	− 0.117851i
$73$	10.0000i	1.17041i	0.810885	+	0.585206i	$0.198986\pi$
$73$	10.0000i	1.17041i	−0.810885	+	0.585206i	$0.801014\pi$
$74$	−10.0000	−1.16248
$75$	0	0
$76$	8.00000	0.917663
$77$	0	0
$78$	2.00000i	0.226455i
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	6.00000i	0.662589i
$83$	− 12.0000i	− 1.31717i	−0.752506	−	0.658586i	$-0.771155\pi$
$83$	− 12.0000i	− 1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$	−1.00000	−0.109109
$85$	0	0
$86$	4.00000	0.431331
$87$	6.00000i	0.643268i
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	0	0
$93$	4.00000i	0.414781i
$94$	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	$-0.169494\pi$
$98$	1.00000i	0.101015i
$99$	0	0
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	6.00000i	0.594089i
$103$	− 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
$107$	12.0000i	1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$	12.0000i	1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$	− 1.00000i	− 0.0962250i
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	0	0
$111$	−10.0000	−0.949158
$112$	1.00000i	0.0944911i
$113$	− 6.00000i	− 0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$	− 6.00000i	− 0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$	8.00000	0.749269
$115$	0	0
$116$	6.00000	0.557086
$117$	2.00000i	0.184900i
$118$	− 12.0000i	− 1.10469i
$119$	6.00000	0.550019
$120$	0	0
$121$	−11.0000	−1.00000
$122$	10.0000i	0.905357i
$123$	6.00000i	0.541002i
$124$	4.00000	0.359211
$125$	0	0
$126$	−1.00000	−0.0890871
$127$	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
$129$	4.00000	0.352180
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	0	0
$133$	− 8.00000i	− 0.693688i
$134$	−4.00000	−0.345547
$135$	0	0
$136$	6.00000	0.514496
$137$	6.00000i	0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$	6.00000i	0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$	0	0
$139$	16.0000	1.35710	0.678551	−	0.734553i	$-0.262608\pi$
$139$	16.0000	1.35710	0.678551	+	0.734553i	$0.262608\pi$
$140$	0	0
$141$	0	0
$142$	− 12.0000i	− 1.00702i
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	10.0000	0.827606
$147$	1.00000i	0.0824786i
$148$	10.0000i	0.821995i
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	− 8.00000i	− 0.648886i
$153$	6.00000i	0.485071i
$154$	0	0
$155$	0	0
$156$	2.00000	0.160128
$157$	− 22.0000i	− 1.75579i	−0.478852	−	0.877896i	$-0.658947\pi$
$157$	− 22.0000i	− 1.75579i	0.478852	−	0.877896i	$-0.341053\pi$
$158$	8.00000i	0.636446i
$159$	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	$-0.286445\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	−12.0000	−0.931381
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	1.00000i	0.0771517i
$169$	9.00000	0.692308
$170$	0	0
$171$	8.00000	0.611775
$172$	− 4.00000i	− 0.304997i
$173$	− 6.00000i	− 0.456172i	−0.973641	−	0.228086i	$-0.926753\pi$
$173$	− 6.00000i	− 0.456172i	0.973641	−	0.228086i	$-0.0732467\pi$
$174$	6.00000	0.454859
$175$	0	0
$176$	0	0
$177$	− 12.0000i	− 0.901975i
$178$	− 6.00000i	− 0.449719i
$179$	24.0000	1.79384	0.896922	−	0.442189i	$-0.145798\pi$
$179$	24.0000	1.79384	0.896922	+	0.442189i	$0.145798\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	− 2.00000i	− 0.148250i
$183$	10.0000i	0.739221i
$184$	0	0
$185$	0	0
$186$	4.00000	0.293294
$187$	0	0
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	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
$196$	1.00000	0.0714286
$197$	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
$199$	−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
$203$	− 6.00000i	− 0.421117i
$204$	6.00000	0.420084
$205$	0	0
$206$	−8.00000	−0.557386
$207$	0	0
$208$	− 2.00000i	− 0.138675i
$209$	0	0
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	− 6.00000i	− 0.412082i
$213$	− 12.0000i	− 0.822226i
$214$	12.0000	0.820303
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	− 4.00000i	− 0.271538i
$218$	14.0000i	0.948200i
$219$	10.0000	0.675737
$220$	0	0
$221$	−12.0000	−0.807207
$222$	10.0000i	0.671156i
$223$	− 8.00000i	− 0.535720i	−0.963458	−	0.267860i	$-0.913684\pi$
$223$	− 8.00000i	− 0.535720i	0.963458	−	0.267860i	$-0.0863164\pi$
$224$	1.00000	0.0668153
$225$	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	$0.869623\pi$
$228$	− 8.00000i	− 0.529813i
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	0	0
$232$	− 6.00000i	− 0.393919i
$233$	− 30.0000i	− 1.96537i	−0.185296	−	0.982683i	$-0.559325\pi$
$233$	− 30.0000i	− 1.96537i	0.185296	−	0.982683i	$-0.440675\pi$
$234$	2.00000	0.130744
$235$	0	0
$236$	−12.0000	−0.781133
$237$	8.00000i	0.519656i
$238$	− 6.00000i	− 0.388922i
$239$	12.0000	0.776215	0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	0	0
$241$	26.0000	1.67481	0.837404	−	0.546585i	$-0.184072\pi$
$241$	26.0000	1.67481	0.837404	+	0.546585i	$0.184072\pi$
$242$	11.0000i	0.707107i
$243$	− 1.00000i	− 0.0641500i
$244$	10.0000	0.640184
$245$	0	0
$246$	6.00000	0.382546
$247$	16.0000i	1.01806i
$248$	− 4.00000i	− 0.254000i
$249$	−12.0000	−0.760469
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	1.00000i	0.0629941i
$253$	0	0
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 30.0000i	− 1.87135i	−0.352865	−	0.935674i	$-0.614792\pi$
$257$	− 30.0000i	− 1.87135i	0.352865	−	0.935674i	$-0.385208\pi$
$258$	− 4.00000i	− 0.249029i
$259$	10.0000	0.621370
$260$	0	0
$261$	6.00000	0.371391
$262$	12.0000i	0.741362i
$263$	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
$266$	−8.00000	−0.490511
$267$	− 6.00000i	− 0.367194i
$268$	4.00000i	0.244339i
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	0	0
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	− 6.00000i	− 0.363803i
$273$	− 2.00000i	− 0.121046i
$274$	6.00000	0.362473
$275$	0	0
$276$	0	0
$277$	− 10.0000i	− 0.600842i	−0.953807	−	0.300421i	$-0.902873\pi$
$277$	− 10.0000i	− 0.600842i	0.953807	−	0.300421i	$-0.0971271\pi$
$278$	− 16.0000i	− 0.959616i
$279$	4.00000	0.239474
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	0	0
$283$	4.00000i	0.237775i	0.992908	+	0.118888i	$0.0379328\pi$
$283$	4.00000i	0.237775i	−0.992908	+	0.118888i	$0.962067\pi$
$284$	−12.0000	−0.712069
$285$	0	0
$286$	0	0
$287$	− 6.00000i	− 0.354169i
$288$	1.00000i	0.0589256i
$289$	−19.0000	−1.11765
$290$	0	0
$291$	−10.0000	−0.586210
$292$	− 10.0000i	− 0.585206i
$293$	− 30.0000i	− 1.75262i	−0.481749	−	0.876309i	$-0.659998\pi$
$293$	− 30.0000i	− 1.75262i	0.481749	−	0.876309i	$-0.340002\pi$
$294$	1.00000	0.0583212
$295$	0	0
$296$	10.0000	0.581238
$297$	0	0
$298$	6.00000i	0.347571i
$299$	0	0
$300$	0	0
$301$	−4.00000	−0.230556
$302$	− 8.00000i	− 0.460348i
$303$	− 6.00000i	− 0.344691i
$304$	−8.00000	−0.458831
$305$	0	0
$306$	6.00000	0.342997
$307$	− 4.00000i	− 0.228292i	−0.993464	−	0.114146i	$-0.963587\pi$
$307$	− 4.00000i	− 0.228292i	0.993464	−	0.114146i	$-0.0364132\pi$
$308$	0	0
$309$	−8.00000	−0.455104
$310$	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
$313$	− 14.0000i	− 0.791327i	−0.918396	−	0.395663i	$-0.870515\pi$
$313$	− 14.0000i	− 0.791327i	0.918396	−	0.395663i	$-0.129485\pi$
$314$	−22.0000	−1.24153
$315$	0	0
$316$	8.00000	0.450035
$317$	18.0000i	1.01098i	0.862832	+	0.505490i	$0.168688\pi$
$317$	18.0000i	1.01098i	−0.862832	+	0.505490i	$0.831312\pi$
$318$	− 6.00000i	− 0.336463i
$319$	0	0
$320$	0	0
$321$	12.0000	0.669775
$322$	0	0
$323$	48.0000i	2.67079i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−20.0000	−1.10770
$327$	14.0000i	0.774202i
$328$	− 6.00000i	− 0.331295i
$329$	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	$0.779499\pi$
$332$	12.0000i	0.658586i
$333$	10.0000i	0.547997i
$334$	0	0
$335$	0	0
$336$	1.00000	0.0545545
$337$	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
$342$	− 8.00000i	− 0.432590i
$343$	− 1.00000i	− 0.0539949i
$344$	−4.00000	−0.215666
$345$	0	0
$346$	−6.00000	−0.322562
$347$	− 12.0000i	− 0.644194i	−0.946707	−	0.322097i	$-0.895612\pi$
$347$	− 12.0000i	− 0.644194i	0.946707	−	0.322097i	$-0.104388\pi$
$348$	− 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
$353$	6.00000i	0.319348i	0.987170	+	0.159674i	$0.0510443\pi$
$353$	6.00000i	0.319348i	−0.987170	+	0.159674i	$0.948956\pi$
$354$	−12.0000	−0.637793
$355$	0	0
$356$	−6.00000	−0.317999
$357$	− 6.00000i	− 0.317554i
$358$	− 24.0000i	− 1.26844i
$359$	12.0000	0.633336	0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000	0.633336	0.316668	+	0.948536i	$0.397436\pi$
$360$	0	0
$361$	45.0000	2.36842
$362$	10.0000i	0.525588i
$363$	11.0000i	0.577350i
$364$	−2.00000	−0.104828
$365$	0	0
$366$	10.0000	0.522708
$367$	8.00000i	0.417597i	0.977959	+	0.208798i	$0.0669552\pi$
$367$	8.00000i	0.417597i	−0.977959	+	0.208798i	$0.933045\pi$
$368$	0	0
$369$	6.00000	0.312348
$370$	0	0
$371$	−6.00000	−0.311504
$372$	− 4.00000i	− 0.207390i
$373$	34.0000i	1.76045i	0.474554	+	0.880227i	$0.342610\pi$
$373$	34.0000i	1.76045i	−0.474554	+	0.880227i	$0.657390\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	12.0000i	0.618031i
$378$	1.00000i	0.0514344i
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	0	0
$381$	8.00000	0.409852
$382$	− 12.0000i	− 0.613973i
$383$	− 24.0000i	− 1.22634i	−0.789950	−	0.613171i	$-0.789894\pi$
$383$	− 24.0000i	− 1.22634i	0.789950	−	0.613171i	$-0.210106\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−2.00000	−0.101797
$387$	− 4.00000i	− 0.203331i
$388$	10.0000i	0.507673i
$389$	−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
$391$	0	0
$392$	− 1.00000i	− 0.0505076i
$393$	12.0000i	0.605320i
$394$	18.0000	0.906827
$395$	0	0
$396$	0	0
$397$	2.00000i	0.100377i	0.998740	+	0.0501886i	$0.0159822\pi$
$397$	2.00000i	0.100377i	−0.998740	+	0.0501886i	$0.984018\pi$
$398$	20.0000i	1.00251i
$399$	−8.00000	−0.400501
$400$	0	0
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	4.00000i	0.199502i
$403$	8.00000i	0.398508i
$404$	−6.00000	−0.298511
$405$	0	0
$406$	−6.00000	−0.297775
$407$	0	0
$408$	− 6.00000i	− 0.297044i
$409$	−26.0000	−1.28562	−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000	−1.28562	−0.642809	+	0.766027i	$0.722231\pi$
$410$	0	0
$411$	6.00000	0.295958
$412$	8.00000i	0.394132i
$413$	12.0000i	0.590481i
$414$	0	0
$415$	0	0
$416$	−2.00000	−0.0980581
$417$	− 16.0000i	− 0.783523i
$418$	0	0
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	38.0000	1.85201	0.926003	−	0.377515i	$-0.123221\pi$
$421$	38.0000	1.85201	0.926003	+	0.377515i	$0.123221\pi$
$422$	4.00000i	0.194717i
$423$	0	0
$424$	−6.00000	−0.291386
$425$	0	0
$426$	−12.0000	−0.581402
$427$	− 10.0000i	− 0.483934i
$428$	− 12.0000i	− 0.580042i
$429$	0	0
$430$	0	0
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	1.00000i	0.0481125i
$433$	10.0000i	0.480569i	0.970702	+	0.240285i	$0.0772408\pi$
$433$	10.0000i	0.480569i	−0.970702	+	0.240285i	$0.922759\pi$
$434$	−4.00000	−0.192006
$435$	0	0
$436$	14.0000	0.670478
$437$	0	0
$438$	− 10.0000i	− 0.477818i
$439$	28.0000	1.33637	0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000	1.33637	0.668184	+	0.743996i	$0.267072\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	12.0000i	0.570782i
$443$	12.0000i	0.570137i	0.958507	+	0.285069i	$0.0920164\pi$
$443$	12.0000i	0.570137i	−0.958507	+	0.285069i	$0.907984\pi$
$444$	10.0000	0.474579
$445$	0	0
$446$	−8.00000	−0.378811
$447$	6.00000i	0.283790i
$448$	− 1.00000i	− 0.0472456i
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	0	0
$451$	0	0
$452$	6.00000i	0.282216i
$453$	− 8.00000i	− 0.375873i
$454$	12.0000	0.563188
$455$	0	0
$456$	−8.00000	−0.374634
$457$	2.00000i	0.0935561i	0.998905	+	0.0467780i	$0.0148953\pi$
$457$	2.00000i	0.0935561i	−0.998905	+	0.0467780i	$0.985105\pi$
$458$	14.0000i	0.654177i
$459$	6.00000	0.280056
$460$	0	0
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	0	0
$463$	40.0000i	1.85896i	0.368875	+	0.929479i	$0.379743\pi$
$463$	40.0000i	1.85896i	−0.368875	+	0.929479i	$0.620257\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	−30.0000	−1.38972
$467$	− 36.0000i	− 1.66588i	−0.553362	−	0.832941i	$-0.686655\pi$
$467$	− 36.0000i	− 1.66588i	0.553362	−	0.832941i	$-0.313345\pi$
$468$	− 2.00000i	− 0.0924500i
$469$	4.00000	0.184703
$470$	0	0
$471$	−22.0000	−1.01371
$472$	12.0000i	0.552345i
$473$	0	0
$474$	8.00000	0.367452
$475$	0	0
$476$	−6.00000	−0.275010
$477$	− 6.00000i	− 0.274721i
$478$	− 12.0000i	− 0.548867i
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	−20.0000	−0.911922
$482$	− 26.0000i	− 1.18427i
$483$	0	0
$484$	11.0000	0.500000
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	32.0000i	1.45006i	0.688718	+	0.725029i	$0.258174\pi$
$487$	32.0000i	1.45006i	−0.688718	+	0.725029i	$0.741826\pi$
$488$	− 10.0000i	− 0.452679i
$489$	−20.0000	−0.904431
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	− 6.00000i	− 0.270501i
$493$	36.0000i	1.62136i
$494$	16.0000	0.719874
$495$	0	0
$496$	−4.00000	−0.179605
$497$	12.0000i	0.538274i
$498$	12.0000i	0.537733i
$499$	−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
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	1.00000	0.0445435
$505$	0	0
$506$	0	0
$507$	− 9.00000i	− 0.399704i
$508$	− 8.00000i	− 0.354943i
$509$	−30.0000	−1.32973	−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000	−1.32973	−0.664863	+	0.746965i	$0.731510\pi$
$510$	0	0
$511$	−10.0000	−0.442374
$512$	− 1.00000i	− 0.0441942i
$513$	− 8.00000i	− 0.353209i
$514$	−30.0000	−1.32324
$515$	0	0
$516$	−4.00000	−0.176090
$517$	0	0
$518$	− 10.0000i	− 0.439375i
$519$	−6.00000	−0.263371
$520$	0	0
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	− 6.00000i	− 0.262613i
$523$	28.0000i	1.22435i	0.790721	+	0.612177i	$0.209706\pi$
$523$	28.0000i	1.22435i	−0.790721	+	0.612177i	$0.790294\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	24.0000	1.04645
$527$	24.0000i	1.04546i
$528$	0	0
$529$	23.0000	1.00000
$530$	0	0
$531$	−12.0000	−0.520756
$532$	8.00000i	0.346844i
$533$	12.0000i	0.519778i
$534$	−6.00000	−0.259645
$535$	0	0
$536$	4.00000	0.172774
$537$	− 24.0000i	− 1.03568i
$538$	− 18.0000i	− 0.776035i
$539$	0	0
$540$	0	0
$541$	−34.0000	−1.46177	−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000	−1.46177	−0.730887	+	0.682498i	$0.760893\pi$
$542$	− 20.0000i	− 0.859074i
$543$	10.0000i	0.429141i
$544$	−6.00000	−0.257248
$545$	0	0
$546$	−2.00000	−0.0855921
$547$	44.0000i	1.88130i	0.339372	+	0.940652i	$0.389785\pi$
$547$	44.0000i	1.88130i	−0.339372	+	0.940652i	$0.610215\pi$
$548$	− 6.00000i	− 0.256307i
$549$	10.0000	0.426790
$550$	0	0
$551$	48.0000	2.04487
$552$	0	0
$553$	− 8.00000i	− 0.340195i
$554$	−10.0000	−0.424859
$555$	0	0
$556$	−16.0000	−0.678551
$557$	18.0000i	0.762684i	0.924434	+	0.381342i	$0.124538\pi$
$557$	18.0000i	0.762684i	−0.924434	+	0.381342i	$0.875462\pi$
$558$	− 4.00000i	− 0.169334i
$559$	8.00000	0.338364
$560$	0	0
$561$	0	0
$562$	− 18.0000i	− 0.759284i
$563$	− 36.0000i	− 1.51722i	−0.651546	−	0.758610i	$-0.725879\pi$
$563$	− 36.0000i	− 1.51722i	0.651546	−	0.758610i	$-0.274121\pi$
$564$	0	0
$565$	0	0
$566$	4.00000	0.168133
$567$	1.00000i	0.0419961i
$568$	12.0000i	0.503509i
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	0	0
$573$	− 12.0000i	− 0.501307i
$574$	−6.00000	−0.250435
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 10.0000i	− 0.416305i	−0.978096	−	0.208153i	$-0.933255\pi$
$577$	− 10.0000i	− 0.416305i	0.978096	−	0.208153i	$-0.0667451\pi$
$578$	19.0000i	0.790296i
$579$	−2.00000	−0.0831172
$580$	0	0
$581$	12.0000	0.497844
$582$	10.0000i	0.414513i
$583$	0	0
$584$	−10.0000	−0.413803
$585$	0	0
$586$	−30.0000	−1.23929
$587$	12.0000i	0.495293i	0.968850	+	0.247647i	$0.0796572\pi$
$587$	12.0000i	0.495293i	−0.968850	+	0.247647i	$0.920343\pi$
$588$	− 1.00000i	− 0.0412393i
$589$	32.0000	1.31854
$590$	0	0
$591$	18.0000	0.740421
$592$	− 10.0000i	− 0.410997i
$593$	6.00000i	0.246390i	0.992382	+	0.123195i	$0.0393141\pi$
$593$	6.00000i	0.246390i	−0.992382	+	0.123195i	$0.960686\pi$
$594$	0	0
$595$	0	0
$596$	6.00000	0.245770
$597$	20.0000i	0.818546i
$598$	0	0
$599$	−36.0000	−1.47092	−0.735460	−	0.677568i	$-0.763034\pi$
$599$	−36.0000	−1.47092	−0.735460	+	0.677568i	$0.763034\pi$
$600$	0	0
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	4.00000i	0.163028i
$603$	4.00000i	0.162893i
$604$	−8.00000	−0.325515
$605$	0	0
$606$	−6.00000	−0.243733
$607$	8.00000i	0.324710i	0.986732	+	0.162355i	$0.0519090\pi$
$607$	8.00000i	0.324710i	−0.986732	+	0.162355i	$0.948091\pi$
$608$	8.00000i	0.324443i
$609$	−6.00000	−0.243132
$610$	0	0
$611$	0	0
$612$	− 6.00000i	− 0.242536i
$613$	− 38.0000i	− 1.53481i	−0.641165	−	0.767403i	$-0.721549\pi$
$613$	− 38.0000i	− 1.53481i	0.641165	−	0.767403i	$-0.278451\pi$
$614$	−4.00000	−0.161427
$615$	0	0
$616$	0	0
$617$	− 18.0000i	− 0.724653i	−0.932051	−	0.362326i	$-0.881983\pi$
$617$	− 18.0000i	− 0.724653i	0.932051	−	0.362326i	$-0.118017\pi$
$618$	8.00000i	0.321807i
$619$	−32.0000	−1.28619	−0.643094	−	0.765787i	$-0.722350\pi$
$619$	−32.0000	−1.28619	−0.643094	+	0.765787i	$0.722350\pi$
$620$	0	0
$621$	0	0
$622$	− 24.0000i	− 0.962312i
$623$	6.00000i	0.240385i
$624$	−2.00000	−0.0800641
$625$	0	0
$626$	−14.0000	−0.559553
$627$	0	0
$628$	22.0000i	0.877896i
$629$	−60.0000	−2.39236
$630$	0	0
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	− 8.00000i	− 0.318223i
$633$	4.00000i	0.158986i
$634$	18.0000	0.714871
$635$	0	0
$636$	−6.00000	−0.237915
$637$	2.00000i	0.0792429i
$638$	0	0
$639$	−12.0000	−0.474713
$640$	0	0
$641$	42.0000	1.65890	0.829450	−	0.558581i	$-0.188654\pi$
$641$	42.0000	1.65890	0.829450	+	0.558581i	$0.188654\pi$
$642$	− 12.0000i	− 0.473602i
$643$	4.00000i	0.157745i	0.996885	+	0.0788723i	$0.0251319\pi$
$643$	4.00000i	0.157745i	−0.996885	+	0.0788723i	$0.974868\pi$
$644$	0	0
$645$	0	0
$646$	48.0000	1.88853
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	1.00000i	0.0392837i
$649$	0	0
$650$	0	0
$651$	−4.00000	−0.156772
$652$	20.0000i	0.783260i
$653$	6.00000i	0.234798i	0.993085	+	0.117399i	$0.0374557\pi$
$653$	6.00000i	0.234798i	−0.993085	+	0.117399i	$0.962544\pi$
$654$	14.0000	0.547443
$655$	0	0
$656$	−6.00000	−0.234261
$657$	− 10.0000i	− 0.390137i
$658$	0	0
$659$	−48.0000	−1.86981	−0.934907	−	0.354892i	$-0.884518\pi$
$659$	−48.0000	−1.86981	−0.934907	+	0.354892i	$0.884518\pi$
$660$	0	0
$661$	−34.0000	−1.32245	−0.661223	−	0.750189i	$-0.729962\pi$
$661$	−34.0000	−1.32245	−0.661223	+	0.750189i	$0.729962\pi$
$662$	28.0000i	1.08825i
$663$	12.0000i	0.466041i
$664$	12.0000	0.465690
$665$	0	0
$666$	10.0000	0.387492
$667$	0	0
$668$	0	0
$669$	−8.00000	−0.309298
$670$	0	0
$671$	0	0
$672$	− 1.00000i	− 0.0385758i
$673$	− 26.0000i	− 1.00223i	−0.865382	−	0.501113i	$-0.832924\pi$
$673$	− 26.0000i	− 1.00223i	0.865382	−	0.501113i	$-0.167076\pi$
$674$	2.00000	0.0770371
$675$	0	0
$676$	−9.00000	−0.346154
$677$	− 18.0000i	− 0.691796i	−0.938272	−	0.345898i	$-0.887574\pi$
$677$	− 18.0000i	− 0.691796i	0.938272	−	0.345898i	$-0.112426\pi$
$678$	6.00000i	0.230429i
$679$	10.0000	0.383765
$680$	0	0
$681$	12.0000	0.459841
$682$	0	0
$683$	− 36.0000i	− 1.37750i	−0.724998	−	0.688751i	$-0.758159\pi$
$683$	− 36.0000i	− 1.37750i	0.724998	−	0.688751i	$-0.241841\pi$
$684$	−8.00000	−0.305888
$685$	0	0
$686$	−1.00000	−0.0381802
$687$	14.0000i	0.534133i
$688$	4.00000i	0.152499i
$689$	12.0000	0.457164
$690$	0	0
$691$	−16.0000	−0.608669	−0.304334	−	0.952565i	$-0.598434\pi$
$691$	−16.0000	−0.608669	−0.304334	+	0.952565i	$0.598434\pi$
$692$	6.00000i	0.228086i
$693$	0	0
$694$	−12.0000	−0.455514
$695$	0	0
$696$	−6.00000	−0.227429
$697$	36.0000i	1.36360i
$698$	− 10.0000i	− 0.378506i
$699$	−30.0000	−1.13470
$700$	0	0
$701$	6.00000	0.226617	0.113308	−	0.993560i	$-0.463855\pi$
$701$	6.00000	0.226617	0.113308	+	0.993560i	$0.463855\pi$
$702$	− 2.00000i	− 0.0754851i
$703$	80.0000i	3.01726i
$704$	0	0
$705$	0	0
$706$	6.00000	0.225813
$707$	6.00000i	0.225653i
$708$	12.0000i	0.450988i
$709$	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
$711$	8.00000	0.300023
$712$	6.00000i	0.224860i
$713$	0	0
$714$	−6.00000	−0.224544
$715$	0	0
$716$	−24.0000	−0.896922
$717$	− 12.0000i	− 0.448148i
$718$	− 12.0000i	− 0.447836i
$719$	24.0000	0.895049	0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	− 45.0000i	− 1.67473i
$723$	− 26.0000i	− 0.966950i
$724$	10.0000	0.371647
$725$	0	0
$726$	11.0000	0.408248
$727$	32.0000i	1.18681i	0.804902	+	0.593407i	$0.202218\pi$
$727$	32.0000i	1.18681i	−0.804902	+	0.593407i	$0.797782\pi$
$728$	2.00000i	0.0741249i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	24.0000	0.887672
$732$	− 10.0000i	− 0.369611i
$733$	− 2.00000i	− 0.0738717i	−0.999318	−	0.0369358i	$-0.988240\pi$
$733$	− 2.00000i	− 0.0738717i	0.999318	−	0.0369358i	$-0.0117597\pi$
$734$	8.00000	0.295285
$735$	0	0
$736$	0	0
$737$	0	0
$738$	− 6.00000i	− 0.220863i
$739$	28.0000	1.03000	0.514998	−	0.857191i	$-0.327793\pi$
$739$	28.0000	1.03000	0.514998	+	0.857191i	$0.327793\pi$
$740$	0	0
$741$	16.0000	0.587775
$742$	6.00000i	0.220267i
$743$	24.0000i	0.880475i	0.897881	+	0.440237i	$0.145106\pi$
$743$	24.0000i	0.880475i	−0.897881	+	0.440237i	$0.854894\pi$
$744$	−4.00000	−0.146647
$745$	0	0
$746$	34.0000	1.24483
$747$	12.0000i	0.439057i
$748$	0	0
$749$	−12.0000	−0.438470
$750$	0	0
$751$	8.00000	0.291924	0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	0	0
$753$	12.0000i	0.437304i
$754$	12.0000	0.437014
$755$	0	0
$756$	1.00000	0.0363696
$757$	− 34.0000i	− 1.23575i	−0.786276	−	0.617876i	$-0.787994\pi$
$757$	− 34.0000i	− 1.23575i	0.786276	−	0.617876i	$-0.212006\pi$
$758$	− 4.00000i	− 0.145287i
$759$	0	0
$760$	0	0
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	− 8.00000i	− 0.289809i
$763$	− 14.0000i	− 0.506834i
$764$	−12.0000	−0.434145
$765$	0	0
$766$	−24.0000	−0.867155
$767$	− 24.0000i	− 0.866590i
$768$	− 1.00000i	− 0.0360844i
$769$	−26.0000	−0.937584	−0.468792	−	0.883309i	$-0.655311\pi$
$769$	−26.0000	−0.937584	−0.468792	+	0.883309i	$0.655311\pi$
$770$	0	0
$771$	−30.0000	−1.08042
$772$	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	$0.895071\pi$
$774$	−4.00000	−0.143777
$775$	0	0
$776$	10.0000	0.358979
$777$	− 10.0000i	− 0.358748i
$778$	6.00000i	0.215110i
$779$	48.0000	1.71978
$780$	0	0
$781$	0	0
$782$	0	0
$783$	− 6.00000i	− 0.214423i
$784$	−1.00000	−0.0357143
$785$	0	0
$786$	12.0000	0.428026
$787$	− 4.00000i	− 0.142585i	−0.997455	−	0.0712923i	$-0.977288\pi$
$787$	− 4.00000i	− 0.142585i	0.997455	−	0.0712923i	$-0.0227123\pi$
$788$	− 18.0000i	− 0.641223i
$789$	24.0000	0.854423
$790$	0	0
$791$	6.00000	0.213335
$792$	0	0
$793$	20.0000i	0.710221i
$794$	2.00000	0.0709773
$795$	0	0
$796$	20.0000	0.708881
$797$	− 42.0000i	− 1.48772i	−0.668338	−	0.743858i	$-0.732994\pi$
$797$	− 42.0000i	− 1.48772i	0.668338	−	0.743858i	$-0.267006\pi$
$798$	8.00000i	0.283197i
$799$	0	0
$800$	0	0
$801$	−6.00000	−0.212000
$802$	30.0000i	1.05934i
$803$	0	0
$804$	4.00000	0.141069
$805$	0	0
$806$	8.00000	0.281788
$807$	− 18.0000i	− 0.633630i
$808$	6.00000i	0.211079i
$809$	30.0000	1.05474	0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000	1.05474	0.527372	+	0.849635i	$0.323177\pi$
$810$	0	0
$811$	−16.0000	−0.561836	−0.280918	−	0.959732i	$-0.590639\pi$
$811$	−16.0000	−0.561836	−0.280918	+	0.959732i	$0.590639\pi$
$812$	6.00000i	0.210559i
$813$	− 20.0000i	− 0.701431i
$814$	0	0
$815$	0	0
$816$	−6.00000	−0.210042
$817$	− 32.0000i	− 1.11954i
$818$	26.0000i	0.909069i
$819$	−2.00000	−0.0698857
$820$	0	0
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	− 6.00000i	− 0.209274i
$823$	− 32.0000i	− 1.11545i	−0.830026	−	0.557725i	$-0.811674\pi$
$823$	− 32.0000i	− 1.11545i	0.830026	−	0.557725i	$-0.188326\pi$
$824$	8.00000	0.278693
$825$	0	0
$826$	12.0000	0.417533
$827$	− 36.0000i	− 1.25184i	−0.779886	−	0.625921i	$-0.784723\pi$
$827$	− 36.0000i	− 1.25184i	0.779886	−	0.625921i	$-0.215277\pi$
$828$	0	0
$829$	34.0000	1.18087	0.590434	−	0.807086i	$-0.298956\pi$
$829$	34.0000	1.18087	0.590434	+	0.807086i	$0.298956\pi$
$830$	0	0
$831$	−10.0000	−0.346896
$832$	2.00000i	0.0693375i
$833$	6.00000i	0.207888i
$834$	−16.0000	−0.554035
$835$	0	0
$836$	0	0
$837$	− 4.00000i	− 0.138260i
$838$	− 12.0000i	− 0.414533i
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	− 38.0000i	− 1.30957i
$843$	− 18.0000i	− 0.619953i
$844$	4.00000	0.137686
$845$	0	0
$846$	0	0
$847$	− 11.0000i	− 0.377964i
$848$	6.00000i	0.206041i
$849$	4.00000	0.137280
$850$	0	0
$851$	0	0
$852$	12.0000i	0.411113i
$853$	− 26.0000i	− 0.890223i	−0.895475	−	0.445112i	$-0.853164\pi$
$853$	− 26.0000i	− 0.890223i	0.895475	−	0.445112i	$-0.146836\pi$
$854$	−10.0000	−0.342193
$855$	0	0
$856$	−12.0000	−0.410152
$857$	− 6.00000i	− 0.204956i	−0.994735	−	0.102478i	$-0.967323\pi$
$857$	− 6.00000i	− 0.204956i	0.994735	−	0.102478i	$-0.0326771\pi$
$858$	0	0
$859$	40.0000	1.36478	0.682391	−	0.730987i	$-0.260940\pi$
$859$	40.0000	1.36478	0.682391	+	0.730987i	$0.260940\pi$
$860$	0	0
$861$	−6.00000	−0.204479
$862$	12.0000i	0.408722i
$863$	24.0000i	0.816970i	0.912765	+	0.408485i	$0.133943\pi$
$863$	24.0000i	0.816970i	−0.912765	+	0.408485i	$0.866057\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	10.0000	0.339814
$867$	19.0000i	0.645274i
$868$	4.00000i	0.135769i
$869$	0	0
$870$	0	0
$871$	−8.00000	−0.271070
$872$	− 14.0000i	− 0.474100i
$873$	10.0000i	0.338449i
$874$	0	0
$875$	0	0
$876$	−10.0000	−0.337869
$877$	14.0000i	0.472746i	0.971662	+	0.236373i	$0.0759588\pi$
$877$	14.0000i	0.472746i	−0.971662	+	0.236373i	$0.924041\pi$
$878$	− 28.0000i	− 0.944954i
$879$	−30.0000	−1.01187
$880$	0	0
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	− 1.00000i	− 0.0336718i
$883$	− 44.0000i	− 1.48072i	−0.672212	−	0.740359i	$-0.734656\pi$
$883$	− 44.0000i	− 1.48072i	0.672212	−	0.740359i	$-0.265344\pi$
$884$	12.0000	0.403604
$885$	0	0
$886$	12.0000	0.403148
$887$	24.0000i	0.805841i	0.915235	+	0.402921i	$0.132005\pi$
$887$	24.0000i	0.805841i	−0.915235	+	0.402921i	$0.867995\pi$
$888$	− 10.0000i	− 0.335578i
$889$	−8.00000	−0.268311
$890$	0	0
$891$	0	0
$892$	8.00000i	0.267860i
$893$	0	0
$894$	6.00000	0.200670
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	18.0000i	0.600668i
$899$	24.0000	0.800445
$900$	0	0
$901$	36.0000	1.19933
$902$	0	0
$903$	4.00000i	0.133112i
$904$	6.00000	0.199557
$905$	0	0
$906$	−8.00000	−0.265782
$907$	− 28.0000i	− 0.929725i	−0.885383	−	0.464862i	$-0.846104\pi$
$907$	− 28.0000i	− 0.929725i	0.885383	−	0.464862i	$-0.153896\pi$
$908$	− 12.0000i	− 0.398234i
$909$	−6.00000	−0.199007
$910$	0	0
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	8.00000i	0.264906i
$913$	0	0
$914$	2.00000	0.0661541
$915$	0	0
$916$	14.0000	0.462573
$917$	− 12.0000i	− 0.396275i
$918$	− 6.00000i	− 0.198030i
$919$	−32.0000	−1.05558	−0.527791	−	0.849374i	$-0.676980\pi$
$919$	−32.0000	−1.05558	−0.527791	+	0.849374i	$0.676980\pi$
$920$	0	0
$921$	−4.00000	−0.131804
$922$	18.0000i	0.592798i
$923$	− 24.0000i	− 0.789970i
$924$	0	0
$925$	0	0
$926$	40.0000	1.31448
$927$	8.00000i	0.262754i
$928$	6.00000i	0.196960i
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	8.00000	0.262189
$932$	30.0000i	0.982683i
$933$	− 24.0000i	− 0.785725i
$934$	−36.0000	−1.17796
$935$	0	0
$936$	−2.00000	−0.0653720
$937$	38.0000i	1.24141i	0.784046	+	0.620703i	$0.213153\pi$
$937$	38.0000i	1.24141i	−0.784046	+	0.620703i	$0.786847\pi$
$938$	− 4.00000i	− 0.130605i
$939$	−14.0000	−0.456873
$940$	0	0
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	22.0000i	0.716799i
$943$	0	0
$944$	12.0000	0.390567
$945$	0	0
$946$	0	0
$947$	− 36.0000i	− 1.16984i	−0.811090	−	0.584921i	$-0.801125\pi$
$947$	− 36.0000i	− 1.16984i	0.811090	−	0.584921i	$-0.198875\pi$
$948$	− 8.00000i	− 0.259828i
$949$	20.0000	0.649227
$950$	0	0
$951$	18.0000	0.583690
$952$	6.00000i	0.194461i
$953$	− 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
$956$	−12.0000	−0.388108
$957$	0	0
$958$	0	0
$959$	−6.00000	−0.193750
$960$	0	0
$961$	−15.0000	−0.483871
$962$	20.0000i	0.644826i
$963$	− 12.0000i	− 0.386695i
$964$	−26.0000	−0.837404
$965$	0	0
$966$	0	0
$967$	8.00000i	0.257263i	0.991692	+	0.128631i	$0.0410584\pi$
$967$	8.00000i	0.257263i	−0.991692	+	0.128631i	$0.958942\pi$
$968$	− 11.0000i	− 0.353553i
$969$	48.0000	1.54198
$970$	0	0
$971$	36.0000	1.15529	0.577647	−	0.816286i	$-0.303971\pi$
$971$	36.0000	1.15529	0.577647	+	0.816286i	$0.303971\pi$
$972$	1.00000i	0.0320750i
$973$	16.0000i	0.512936i
$974$	32.0000	1.02535
$975$	0	0
$976$	−10.0000	−0.320092
$977$	6.00000i	0.191957i	0.995383	+	0.0959785i	$0.0305980\pi$
$977$	6.00000i	0.191957i	−0.995383	+	0.0959785i	$0.969402\pi$
$978$	20.0000i	0.639529i
$979$	0	0
$980$	0	0
$981$	14.0000	0.446986
$982$	0	0
$983$	− 24.0000i	− 0.765481i	−0.923856	−	0.382741i	$-0.874980\pi$
$983$	− 24.0000i	− 0.765481i	0.923856	−	0.382741i	$-0.125020\pi$
$984$	−6.00000	−0.191273
$985$	0	0
$986$	36.0000	1.14647
$987$	0	0
$988$	− 16.0000i	− 0.509028i
$989$	0	0
$990$	0	0
$991$	8.00000	0.254128	0.127064	−	0.991894i	$-0.459445\pi$
$991$	8.00000	0.254128	0.127064	+	0.991894i	$0.459445\pi$
$992$	4.00000i	0.127000i
$993$	28.0000i	0.888553i
$994$	12.0000	0.380617
$995$	0	0
$996$	12.0000	0.380235
$997$	26.0000i	0.823428i	0.911313	+	0.411714i	$0.135070\pi$
$997$	26.0000i	0.823428i	−0.911313	+	0.411714i	$0.864930\pi$
$998$	20.0000i	0.633089i
$999$	10.0000	0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1050.2.g.c.799.1		2
3.2	odd	2		3150.2.g.i.2899.2		2
5.2	odd	4		1050.2.a.k.1.1		1
5.3	odd	4		210.2.a.b.1.1	✓	1
5.4	even	2	inner	1050.2.g.c.799.2		2
15.2	even	4		3150.2.a.f.1.1		1
15.8	even	4		630.2.a.h.1.1		1
15.14	odd	2		3150.2.g.i.2899.1		2
20.3	even	4		1680.2.a.g.1.1		1
20.7	even	4		8400.2.a.cm.1.1		1
35.3	even	12		1470.2.i.s.961.1		2
35.13	even	4		1470.2.a.b.1.1		1
35.18	odd	12		1470.2.i.l.961.1		2
35.23	odd	12		1470.2.i.l.361.1		2
35.27	even	4		7350.2.a.cs.1.1		1
35.33	even	12		1470.2.i.s.361.1		2
40.3	even	4		6720.2.a.bi.1.1		1
40.13	odd	4		6720.2.a.n.1.1		1
60.23	odd	4		5040.2.a.g.1.1		1
105.83	odd	4		4410.2.a.bi.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.2.a.b.1.1	✓	1	5.3	odd	4
630.2.a.h.1.1		1	15.8	even	4
1050.2.a.k.1.1		1	5.2	odd	4
1050.2.g.c.799.1		2	1.1	even	1	trivial
1050.2.g.c.799.2		2	5.4	even	2	inner
1470.2.a.b.1.1		1	35.13	even	4
1470.2.i.l.361.1		2	35.23	odd	12
1470.2.i.l.961.1		2	35.18	odd	12
1470.2.i.s.361.1		2	35.33	even	12
1470.2.i.s.961.1		2	35.3	even	12
1680.2.a.g.1.1		1	20.3	even	4
3150.2.a.f.1.1		1	15.2	even	4
3150.2.g.i.2899.1		2	15.14	odd	2
3150.2.g.i.2899.2		2	3.2	odd	2
4410.2.a.bi.1.1		1	105.83	odd	4
5040.2.a.g.1.1		1	60.23	odd	4
6720.2.a.n.1.1		1	40.13	odd	4
6720.2.a.bi.1.1		1	40.3	even	4
7350.2.a.cs.1.1		1	35.27	even	4
8400.2.a.cm.1.1		1	20.7	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:	\( N \)	\(=\)	\( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1050.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +1.00000i q^{12} -2.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} -6.00000i q^{17} +1.00000i q^{18} -8.00000 q^{19} +1.00000 q^{21} +1.00000 q^{24} -2.00000 q^{26} +1.00000i q^{27} -1.00000i q^{28} -6.00000 q^{29} -4.00000 q^{31} -1.00000i q^{32} -6.00000 q^{34} +1.00000 q^{36} -10.0000i q^{37} +8.00000i q^{38} -2.00000 q^{39} -6.00000 q^{41} -1.00000i q^{42} +4.00000i q^{43} -1.00000i q^{48} -1.00000 q^{49} -6.00000 q^{51} +2.00000i q^{52} +6.00000i q^{53} +1.00000 q^{54} -1.00000 q^{56} +8.00000i q^{57} +6.00000i q^{58} +12.0000 q^{59} -10.0000 q^{61} +4.00000i q^{62} -1.00000i q^{63} -1.00000 q^{64} -4.00000i q^{67} +6.00000i q^{68} +12.0000 q^{71} -1.00000i q^{72} +10.0000i q^{73} -10.0000 q^{74} +8.00000 q^{76} +2.00000i q^{78} -8.00000 q^{79} +1.00000 q^{81} +6.00000i q^{82} -12.0000i q^{83} -1.00000 q^{84} +4.00000 q^{86} +6.00000i q^{87} +6.00000 q^{89} +2.00000 q^{91} +4.00000i q^{93} -1.00000 q^{96} -10.0000i q^{97} +1.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + 2 q^{14} + 2 q^{16} - 16 q^{19} + 2 q^{21} + 2 q^{24} - 4 q^{26} - 12 q^{29} - 8 q^{31} - 12 q^{34} + 2 q^{36} - 4 q^{39} - 12 q^{41} - 2 q^{49} - 12 q^{51} + 2 q^{54} - 2 q^{56} + 24 q^{59} - 20 q^{61} - 2 q^{64} + 24 q^{71} - 20 q^{74} + 16 q^{76} - 16 q^{79} + 2 q^{81} - 2 q^{84} + 8 q^{86} + 12 q^{89} + 4 q^{91} - 2 q^{96}+O(q^{100}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 + 2 * q^14 + 2 * q^16 - 16 * q^19 + 2 * q^21 + 2 * q^24 - 4 * q^26 - 12 * q^29 - 8 * q^31 - 12 * q^34 + 2 * q^36 - 4 * q^39 - 12 * q^41 - 2 * q^49 - 12 * q^51 + 2 * q^54 - 2 * q^56 + 24 * q^59 - 20 * q^61 - 2 * q^64 + 24 * q^71 - 20 * q^74 + 16 * q^76 - 16 * q^79 + 2 * q^81 - 2 * q^84 + 8 * q^86 + 12 * q^89 + 4 * q^91 - 2 * q^96

Introduction

L-functions

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