LMFDB - Embedded newform 1650.2.c.a.199.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1650,2,Mod(199,1650)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1650, 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("1650.199");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1650 = 2 \cdot 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1650.c (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:	$13.1753163335$
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 330)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1650.199
Dual form			1650.2.c.a.199.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -1.00000 q^{11} -1.00000i q^{12} -2.00000i q^{13} -4.00000 q^{14} +1.00000 q^{16} +2.00000i q^{17} -1.00000i q^{18} -4.00000 q^{19} -4.00000 q^{21} -1.00000i q^{22} +4.00000i q^{23} +1.00000 q^{24} +2.00000 q^{26} -1.00000i q^{27} -4.00000i q^{28} -6.00000 q^{29} +1.00000i q^{32} -1.00000i q^{33} -2.00000 q^{34} +1.00000 q^{36} -10.0000i q^{37} -4.00000i q^{38} +2.00000 q^{39} -6.00000 q^{41} -4.00000i q^{42} +12.0000i q^{43} +1.00000 q^{44} -4.00000 q^{46} -4.00000i q^{47} +1.00000i q^{48} -9.00000 q^{49} -2.00000 q^{51} +2.00000i q^{52} +6.00000i q^{53} +1.00000 q^{54} +4.00000 q^{56} -4.00000i q^{57} -6.00000i q^{58} +4.00000 q^{59} +10.0000 q^{61} -4.00000i q^{63} -1.00000 q^{64} +1.00000 q^{66} -12.0000i q^{67} -2.00000i q^{68} -4.00000 q^{69} -4.00000 q^{71} +1.00000i q^{72} -10.0000i q^{73} +10.0000 q^{74} +4.00000 q^{76} -4.00000i q^{77} +2.00000i q^{78} -4.00000 q^{79} +1.00000 q^{81} -6.00000i q^{82} -4.00000i q^{83} +4.00000 q^{84} -12.0000 q^{86} -6.00000i q^{87} +1.00000i q^{88} -10.0000 q^{89} +8.00000 q^{91} -4.00000i q^{92} +4.00000 q^{94} -1.00000 q^{96} +18.0000i q^{97} -9.00000i q^{98} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} - 2 q^{11} - 8 q^{14} + 2 q^{16} - 8 q^{19} - 8 q^{21} + 2 q^{24} + 4 q^{26} - 12 q^{29} - 4 q^{34} + 2 q^{36} + 4 q^{39} - 12 q^{41} + 2 q^{44} - 8 q^{46} - 18 q^{49}+ \cdots + 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 - 2 * q^11 - 8 * q^14 + 2 * q^16 - 8 * q^19 - 8 * q^21 + 2 * q^24 + 4 * q^26 - 12 * q^29 - 4 * q^34 + 2 * q^36 + 4 * q^39 - 12 * q^41 + 2 * q^44 - 8 * q^46 - 18 * q^49 - 4 * q^51 + 2 * q^54 + 8 * q^56 + 8 * q^59 + 20 * q^61 - 2 * q^64 + 2 * q^66 - 8 * q^69 - 8 * q^71 + 20 * q^74 + 8 * q^76 - 8 * q^79 + 2 * q^81 + 8 * q^84 - 24 * q^86 - 20 * q^89 + 16 * q^91 + 8 * q^94 - 2 * q^96 + 2 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1650\mathbb{Z}\right)^\times$.

$n$	$551$	$727$	$1201$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	4.00000i	1.51186i	0.654654	+	0.755929i	$0.272814\pi$
$7$	4.00000i	1.51186i	−0.654654	+	0.755929i	$0.727186\pi$
$8$	− 1.00000i	− 0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	− 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	$-0.0894562\pi$
$14$	−4.00000	−1.06904
$15$	0	0
$16$	1.00000	0.250000
$17$	2.00000i	0.485071i	0.970143	+	0.242536i	$0.0779791\pi$
$17$	2.00000i	0.485071i	−0.970143	+	0.242536i	$0.922021\pi$
$18$	− 1.00000i	− 0.235702i
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	−4.00000	−0.872872
$22$	− 1.00000i	− 0.213201i
$23$	4.00000i	0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$	4.00000i	0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	2.00000	0.392232
$27$	− 1.00000i	− 0.192450i
$28$	− 4.00000i	− 0.755929i
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	1.00000i	0.176777i
$33$	− 1.00000i	− 0.174078i
$34$	−2.00000	−0.342997
$35$	0	0
$36$	1.00000	0.166667
$37$	− 10.0000i	− 1.64399i	−0.569495	−	0.821995i	$-0.692861\pi$
$37$	− 10.0000i	− 1.64399i	0.569495	−	0.821995i	$-0.307139\pi$
$38$	− 4.00000i	− 0.648886i
$39$	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	$0.655213\pi$
$42$	− 4.00000i	− 0.617213i
$43$	12.0000i	1.82998i	0.403473	+	0.914991i	$0.367803\pi$
$43$	12.0000i	1.82998i	−0.403473	+	0.914991i	$0.632197\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	−4.00000	−0.589768
$47$	− 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
$49$	−9.00000	−1.28571
$50$	0	0
$51$	−2.00000	−0.280056
$52$	2.00000i	0.277350i
$53$	6.00000i	0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$	6.00000i	0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	4.00000	0.534522
$57$	− 4.00000i	− 0.529813i
$58$	− 6.00000i	− 0.787839i
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	0	0
$63$	− 4.00000i	− 0.503953i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	1.00000	0.123091
$67$	− 12.0000i	− 1.46603i	−0.680211	−	0.733017i	$-0.738112\pi$
$67$	− 12.0000i	− 1.46603i	0.680211	−	0.733017i	$-0.261888\pi$
$68$	− 2.00000i	− 0.242536i
$69$	−4.00000	−0.481543
$70$	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	$-0.801014\pi$
$73$	− 10.0000i	− 1.17041i	0.810885	−	0.585206i	$-0.198986\pi$
$74$	10.0000	1.16248
$75$	0	0
$76$	4.00000	0.458831
$77$	− 4.00000i	− 0.455842i
$78$	2.00000i	0.226455i
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 6.00000i	− 0.662589i
$83$	− 4.00000i	− 0.439057i	−0.975606	−	0.219529i	$-0.929548\pi$
$83$	− 4.00000i	− 0.439057i	0.975606	−	0.219529i	$-0.0704519\pi$
$84$	4.00000	0.436436
$85$	0	0
$86$	−12.0000	−1.29399
$87$	− 6.00000i	− 0.643268i
$88$	1.00000i	0.106600i
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	8.00000	0.838628
$92$	− 4.00000i	− 0.417029i
$93$	0	0
$94$	4.00000	0.412568
$95$	0	0
$96$	−1.00000	−0.102062
$97$	18.0000i	1.82762i	0.406138	+	0.913812i	$0.366875\pi$
$97$	18.0000i	1.82762i	−0.406138	+	0.913812i	$0.633125\pi$
$98$	− 9.00000i	− 0.909137i
$99$	1.00000	0.100504
$100$	0	0
$101$	−18.0000	−1.79107	−0.895533	−	0.444994i	$-0.853206\pi$
$101$	−18.0000	−1.79107	−0.895533	+	0.444994i	$0.853206\pi$
$102$	− 2.00000i	− 0.198030i
$103$	16.0000i	1.57653i	0.615338	+	0.788263i	$0.289020\pi$
$103$	16.0000i	1.57653i	−0.615338	+	0.788263i	$0.710980\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	−6.00000	−0.582772
$107$	− 12.0000i	− 1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$	− 12.0000i	− 1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$	1.00000i	0.0962250i
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	10.0000	0.949158
$112$	4.00000i	0.377964i
$113$	− 18.0000i	− 1.69330i	−0.532152	−	0.846649i	$-0.678617\pi$
$113$	− 18.0000i	− 1.69330i	0.532152	−	0.846649i	$-0.321383\pi$
$114$	4.00000	0.374634
$115$	0	0
$116$	6.00000	0.557086
$117$	2.00000i	0.184900i
$118$	4.00000i	0.368230i
$119$	−8.00000	−0.733359
$120$	0	0
$121$	1.00000	0.0909091
$122$	10.0000i	0.905357i
$123$	− 6.00000i	− 0.541002i
$124$	0	0
$125$	0	0
$126$	4.00000	0.356348
$127$	4.00000i	0.354943i	0.984126	+	0.177471i	$0.0567917\pi$
$127$	4.00000i	0.354943i	−0.984126	+	0.177471i	$0.943208\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	−12.0000	−1.05654
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	1.00000i	0.0870388i
$133$	− 16.0000i	− 1.38738i
$134$	12.0000	1.03664
$135$	0	0
$136$	2.00000	0.171499
$137$	2.00000i	0.170872i	0.996344	+	0.0854358i	$0.0272282\pi$
$137$	2.00000i	0.170872i	−0.996344	+	0.0854358i	$0.972772\pi$
$138$	− 4.00000i	− 0.340503i
$139$	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
$141$	4.00000	0.336861
$142$	− 4.00000i	− 0.335673i
$143$	2.00000i	0.167248i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	10.0000	0.827606
$147$	− 9.00000i	− 0.742307i
$148$	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
$151$	12.0000	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	4.00000i	0.324443i
$153$	− 2.00000i	− 0.161690i
$154$	4.00000	0.322329
$155$	0	0
$156$	−2.00000	−0.160128
$157$	− 10.0000i	− 0.798087i	−0.916932	−	0.399043i	$-0.869342\pi$
$157$	− 10.0000i	− 0.798087i	0.916932	−	0.399043i	$-0.130658\pi$
$158$	− 4.00000i	− 0.318223i
$159$	−6.00000	−0.475831
$160$	0	0
$161$	−16.0000	−1.26098
$162$	1.00000i	0.0785674i
$163$	20.0000i	1.56652i	0.621694	+	0.783260i	$0.286445\pi$
$163$	20.0000i	1.56652i	−0.621694	+	0.783260i	$0.713555\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	4.00000	0.310460
$167$	16.0000i	1.23812i	0.785345	+	0.619059i	$0.212486\pi$
$167$	16.0000i	1.23812i	−0.785345	+	0.619059i	$0.787514\pi$
$168$	4.00000i	0.308607i
$169$	9.00000	0.692308
$170$	0	0
$171$	4.00000	0.305888
$172$	− 12.0000i	− 0.914991i
$173$	− 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
$176$	−1.00000	−0.0753778
$177$	4.00000i	0.300658i
$178$	− 10.0000i	− 0.749532i
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	8.00000i	0.592999i
$183$	10.0000i	0.739221i
$184$	4.00000	0.294884
$185$	0	0
$186$	0	0
$187$	− 2.00000i	− 0.146254i
$188$	4.00000i	0.291730i
$189$	4.00000	0.290957
$190$	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
$193$	22.0000i	1.58359i	0.610784	+	0.791797i	$0.290854\pi$
$193$	22.0000i	1.58359i	−0.610784	+	0.791797i	$0.709146\pi$
$194$	−18.0000	−1.29232
$195$	0	0
$196$	9.00000	0.642857
$197$	− 18.0000i	− 1.28245i	−0.767354	−	0.641223i	$-0.778427\pi$
$197$	− 18.0000i	− 1.28245i	0.767354	−	0.641223i	$-0.221573\pi$
$198$	1.00000i	0.0710669i
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	12.0000	0.846415
$202$	− 18.0000i	− 1.26648i
$203$	− 24.0000i	− 1.68447i
$204$	2.00000	0.140028
$205$	0	0
$206$	−16.0000	−1.11477
$207$	− 4.00000i	− 0.278019i
$208$	− 2.00000i	− 0.138675i
$209$	4.00000	0.276686
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	− 6.00000i	− 0.412082i
$213$	− 4.00000i	− 0.274075i
$214$	12.0000	0.820303
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	− 2.00000i	− 0.135457i
$219$	10.0000	0.675737
$220$	0	0
$221$	4.00000	0.269069
$222$	10.0000i	0.671156i
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	−4.00000	−0.267261
$225$	0	0
$226$	18.0000	1.19734
$227$	28.0000i	1.85843i	0.369546	+	0.929213i	$0.379513\pi$
$227$	28.0000i	1.85843i	−0.369546	+	0.929213i	$0.620487\pi$
$228$	4.00000i	0.264906i
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	0	0
$231$	4.00000	0.263181
$232$	6.00000i	0.393919i
$233$	22.0000i	1.44127i	0.693316	+	0.720634i	$0.256149\pi$
$233$	22.0000i	1.44127i	−0.693316	+	0.720634i	$0.743851\pi$
$234$	−2.00000	−0.130744
$235$	0	0
$236$	−4.00000	−0.260378
$237$	− 4.00000i	− 0.259828i
$238$	− 8.00000i	− 0.518563i
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	0	0
$241$	−6.00000	−0.386494	−0.193247	−	0.981150i	$-0.561902\pi$
$241$	−6.00000	−0.386494	−0.193247	+	0.981150i	$0.561902\pi$
$242$	1.00000i	0.0642824i
$243$	1.00000i	0.0641500i
$244$	−10.0000	−0.640184
$245$	0	0
$246$	6.00000	0.382546
$247$	8.00000i	0.509028i
$248$	0	0
$249$	4.00000	0.253490
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	4.00000i	0.251976i
$253$	− 4.00000i	− 0.251478i
$254$	−4.00000	−0.250982
$255$	0	0
$256$	1.00000	0.0625000
$257$	18.0000i	1.12281i	0.827541	+	0.561405i	$0.189739\pi$
$257$	18.0000i	1.12281i	−0.827541	+	0.561405i	$0.810261\pi$
$258$	− 12.0000i	− 0.747087i
$259$	40.0000	2.48548
$260$	0	0
$261$	6.00000	0.371391
$262$	− 12.0000i	− 0.741362i
$263$	24.0000i	1.47990i	0.672660	+	0.739952i	$0.265152\pi$
$263$	24.0000i	1.47990i	−0.672660	+	0.739952i	$0.734848\pi$
$264$	−1.00000	−0.0615457
$265$	0	0
$266$	16.0000	0.981023
$267$	− 10.0000i	− 0.611990i
$268$	12.0000i	0.733017i
$269$	−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	$-0.707810\pi$
$271$	−20.0000	−1.21491	−0.607457	+	0.794353i	$0.707810\pi$
$272$	2.00000i	0.121268i
$273$	8.00000i	0.484182i
$274$	−2.00000	−0.120824
$275$	0	0
$276$	4.00000	0.240772
$277$	18.0000i	1.08152i	0.841178	+	0.540758i	$0.181862\pi$
$277$	18.0000i	1.08152i	−0.841178	+	0.540758i	$0.818138\pi$
$278$	4.00000i	0.239904i
$279$	0	0
$280$	0	0
$281$	−22.0000	−1.31241	−0.656205	−	0.754583i	$-0.727839\pi$
$281$	−22.0000	−1.31241	−0.656205	+	0.754583i	$0.727839\pi$
$282$	4.00000i	0.238197i
$283$	12.0000i	0.713326i	0.934233	+	0.356663i	$0.116086\pi$
$283$	12.0000i	0.713326i	−0.934233	+	0.356663i	$0.883914\pi$
$284$	4.00000	0.237356
$285$	0	0
$286$	−2.00000	−0.118262
$287$	− 24.0000i	− 1.41668i
$288$	− 1.00000i	− 0.0589256i
$289$	13.0000	0.764706
$290$	0	0
$291$	−18.0000	−1.05518
$292$	10.0000i	0.585206i
$293$	10.0000i	0.584206i	0.956387	+	0.292103i	$0.0943550\pi$
$293$	10.0000i	0.584206i	−0.956387	+	0.292103i	$0.905645\pi$
$294$	9.00000	0.524891
$295$	0	0
$296$	−10.0000	−0.581238
$297$	1.00000i	0.0580259i
$298$	− 6.00000i	− 0.347571i
$299$	8.00000	0.462652
$300$	0	0
$301$	−48.0000	−2.76667
$302$	12.0000i	0.690522i
$303$	− 18.0000i	− 1.03407i
$304$	−4.00000	−0.229416
$305$	0	0
$306$	2.00000	0.114332
$307$	− 28.0000i	− 1.59804i	−0.601302	−	0.799022i	$-0.705351\pi$
$307$	− 28.0000i	− 1.59804i	0.601302	−	0.799022i	$-0.294649\pi$
$308$	4.00000i	0.227921i
$309$	−16.0000	−0.910208
$310$	0	0
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	− 2.00000i	− 0.113228i
$313$	6.00000i	0.339140i	0.985518	+	0.169570i	$0.0542379\pi$
$313$	6.00000i	0.339140i	−0.985518	+	0.169570i	$0.945762\pi$
$314$	10.0000	0.564333
$315$	0	0
$316$	4.00000	0.225018
$317$	26.0000i	1.46031i	0.683284	+	0.730153i	$0.260551\pi$
$317$	26.0000i	1.46031i	−0.683284	+	0.730153i	$0.739449\pi$
$318$	− 6.00000i	− 0.336463i
$319$	6.00000	0.335936
$320$	0	0
$321$	12.0000	0.669775
$322$	− 16.0000i	− 0.891645i
$323$	− 8.00000i	− 0.445132i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−20.0000	−1.10770
$327$	− 2.00000i	− 0.110600i
$328$	6.00000i	0.331295i
$329$	16.0000	0.882109
$330$	0	0
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	4.00000i	0.219529i
$333$	10.0000i	0.547997i
$334$	−16.0000	−0.875481
$335$	0	0
$336$	−4.00000	−0.218218
$337$	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
$339$	18.0000	0.977626
$340$	0	0
$341$	0	0
$342$	4.00000i	0.216295i
$343$	− 8.00000i	− 0.431959i
$344$	12.0000	0.646997
$345$	0	0
$346$	6.00000	0.322562
$347$	28.0000i	1.50312i	0.659665	+	0.751559i	$0.270698\pi$
$347$	28.0000i	1.50312i	−0.659665	+	0.751559i	$0.729302\pi$
$348$	6.00000i	0.321634i
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	− 1.00000i	− 0.0533002i
$353$	30.0000i	1.59674i	0.602168	+	0.798369i	$0.294304\pi$
$353$	30.0000i	1.59674i	−0.602168	+	0.798369i	$0.705696\pi$
$354$	−4.00000	−0.212598
$355$	0	0
$356$	10.0000	0.529999
$357$	− 8.00000i	− 0.423405i
$358$	− 4.00000i	− 0.211407i
$359$	16.0000	0.844448	0.422224	−	0.906492i	$-0.361250\pi$
$359$	16.0000	0.844448	0.422224	+	0.906492i	$0.361250\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	14.0000i	0.735824i
$363$	1.00000i	0.0524864i
$364$	−8.00000	−0.419314
$365$	0	0
$366$	−10.0000	−0.522708
$367$	16.0000i	0.835193i	0.908633	+	0.417597i	$0.137127\pi$
$367$	16.0000i	0.835193i	−0.908633	+	0.417597i	$0.862873\pi$
$368$	4.00000i	0.208514i
$369$	6.00000	0.312348
$370$	0	0
$371$	−24.0000	−1.24602
$372$	0	0
$373$	6.00000i	0.310668i	0.987862	+	0.155334i	$0.0496454\pi$
$373$	6.00000i	0.310668i	−0.987862	+	0.155334i	$0.950355\pi$
$374$	2.00000	0.103418
$375$	0	0
$376$	−4.00000	−0.206284
$377$	12.0000i	0.618031i
$378$	4.00000i	0.205738i
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	0	0
$381$	−4.00000	−0.204926
$382$	12.0000i	0.613973i
$383$	− 20.0000i	− 1.02195i	−0.859595	−	0.510976i	$-0.829284\pi$
$383$	− 20.0000i	− 1.02195i	0.859595	−	0.510976i	$-0.170716\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−22.0000	−1.11977
$387$	− 12.0000i	− 0.609994i
$388$	− 18.0000i	− 0.913812i
$389$	−26.0000	−1.31825	−0.659126	−	0.752032i	$-0.729074\pi$
$389$	−26.0000	−1.31825	−0.659126	+	0.752032i	$0.729074\pi$
$390$	0	0
$391$	−8.00000	−0.404577
$392$	9.00000i	0.454569i
$393$	− 12.0000i	− 0.605320i
$394$	18.0000	0.906827
$395$	0	0
$396$	−1.00000	−0.0502519
$397$	22.0000i	1.10415i	0.833795	+	0.552074i	$0.186163\pi$
$397$	22.0000i	1.10415i	−0.833795	+	0.552074i	$0.813837\pi$
$398$	0	0
$399$	16.0000	0.801002
$400$	0	0
$401$	−22.0000	−1.09863	−0.549314	−	0.835616i	$-0.685111\pi$
$401$	−22.0000	−1.09863	−0.549314	+	0.835616i	$0.685111\pi$
$402$	12.0000i	0.598506i
$403$	0	0
$404$	18.0000	0.895533
$405$	0	0
$406$	24.0000	1.19110
$407$	10.0000i	0.495682i
$408$	2.00000i	0.0990148i
$409$	−2.00000	−0.0988936	−0.0494468	−	0.998777i	$-0.515746\pi$
$409$	−2.00000	−0.0988936	−0.0494468	+	0.998777i	$0.515746\pi$
$410$	0	0
$411$	−2.00000	−0.0986527
$412$	− 16.0000i	− 0.788263i
$413$	16.0000i	0.787309i
$414$	4.00000	0.196589
$415$	0	0
$416$	2.00000	0.0980581
$417$	4.00000i	0.195881i
$418$	4.00000i	0.195646i
$419$	28.0000	1.36789	0.683945	−	0.729534i	$-0.260263\pi$
$419$	28.0000	1.36789	0.683945	+	0.729534i	$0.260263\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	12.0000i	0.584151i
$423$	4.00000i	0.194487i
$424$	6.00000	0.291386
$425$	0	0
$426$	4.00000	0.193801
$427$	40.0000i	1.93574i
$428$	12.0000i	0.580042i
$429$	−2.00000	−0.0965609
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	− 34.0000i	− 1.63394i	−0.576683	−	0.816968i	$-0.695653\pi$
$433$	− 34.0000i	− 1.63394i	0.576683	−	0.816968i	$-0.304347\pi$
$434$	0	0
$435$	0	0
$436$	2.00000	0.0957826
$437$	− 16.0000i	− 0.765384i
$438$	10.0000i	0.477818i
$439$	−12.0000	−0.572729	−0.286364	−	0.958121i	$-0.592447\pi$
$439$	−12.0000	−0.572729	−0.286364	+	0.958121i	$0.592447\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	4.00000i	0.190261i
$443$	− 12.0000i	− 0.570137i	−0.958507	−	0.285069i	$-0.907984\pi$
$443$	− 12.0000i	− 0.570137i	0.958507	−	0.285069i	$-0.0920164\pi$
$444$	−10.0000	−0.474579
$445$	0	0
$446$	0	0
$447$	− 6.00000i	− 0.283790i
$448$	− 4.00000i	− 0.188982i
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	6.00000	0.282529
$452$	18.0000i	0.846649i
$453$	12.0000i	0.563809i
$454$	−28.0000	−1.31411
$455$	0	0
$456$	−4.00000	−0.187317
$457$	34.0000i	1.59045i	0.606313	+	0.795226i	$0.292648\pi$
$457$	34.0000i	1.59045i	−0.606313	+	0.795226i	$0.707352\pi$
$458$	2.00000i	0.0934539i
$459$	2.00000	0.0933520
$460$	0	0
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	4.00000i	0.186097i
$463$	− 24.0000i	− 1.11537i	−0.830051	−	0.557687i	$-0.811689\pi$
$463$	− 24.0000i	− 1.11537i	0.830051	−	0.557687i	$-0.188311\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	−22.0000	−1.01913
$467$	− 20.0000i	− 0.925490i	−0.886492	−	0.462745i	$-0.846865\pi$
$467$	− 20.0000i	− 0.925490i	0.886492	−	0.462745i	$-0.153135\pi$
$468$	− 2.00000i	− 0.0924500i
$469$	48.0000	2.21643
$470$	0	0
$471$	10.0000	0.460776
$472$	− 4.00000i	− 0.184115i
$473$	− 12.0000i	− 0.551761i
$474$	4.00000	0.183726
$475$	0	0
$476$	8.00000	0.366679
$477$	− 6.00000i	− 0.274721i
$478$	− 24.0000i	− 1.09773i
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	−20.0000	−0.911922
$482$	− 6.00000i	− 0.273293i
$483$	− 16.0000i	− 0.728025i
$484$	−1.00000	−0.0454545
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	− 32.0000i	− 1.45006i	−0.688718	−	0.725029i	$-0.741826\pi$
$487$	− 32.0000i	− 1.45006i	0.688718	−	0.725029i	$-0.258174\pi$
$488$	− 10.0000i	− 0.452679i
$489$	−20.0000	−0.904431
$490$	0	0
$491$	36.0000	1.62466	0.812329	−	0.583200i	$-0.198200\pi$
$491$	36.0000	1.62466	0.812329	+	0.583200i	$0.198200\pi$
$492$	6.00000i	0.270501i
$493$	− 12.0000i	− 0.540453i
$494$	−8.00000	−0.359937
$495$	0	0
$496$	0	0
$497$	− 16.0000i	− 0.717698i
$498$	4.00000i	0.179244i
$499$	−12.0000	−0.537194	−0.268597	−	0.963253i	$-0.586560\pi$
$499$	−12.0000	−0.537194	−0.268597	+	0.963253i	$0.586560\pi$
$500$	0	0
$501$	−16.0000	−0.714827
$502$	12.0000i	0.535586i
$503$	− 16.0000i	− 0.713405i	−0.934218	−	0.356702i	$-0.883901\pi$
$503$	− 16.0000i	− 0.713405i	0.934218	−	0.356702i	$-0.116099\pi$
$504$	−4.00000	−0.178174
$505$	0	0
$506$	4.00000	0.177822
$507$	9.00000i	0.399704i
$508$	− 4.00000i	− 0.177471i
$509$	−34.0000	−1.50702	−0.753512	−	0.657434i	$-0.771642\pi$
$509$	−34.0000	−1.50702	−0.753512	+	0.657434i	$0.771642\pi$
$510$	0	0
$511$	40.0000	1.76950
$512$	1.00000i	0.0441942i
$513$	4.00000i	0.176604i
$514$	−18.0000	−0.793946
$515$	0	0
$516$	12.0000	0.528271
$517$	4.00000i	0.175920i
$518$	40.0000i	1.75750i
$519$	6.00000	0.263371
$520$	0	0
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	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
$527$	0	0
$528$	− 1.00000i	− 0.0435194i
$529$	7.00000	0.304348
$530$	0	0
$531$	−4.00000	−0.173585
$532$	16.0000i	0.693688i
$533$	12.0000i	0.519778i
$534$	10.0000	0.432742
$535$	0	0
$536$	−12.0000	−0.518321
$537$	− 4.00000i	− 0.172613i
$538$	− 18.0000i	− 0.776035i
$539$	9.00000	0.387657
$540$	0	0
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	− 20.0000i	− 0.859074i
$543$	14.0000i	0.600798i
$544$	−2.00000	−0.0857493
$545$	0	0
$546$	−8.00000	−0.342368
$547$	4.00000i	0.171028i	0.996337	+	0.0855138i	$0.0272532\pi$
$547$	4.00000i	0.171028i	−0.996337	+	0.0855138i	$0.972747\pi$
$548$	− 2.00000i	− 0.0854358i
$549$	−10.0000	−0.426790
$550$	0	0
$551$	24.0000	1.02243
$552$	4.00000i	0.170251i
$553$	− 16.0000i	− 0.680389i
$554$	−18.0000	−0.764747
$555$	0	0
$556$	−4.00000	−0.169638
$557$	30.0000i	1.27114i	0.772043	+	0.635570i	$0.219235\pi$
$557$	30.0000i	1.27114i	−0.772043	+	0.635570i	$0.780765\pi$
$558$	0	0
$559$	24.0000	1.01509
$560$	0	0
$561$	2.00000	0.0844401
$562$	− 22.0000i	− 0.928014i
$563$	4.00000i	0.168580i	0.996441	+	0.0842900i	$0.0268622\pi$
$563$	4.00000i	0.168580i	−0.996441	+	0.0842900i	$0.973138\pi$
$564$	−4.00000	−0.168430
$565$	0	0
$566$	−12.0000	−0.504398
$567$	4.00000i	0.167984i
$568$	4.00000i	0.167836i
$569$	22.0000	0.922288	0.461144	−	0.887325i	$-0.347439\pi$
$569$	22.0000	0.922288	0.461144	+	0.887325i	$0.347439\pi$
$570$	0	0
$571$	44.0000	1.84134	0.920671	−	0.390339i	$-0.127642\pi$
$571$	44.0000	1.84134	0.920671	+	0.390339i	$0.127642\pi$
$572$	− 2.00000i	− 0.0836242i
$573$	12.0000i	0.501307i
$574$	24.0000	1.00174
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 14.0000i	− 0.582828i	−0.956597	−	0.291414i	$-0.905874\pi$
$577$	− 14.0000i	− 0.582828i	0.956597	−	0.291414i	$-0.0941257\pi$
$578$	13.0000i	0.540729i
$579$	−22.0000	−0.914289
$580$	0	0
$581$	16.0000	0.663792
$582$	− 18.0000i	− 0.746124i
$583$	− 6.00000i	− 0.248495i
$584$	−10.0000	−0.413803
$585$	0	0
$586$	−10.0000	−0.413096
$587$	12.0000i	0.495293i	0.968850	+	0.247647i	$0.0796572\pi$
$587$	12.0000i	0.495293i	−0.968850	+	0.247647i	$0.920343\pi$
$588$	9.00000i	0.371154i
$589$	0	0
$590$	0	0
$591$	18.0000	0.740421
$592$	− 10.0000i	− 0.410997i
$593$	− 2.00000i	− 0.0821302i	−0.999156	−	0.0410651i	$-0.986925\pi$
$593$	− 2.00000i	− 0.0821302i	0.999156	−	0.0410651i	$-0.0130751\pi$
$594$	−1.00000	−0.0410305
$595$	0	0
$596$	6.00000	0.245770
$597$	0	0
$598$	8.00000i	0.327144i
$599$	−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
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	− 48.0000i	− 1.95633i
$603$	12.0000i	0.488678i
$604$	−12.0000	−0.488273
$605$	0	0
$606$	18.0000	0.731200
$607$	20.0000i	0.811775i	0.913923	+	0.405887i	$0.133038\pi$
$607$	20.0000i	0.811775i	−0.913923	+	0.405887i	$0.866962\pi$
$608$	− 4.00000i	− 0.162221i
$609$	24.0000	0.972529
$610$	0	0
$611$	−8.00000	−0.323645
$612$	2.00000i	0.0808452i
$613$	14.0000i	0.565455i	0.959200	+	0.282727i	$0.0912392\pi$
$613$	14.0000i	0.565455i	−0.959200	+	0.282727i	$0.908761\pi$
$614$	28.0000	1.12999
$615$	0	0
$616$	−4.00000	−0.161165
$617$	− 6.00000i	− 0.241551i	−0.992680	−	0.120775i	$-0.961462\pi$
$617$	− 6.00000i	− 0.241551i	0.992680	−	0.120775i	$-0.0385381\pi$
$618$	− 16.0000i	− 0.643614i
$619$	36.0000	1.44696	0.723481	−	0.690344i	$-0.242541\pi$
$619$	36.0000	1.44696	0.723481	+	0.690344i	$0.242541\pi$
$620$	0	0
$621$	4.00000	0.160514
$622$	12.0000i	0.481156i
$623$	− 40.0000i	− 1.60257i
$624$	2.00000	0.0800641
$625$	0	0
$626$	−6.00000	−0.239808
$627$	4.00000i	0.159745i
$628$	10.0000i	0.399043i
$629$	20.0000	0.797452
$630$	0	0
$631$	24.0000	0.955425	0.477712	−	0.878516i	$-0.341466\pi$
$631$	24.0000	0.955425	0.477712	+	0.878516i	$0.341466\pi$
$632$	4.00000i	0.159111i
$633$	12.0000i	0.476957i
$634$	−26.0000	−1.03259
$635$	0	0
$636$	6.00000	0.237915
$637$	18.0000i	0.713186i
$638$	6.00000i	0.237542i
$639$	4.00000	0.158238
$640$	0	0
$641$	26.0000	1.02694	0.513469	−	0.858108i	$-0.328360\pi$
$641$	26.0000	1.02694	0.513469	+	0.858108i	$0.328360\pi$
$642$	12.0000i	0.473602i
$643$	− 20.0000i	− 0.788723i	−0.918955	−	0.394362i	$-0.870966\pi$
$643$	− 20.0000i	− 0.788723i	0.918955	−	0.394362i	$-0.129034\pi$
$644$	16.0000	0.630488
$645$	0	0
$646$	8.00000	0.314756
$647$	− 36.0000i	− 1.41531i	−0.706560	−	0.707653i	$-0.749754\pi$
$647$	− 36.0000i	− 1.41531i	0.706560	−	0.707653i	$-0.250246\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	−4.00000	−0.157014
$650$	0	0
$651$	0	0
$652$	− 20.0000i	− 0.783260i
$653$	− 26.0000i	− 1.01746i	−0.860927	−	0.508729i	$-0.830115\pi$
$653$	− 26.0000i	− 1.01746i	0.860927	−	0.508729i	$-0.169885\pi$
$654$	2.00000	0.0782062
$655$	0	0
$656$	−6.00000	−0.234261
$657$	10.0000i	0.390137i
$658$	16.0000i	0.623745i
$659$	−28.0000	−1.09073	−0.545363	−	0.838200i	$-0.683608\pi$
$659$	−28.0000	−1.09073	−0.545363	+	0.838200i	$0.683608\pi$
$660$	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
$663$	4.00000i	0.155347i
$664$	−4.00000	−0.155230
$665$	0	0
$666$	−10.0000	−0.387492
$667$	− 24.0000i	− 0.929284i
$668$	− 16.0000i	− 0.619059i
$669$	0	0
$670$	0	0
$671$	−10.0000	−0.386046
$672$	− 4.00000i	− 0.154303i
$673$	− 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
$677$	30.0000i	1.15299i	0.817099	+	0.576497i	$0.195581\pi$
$677$	30.0000i	1.15299i	−0.817099	+	0.576497i	$0.804419\pi$
$678$	18.0000i	0.691286i
$679$	−72.0000	−2.76311
$680$	0	0
$681$	−28.0000	−1.07296
$682$	0	0
$683$	4.00000i	0.153056i	0.997067	+	0.0765279i	$0.0243834\pi$
$683$	4.00000i	0.153056i	−0.997067	+	0.0765279i	$0.975617\pi$
$684$	−4.00000	−0.152944
$685$	0	0
$686$	8.00000	0.305441
$687$	2.00000i	0.0763048i
$688$	12.0000i	0.457496i
$689$	12.0000	0.457164
$690$	0	0
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	6.00000i	0.228086i
$693$	4.00000i	0.151947i
$694$	−28.0000	−1.06287
$695$	0	0
$696$	−6.00000	−0.227429
$697$	− 12.0000i	− 0.454532i
$698$	− 2.00000i	− 0.0757011i
$699$	−22.0000	−0.832116
$700$	0	0
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	− 2.00000i	− 0.0754851i
$703$	40.0000i	1.50863i
$704$	1.00000	0.0376889
$705$	0	0
$706$	−30.0000	−1.12906
$707$	− 72.0000i	− 2.70784i
$708$	− 4.00000i	− 0.150329i
$709$	34.0000	1.27690	0.638448	−	0.769665i	$-0.279577\pi$
$709$	34.0000	1.27690	0.638448	+	0.769665i	$0.279577\pi$
$710$	0	0
$711$	4.00000	0.150012
$712$	10.0000i	0.374766i
$713$	0	0
$714$	8.00000	0.299392
$715$	0	0
$716$	4.00000	0.149487
$717$	− 24.0000i	− 0.896296i
$718$	16.0000i	0.597115i
$719$	−20.0000	−0.745874	−0.372937	−	0.927857i	$-0.621649\pi$
$719$	−20.0000	−0.745874	−0.372937	+	0.927857i	$0.621649\pi$
$720$	0	0
$721$	−64.0000	−2.38348
$722$	− 3.00000i	− 0.111648i
$723$	− 6.00000i	− 0.223142i
$724$	−14.0000	−0.520306
$725$	0	0
$726$	−1.00000	−0.0371135
$727$	16.0000i	0.593407i	0.954970	+	0.296704i	$0.0958873\pi$
$727$	16.0000i	0.593407i	−0.954970	+	0.296704i	$0.904113\pi$
$728$	− 8.00000i	− 0.296500i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−24.0000	−0.887672
$732$	− 10.0000i	− 0.369611i
$733$	14.0000i	0.517102i	0.965998	+	0.258551i	$0.0832450\pi$
$733$	14.0000i	0.517102i	−0.965998	+	0.258551i	$0.916755\pi$
$734$	−16.0000	−0.590571
$735$	0	0
$736$	−4.00000	−0.147442
$737$	12.0000i	0.442026i
$738$	6.00000i	0.220863i
$739$	36.0000	1.32428	0.662141	−	0.749380i	$-0.269648\pi$
$739$	36.0000	1.32428	0.662141	+	0.749380i	$0.269648\pi$
$740$	0	0
$741$	−8.00000	−0.293887
$742$	− 24.0000i	− 0.881068i
$743$	− 16.0000i	− 0.586983i	−0.955962	−	0.293492i	$-0.905183\pi$
$743$	− 16.0000i	− 0.586983i	0.955962	−	0.293492i	$-0.0948173\pi$
$744$	0	0
$745$	0	0
$746$	−6.00000	−0.219676
$747$	4.00000i	0.146352i
$748$	2.00000i	0.0731272i
$749$	48.0000	1.75388
$750$	0	0
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	− 4.00000i	− 0.145865i
$753$	12.0000i	0.437304i
$754$	−12.0000	−0.437014
$755$	0	0
$756$	−4.00000	−0.145479
$757$	22.0000i	0.799604i	0.916602	+	0.399802i	$0.130921\pi$
$757$	22.0000i	0.799604i	−0.916602	+	0.399802i	$0.869079\pi$
$758$	− 20.0000i	− 0.726433i
$759$	4.00000	0.145191
$760$	0	0
$761$	10.0000	0.362500	0.181250	−	0.983437i	$-0.441986\pi$
$761$	10.0000	0.362500	0.181250	+	0.983437i	$0.441986\pi$
$762$	− 4.00000i	− 0.144905i
$763$	− 8.00000i	− 0.289619i
$764$	−12.0000	−0.434145
$765$	0	0
$766$	20.0000	0.722629
$767$	− 8.00000i	− 0.288863i
$768$	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
$771$	−18.0000	−0.648254
$772$	− 22.0000i	− 0.791797i
$773$	− 10.0000i	− 0.359675i	−0.983696	−	0.179838i	$-0.942443\pi$
$773$	− 10.0000i	− 0.359675i	0.983696	−	0.179838i	$-0.0575572\pi$
$774$	12.0000	0.431331
$775$	0	0
$776$	18.0000	0.646162
$777$	40.0000i	1.43499i
$778$	− 26.0000i	− 0.932145i
$779$	24.0000	0.859889
$780$	0	0
$781$	4.00000	0.143131
$782$	− 8.00000i	− 0.286079i
$783$	6.00000i	0.214423i
$784$	−9.00000	−0.321429
$785$	0	0
$786$	12.0000	0.428026
$787$	− 36.0000i	− 1.28326i	−0.767014	−	0.641631i	$-0.778258\pi$
$787$	− 36.0000i	− 1.28326i	0.767014	−	0.641631i	$-0.221742\pi$
$788$	18.0000i	0.641223i
$789$	−24.0000	−0.854423
$790$	0	0
$791$	72.0000	2.56003
$792$	− 1.00000i	− 0.0355335i
$793$	− 20.0000i	− 0.710221i
$794$	−22.0000	−0.780751
$795$	0	0
$796$	0	0
$797$	− 30.0000i	− 1.06265i	−0.847167	−	0.531327i	$-0.821693\pi$
$797$	− 30.0000i	− 1.06265i	0.847167	−	0.531327i	$-0.178307\pi$
$798$	16.0000i	0.566394i
$799$	8.00000	0.283020
$800$	0	0
$801$	10.0000	0.353333
$802$	− 22.0000i	− 0.776847i
$803$	10.0000i	0.352892i
$804$	−12.0000	−0.423207
$805$	0	0
$806$	0	0
$807$	− 18.0000i	− 0.633630i
$808$	18.0000i	0.633238i
$809$	22.0000	0.773479	0.386739	−	0.922189i	$-0.373601\pi$
$809$	22.0000	0.773479	0.386739	+	0.922189i	$0.373601\pi$
$810$	0	0
$811$	52.0000	1.82597	0.912983	−	0.407997i	$-0.133772\pi$
$811$	52.0000	1.82597	0.912983	+	0.407997i	$0.133772\pi$
$812$	24.0000i	0.842235i
$813$	− 20.0000i	− 0.701431i
$814$	−10.0000	−0.350500
$815$	0	0
$816$	−2.00000	−0.0700140
$817$	− 48.0000i	− 1.67931i
$818$	− 2.00000i	− 0.0699284i
$819$	−8.00000	−0.279543
$820$	0	0
$821$	−2.00000	−0.0698005	−0.0349002	−	0.999391i	$-0.511111\pi$
$821$	−2.00000	−0.0698005	−0.0349002	+	0.999391i	$0.511111\pi$
$822$	− 2.00000i	− 0.0697580i
$823$	40.0000i	1.39431i	0.716919	+	0.697156i	$0.245552\pi$
$823$	40.0000i	1.39431i	−0.716919	+	0.697156i	$0.754448\pi$
$824$	16.0000	0.557386
$825$	0	0
$826$	−16.0000	−0.556711
$827$	12.0000i	0.417281i	0.977992	+	0.208640i	$0.0669038\pi$
$827$	12.0000i	0.417281i	−0.977992	+	0.208640i	$0.933096\pi$
$828$	4.00000i	0.139010i
$829$	−46.0000	−1.59765	−0.798823	−	0.601566i	$-0.794544\pi$
$829$	−46.0000	−1.59765	−0.798823	+	0.601566i	$0.794544\pi$
$830$	0	0
$831$	−18.0000	−0.624413
$832$	2.00000i	0.0693375i
$833$	− 18.0000i	− 0.623663i
$834$	−4.00000	−0.138509
$835$	0	0
$836$	−4.00000	−0.138343
$837$	0	0
$838$	28.0000i	0.967244i
$839$	−36.0000	−1.24286	−0.621429	−	0.783470i	$-0.713448\pi$
$839$	−36.0000	−1.24286	−0.621429	+	0.783470i	$0.713448\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	22.0000i	0.758170i
$843$	− 22.0000i	− 0.757720i
$844$	−12.0000	−0.413057
$845$	0	0
$846$	−4.00000	−0.137523
$847$	4.00000i	0.137442i
$848$	6.00000i	0.206041i
$849$	−12.0000	−0.411839
$850$	0	0
$851$	40.0000	1.37118
$852$	4.00000i	0.137038i
$853$	− 18.0000i	− 0.616308i	−0.951336	−	0.308154i	$-0.900289\pi$
$853$	− 18.0000i	− 0.616308i	0.951336	−	0.308154i	$-0.0997113\pi$
$854$	−40.0000	−1.36877
$855$	0	0
$856$	−12.0000	−0.410152
$857$	− 38.0000i	− 1.29806i	−0.760765	−	0.649028i	$-0.775176\pi$
$857$	− 38.0000i	− 1.29806i	0.760765	−	0.649028i	$-0.224824\pi$
$858$	− 2.00000i	− 0.0682789i
$859$	44.0000	1.50126	0.750630	−	0.660722i	$-0.229750\pi$
$859$	44.0000	1.50126	0.750630	+	0.660722i	$0.229750\pi$
$860$	0	0
$861$	24.0000	0.817918
$862$	0	0
$863$	36.0000i	1.22545i	0.790295	+	0.612727i	$0.209928\pi$
$863$	36.0000i	1.22545i	−0.790295	+	0.612727i	$0.790072\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	34.0000	1.15537
$867$	13.0000i	0.441503i
$868$	0	0
$869$	4.00000	0.135691
$870$	0	0
$871$	−24.0000	−0.813209
$872$	2.00000i	0.0677285i
$873$	− 18.0000i	− 0.609208i
$874$	16.0000	0.541208
$875$	0	0
$876$	−10.0000	−0.337869
$877$	− 14.0000i	− 0.472746i	−0.971662	−	0.236373i	$-0.924041\pi$
$877$	− 14.0000i	− 0.472746i	0.971662	−	0.236373i	$-0.0759588\pi$
$878$	− 12.0000i	− 0.404980i
$879$	−10.0000	−0.337292
$880$	0	0
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	9.00000i	0.303046i
$883$	− 20.0000i	− 0.673054i	−0.941674	−	0.336527i	$-0.890748\pi$
$883$	− 20.0000i	− 0.673054i	0.941674	−	0.336527i	$-0.109252\pi$
$884$	−4.00000	−0.134535
$885$	0	0
$886$	12.0000	0.403148
$887$	48.0000i	1.61168i	0.592132	+	0.805841i	$0.298286\pi$
$887$	48.0000i	1.61168i	−0.592132	+	0.805841i	$0.701714\pi$
$888$	− 10.0000i	− 0.335578i
$889$	−16.0000	−0.536623
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	16.0000i	0.535420i
$894$	6.00000	0.200670
$895$	0	0
$896$	4.00000	0.133631
$897$	8.00000i	0.267112i
$898$	6.00000i	0.200223i
$899$	0	0
$900$	0	0
$901$	−12.0000	−0.399778
$902$	6.00000i	0.199778i
$903$	− 48.0000i	− 1.59734i
$904$	−18.0000	−0.598671
$905$	0	0
$906$	−12.0000	−0.398673
$907$	− 52.0000i	− 1.72663i	−0.504664	−	0.863316i	$-0.668384\pi$
$907$	− 52.0000i	− 1.72663i	0.504664	−	0.863316i	$-0.331616\pi$
$908$	− 28.0000i	− 0.929213i
$909$	18.0000	0.597022
$910$	0	0
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	− 4.00000i	− 0.132453i
$913$	4.00000i	0.132381i
$914$	−34.0000	−1.12462
$915$	0	0
$916$	−2.00000	−0.0660819
$917$	− 48.0000i	− 1.58510i
$918$	2.00000i	0.0660098i
$919$	12.0000	0.395843	0.197922	−	0.980218i	$-0.436581\pi$
$919$	12.0000	0.395843	0.197922	+	0.980218i	$0.436581\pi$
$920$	0	0
$921$	28.0000	0.922631
$922$	30.0000i	0.987997i
$923$	8.00000i	0.263323i
$924$	−4.00000	−0.131590
$925$	0	0
$926$	24.0000	0.788689
$927$	− 16.0000i	− 0.525509i
$928$	− 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
$931$	36.0000	1.17985
$932$	− 22.0000i	− 0.720634i
$933$	12.0000i	0.392862i
$934$	20.0000	0.654420
$935$	0	0
$936$	2.00000	0.0653720
$937$	− 22.0000i	− 0.718709i	−0.933201	−	0.359354i	$-0.882997\pi$
$937$	− 22.0000i	− 0.718709i	0.933201	−	0.359354i	$-0.117003\pi$
$938$	48.0000i	1.56726i
$939$	−6.00000	−0.195803
$940$	0	0
$941$	22.0000	0.717180	0.358590	−	0.933495i	$-0.383258\pi$
$941$	22.0000	0.717180	0.358590	+	0.933495i	$0.383258\pi$
$942$	10.0000i	0.325818i
$943$	− 24.0000i	− 0.781548i
$944$	4.00000	0.130189
$945$	0	0
$946$	12.0000	0.390154
$947$	− 52.0000i	− 1.68977i	−0.534946	−	0.844886i	$-0.679668\pi$
$947$	− 52.0000i	− 1.68977i	0.534946	−	0.844886i	$-0.320332\pi$
$948$	4.00000i	0.129914i
$949$	−20.0000	−0.649227
$950$	0	0
$951$	−26.0000	−0.843108
$952$	8.00000i	0.259281i
$953$	6.00000i	0.194359i	0.995267	+	0.0971795i	$0.0309821\pi$
$953$	6.00000i	0.194359i	−0.995267	+	0.0971795i	$0.969018\pi$
$954$	6.00000	0.194257
$955$	0	0
$956$	24.0000	0.776215
$957$	6.00000i	0.193952i
$958$	0	0
$959$	−8.00000	−0.258333
$960$	0	0
$961$	−31.0000	−1.00000
$962$	− 20.0000i	− 0.644826i
$963$	12.0000i	0.386695i
$964$	6.00000	0.193247
$965$	0	0
$966$	16.0000	0.514792
$967$	12.0000i	0.385894i	0.981209	+	0.192947i	$0.0618045\pi$
$967$	12.0000i	0.385894i	−0.981209	+	0.192947i	$0.938195\pi$
$968$	− 1.00000i	− 0.0321412i
$969$	8.00000	0.256997
$970$	0	0
$971$	−4.00000	−0.128366	−0.0641831	−	0.997938i	$-0.520444\pi$
$971$	−4.00000	−0.128366	−0.0641831	+	0.997938i	$0.520444\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	16.0000i	0.512936i
$974$	32.0000	1.02535
$975$	0	0
$976$	10.0000	0.320092
$977$	− 38.0000i	− 1.21573i	−0.794041	−	0.607864i	$-0.792027\pi$
$977$	− 38.0000i	− 1.21573i	0.794041	−	0.607864i	$-0.207973\pi$
$978$	− 20.0000i	− 0.639529i
$979$	10.0000	0.319601
$980$	0	0
$981$	2.00000	0.0638551
$982$	36.0000i	1.14881i
$983$	− 28.0000i	− 0.893061i	−0.894768	−	0.446531i	$-0.852659\pi$
$983$	− 28.0000i	− 0.893061i	0.894768	−	0.446531i	$-0.147341\pi$
$984$	−6.00000	−0.191273
$985$	0	0
$986$	12.0000	0.382158
$987$	16.0000i	0.509286i
$988$	− 8.00000i	− 0.254514i
$989$	−48.0000	−1.52631
$990$	0	0
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	0	0
$993$	− 28.0000i	− 0.888553i
$994$	16.0000	0.507489
$995$	0	0
$996$	−4.00000	−0.126745
$997$	− 38.0000i	− 1.20347i	−0.798695	−	0.601736i	$-0.794476\pi$
$997$	− 38.0000i	− 1.20347i	0.798695	−	0.601736i	$-0.205524\pi$
$998$	− 12.0000i	− 0.379853i
$999$	−10.0000	−0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1650.2.c.a.199.2		2
3.2	odd	2		4950.2.c.y.199.1		2
5.2	odd	4		1650.2.a.e.1.1		1
5.3	odd	4		330.2.a.c.1.1	✓	1
5.4	even	2	inner	1650.2.c.a.199.1		2
15.2	even	4		4950.2.a.x.1.1		1
15.8	even	4		990.2.a.g.1.1		1
15.14	odd	2		4950.2.c.y.199.2		2
20.3	even	4		2640.2.a.j.1.1		1
55.43	even	4		3630.2.a.a.1.1		1
60.23	odd	4		7920.2.a.t.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
330.2.a.c.1.1	✓	1	5.3	odd	4
990.2.a.g.1.1		1	15.8	even	4
1650.2.a.e.1.1		1	5.2	odd	4
1650.2.c.a.199.1		2	5.4	even	2	inner
1650.2.c.a.199.2		2	1.1	even	1	trivial
2640.2.a.j.1.1		1	20.3	even	4
3630.2.a.a.1.1		1	55.43	even	4
4950.2.a.x.1.1		1	15.2	even	4
4950.2.c.y.199.1		2	3.2	odd	2
4950.2.c.y.199.2		2	15.14	odd	2
7920.2.a.t.1.1		1	60.23	odd	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:	\( N \)	\(=\)	\( 1650 = 2 \cdot 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1650.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -1.00000 q^{11} -1.00000i q^{12} -2.00000i q^{13} -4.00000 q^{14} +1.00000 q^{16} +2.00000i q^{17} -1.00000i q^{18} -4.00000 q^{19} -4.00000 q^{21} -1.00000i q^{22} +4.00000i q^{23} +1.00000 q^{24} +2.00000 q^{26} -1.00000i q^{27} -4.00000i q^{28} -6.00000 q^{29} +1.00000i q^{32} -1.00000i q^{33} -2.00000 q^{34} +1.00000 q^{36} -10.0000i q^{37} -4.00000i q^{38} +2.00000 q^{39} -6.00000 q^{41} -4.00000i q^{42} +12.0000i q^{43} +1.00000 q^{44} -4.00000 q^{46} -4.00000i q^{47} +1.00000i q^{48} -9.00000 q^{49} -2.00000 q^{51} +2.00000i q^{52} +6.00000i q^{53} +1.00000 q^{54} +4.00000 q^{56} -4.00000i q^{57} -6.00000i q^{58} +4.00000 q^{59} +10.0000 q^{61} -4.00000i q^{63} -1.00000 q^{64} +1.00000 q^{66} -12.0000i q^{67} -2.00000i q^{68} -4.00000 q^{69} -4.00000 q^{71} +1.00000i q^{72} -10.0000i q^{73} +10.0000 q^{74} +4.00000 q^{76} -4.00000i q^{77} +2.00000i q^{78} -4.00000 q^{79} +1.00000 q^{81} -6.00000i q^{82} -4.00000i q^{83} +4.00000 q^{84} -12.0000 q^{86} -6.00000i q^{87} +1.00000i q^{88} -10.0000 q^{89} +8.00000 q^{91} -4.00000i q^{92} +4.00000 q^{94} -1.00000 q^{96} +18.0000i q^{97} -9.00000i q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} - 2 q^{11} - 8 q^{14} + 2 q^{16} - 8 q^{19} - 8 q^{21} + 2 q^{24} + 4 q^{26} - 12 q^{29} - 4 q^{34} + 2 q^{36} + 4 q^{39} - 12 q^{41} + 2 q^{44} - 8 q^{46} - 18 q^{49}+ \cdots + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 - 2 * q^11 - 8 * q^14 + 2 * q^16 - 8 * q^19 - 8 * q^21 + 2 * q^24 + 4 * q^26 - 12 * q^29 - 4 * q^34 + 2 * q^36 + 4 * q^39 - 12 * q^41 + 2 * q^44 - 8 * q^46 - 18 * q^49 - 4 * q^51 + 2 * q^54 + 8 * q^56 + 8 * q^59 + 20 * q^61 - 2 * q^64 + 2 * q^66 - 8 * q^69 - 8 * q^71 + 20 * q^74 + 8 * q^76 - 8 * q^79 + 2 * q^81 + 8 * q^84 - 24 * q^86 - 20 * q^89 + 16 * q^91 + 8 * q^94 - 2 * q^96 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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