LMFDB - Embedded newform 1050.2.g.b.799.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1050,2,Mod(799,1050)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1050.799"); S:= CuspForms(chi, 2); N := Newforms(S);

sage: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")

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:traces = [2,0,0,-2,0,-2,0,0,-2,0,-4,0,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)

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:	yes
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.b.799.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -2.00000 q^{11} +1.00000i q^{12} +7.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} +7.00000i q^{17} +1.00000i q^{18} -8.00000 q^{19} -1.00000 q^{21} +2.00000i q^{22} -5.00000i q^{23} +1.00000 q^{24} +7.00000 q^{26} +1.00000i q^{27} +1.00000i q^{28} -9.00000 q^{29} +1.00000 q^{31} -1.00000i q^{32} +2.00000i q^{33} +7.00000 q^{34} +1.00000 q^{36} +2.00000i q^{37} +8.00000i q^{38} +7.00000 q^{39} +11.0000 q^{41} +1.00000i q^{42} +3.00000i q^{43} +2.00000 q^{44} -5.00000 q^{46} -4.00000i q^{47} -1.00000i q^{48} -1.00000 q^{49} +7.00000 q^{51} -7.00000i q^{52} +3.00000i q^{53} +1.00000 q^{54} +1.00000 q^{56} +8.00000i q^{57} +9.00000i q^{58} -7.00000 q^{59} -5.00000 q^{61} -1.00000i q^{62} +1.00000i q^{63} -1.00000 q^{64} +2.00000 q^{66} +12.0000i q^{67} -7.00000i q^{68} -5.00000 q^{69} -4.00000 q^{71} -1.00000i q^{72} +10.0000i q^{73} +2.00000 q^{74} +8.00000 q^{76} +2.00000i q^{77} -7.00000i q^{78} +6.00000 q^{79} +1.00000 q^{81} -11.0000i q^{82} -9.00000i q^{83} +1.00000 q^{84} +3.00000 q^{86} +9.00000i q^{87} -2.00000i q^{88} +10.0000 q^{89} +7.00000 q^{91} +5.00000i q^{92} -1.00000i q^{93} -4.00000 q^{94} -1.00000 q^{96} -10.0000i q^{97} +1.00000i q^{98} +2.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} - 4 q^{11} - 2 q^{14} + 2 q^{16} - 16 q^{19} - 2 q^{21} + 2 q^{24} + 14 q^{26} - 18 q^{29} + 2 q^{31} + 14 q^{34} + 2 q^{36} + 14 q^{39} + 22 q^{41} + 4 q^{44} - 10 q^{46}+ \cdots + 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 - 4 * q^11 - 2 * q^14 + 2 * q^16 - 16 * q^19 - 2 * q^21 + 2 * q^24 + 14 * q^26 - 18 * q^29 + 2 * q^31 + 14 * q^34 + 2 * q^36 + 14 * q^39 + 22 * q^41 + 4 * q^44 - 10 * q^46 - 2 * q^49 + 14 * q^51 + 2 * q^54 + 2 * q^56 - 14 * q^59 - 10 * q^61 - 2 * q^64 + 4 * q^66 - 10 * q^69 - 8 * q^71 + 4 * q^74 + 16 * q^76 + 12 * q^79 + 2 * q^81 + 2 * q^84 + 6 * q^86 + 20 * q^89 + 14 * q^91 - 8 * q^94 - 2 * q^96 + 4 * q^99

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
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	1.00000i	0.288675i
$13$	7.00000i	1.94145i	0.240192	+	0.970725i	$0.422790\pi$
$13$	7.00000i	1.94145i	−0.240192	+	0.970725i	$0.577210\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	7.00000i	1.69775i	0.528594	+	0.848875i	$0.322719\pi$
$17$	7.00000i	1.69775i	−0.528594	+	0.848875i	$0.677281\pi$
$18$	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
$22$	2.00000i	0.426401i
$23$	− 5.00000i	− 1.04257i	−0.853382	−	0.521286i	$-0.825452\pi$
$23$	− 5.00000i	− 1.04257i	0.853382	−	0.521286i	$-0.174548\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	7.00000	1.37281
$27$	1.00000i	0.192450i
$28$	1.00000i	0.188982i
$29$	−9.00000	−1.67126	−0.835629	−	0.549294i	$-0.814897\pi$
$29$	−9.00000	−1.67126	−0.835629	+	0.549294i	$0.814897\pi$
$30$	0	0
$31$	1.00000	0.179605	0.0898027	−	0.995960i	$-0.471376\pi$
$31$	1.00000	0.179605	0.0898027	+	0.995960i	$0.471376\pi$
$32$	− 1.00000i	− 0.176777i
$33$	2.00000i	0.348155i
$34$	7.00000	1.20049
$35$	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	$0.947432\pi$
$38$	8.00000i	1.29777i
$39$	7.00000	1.12090
$40$	0	0
$41$	11.0000	1.71791	0.858956	−	0.512050i	$-0.171114\pi$
$41$	11.0000	1.71791	0.858956	+	0.512050i	$0.171114\pi$
$42$	1.00000i	0.154303i
$43$	3.00000i	0.457496i	0.973486	+	0.228748i	$0.0734631\pi$
$43$	3.00000i	0.457496i	−0.973486	+	0.228748i	$0.926537\pi$
$44$	2.00000	0.301511
$45$	0	0
$46$	−5.00000	−0.737210
$47$	− 4.00000i	− 0.583460i	−0.956501	−	0.291730i	$-0.905769\pi$
$47$	− 4.00000i	− 0.583460i	0.956501	−	0.291730i	$-0.0942309\pi$
$48$	− 1.00000i	− 0.144338i
$49$	−1.00000	−0.142857
$50$	0	0
$51$	7.00000	0.980196
$52$	− 7.00000i	− 0.970725i
$53$	3.00000i	0.412082i	0.978543	+	0.206041i	$0.0660580\pi$
$53$	3.00000i	0.412082i	−0.978543	+	0.206041i	$0.933942\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	1.00000	0.133631
$57$	8.00000i	1.05963i
$58$	9.00000i	1.18176i
$59$	−7.00000	−0.911322	−0.455661	−	0.890153i	$-0.650597\pi$
$59$	−7.00000	−0.911322	−0.455661	+	0.890153i	$0.650597\pi$
$60$	0	0
$61$	−5.00000	−0.640184	−0.320092	−	0.947386i	$-0.603714\pi$
$61$	−5.00000	−0.640184	−0.320092	+	0.947386i	$0.603714\pi$
$62$	− 1.00000i	− 0.127000i
$63$	1.00000i	0.125988i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	2.00000	0.246183
$67$	12.0000i	1.46603i	0.680211	+	0.733017i	$0.261888\pi$
$67$	12.0000i	1.46603i	−0.680211	+	0.733017i	$0.738112\pi$
$68$	− 7.00000i	− 0.848875i
$69$	−5.00000	−0.601929
$70$	0	0
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	− 1.00000i	− 0.117851i
$73$	10.0000i	1.17041i	0.810885	+	0.585206i	$0.198986\pi$
$73$	10.0000i	1.17041i	−0.810885	+	0.585206i	$0.801014\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	8.00000	0.917663
$77$	2.00000i	0.227921i
$78$	− 7.00000i	− 0.792594i
$79$	6.00000	0.675053	0.337526	−	0.941316i	$-0.390410\pi$
$79$	6.00000	0.675053	0.337526	+	0.941316i	$0.390410\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 11.0000i	− 1.21475i
$83$	− 9.00000i	− 0.987878i	−0.869496	−	0.493939i	$-0.835557\pi$
$83$	− 9.00000i	− 0.987878i	0.869496	−	0.493939i	$-0.164443\pi$
$84$	1.00000	0.109109
$85$	0	0
$86$	3.00000	0.323498
$87$	9.00000i	0.964901i
$88$	− 2.00000i	− 0.213201i
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	7.00000	0.733799
$92$	5.00000i	0.521286i
$93$	− 1.00000i	− 0.103695i
$94$	−4.00000	−0.412568
$95$	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
$99$	2.00000	0.201008
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	− 7.00000i	− 0.693103i
$103$	5.00000i	0.492665i	0.969185	+	0.246332i	$0.0792255\pi$
$103$	5.00000i	0.492665i	−0.969185	+	0.246332i	$0.920775\pi$
$104$	−7.00000	−0.686406
$105$	0	0
$106$	3.00000	0.291386
$107$	2.00000i	0.193347i	0.995316	+	0.0966736i	$0.0308203\pi$
$107$	2.00000i	0.193347i	−0.995316	+	0.0966736i	$0.969180\pi$
$108$	− 1.00000i	− 0.0962250i
$109$	−4.00000	−0.383131	−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000	−0.383131	−0.191565	+	0.981480i	$0.561356\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	− 1.00000i	− 0.0944911i
$113$	− 14.0000i	− 1.31701i	−0.752577	−	0.658505i	$-0.771189\pi$
$113$	− 14.0000i	− 1.31701i	0.752577	−	0.658505i	$-0.228811\pi$
$114$	8.00000	0.749269
$115$	0	0
$116$	9.00000	0.835629
$117$	− 7.00000i	− 0.647150i
$118$	7.00000i	0.644402i
$119$	7.00000	0.641689
$120$	0	0
$121$	−7.00000	−0.636364
$122$	5.00000i	0.452679i
$123$	− 11.0000i	− 0.991837i
$124$	−1.00000	−0.0898027
$125$	0	0
$126$	1.00000	0.0890871
$127$	18.0000i	1.59724i	0.601834	+	0.798621i	$0.294437\pi$
$127$	18.0000i	1.59724i	−0.601834	+	0.798621i	$0.705563\pi$
$128$	1.00000i	0.0883883i
$129$	3.00000	0.264135
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	− 2.00000i	− 0.174078i
$133$	8.00000i	0.693688i
$134$	12.0000	1.03664
$135$	0	0
$136$	−7.00000	−0.600245
$137$	− 16.0000i	− 1.36697i	−0.729964	−	0.683486i	$-0.760463\pi$
$137$	− 16.0000i	− 1.36697i	0.729964	−	0.683486i	$-0.239537\pi$
$138$	5.00000i	0.425628i
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	0	0
$141$	−4.00000	−0.336861
$142$	4.00000i	0.335673i
$143$	− 14.0000i	− 1.17074i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	10.0000	0.827606
$147$	1.00000i	0.0824786i
$148$	− 2.00000i	− 0.164399i
$149$	3.00000	0.245770	0.122885	−	0.992421i	$-0.460785\pi$
$149$	3.00000	0.245770	0.122885	+	0.992421i	$0.460785\pi$
$150$	0	0
$151$	−22.0000	−1.79033	−0.895167	−	0.445730i	$-0.852944\pi$
$151$	−22.0000	−1.79033	−0.895167	+	0.445730i	$0.852944\pi$
$152$	− 8.00000i	− 0.648886i
$153$	− 7.00000i	− 0.565916i
$154$	2.00000	0.161165
$155$	0	0
$156$	−7.00000	−0.560449
$157$	10.0000i	0.798087i	0.916932	+	0.399043i	$0.130658\pi$
$157$	10.0000i	0.798087i	−0.916932	+	0.399043i	$0.869342\pi$
$158$	− 6.00000i	− 0.477334i
$159$	3.00000	0.237915
$160$	0	0
$161$	−5.00000	−0.394055
$162$	− 1.00000i	− 0.0785674i
$163$	− 15.0000i	− 1.17489i	−0.809264	−	0.587445i	$-0.800134\pi$
$163$	− 15.0000i	− 1.17489i	0.809264	−	0.587445i	$-0.199866\pi$
$164$	−11.0000	−0.858956
$165$	0	0
$166$	−9.00000	−0.698535
$167$	− 2.00000i	− 0.154765i	−0.997001	−	0.0773823i	$-0.975344\pi$
$167$	− 2.00000i	− 0.154765i	0.997001	−	0.0773823i	$-0.0246562\pi$
$168$	− 1.00000i	− 0.0771517i
$169$	−36.0000	−2.76923
$170$	0	0
$171$	8.00000	0.611775
$172$	− 3.00000i	− 0.228748i
$173$	− 16.0000i	− 1.21646i	−0.793762	−	0.608229i	$-0.791880\pi$
$173$	− 16.0000i	− 1.21646i	0.793762	−	0.608229i	$-0.208120\pi$
$174$	9.00000	0.682288
$175$	0	0
$176$	−2.00000	−0.150756
$177$	7.00000i	0.526152i
$178$	− 10.0000i	− 0.749532i
$179$	2.00000	0.149487	0.0747435	−	0.997203i	$-0.476186\pi$
$179$	2.00000	0.149487	0.0747435	+	0.997203i	$0.476186\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	− 7.00000i	− 0.518875i
$183$	5.00000i	0.369611i
$184$	5.00000	0.368605
$185$	0	0
$186$	−1.00000	−0.0733236
$187$	− 14.0000i	− 1.02378i
$188$	4.00000i	0.291730i
$189$	1.00000	0.0727393
$190$	0	0
$191$	−1.00000	−0.0723575	−0.0361787	−	0.999345i	$-0.511519\pi$
$191$	−1.00000	−0.0723575	−0.0361787	+	0.999345i	$0.511519\pi$
$192$	1.00000i	0.0721688i
$193$	10.0000i	0.719816i	0.932988	+	0.359908i	$0.117192\pi$
$193$	10.0000i	0.719816i	−0.932988	+	0.359908i	$0.882808\pi$
$194$	−10.0000	−0.717958
$195$	0	0
$196$	1.00000	0.0714286
$197$	21.0000i	1.49619i	0.663593	+	0.748094i	$0.269031\pi$
$197$	21.0000i	1.49619i	−0.663593	+	0.748094i	$0.730969\pi$
$198$	− 2.00000i	− 0.142134i
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	12.0000	0.846415
$202$	0	0
$203$	9.00000i	0.631676i
$204$	−7.00000	−0.490098
$205$	0	0
$206$	5.00000	0.348367
$207$	5.00000i	0.347524i
$208$	7.00000i	0.485363i
$209$	16.0000	1.10674
$210$	0	0
$211$	−13.0000	−0.894957	−0.447478	−	0.894295i	$-0.647678\pi$
$211$	−13.0000	−0.894957	−0.447478	+	0.894295i	$0.647678\pi$
$212$	− 3.00000i	− 0.206041i
$213$	4.00000i	0.274075i
$214$	2.00000	0.136717
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	− 1.00000i	− 0.0678844i
$218$	4.00000i	0.270914i
$219$	10.0000	0.675737
$220$	0	0
$221$	−49.0000	−3.29610
$222$	− 2.00000i	− 0.134231i
$223$	9.00000i	0.602685i	0.953516	+	0.301342i	$0.0974347\pi$
$223$	9.00000i	0.602685i	−0.953516	+	0.301342i	$0.902565\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	−14.0000	−0.931266
$227$	3.00000i	0.199117i	0.995032	+	0.0995585i	$0.0317430\pi$
$227$	3.00000i	0.199117i	−0.995032	+	0.0995585i	$0.968257\pi$
$228$	− 8.00000i	− 0.529813i
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	0	0
$231$	2.00000	0.131590
$232$	− 9.00000i	− 0.590879i
$233$	16.0000i	1.04819i	0.851658	+	0.524097i	$0.175597\pi$
$233$	16.0000i	1.04819i	−0.851658	+	0.524097i	$0.824403\pi$
$234$	−7.00000	−0.457604
$235$	0	0
$236$	7.00000	0.455661
$237$	− 6.00000i	− 0.389742i
$238$	− 7.00000i	− 0.453743i
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	0	0
$241$	−30.0000	−1.93247	−0.966235	−	0.257663i	$-0.917048\pi$
$241$	−30.0000	−1.93247	−0.966235	+	0.257663i	$0.917048\pi$
$242$	7.00000i	0.449977i
$243$	− 1.00000i	− 0.0641500i
$244$	5.00000	0.320092
$245$	0	0
$246$	−11.0000	−0.701334
$247$	− 56.0000i	− 3.56319i
$248$	1.00000i	0.0635001i
$249$	−9.00000	−0.570352
$250$	0	0
$251$	−5.00000	−0.315597	−0.157799	−	0.987471i	$-0.550440\pi$
$251$	−5.00000	−0.315597	−0.157799	+	0.987471i	$0.550440\pi$
$252$	− 1.00000i	− 0.0629941i
$253$	10.0000i	0.628695i
$254$	18.0000	1.12942
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 5.00000i	− 0.311891i	−0.987766	−	0.155946i	$-0.950158\pi$
$257$	− 5.00000i	− 0.311891i	0.987766	−	0.155946i	$-0.0498425\pi$
$258$	− 3.00000i	− 0.186772i
$259$	2.00000	0.124274
$260$	0	0
$261$	9.00000	0.557086
$262$	0	0
$263$	− 9.00000i	− 0.554964i	−0.960731	−	0.277482i	$-0.910500\pi$
$263$	− 9.00000i	− 0.554964i	0.960731	−	0.277482i	$-0.0894999\pi$
$264$	−2.00000	−0.123091
$265$	0	0
$266$	8.00000	0.490511
$267$	− 10.0000i	− 0.611990i
$268$	− 12.0000i	− 0.733017i
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	0	0
$271$	12.0000	0.728948	0.364474	−	0.931214i	$-0.381249\pi$
$271$	12.0000	0.728948	0.364474	+	0.931214i	$0.381249\pi$
$272$	7.00000i	0.424437i
$273$	− 7.00000i	− 0.423659i
$274$	−16.0000	−0.966595
$275$	0	0
$276$	5.00000	0.300965
$277$	− 2.00000i	− 0.120168i	−0.998193	−	0.0600842i	$-0.980863\pi$
$277$	− 2.00000i	− 0.120168i	0.998193	−	0.0600842i	$-0.0191369\pi$
$278$	8.00000i	0.479808i
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	−8.00000	−0.477240	−0.238620	−	0.971113i	$-0.576695\pi$
$281$	−8.00000	−0.477240	−0.238620	+	0.971113i	$0.576695\pi$
$282$	4.00000i	0.238197i
$283$	− 16.0000i	− 0.951101i	−0.879688	−	0.475551i	$-0.842249\pi$
$283$	− 16.0000i	− 0.951101i	0.879688	−	0.475551i	$-0.157751\pi$
$284$	4.00000	0.237356
$285$	0	0
$286$	−14.0000	−0.827837
$287$	− 11.0000i	− 0.649309i
$288$	1.00000i	0.0589256i
$289$	−32.0000	−1.88235
$290$	0	0
$291$	−10.0000	−0.586210
$292$	− 10.0000i	− 0.585206i
$293$	24.0000i	1.40209i	0.713115	+	0.701047i	$0.247284\pi$
$293$	24.0000i	1.40209i	−0.713115	+	0.701047i	$0.752716\pi$
$294$	1.00000	0.0583212
$295$	0	0
$296$	−2.00000	−0.116248
$297$	− 2.00000i	− 0.116052i
$298$	− 3.00000i	− 0.173785i
$299$	35.0000	2.02410
$300$	0	0
$301$	3.00000	0.172917
$302$	22.0000i	1.26596i
$303$	0	0
$304$	−8.00000	−0.458831
$305$	0	0
$306$	−7.00000	−0.400163
$307$	22.0000i	1.25561i	0.778372	+	0.627803i	$0.216046\pi$
$307$	22.0000i	1.25561i	−0.778372	+	0.627803i	$0.783954\pi$
$308$	− 2.00000i	− 0.113961i
$309$	5.00000	0.284440
$310$	0	0
$311$	2.00000	0.113410	0.0567048	−	0.998391i	$-0.481941\pi$
$311$	2.00000	0.113410	0.0567048	+	0.998391i	$0.481941\pi$
$312$	7.00000i	0.396297i
$313$	10.0000i	0.565233i	0.959233	+	0.282617i	$0.0912024\pi$
$313$	10.0000i	0.565233i	−0.959233	+	0.282617i	$0.908798\pi$
$314$	10.0000	0.564333
$315$	0	0
$316$	−6.00000	−0.337526
$317$	− 23.0000i	− 1.29181i	−0.763418	−	0.645904i	$-0.776480\pi$
$317$	− 23.0000i	− 1.29181i	0.763418	−	0.645904i	$-0.223520\pi$
$318$	− 3.00000i	− 0.168232i
$319$	18.0000	1.00781
$320$	0	0
$321$	2.00000	0.111629
$322$	5.00000i	0.278639i
$323$	− 56.0000i	− 3.11592i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−15.0000	−0.830773
$327$	4.00000i	0.221201i
$328$	11.0000i	0.607373i
$329$	−4.00000	−0.220527
$330$	0	0
$331$	25.0000	1.37412	0.687062	−	0.726599i	$-0.258900\pi$
$331$	25.0000	1.37412	0.687062	+	0.726599i	$0.258900\pi$
$332$	9.00000i	0.493939i
$333$	− 2.00000i	− 0.109599i
$334$	−2.00000	−0.109435
$335$	0	0
$336$	−1.00000	−0.0545545
$337$	13.0000i	0.708155i	0.935216	+	0.354078i	$0.115205\pi$
$337$	13.0000i	0.708155i	−0.935216	+	0.354078i	$0.884795\pi$
$338$	36.0000i	1.95814i
$339$	−14.0000	−0.760376
$340$	0	0
$341$	−2.00000	−0.108306
$342$	− 8.00000i	− 0.432590i
$343$	1.00000i	0.0539949i
$344$	−3.00000	−0.161749
$345$	0	0
$346$	−16.0000	−0.860165
$347$	6.00000i	0.322097i	0.986947	+	0.161048i	$0.0514875\pi$
$347$	6.00000i	0.322097i	−0.986947	+	0.161048i	$0.948512\pi$
$348$	− 9.00000i	− 0.482451i
$349$	7.00000	0.374701	0.187351	−	0.982293i	$-0.440010\pi$
$349$	7.00000	0.374701	0.187351	+	0.982293i	$0.440010\pi$
$350$	0	0
$351$	−7.00000	−0.373632
$352$	2.00000i	0.106600i
$353$	30.0000i	1.59674i	0.602168	+	0.798369i	$0.294304\pi$
$353$	30.0000i	1.59674i	−0.602168	+	0.798369i	$0.705696\pi$
$354$	7.00000	0.372046
$355$	0	0
$356$	−10.0000	−0.529999
$357$	− 7.00000i	− 0.370479i
$358$	− 2.00000i	− 0.105703i
$359$	−19.0000	−1.00278	−0.501391	−	0.865221i	$-0.667178\pi$
$359$	−19.0000	−1.00278	−0.501391	+	0.865221i	$0.667178\pi$
$360$	0	0
$361$	45.0000	2.36842
$362$	− 2.00000i	− 0.105118i
$363$	7.00000i	0.367405i
$364$	−7.00000	−0.366900
$365$	0	0
$366$	5.00000	0.261354
$367$	17.0000i	0.887393i	0.896177	+	0.443696i	$0.146333\pi$
$367$	17.0000i	0.887393i	−0.896177	+	0.443696i	$0.853667\pi$
$368$	− 5.00000i	− 0.260643i
$369$	−11.0000	−0.572637
$370$	0	0
$371$	3.00000	0.155752
$372$	1.00000i	0.0518476i
$373$	− 4.00000i	− 0.207112i	−0.994624	−	0.103556i	$-0.966978\pi$
$373$	− 4.00000i	− 0.207112i	0.994624	−	0.103556i	$-0.0330221\pi$
$374$	−14.0000	−0.723923
$375$	0	0
$376$	4.00000	0.206284
$377$	− 63.0000i	− 3.24467i
$378$	− 1.00000i	− 0.0514344i
$379$	−7.00000	−0.359566	−0.179783	−	0.983706i	$-0.557540\pi$
$379$	−7.00000	−0.359566	−0.179783	+	0.983706i	$0.557540\pi$
$380$	0	0
$381$	18.0000	0.922168
$382$	1.00000i	0.0511645i
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	10.0000	0.508987
$387$	− 3.00000i	− 0.152499i
$388$	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
$391$	35.0000	1.77003
$392$	− 1.00000i	− 0.0505076i
$393$	0	0
$394$	21.0000	1.05796
$395$	0	0
$396$	−2.00000	−0.100504
$397$	− 37.0000i	− 1.85698i	−0.371361	−	0.928488i	$-0.621109\pi$
$397$	− 37.0000i	− 1.85698i	0.371361	−	0.928488i	$-0.378891\pi$
$398$	− 16.0000i	− 0.802008i
$399$	8.00000	0.400501
$400$	0	0
$401$	−12.0000	−0.599251	−0.299626	−	0.954057i	$-0.596862\pi$
$401$	−12.0000	−0.599251	−0.299626	+	0.954057i	$0.596862\pi$
$402$	− 12.0000i	− 0.598506i
$403$	7.00000i	0.348695i
$404$	0	0
$405$	0	0
$406$	9.00000	0.446663
$407$	− 4.00000i	− 0.198273i
$408$	7.00000i	0.346552i
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	0	0
$411$	−16.0000	−0.789222
$412$	− 5.00000i	− 0.246332i
$413$	7.00000i	0.344447i
$414$	5.00000	0.245737
$415$	0	0
$416$	7.00000	0.343203
$417$	8.00000i	0.391762i
$418$	− 16.0000i	− 0.782586i
$419$	3.00000	0.146560	0.0732798	−	0.997311i	$-0.476653\pi$
$419$	3.00000	0.146560	0.0732798	+	0.997311i	$0.476653\pi$
$420$	0	0
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	13.0000i	0.632830i
$423$	4.00000i	0.194487i
$424$	−3.00000	−0.145693
$425$	0	0
$426$	4.00000	0.193801
$427$	5.00000i	0.241967i
$428$	− 2.00000i	− 0.0966736i
$429$	−14.0000	−0.675926
$430$	0	0
$431$	39.0000	1.87856	0.939282	−	0.343146i	$-0.111493\pi$
$431$	39.0000	1.87856	0.939282	+	0.343146i	$0.111493\pi$
$432$	1.00000i	0.0481125i
$433$	6.00000i	0.288342i	0.989553	+	0.144171i	$0.0460515\pi$
$433$	6.00000i	0.288342i	−0.989553	+	0.144171i	$0.953949\pi$
$434$	−1.00000	−0.0480015
$435$	0	0
$436$	4.00000	0.191565
$437$	40.0000i	1.91346i
$438$	− 10.0000i	− 0.477818i
$439$	19.0000	0.906821	0.453410	−	0.891302i	$-0.350207\pi$
$439$	19.0000	0.906821	0.453410	+	0.891302i	$0.350207\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	49.0000i	2.33069i
$443$	− 4.00000i	− 0.190046i	−0.995475	−	0.0950229i	$-0.969708\pi$
$443$	− 4.00000i	− 0.190046i	0.995475	−	0.0950229i	$-0.0302924\pi$
$444$	−2.00000	−0.0949158
$445$	0	0
$446$	9.00000	0.426162
$447$	− 3.00000i	− 0.141895i
$448$	1.00000i	0.0472456i
$449$	20.0000	0.943858	0.471929	−	0.881636i	$-0.343558\pi$
$449$	20.0000	0.943858	0.471929	+	0.881636i	$0.343558\pi$
$450$	0	0
$451$	−22.0000	−1.03594
$452$	14.0000i	0.658505i
$453$	22.0000i	1.03365i
$454$	3.00000	0.140797
$455$	0	0
$456$	−8.00000	−0.374634
$457$	− 7.00000i	− 0.327446i	−0.986506	−	0.163723i	$-0.947650\pi$
$457$	− 7.00000i	− 0.327446i	0.986506	−	0.163723i	$-0.0523504\pi$
$458$	− 14.0000i	− 0.654177i
$459$	−7.00000	−0.326732
$460$	0	0
$461$	8.00000	0.372597	0.186299	−	0.982493i	$-0.440351\pi$
$461$	8.00000	0.372597	0.186299	+	0.982493i	$0.440351\pi$
$462$	− 2.00000i	− 0.0930484i
$463$	8.00000i	0.371792i	0.982569	+	0.185896i	$0.0595187\pi$
$463$	8.00000i	0.371792i	−0.982569	+	0.185896i	$0.940481\pi$
$464$	−9.00000	−0.417815
$465$	0	0
$466$	16.0000	0.741186
$467$	29.0000i	1.34196i	0.741475	+	0.670980i	$0.234126\pi$
$467$	29.0000i	1.34196i	−0.741475	+	0.670980i	$0.765874\pi$
$468$	7.00000i	0.323575i
$469$	12.0000	0.554109
$470$	0	0
$471$	10.0000	0.460776
$472$	− 7.00000i	− 0.322201i
$473$	− 6.00000i	− 0.275880i
$474$	−6.00000	−0.275589
$475$	0	0
$476$	−7.00000	−0.320844
$477$	− 3.00000i	− 0.137361i
$478$	8.00000i	0.365911i
$479$	12.0000	0.548294	0.274147	−	0.961688i	$-0.411605\pi$
$479$	12.0000	0.548294	0.274147	+	0.961688i	$0.411605\pi$
$480$	0	0
$481$	−14.0000	−0.638345
$482$	30.0000i	1.36646i
$483$	5.00000i	0.227508i
$484$	7.00000	0.318182
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	− 6.00000i	− 0.271886i	−0.990717	−	0.135943i	$-0.956594\pi$
$487$	− 6.00000i	− 0.271886i	0.990717	−	0.135943i	$-0.0434064\pi$
$488$	− 5.00000i	− 0.226339i
$489$	−15.0000	−0.678323
$490$	0	0
$491$	−42.0000	−1.89543	−0.947717	−	0.319113i	$-0.896615\pi$
$491$	−42.0000	−1.89543	−0.947717	+	0.319113i	$0.896615\pi$
$492$	11.0000i	0.495918i
$493$	− 63.0000i	− 2.83738i
$494$	−56.0000	−2.51956
$495$	0	0
$496$	1.00000	0.0449013
$497$	4.00000i	0.179425i
$498$	9.00000i	0.403300i
$499$	23.0000	1.02962	0.514811	−	0.857304i	$-0.327862\pi$
$499$	23.0000	1.02962	0.514811	+	0.857304i	$0.327862\pi$
$500$	0	0
$501$	−2.00000	−0.0893534
$502$	5.00000i	0.223161i
$503$	− 26.0000i	− 1.15928i	−0.814872	−	0.579641i	$-0.803193\pi$
$503$	− 26.0000i	− 1.15928i	0.814872	−	0.579641i	$-0.196807\pi$
$504$	−1.00000	−0.0445435
$505$	0	0
$506$	10.0000	0.444554
$507$	36.0000i	1.59882i
$508$	− 18.0000i	− 0.798621i
$509$	28.0000	1.24108	0.620539	−	0.784176i	$-0.286914\pi$
$509$	28.0000	1.24108	0.620539	+	0.784176i	$0.286914\pi$
$510$	0	0
$511$	10.0000	0.442374
$512$	− 1.00000i	− 0.0441942i
$513$	− 8.00000i	− 0.353209i
$514$	−5.00000	−0.220541
$515$	0	0
$516$	−3.00000	−0.132068
$517$	8.00000i	0.351840i
$518$	− 2.00000i	− 0.0878750i
$519$	−16.0000	−0.702322
$520$	0	0
$521$	9.00000	0.394297	0.197149	−	0.980374i	$-0.436832\pi$
$521$	9.00000	0.394297	0.197149	+	0.980374i	$0.436832\pi$
$522$	− 9.00000i	− 0.393919i
$523$	8.00000i	0.349816i	0.984585	+	0.174908i	$0.0559627\pi$
$523$	8.00000i	0.349816i	−0.984585	+	0.174908i	$0.944037\pi$
$524$	0	0
$525$	0	0
$526$	−9.00000	−0.392419
$527$	7.00000i	0.304925i
$528$	2.00000i	0.0870388i
$529$	−2.00000	−0.0869565
$530$	0	0
$531$	7.00000	0.303774
$532$	− 8.00000i	− 0.346844i
$533$	77.0000i	3.33524i
$534$	−10.0000	−0.432742
$535$	0	0
$536$	−12.0000	−0.518321
$537$	− 2.00000i	− 0.0863064i
$538$	− 6.00000i	− 0.258678i
$539$	2.00000	0.0861461
$540$	0	0
$541$	8.00000	0.343947	0.171973	−	0.985102i	$-0.444986\pi$
$541$	8.00000	0.343947	0.171973	+	0.985102i	$0.444986\pi$
$542$	− 12.0000i	− 0.515444i
$543$	− 2.00000i	− 0.0858282i
$544$	7.00000	0.300123
$545$	0	0
$546$	−7.00000	−0.299572
$547$	35.0000i	1.49649i	0.663421	+	0.748246i	$0.269104\pi$
$547$	35.0000i	1.49649i	−0.663421	+	0.748246i	$0.730896\pi$
$548$	16.0000i	0.683486i
$549$	5.00000	0.213395
$550$	0	0
$551$	72.0000	3.06730
$552$	− 5.00000i	− 0.212814i
$553$	− 6.00000i	− 0.255146i
$554$	−2.00000	−0.0849719
$555$	0	0
$556$	8.00000	0.339276
$557$	14.0000i	0.593199i	0.955002	+	0.296600i	$0.0958526\pi$
$557$	14.0000i	0.593199i	−0.955002	+	0.296600i	$0.904147\pi$
$558$	1.00000i	0.0423334i
$559$	−21.0000	−0.888205
$560$	0	0
$561$	−14.0000	−0.591080
$562$	8.00000i	0.337460i
$563$	27.0000i	1.13791i	0.822367	+	0.568957i	$0.192653\pi$
$563$	27.0000i	1.13791i	−0.822367	+	0.568957i	$0.807347\pi$
$564$	4.00000	0.168430
$565$	0	0
$566$	−16.0000	−0.672530
$567$	− 1.00000i	− 0.0419961i
$568$	− 4.00000i	− 0.167836i
$569$	−18.0000	−0.754599	−0.377300	−	0.926091i	$-0.623147\pi$
$569$	−18.0000	−0.754599	−0.377300	+	0.926091i	$0.623147\pi$
$570$	0	0
$571$	31.0000	1.29731	0.648655	−	0.761083i	$-0.275332\pi$
$571$	31.0000	1.29731	0.648655	+	0.761083i	$0.275332\pi$
$572$	14.0000i	0.585369i
$573$	1.00000i	0.0417756i
$574$	−11.0000	−0.459131
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 38.0000i	− 1.58196i	−0.611842	−	0.790980i	$-0.709571\pi$
$577$	− 38.0000i	− 1.58196i	0.611842	−	0.790980i	$-0.290429\pi$
$578$	32.0000i	1.33102i
$579$	10.0000	0.415586
$580$	0	0
$581$	−9.00000	−0.373383
$582$	10.0000i	0.414513i
$583$	− 6.00000i	− 0.248495i
$584$	−10.0000	−0.413803
$585$	0	0
$586$	24.0000	0.991431
$587$	− 39.0000i	− 1.60970i	−0.593477	−	0.804851i	$-0.702245\pi$
$587$	− 39.0000i	− 1.60970i	0.593477	−	0.804851i	$-0.297755\pi$
$588$	− 1.00000i	− 0.0412393i
$589$	−8.00000	−0.329634
$590$	0	0
$591$	21.0000	0.863825
$592$	2.00000i	0.0821995i
$593$	18.0000i	0.739171i	0.929197	+	0.369586i	$0.120500\pi$
$593$	18.0000i	0.739171i	−0.929197	+	0.369586i	$0.879500\pi$
$594$	−2.00000	−0.0820610
$595$	0	0
$596$	−3.00000	−0.122885
$597$	− 16.0000i	− 0.654836i
$598$	− 35.0000i	− 1.43126i
$599$	1.00000	0.0408589	0.0204294	−	0.999791i	$-0.493497\pi$
$599$	1.00000	0.0408589	0.0204294	+	0.999791i	$0.493497\pi$
$600$	0	0
$601$	8.00000	0.326327	0.163163	−	0.986599i	$-0.447830\pi$
$601$	8.00000	0.326327	0.163163	+	0.986599i	$0.447830\pi$
$602$	− 3.00000i	− 0.122271i
$603$	− 12.0000i	− 0.488678i
$604$	22.0000	0.895167
$605$	0	0
$606$	0	0
$607$	36.0000i	1.46119i	0.682808	+	0.730597i	$0.260758\pi$
$607$	36.0000i	1.46119i	−0.682808	+	0.730597i	$0.739242\pi$
$608$	8.00000i	0.324443i
$609$	9.00000	0.364698
$610$	0	0
$611$	28.0000	1.13276
$612$	7.00000i	0.282958i
$613$	2.00000i	0.0807792i	0.999184	+	0.0403896i	$0.0128599\pi$
$613$	2.00000i	0.0807792i	−0.999184	+	0.0403896i	$0.987140\pi$
$614$	22.0000	0.887848
$615$	0	0
$616$	−2.00000	−0.0805823
$617$	38.0000i	1.52982i	0.644136	+	0.764911i	$0.277217\pi$
$617$	38.0000i	1.52982i	−0.644136	+	0.764911i	$0.722783\pi$
$618$	− 5.00000i	− 0.201129i
$619$	38.0000	1.52735	0.763674	−	0.645601i	$-0.223393\pi$
$619$	38.0000	1.52735	0.763674	+	0.645601i	$0.223393\pi$
$620$	0	0
$621$	5.00000	0.200643
$622$	− 2.00000i	− 0.0801927i
$623$	− 10.0000i	− 0.400642i
$624$	7.00000	0.280224
$625$	0	0
$626$	10.0000	0.399680
$627$	− 16.0000i	− 0.638978i
$628$	− 10.0000i	− 0.399043i
$629$	−14.0000	−0.558217
$630$	0	0
$631$	42.0000	1.67199	0.835997	−	0.548734i	$-0.184890\pi$
$631$	42.0000	1.67199	0.835997	+	0.548734i	$0.184890\pi$
$632$	6.00000i	0.238667i
$633$	13.0000i	0.516704i
$634$	−23.0000	−0.913447
$635$	0	0
$636$	−3.00000	−0.118958
$637$	− 7.00000i	− 0.277350i
$638$	− 18.0000i	− 0.712627i
$639$	4.00000	0.158238
$640$	0	0
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	− 2.00000i	− 0.0789337i
$643$	10.0000i	0.394362i	0.980367	+	0.197181i	$0.0631786\pi$
$643$	10.0000i	0.394362i	−0.980367	+	0.197181i	$0.936821\pi$
$644$	5.00000	0.197028
$645$	0	0
$646$	−56.0000	−2.20329
$647$	− 46.0000i	− 1.80845i	−0.427060	−	0.904223i	$-0.640451\pi$
$647$	− 46.0000i	− 1.80845i	0.427060	−	0.904223i	$-0.359549\pi$
$648$	1.00000i	0.0392837i
$649$	14.0000	0.549548
$650$	0	0
$651$	−1.00000	−0.0391931
$652$	15.0000i	0.587445i
$653$	− 34.0000i	− 1.33052i	−0.746611	−	0.665261i	$-0.768320\pi$
$653$	− 34.0000i	− 1.33052i	0.746611	−	0.665261i	$-0.231680\pi$
$654$	4.00000	0.156412
$655$	0	0
$656$	11.0000	0.429478
$657$	− 10.0000i	− 0.390137i
$658$	4.00000i	0.155936i
$659$	−34.0000	−1.32445	−0.662226	−	0.749304i	$-0.730388\pi$
$659$	−34.0000	−1.32445	−0.662226	+	0.749304i	$0.730388\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	− 25.0000i	− 0.971653i
$663$	49.0000i	1.90300i
$664$	9.00000	0.349268
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	45.0000i	1.74241i
$668$	2.00000i	0.0773823i
$669$	9.00000	0.347960
$670$	0	0
$671$	10.0000	0.386046
$672$	1.00000i	0.0385758i
$673$	− 13.0000i	− 0.501113i	−0.968102	−	0.250557i	$-0.919386\pi$
$673$	− 13.0000i	− 0.501113i	0.968102	−	0.250557i	$-0.0806136\pi$
$674$	13.0000	0.500741
$675$	0	0
$676$	36.0000	1.38462
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	14.0000i	0.537667i
$679$	−10.0000	−0.383765
$680$	0	0
$681$	3.00000	0.114960
$682$	2.00000i	0.0765840i
$683$	38.0000i	1.45403i	0.686622	+	0.727015i	$0.259093\pi$
$683$	38.0000i	1.45403i	−0.686622	+	0.727015i	$0.740907\pi$
$684$	−8.00000	−0.305888
$685$	0	0
$686$	1.00000	0.0381802
$687$	− 14.0000i	− 0.534133i
$688$	3.00000i	0.114374i
$689$	−21.0000	−0.800036
$690$	0	0
$691$	−14.0000	−0.532585	−0.266293	−	0.963892i	$-0.585799\pi$
$691$	−14.0000	−0.532585	−0.266293	+	0.963892i	$0.585799\pi$
$692$	16.0000i	0.608229i
$693$	− 2.00000i	− 0.0759737i
$694$	6.00000	0.227757
$695$	0	0
$696$	−9.00000	−0.341144
$697$	77.0000i	2.91658i
$698$	− 7.00000i	− 0.264954i
$699$	16.0000	0.605176
$700$	0	0
$701$	−15.0000	−0.566542	−0.283271	−	0.959040i	$-0.591420\pi$
$701$	−15.0000	−0.566542	−0.283271	+	0.959040i	$0.591420\pi$
$702$	7.00000i	0.264198i
$703$	− 16.0000i	− 0.603451i
$704$	2.00000	0.0753778
$705$	0	0
$706$	30.0000	1.12906
$707$	0	0
$708$	− 7.00000i	− 0.263076i
$709$	8.00000	0.300446	0.150223	−	0.988652i	$-0.452001\pi$
$709$	8.00000	0.300446	0.150223	+	0.988652i	$0.452001\pi$
$710$	0	0
$711$	−6.00000	−0.225018
$712$	10.0000i	0.374766i
$713$	− 5.00000i	− 0.187251i
$714$	−7.00000	−0.261968
$715$	0	0
$716$	−2.00000	−0.0747435
$717$	8.00000i	0.298765i
$718$	19.0000i	0.709074i
$719$	−30.0000	−1.11881	−0.559406	−	0.828894i	$-0.688971\pi$
$719$	−30.0000	−1.11881	−0.559406	+	0.828894i	$0.688971\pi$
$720$	0	0
$721$	5.00000	0.186210
$722$	− 45.0000i	− 1.67473i
$723$	30.0000i	1.11571i
$724$	−2.00000	−0.0743294
$725$	0	0
$726$	7.00000	0.259794
$727$	− 21.0000i	− 0.778847i	−0.921059	−	0.389423i	$-0.872674\pi$
$727$	− 21.0000i	− 0.778847i	0.921059	−	0.389423i	$-0.127326\pi$
$728$	7.00000i	0.259437i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−21.0000	−0.776713
$732$	− 5.00000i	− 0.184805i
$733$	45.0000i	1.66211i	0.556188	+	0.831056i	$0.312263\pi$
$733$	45.0000i	1.66211i	−0.556188	+	0.831056i	$0.687737\pi$
$734$	17.0000	0.627481
$735$	0	0
$736$	−5.00000	−0.184302
$737$	− 24.0000i	− 0.884051i
$738$	11.0000i	0.404916i
$739$	−15.0000	−0.551784	−0.275892	−	0.961189i	$-0.588973\pi$
$739$	−15.0000	−0.551784	−0.275892	+	0.961189i	$0.588973\pi$
$740$	0	0
$741$	−56.0000	−2.05721
$742$	− 3.00000i	− 0.110133i
$743$	− 39.0000i	− 1.43077i	−0.698730	−	0.715386i	$-0.746251\pi$
$743$	− 39.0000i	− 1.43077i	0.698730	−	0.715386i	$-0.253749\pi$
$744$	1.00000	0.0366618
$745$	0	0
$746$	−4.00000	−0.146450
$747$	9.00000i	0.329293i
$748$	14.0000i	0.511891i
$749$	2.00000	0.0730784
$750$	0	0
$751$	4.00000	0.145962	0.0729810	−	0.997333i	$-0.476749\pi$
$751$	4.00000	0.145962	0.0729810	+	0.997333i	$0.476749\pi$
$752$	− 4.00000i	− 0.145865i
$753$	5.00000i	0.182210i
$754$	−63.0000	−2.29432
$755$	0	0
$756$	−1.00000	−0.0363696
$757$	− 26.0000i	− 0.944986i	−0.881334	−	0.472493i	$-0.843354\pi$
$757$	− 26.0000i	− 0.944986i	0.881334	−	0.472493i	$-0.156646\pi$
$758$	7.00000i	0.254251i
$759$	10.0000	0.362977
$760$	0	0
$761$	−34.0000	−1.23250	−0.616250	−	0.787551i	$-0.711349\pi$
$761$	−34.0000	−1.23250	−0.616250	+	0.787551i	$0.711349\pi$
$762$	− 18.0000i	− 0.652071i
$763$	4.00000i	0.144810i
$764$	1.00000	0.0361787
$765$	0	0
$766$	0	0
$767$	− 49.0000i	− 1.76929i
$768$	− 1.00000i	− 0.0360844i
$769$	32.0000	1.15395	0.576975	−	0.816762i	$-0.304233\pi$
$769$	32.0000	1.15395	0.576975	+	0.816762i	$0.304233\pi$
$770$	0	0
$771$	−5.00000	−0.180071
$772$	− 10.0000i	− 0.359908i
$773$	28.0000i	1.00709i	0.863969	+	0.503545i	$0.167971\pi$
$773$	28.0000i	1.00709i	−0.863969	+	0.503545i	$0.832029\pi$
$774$	−3.00000	−0.107833
$775$	0	0
$776$	10.0000	0.358979
$777$	− 2.00000i	− 0.0717496i
$778$	6.00000i	0.215110i
$779$	−88.0000	−3.15293
$780$	0	0
$781$	8.00000	0.286263
$782$	− 35.0000i	− 1.25160i
$783$	− 9.00000i	− 0.321634i
$784$	−1.00000	−0.0357143
$785$	0	0
$786$	0	0
$787$	− 26.0000i	− 0.926800i	−0.886149	−	0.463400i	$-0.846629\pi$
$787$	− 26.0000i	− 0.926800i	0.886149	−	0.463400i	$-0.153371\pi$
$788$	− 21.0000i	− 0.748094i
$789$	−9.00000	−0.320408
$790$	0	0
$791$	−14.0000	−0.497783
$792$	2.00000i	0.0710669i
$793$	− 35.0000i	− 1.24289i
$794$	−37.0000	−1.31308
$795$	0	0
$796$	−16.0000	−0.567105
$797$	30.0000i	1.06265i	0.847167	+	0.531327i	$0.178307\pi$
$797$	30.0000i	1.06265i	−0.847167	+	0.531327i	$0.821693\pi$
$798$	− 8.00000i	− 0.283197i
$799$	28.0000	0.990569
$800$	0	0
$801$	−10.0000	−0.353333
$802$	12.0000i	0.423735i
$803$	− 20.0000i	− 0.705785i
$804$	−12.0000	−0.423207
$805$	0	0
$806$	7.00000	0.246564
$807$	− 6.00000i	− 0.211210i
$808$	0	0
$809$	44.0000	1.54696	0.773479	−	0.633822i	$-0.218515\pi$
$809$	44.0000	1.54696	0.773479	+	0.633822i	$0.218515\pi$
$810$	0	0
$811$	−6.00000	−0.210688	−0.105344	−	0.994436i	$-0.533594\pi$
$811$	−6.00000	−0.210688	−0.105344	+	0.994436i	$0.533594\pi$
$812$	− 9.00000i	− 0.315838i
$813$	− 12.0000i	− 0.420858i
$814$	−4.00000	−0.140200
$815$	0	0
$816$	7.00000	0.245049
$817$	− 24.0000i	− 0.839654i
$818$	10.0000i	0.349642i
$819$	−7.00000	−0.244600
$820$	0	0
$821$	34.0000	1.18661	0.593304	−	0.804978i	$-0.297823\pi$
$821$	34.0000	1.18661	0.593304	+	0.804978i	$0.297823\pi$
$822$	16.0000i	0.558064i
$823$	− 14.0000i	− 0.488009i	−0.969774	−	0.244005i	$-0.921539\pi$
$823$	− 14.0000i	− 0.488009i	0.969774	−	0.244005i	$-0.0784612\pi$
$824$	−5.00000	−0.174183
$825$	0	0
$826$	7.00000	0.243561
$827$	6.00000i	0.208640i	0.994544	+	0.104320i	$0.0332667\pi$
$827$	6.00000i	0.208640i	−0.994544	+	0.104320i	$0.966733\pi$
$828$	− 5.00000i	− 0.173762i
$829$	19.0000	0.659897	0.329949	−	0.943999i	$-0.392969\pi$
$829$	19.0000	0.659897	0.329949	+	0.943999i	$0.392969\pi$
$830$	0	0
$831$	−2.00000	−0.0693792
$832$	− 7.00000i	− 0.242681i
$833$	− 7.00000i	− 0.242536i
$834$	8.00000	0.277017
$835$	0	0
$836$	−16.0000	−0.553372
$837$	1.00000i	0.0345651i
$838$	− 3.00000i	− 0.103633i
$839$	36.0000	1.24286	0.621429	−	0.783470i	$-0.286552\pi$
$839$	36.0000	1.24286	0.621429	+	0.783470i	$0.286552\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	− 10.0000i	− 0.344623i
$843$	8.00000i	0.275535i
$844$	13.0000	0.447478
$845$	0	0
$846$	4.00000	0.137523
$847$	7.00000i	0.240523i
$848$	3.00000i	0.103020i
$849$	−16.0000	−0.549119
$850$	0	0
$851$	10.0000	0.342796
$852$	− 4.00000i	− 0.137038i
$853$	− 35.0000i	− 1.19838i	−0.800608	−	0.599189i	$-0.795490\pi$
$853$	− 35.0000i	− 1.19838i	0.800608	−	0.599189i	$-0.204510\pi$
$854$	5.00000	0.171096
$855$	0	0
$856$	−2.00000	−0.0683586
$857$	10.0000i	0.341593i	0.985306	+	0.170797i	$0.0546341\pi$
$857$	10.0000i	0.341593i	−0.985306	+	0.170797i	$0.945366\pi$
$858$	14.0000i	0.477952i
$859$	16.0000	0.545913	0.272956	−	0.962026i	$-0.411998\pi$
$859$	16.0000	0.545913	0.272956	+	0.962026i	$0.411998\pi$
$860$	0	0
$861$	−11.0000	−0.374879
$862$	− 39.0000i	− 1.32835i
$863$	48.0000i	1.63394i	0.576681	+	0.816970i	$0.304348\pi$
$863$	48.0000i	1.63394i	−0.576681	+	0.816970i	$0.695652\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	6.00000	0.203888
$867$	32.0000i	1.08678i
$868$	1.00000i	0.0339422i
$869$	−12.0000	−0.407072
$870$	0	0
$871$	−84.0000	−2.84623
$872$	− 4.00000i	− 0.135457i
$873$	10.0000i	0.338449i
$874$	40.0000	1.35302
$875$	0	0
$876$	−10.0000	−0.337869
$877$	− 32.0000i	− 1.08056i	−0.841484	−	0.540282i	$-0.818318\pi$
$877$	− 32.0000i	− 1.08056i	0.841484	−	0.540282i	$-0.181682\pi$
$878$	− 19.0000i	− 0.641219i
$879$	24.0000	0.809500
$880$	0	0
$881$	−21.0000	−0.707508	−0.353754	−	0.935339i	$-0.615095\pi$
$881$	−21.0000	−0.707508	−0.353754	+	0.935339i	$0.615095\pi$
$882$	− 1.00000i	− 0.0336718i
$883$	− 23.0000i	− 0.774012i	−0.922077	−	0.387006i	$-0.873509\pi$
$883$	− 23.0000i	− 0.774012i	0.922077	−	0.387006i	$-0.126491\pi$
$884$	49.0000	1.64805
$885$	0	0
$886$	−4.00000	−0.134383
$887$	14.0000i	0.470074i	0.971986	+	0.235037i	$0.0755211\pi$
$887$	14.0000i	0.470074i	−0.971986	+	0.235037i	$0.924479\pi$
$888$	2.00000i	0.0671156i
$889$	18.0000	0.603701
$890$	0	0
$891$	−2.00000	−0.0670025
$892$	− 9.00000i	− 0.301342i
$893$	32.0000i	1.07084i
$894$	−3.00000	−0.100335
$895$	0	0
$896$	1.00000	0.0334077
$897$	− 35.0000i	− 1.16862i
$898$	− 20.0000i	− 0.667409i
$899$	−9.00000	−0.300167
$900$	0	0
$901$	−21.0000	−0.699611
$902$	22.0000i	0.732520i
$903$	− 3.00000i	− 0.0998337i
$904$	14.0000	0.465633
$905$	0	0
$906$	22.0000	0.730901
$907$	39.0000i	1.29497i	0.762077	+	0.647487i	$0.224180\pi$
$907$	39.0000i	1.29497i	−0.762077	+	0.647487i	$0.775820\pi$
$908$	− 3.00000i	− 0.0995585i
$909$	0	0
$910$	0	0
$911$	−39.0000	−1.29213	−0.646064	−	0.763283i	$-0.723586\pi$
$911$	−39.0000	−1.29213	−0.646064	+	0.763283i	$0.723586\pi$
$912$	8.00000i	0.264906i
$913$	18.0000i	0.595713i
$914$	−7.00000	−0.231539
$915$	0	0
$916$	−14.0000	−0.462573
$917$	0	0
$918$	7.00000i	0.231034i
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	22.0000	0.724925
$922$	− 8.00000i	− 0.263466i
$923$	− 28.0000i	− 0.921631i
$924$	−2.00000	−0.0657952
$925$	0	0
$926$	8.00000	0.262896
$927$	− 5.00000i	− 0.164222i
$928$	9.00000i	0.295439i
$929$	−7.00000	−0.229663	−0.114831	−	0.993385i	$-0.536633\pi$
$929$	−7.00000	−0.229663	−0.114831	+	0.993385i	$0.536633\pi$
$930$	0	0
$931$	8.00000	0.262189
$932$	− 16.0000i	− 0.524097i
$933$	− 2.00000i	− 0.0654771i
$934$	29.0000	0.948909
$935$	0	0
$936$	7.00000	0.228802
$937$	6.00000i	0.196011i	0.995186	+	0.0980057i	$0.0312463\pi$
$937$	6.00000i	0.196011i	−0.995186	+	0.0980057i	$0.968754\pi$
$938$	− 12.0000i	− 0.391814i
$939$	10.0000	0.326338
$940$	0	0
$941$	24.0000	0.782378	0.391189	−	0.920310i	$-0.372064\pi$
$941$	24.0000	0.782378	0.391189	+	0.920310i	$0.372064\pi$
$942$	− 10.0000i	− 0.325818i
$943$	− 55.0000i	− 1.79105i
$944$	−7.00000	−0.227831
$945$	0	0
$946$	−6.00000	−0.195077
$947$	− 12.0000i	− 0.389948i	−0.980808	−	0.194974i	$-0.937538\pi$
$947$	− 12.0000i	− 0.389948i	0.980808	−	0.194974i	$-0.0624622\pi$
$948$	6.00000i	0.194871i
$949$	−70.0000	−2.27230
$950$	0	0
$951$	−23.0000	−0.745826
$952$	7.00000i	0.226871i
$953$	− 48.0000i	− 1.55487i	−0.628962	−	0.777436i	$-0.716520\pi$
$953$	− 48.0000i	− 1.55487i	0.628962	−	0.777436i	$-0.283480\pi$
$954$	−3.00000	−0.0971286
$955$	0	0
$956$	8.00000	0.258738
$957$	− 18.0000i	− 0.581857i
$958$	− 12.0000i	− 0.387702i
$959$	−16.0000	−0.516667
$960$	0	0
$961$	−30.0000	−0.967742
$962$	14.0000i	0.451378i
$963$	− 2.00000i	− 0.0644491i
$964$	30.0000	0.966235
$965$	0	0
$966$	5.00000	0.160872
$967$	− 16.0000i	− 0.514525i	−0.966342	−	0.257263i	$-0.917179\pi$
$967$	− 16.0000i	− 0.514525i	0.966342	−	0.257263i	$-0.0828206\pi$
$968$	− 7.00000i	− 0.224989i
$969$	−56.0000	−1.79898
$970$	0	0
$971$	−36.0000	−1.15529	−0.577647	−	0.816286i	$-0.696029\pi$
$971$	−36.0000	−1.15529	−0.577647	+	0.816286i	$0.696029\pi$
$972$	1.00000i	0.0320750i
$973$	8.00000i	0.256468i
$974$	−6.00000	−0.192252
$975$	0	0
$976$	−5.00000	−0.160046
$977$	18.0000i	0.575871i	0.957650	+	0.287936i	$0.0929689\pi$
$977$	18.0000i	0.575871i	−0.957650	+	0.287936i	$0.907031\pi$
$978$	15.0000i	0.479647i
$979$	−20.0000	−0.639203
$980$	0	0
$981$	4.00000	0.127710
$982$	42.0000i	1.34027i
$983$	32.0000i	1.02064i	0.859984	+	0.510321i	$0.170473\pi$
$983$	32.0000i	1.02064i	−0.859984	+	0.510321i	$0.829527\pi$
$984$	11.0000	0.350667
$985$	0	0
$986$	−63.0000	−2.00633
$987$	4.00000i	0.127321i
$988$	56.0000i	1.78160i
$989$	15.0000	0.476972
$990$	0	0
$991$	−20.0000	−0.635321	−0.317660	−	0.948205i	$-0.602897\pi$
$991$	−20.0000	−0.635321	−0.317660	+	0.948205i	$0.602897\pi$
$992$	− 1.00000i	− 0.0317500i
$993$	− 25.0000i	− 0.793351i
$994$	4.00000	0.126872
$995$	0	0
$996$	9.00000	0.285176
$997$	− 2.00000i	− 0.0633406i	−0.999498	−	0.0316703i	$-0.989917\pi$
$997$	− 2.00000i	− 0.0633406i	0.999498	−	0.0316703i	$-0.0100827\pi$
$998$	− 23.0000i	− 0.728052i
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1050.2.g.b.799.1		2
3.2	odd	2		3150.2.g.p.2899.2		2
5.2	odd	4		1050.2.a.n.1.1	yes	1
5.3	odd	4		1050.2.a.f.1.1	✓	1
5.4	even	2	inner	1050.2.g.b.799.2		2
15.2	even	4		3150.2.a.r.1.1		1
15.8	even	4		3150.2.a.bd.1.1		1
15.14	odd	2		3150.2.g.p.2899.1		2
20.3	even	4		8400.2.a.bb.1.1		1
20.7	even	4		8400.2.a.cb.1.1		1
35.13	even	4		7350.2.a.i.1.1		1
35.27	even	4		7350.2.a.cm.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1050.2.a.f.1.1	✓	1	5.3	odd	4
1050.2.a.n.1.1	yes	1	5.2	odd	4
1050.2.g.b.799.1		2	1.1	even	1	trivial
1050.2.g.b.799.2		2	5.4	even	2	inner
3150.2.a.r.1.1		1	15.2	even	4
3150.2.a.bd.1.1		1	15.8	even	4
3150.2.g.p.2899.1		2	15.14	odd	2
3150.2.g.p.2899.2		2	3.2	odd	2
7350.2.a.i.1.1		1	35.13	even	4
7350.2.a.cm.1.1		1	35.27	even	4
8400.2.a.bb.1.1		1	20.3	even	4
8400.2.a.cb.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 $\F_{q}$
Belyi maps

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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -2.00000 q^{11} +1.00000i q^{12} +7.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} +7.00000i q^{17} +1.00000i q^{18} -8.00000 q^{19} -1.00000 q^{21} +2.00000i q^{22} -5.00000i q^{23} +1.00000 q^{24} +7.00000 q^{26} +1.00000i q^{27} +1.00000i q^{28} -9.00000 q^{29} +1.00000 q^{31} -1.00000i q^{32} +2.00000i q^{33} +7.00000 q^{34} +1.00000 q^{36} +2.00000i q^{37} +8.00000i q^{38} +7.00000 q^{39} +11.0000 q^{41} +1.00000i q^{42} +3.00000i q^{43} +2.00000 q^{44} -5.00000 q^{46} -4.00000i q^{47} -1.00000i q^{48} -1.00000 q^{49} +7.00000 q^{51} -7.00000i q^{52} +3.00000i q^{53} +1.00000 q^{54} +1.00000 q^{56} +8.00000i q^{57} +9.00000i q^{58} -7.00000 q^{59} -5.00000 q^{61} -1.00000i q^{62} +1.00000i q^{63} -1.00000 q^{64} +2.00000 q^{66} +12.0000i q^{67} -7.00000i q^{68} -5.00000 q^{69} -4.00000 q^{71} -1.00000i q^{72} +10.0000i q^{73} +2.00000 q^{74} +8.00000 q^{76} +2.00000i q^{77} -7.00000i q^{78} +6.00000 q^{79} +1.00000 q^{81} -11.0000i q^{82} -9.00000i q^{83} +1.00000 q^{84} +3.00000 q^{86} +9.00000i q^{87} -2.00000i q^{88} +10.0000 q^{89} +7.00000 q^{91} +5.00000i q^{92} -1.00000i q^{93} -4.00000 q^{94} -1.00000 q^{96} -10.0000i q^{97} +1.00000i q^{98} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} - 4 q^{11} - 2 q^{14} + 2 q^{16} - 16 q^{19} - 2 q^{21} + 2 q^{24} + 14 q^{26} - 18 q^{29} + 2 q^{31} + 14 q^{34} + 2 q^{36} + 14 q^{39} + 22 q^{41} + 4 q^{44} - 10 q^{46}+ \cdots + 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 - 4 * q^11 - 2 * q^14 + 2 * q^16 - 16 * q^19 - 2 * q^21 + 2 * q^24 + 14 * q^26 - 18 * q^29 + 2 * q^31 + 14 * q^34 + 2 * q^36 + 14 * q^39 + 22 * q^41 + 4 * q^44 - 10 * q^46 - 2 * q^49 + 14 * q^51 + 2 * q^54 + 2 * q^56 - 14 * q^59 - 10 * q^61 - 2 * q^64 + 4 * q^66 - 10 * q^69 - 8 * q^71 + 4 * q^74 + 16 * q^76 + 12 * q^79 + 2 * q^81 + 2 * q^84 + 6 * q^86 + 20 * q^89 + 14 * q^91 - 8 * q^94 - 2 * q^96 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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