LMFDB - Embedded newform 3800.2.d.f.3649.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3800,2,Mod(3649,3800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3800.3649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3800 = 2^{3} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3800.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$30.3431527681$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3649.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	3800.3649
Dual form			3800.2.d.f.3649.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +3.00000i q^{7} +2.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +3.00000i q^{7} +2.00000 q^{9} +2.00000 q^{11} -1.00000i q^{13} -5.00000i q^{17} -1.00000 q^{19} +3.00000 q^{21} +1.00000i q^{23} -5.00000i q^{27} +3.00000 q^{29} +4.00000 q^{31} -2.00000i q^{33} +2.00000i q^{37} -1.00000 q^{39} -8.00000 q^{41} +8.00000i q^{43} -8.00000i q^{47} -2.00000 q^{49} -5.00000 q^{51} -9.00000i q^{53} +1.00000i q^{57} -1.00000 q^{59} +14.0000 q^{61} +6.00000i q^{63} +13.0000i q^{67} +1.00000 q^{69} +10.0000 q^{71} -9.00000i q^{73} +6.00000i q^{77} +10.0000 q^{79} +1.00000 q^{81} -10.0000i q^{83} -3.00000i q^{87} +12.0000 q^{89} +3.00000 q^{91} -4.00000i q^{93} +14.0000i q^{97} +4.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{9}+O(q^{10}) $ 2 * q + 4 * q^9	$ 2 q + 4 q^{9} + 4 q^{11} - 2 q^{19} + 6 q^{21} + 6 q^{29} + 8 q^{31} - 2 q^{39} - 16 q^{41} - 4 q^{49} - 10 q^{51} - 2 q^{59} + 28 q^{61} + 2 q^{69} + 20 q^{71} + 20 q^{79} + 2 q^{81} + 24 q^{89} + 6 q^{91} + 8 q^{99}+O(q^{100}) $ 2 * q + 4 * q^9 + 4 * q^11 - 2 * q^19 + 6 * q^21 + 6 * q^29 + 8 * q^31 - 2 * q^39 - 16 * q^41 - 4 * q^49 - 10 * q^51 - 2 * q^59 + 28 * q^61 + 2 * q^69 + 20 * q^71 + 20 * q^79 + 2 * q^81 + 24 * q^89 + 6 * q^91 + 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times$.

$n$	$401$	$951$	$1901$	$1977$
$\chi(n)$	$1$	$1$	$1$	$-1$

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 $

$2$	0	0
$2$	0	0
$3$	− 1.00000i	− 0.577350i	−0.957427	−	0.288675i	$-0.906785\pi$
$3$	− 1.00000i	− 0.577350i	0.957427	−	0.288675i	$-0.0932147\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.00000i	1.13389i	0.823754	+	0.566947i	$0.191875\pi$
$7$	3.00000i	1.13389i	−0.823754	+	0.566947i	$0.808125\pi$
$8$	0	0
$9$	2.00000	0.666667
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i	−0.990338	−	0.138675i	$-0.955716\pi$
$13$	− 1.00000i	− 0.277350i	0.990338	−	0.138675i	$-0.0442844\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 5.00000i	− 1.21268i	−0.795206	−	0.606339i	$-0.792637\pi$
$17$	− 5.00000i	− 1.21268i	0.795206	−	0.606339i	$-0.207363\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	3.00000	0.654654
$22$	0	0
$23$	1.00000i	0.208514i	0.994550	+	0.104257i	$0.0332465\pi$
$23$	1.00000i	0.208514i	−0.994550	+	0.104257i	$0.966753\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 5.00000i	− 0.962250i
$28$	0	0
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	0	0
$33$	− 2.00000i	− 0.348155i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−8.00000	−1.24939	−0.624695	−	0.780869i	$-0.714777\pi$
$41$	−8.00000	−1.24939	−0.624695	+	0.780869i	$0.714777\pi$
$42$	0	0
$43$	8.00000i	1.21999i	0.792406	+	0.609994i	$0.208828\pi$
$43$	8.00000i	1.21999i	−0.792406	+	0.609994i	$0.791172\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 8.00000i	− 1.16692i	−0.812142	−	0.583460i	$-0.801699\pi$
$47$	− 8.00000i	− 1.16692i	0.812142	−	0.583460i	$-0.198301\pi$
$48$	0	0
$49$	−2.00000	−0.285714
$50$	0	0
$51$	−5.00000	−0.700140
$52$	0	0
$53$	− 9.00000i	− 1.23625i	−0.786082	−	0.618123i	$-0.787894\pi$
$53$	− 9.00000i	− 1.23625i	0.786082	−	0.618123i	$-0.212106\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.00000i	0.132453i
$58$	0	0
$59$	−1.00000	−0.130189	−0.0650945	−	0.997879i	$-0.520735\pi$
$59$	−1.00000	−0.130189	−0.0650945	+	0.997879i	$0.520735\pi$
$60$	0	0
$61$	14.0000	1.79252	0.896258	−	0.443533i	$-0.146275\pi$
$61$	14.0000	1.79252	0.896258	+	0.443533i	$0.146275\pi$
$62$	0	0
$63$	6.00000i	0.755929i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	13.0000i	1.58820i	0.607785	+	0.794101i	$0.292058\pi$
$67$	13.0000i	1.58820i	−0.607785	+	0.794101i	$0.707942\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	10.0000	1.18678	0.593391	−	0.804914i	$-0.297789\pi$
$71$	10.0000	1.18678	0.593391	+	0.804914i	$0.297789\pi$
$72$	0	0
$73$	− 9.00000i	− 1.05337i	−0.850060	−	0.526685i	$-0.823435\pi$
$73$	− 9.00000i	− 1.05337i	0.850060	−	0.526685i	$-0.176565\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	6.00000i	0.683763i
$78$	0	0
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 10.0000i	− 1.09764i	−0.835940	−	0.548821i	$-0.815077\pi$
$83$	− 10.0000i	− 1.09764i	0.835940	−	0.548821i	$-0.184923\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 3.00000i	− 0.321634i
$88$	0	0
$89$	12.0000	1.27200	0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000	1.27200	0.635999	+	0.771690i	$0.280588\pi$
$90$	0	0
$91$	3.00000	0.314485
$92$	0	0
$93$	− 4.00000i	− 0.414781i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	14.0000i	1.42148i	0.703452	+	0.710742i	$0.251641\pi$
$97$	14.0000i	1.42148i	−0.703452	+	0.710742i	$0.748359\pi$
$98$	0	0
$99$	4.00000	0.402015
$100$	0	0
$101$	−14.0000	−1.39305	−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$102$	0	0
$103$	− 6.00000i	− 0.591198i	−0.955312	−	0.295599i	$-0.904481\pi$
$103$	− 6.00000i	− 0.591198i	0.955312	−	0.295599i	$-0.0955191\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	15.0000i	1.45010i	0.688694	+	0.725052i	$0.258184\pi$
$107$	15.0000i	1.45010i	−0.688694	+	0.725052i	$0.741816\pi$
$108$	0	0
$109$	−7.00000	−0.670478	−0.335239	−	0.942133i	$-0.608817\pi$
$109$	−7.00000	−0.670478	−0.335239	+	0.942133i	$0.608817\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	0	0
$113$	18.0000i	1.69330i	0.532152	+	0.846649i	$0.321383\pi$
$113$	18.0000i	1.69330i	−0.532152	+	0.846649i	$0.678617\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 2.00000i	− 0.184900i
$118$	0	0
$119$	15.0000	1.37505
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	8.00000i	0.721336i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 6.00000i	− 0.532414i	−0.963916	−	0.266207i	$-0.914230\pi$
$127$	− 6.00000i	− 0.532414i	0.963916	−	0.266207i	$-0.0857705\pi$
$128$	0	0
$129$	8.00000	0.704361
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	− 3.00000i	− 0.260133i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	7.00000i	0.598050i	0.954245	+	0.299025i	$0.0966615\pi$
$137$	7.00000i	0.598050i	−0.954245	+	0.299025i	$0.903339\pi$
$138$	0	0
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	0	0
$143$	− 2.00000i	− 0.167248i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	2.00000i	0.164957i
$148$	0	0
$149$	8.00000	0.655386	0.327693	−	0.944784i	$-0.393729\pi$
$149$	8.00000	0.655386	0.327693	+	0.944784i	$0.393729\pi$
$150$	0	0
$151$	22.0000	1.79033	0.895167	−	0.445730i	$-0.147056\pi$
$151$	22.0000	1.79033	0.895167	+	0.445730i	$0.147056\pi$
$152$	0	0
$153$	− 10.0000i	− 0.808452i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 22.0000i	− 1.75579i	−0.478852	−	0.877896i	$-0.658947\pi$
$157$	− 22.0000i	− 1.75579i	0.478852	−	0.877896i	$-0.341053\pi$
$158$	0	0
$159$	−9.00000	−0.713746
$160$	0	0
$161$	−3.00000	−0.236433
$162$	0	0
$163$	4.00000i	0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$	4.00000i	0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	4.00000i	0.309529i	0.987951	+	0.154765i	$0.0494619\pi$
$167$	4.00000i	0.309529i	−0.987951	+	0.154765i	$0.950538\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	−2.00000	−0.152944
$172$	0	0
$173$	− 22.0000i	− 1.67263i	−0.548250	−	0.836315i	$-0.684706\pi$
$173$	− 22.0000i	− 1.67263i	0.548250	−	0.836315i	$-0.315294\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1.00000i	0.0751646i
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	0	0
$183$	− 14.0000i	− 1.03491i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 10.0000i	− 0.731272i
$188$	0	0
$189$	15.0000	1.09109
$190$	0	0
$191$	23.0000	1.66422	0.832111	−	0.554609i	$-0.187132\pi$
$191$	23.0000	1.66422	0.832111	+	0.554609i	$0.187132\pi$
$192$	0	0
$193$	− 6.00000i	− 0.431889i	−0.976406	−	0.215945i	$-0.930717\pi$
$193$	− 6.00000i	− 0.431889i	0.976406	−	0.215945i	$-0.0692831\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 8.00000i	− 0.569976i	−0.958531	−	0.284988i	$-0.908010\pi$
$197$	− 8.00000i	− 0.569976i	0.958531	−	0.284988i	$-0.0919897\pi$
$198$	0	0
$199$	9.00000	0.637993	0.318997	−	0.947756i	$-0.396654\pi$
$199$	9.00000	0.637993	0.318997	+	0.947756i	$0.396654\pi$
$200$	0	0
$201$	13.0000	0.916949
$202$	0	0
$203$	9.00000i	0.631676i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	2.00000i	0.139010i
$208$	0	0
$209$	−2.00000	−0.138343
$210$	0	0
$211$	21.0000	1.44570	0.722850	−	0.691005i	$-0.242832\pi$
$211$	21.0000	1.44570	0.722850	+	0.691005i	$0.242832\pi$
$212$	0	0
$213$	− 10.0000i	− 0.685189i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	12.0000i	0.814613i
$218$	0	0
$219$	−9.00000	−0.608164
$220$	0	0
$221$	−5.00000	−0.336336
$222$	0	0
$223$	14.0000i	0.937509i	0.883328	+	0.468755i	$0.155297\pi$
$223$	14.0000i	0.937509i	−0.883328	+	0.468755i	$0.844703\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	9.00000i	0.597351i	0.954355	+	0.298675i	$0.0965448\pi$
$227$	9.00000i	0.597351i	−0.954355	+	0.298675i	$0.903455\pi$
$228$	0	0
$229$	−2.00000	−0.132164	−0.0660819	−	0.997814i	$-0.521050\pi$
$229$	−2.00000	−0.132164	−0.0660819	+	0.997814i	$0.521050\pi$
$230$	0	0
$231$	6.00000	0.394771
$232$	0	0
$233$	− 26.0000i	− 1.70332i	−0.524097	−	0.851658i	$-0.675597\pi$
$233$	− 26.0000i	− 1.70332i	0.524097	−	0.851658i	$-0.324403\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 10.0000i	− 0.649570i
$238$	0	0
$239$	9.00000	0.582162	0.291081	−	0.956698i	$-0.405985\pi$
$239$	9.00000	0.582162	0.291081	+	0.956698i	$0.405985\pi$
$240$	0	0
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	0	0
$243$	− 16.0000i	− 1.02640i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.00000i	0.0636285i
$248$	0	0
$249$	−10.0000	−0.633724
$250$	0	0
$251$	−18.0000	−1.13615	−0.568075	−	0.822977i	$-0.692312\pi$
$251$	−18.0000	−1.13615	−0.568075	+	0.822977i	$0.692312\pi$
$252$	0	0
$253$	2.00000i	0.125739i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 4.00000i	− 0.249513i	−0.992187	−	0.124757i	$-0.960185\pi$
$257$	− 4.00000i	− 0.249513i	0.992187	−	0.124757i	$-0.0398150\pi$
$258$	0	0
$259$	−6.00000	−0.372822
$260$	0	0
$261$	6.00000	0.371391
$262$	0	0
$263$	− 32.0000i	− 1.97320i	−0.163144	−	0.986602i	$-0.552164\pi$
$263$	− 32.0000i	− 1.97320i	0.163144	−	0.986602i	$-0.447836\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 12.0000i	− 0.734388i
$268$	0	0
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	0	0
$271$	−25.0000	−1.51864	−0.759321	−	0.650716i	$-0.774469\pi$
$271$	−25.0000	−1.51864	−0.759321	+	0.650716i	$0.774469\pi$
$272$	0	0
$273$	− 3.00000i	− 0.181568i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	8.00000	0.478947
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	6.00000i	0.356663i	0.983970	+	0.178331i	$0.0570699\pi$
$283$	6.00000i	0.356663i	−0.983970	+	0.178331i	$0.942930\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 24.0000i	− 1.41668i
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	14.0000	0.820695
$292$	0	0
$293$	− 7.00000i	− 0.408944i	−0.978872	−	0.204472i	$-0.934452\pi$
$293$	− 7.00000i	− 0.408944i	0.978872	−	0.204472i	$-0.0655478\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 10.0000i	− 0.580259i
$298$	0	0
$299$	1.00000	0.0578315
$300$	0	0
$301$	−24.0000	−1.38334
$302$	0	0
$303$	14.0000i	0.804279i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 12.0000i	− 0.684876i	−0.939540	−	0.342438i	$-0.888747\pi$
$307$	− 12.0000i	− 0.684876i	0.939540	−	0.342438i	$-0.111253\pi$
$308$	0	0
$309$	−6.00000	−0.341328
$310$	0	0
$311$	−9.00000	−0.510343	−0.255172	−	0.966896i	$-0.582132\pi$
$311$	−9.00000	−0.510343	−0.255172	+	0.966896i	$0.582132\pi$
$312$	0	0
$313$	− 13.0000i	− 0.734803i	−0.930062	−	0.367402i	$-0.880247\pi$
$313$	− 13.0000i	− 0.734803i	0.930062	−	0.367402i	$-0.119753\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3.00000i	0.168497i	0.996445	+	0.0842484i	$0.0268489\pi$
$317$	3.00000i	0.168497i	−0.996445	+	0.0842484i	$0.973151\pi$
$318$	0	0
$319$	6.00000	0.335936
$320$	0	0
$321$	15.0000	0.837218
$322$	0	0
$323$	5.00000i	0.278207i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	7.00000i	0.387101i
$328$	0	0
$329$	24.0000	1.32316
$330$	0	0
$331$	−25.0000	−1.37412	−0.687062	−	0.726599i	$-0.741100\pi$
$331$	−25.0000	−1.37412	−0.687062	+	0.726599i	$0.741100\pi$
$332$	0	0
$333$	4.00000i	0.219199i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	12.0000i	0.653682i	0.945079	+	0.326841i	$0.105984\pi$
$337$	12.0000i	0.653682i	−0.945079	+	0.326841i	$0.894016\pi$
$338$	0	0
$339$	18.0000	0.977626
$340$	0	0
$341$	8.00000	0.433224
$342$	0	0
$343$	15.0000i	0.809924i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.00000i	0.107366i	0.998558	+	0.0536828i	$0.0170960\pi$
$347$	2.00000i	0.107366i	−0.998558	+	0.0536828i	$0.982904\pi$
$348$	0	0
$349$	26.0000	1.39175	0.695874	−	0.718164i	$-0.255017\pi$
$349$	26.0000	1.39175	0.695874	+	0.718164i	$0.255017\pi$
$350$	0	0
$351$	−5.00000	−0.266880
$352$	0	0
$353$	7.00000i	0.372572i	0.982496	+	0.186286i	$0.0596452\pi$
$353$	7.00000i	0.372572i	−0.982496	+	0.186286i	$0.940355\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 15.0000i	− 0.793884i
$358$	0	0
$359$	−17.0000	−0.897226	−0.448613	−	0.893726i	$-0.648082\pi$
$359$	−17.0000	−0.897226	−0.448613	+	0.893726i	$0.648082\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	7.00000i	0.367405i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	28.0000i	1.46159i	0.682598	+	0.730794i	$0.260850\pi$
$367$	28.0000i	1.46159i	−0.682598	+	0.730794i	$0.739150\pi$
$368$	0	0
$369$	−16.0000	−0.832927
$370$	0	0
$371$	27.0000	1.40177
$372$	0	0
$373$	13.0000i	0.673114i	0.941663	+	0.336557i	$0.109263\pi$
$373$	13.0000i	0.673114i	−0.941663	+	0.336557i	$0.890737\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 3.00000i	− 0.154508i
$378$	0	0
$379$	−9.00000	−0.462299	−0.231149	−	0.972918i	$-0.574249\pi$
$379$	−9.00000	−0.462299	−0.231149	+	0.972918i	$0.574249\pi$
$380$	0	0
$381$	−6.00000	−0.307389
$382$	0	0
$383$	6.00000i	0.306586i	0.988181	+	0.153293i	$0.0489878\pi$
$383$	6.00000i	0.306586i	−0.988181	+	0.153293i	$0.951012\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	16.0000i	0.813326i
$388$	0	0
$389$	−34.0000	−1.72387	−0.861934	−	0.507020i	$-0.830747\pi$
$389$	−34.0000	−1.72387	−0.861934	+	0.507020i	$0.830747\pi$
$390$	0	0
$391$	5.00000	0.252861
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 4.00000i	− 0.200754i	−0.994949	−	0.100377i	$-0.967995\pi$
$397$	− 4.00000i	− 0.200754i	0.994949	−	0.100377i	$-0.0320049\pi$
$398$	0	0
$399$	−3.00000	−0.150188
$400$	0	0
$401$	−16.0000	−0.799002	−0.399501	−	0.916733i	$-0.630817\pi$
$401$	−16.0000	−0.799002	−0.399501	+	0.916733i	$0.630817\pi$
$402$	0	0
$403$	− 4.00000i	− 0.199254i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4.00000i	0.198273i
$408$	0	0
$409$	8.00000	0.395575	0.197787	−	0.980245i	$-0.436624\pi$
$409$	8.00000	0.395575	0.197787	+	0.980245i	$0.436624\pi$
$410$	0	0
$411$	7.00000	0.345285
$412$	0	0
$413$	− 3.00000i	− 0.147620i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 12.0000i	− 0.587643i
$418$	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
$421$	−35.0000	−1.70580	−0.852898	−	0.522078i	$-0.825157\pi$
$421$	−35.0000	−1.70580	−0.852898	+	0.522078i	$0.825157\pi$
$422$	0	0
$423$	− 16.0000i	− 0.777947i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	42.0000i	2.03252i
$428$	0	0
$429$	−2.00000	−0.0965609
$430$	0	0
$431$	30.0000	1.44505	0.722525	−	0.691345i	$-0.242982\pi$
$431$	30.0000	1.44505	0.722525	+	0.691345i	$0.242982\pi$
$432$	0	0
$433$	− 2.00000i	− 0.0961139i	−0.998845	−	0.0480569i	$-0.984697\pi$
$433$	− 2.00000i	− 0.0961139i	0.998845	−	0.0480569i	$-0.0153029\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 1.00000i	− 0.0478365i
$438$	0	0
$439$	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
$441$	−4.00000	−0.190476
$442$	0	0
$443$	− 6.00000i	− 0.285069i	−0.989790	−	0.142534i	$-0.954475\pi$
$443$	− 6.00000i	− 0.285069i	0.989790	−	0.142534i	$-0.0455251\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 8.00000i	− 0.378387i
$448$	0	0
$449$	26.0000	1.22702	0.613508	−	0.789689i	$-0.289758\pi$
$449$	26.0000	1.22702	0.613508	+	0.789689i	$0.289758\pi$
$450$	0	0
$451$	−16.0000	−0.753411
$452$	0	0
$453$	− 22.0000i	− 1.03365i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1.00000i	0.0467780i	0.999726	+	0.0233890i	$0.00744563\pi$
$457$	1.00000i	0.0467780i	−0.999726	+	0.0233890i	$0.992554\pi$
$458$	0	0
$459$	−25.0000	−1.16690
$460$	0	0
$461$	20.0000	0.931493	0.465746	−	0.884918i	$-0.345786\pi$
$461$	20.0000	0.931493	0.465746	+	0.884918i	$0.345786\pi$
$462$	0	0
$463$	− 4.00000i	− 0.185896i	−0.995671	−	0.0929479i	$-0.970371\pi$
$463$	− 4.00000i	− 0.185896i	0.995671	−	0.0929479i	$-0.0296290\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 6.00000i	− 0.277647i	−0.990317	−	0.138823i	$-0.955668\pi$
$467$	− 6.00000i	− 0.277647i	0.990317	−	0.138823i	$-0.0443321\pi$
$468$	0	0
$469$	−39.0000	−1.80085
$470$	0	0
$471$	−22.0000	−1.01371
$472$	0	0
$473$	16.0000i	0.735681i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 18.0000i	− 0.824163i
$478$	0	0
$479$	−12.0000	−0.548294	−0.274147	−	0.961688i	$-0.588395\pi$
$479$	−12.0000	−0.548294	−0.274147	+	0.961688i	$0.588395\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	0	0
$483$	3.00000i	0.136505i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	26.0000i	1.17817i	0.808070	+	0.589086i	$0.200512\pi$
$487$	26.0000i	1.17817i	−0.808070	+	0.589086i	$0.799488\pi$
$488$	0	0
$489$	4.00000	0.180886
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	0	0
$493$	− 15.0000i	− 0.675566i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	30.0000i	1.34568i
$498$	0	0
$499$	−28.0000	−1.25345	−0.626726	−	0.779240i	$-0.715605\pi$
$499$	−28.0000	−1.25345	−0.626726	+	0.779240i	$0.715605\pi$
$500$	0	0
$501$	4.00000	0.178707
$502$	0	0
$503$	33.0000i	1.47140i	0.677309	+	0.735699i	$0.263146\pi$
$503$	33.0000i	1.47140i	−0.677309	+	0.735699i	$0.736854\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 12.0000i	− 0.532939i
$508$	0	0
$509$	2.00000	0.0886484	0.0443242	−	0.999017i	$-0.485887\pi$
$509$	2.00000	0.0886484	0.0443242	+	0.999017i	$0.485887\pi$
$510$	0	0
$511$	27.0000	1.19441
$512$	0	0
$513$	5.00000i	0.220755i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 16.0000i	− 0.703679i
$518$	0	0
$519$	−22.0000	−0.965693
$520$	0	0
$521$	−12.0000	−0.525730	−0.262865	−	0.964833i	$-0.584667\pi$
$521$	−12.0000	−0.525730	−0.262865	+	0.964833i	$0.584667\pi$
$522$	0	0
$523$	5.00000i	0.218635i	0.994007	+	0.109317i	$0.0348665\pi$
$523$	5.00000i	0.218635i	−0.994007	+	0.109317i	$0.965134\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 20.0000i	− 0.871214i
$528$	0	0
$529$	22.0000	0.956522
$530$	0	0
$531$	−2.00000	−0.0867926
$532$	0	0
$533$	8.00000i	0.346518i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−4.00000	−0.172292
$540$	0	0
$541$	26.0000	1.11783	0.558914	−	0.829226i	$-0.311218\pi$
$541$	26.0000	1.11783	0.558914	+	0.829226i	$0.311218\pi$
$542$	0	0
$543$	14.0000i	0.600798i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	28.0000i	1.19719i	0.801050	+	0.598597i	$0.204275\pi$
$547$	28.0000i	1.19719i	−0.801050	+	0.598597i	$0.795725\pi$
$548$	0	0
$549$	28.0000	1.19501
$550$	0	0
$551$	−3.00000	−0.127804
$552$	0	0
$553$	30.0000i	1.27573i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	24.0000i	1.01691i	0.861088	+	0.508456i	$0.169784\pi$
$557$	24.0000i	1.01691i	−0.861088	+	0.508456i	$0.830216\pi$
$558$	0	0
$559$	8.00000	0.338364
$560$	0	0
$561$	−10.0000	−0.422200
$562$	0	0
$563$	44.0000i	1.85438i	0.374593	+	0.927189i	$0.377783\pi$
$563$	44.0000i	1.85438i	−0.374593	+	0.927189i	$0.622217\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	3.00000i	0.125988i
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	0	0
$573$	− 23.0000i	− 0.960839i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 29.0000i	− 1.20729i	−0.797255	−	0.603643i	$-0.793715\pi$
$577$	− 29.0000i	− 1.20729i	0.797255	−	0.603643i	$-0.206285\pi$
$578$	0	0
$579$	−6.00000	−0.249351
$580$	0	0
$581$	30.0000	1.24461
$582$	0	0
$583$	− 18.0000i	− 0.745484i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 44.0000i	− 1.81607i	−0.418890	−	0.908037i	$-0.637581\pi$
$587$	− 44.0000i	− 1.81607i	0.418890	−	0.908037i	$-0.362419\pi$
$588$	0	0
$589$	−4.00000	−0.164817
$590$	0	0
$591$	−8.00000	−0.329076
$592$	0	0
$593$	− 2.00000i	− 0.0821302i	−0.999156	−	0.0410651i	$-0.986925\pi$
$593$	− 2.00000i	− 0.0821302i	0.999156	−	0.0410651i	$-0.0130751\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 9.00000i	− 0.368345i
$598$	0	0
$599$	40.0000	1.63436	0.817178	−	0.576386i	$-0.195537\pi$
$599$	40.0000	1.63436	0.817178	+	0.576386i	$0.195537\pi$
$600$	0	0
$601$	−4.00000	−0.163163	−0.0815817	−	0.996667i	$-0.525997\pi$
$601$	−4.00000	−0.163163	−0.0815817	+	0.996667i	$0.525997\pi$
$602$	0	0
$603$	26.0000i	1.05880i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	42.0000i	1.70473i	0.522949	+	0.852364i	$0.324832\pi$
$607$	42.0000i	1.70473i	−0.522949	+	0.852364i	$0.675168\pi$
$608$	0	0
$609$	9.00000	0.364698
$610$	0	0
$611$	−8.00000	−0.323645
$612$	0	0
$613$	− 26.0000i	− 1.05013i	−0.851062	−	0.525065i	$-0.824041\pi$
$613$	− 26.0000i	− 1.05013i	0.851062	−	0.525065i	$-0.175959\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 6.00000i	− 0.241551i	−0.992680	−	0.120775i	$-0.961462\pi$
$617$	− 6.00000i	− 0.241551i	0.992680	−	0.120775i	$-0.0385381\pi$
$618$	0	0
$619$	−22.0000	−0.884255	−0.442127	−	0.896952i	$-0.645776\pi$
$619$	−22.0000	−0.884255	−0.442127	+	0.896952i	$0.645776\pi$
$620$	0	0
$621$	5.00000	0.200643
$622$	0	0
$623$	36.0000i	1.44231i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	2.00000i	0.0798723i
$628$	0	0
$629$	10.0000	0.398726
$630$	0	0
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	0	0
$633$	− 21.0000i	− 0.834675i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	2.00000i	0.0792429i
$638$	0	0
$639$	20.0000	0.791188
$640$	0	0
$641$	−42.0000	−1.65890	−0.829450	−	0.558581i	$-0.811346\pi$
$641$	−42.0000	−1.65890	−0.829450	+	0.558581i	$0.811346\pi$
$642$	0	0
$643$	46.0000i	1.81406i	0.421063	+	0.907031i	$0.361657\pi$
$643$	46.0000i	1.81406i	−0.421063	+	0.907031i	$0.638343\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 9.00000i	− 0.353827i	−0.984226	−	0.176913i	$-0.943389\pi$
$647$	− 9.00000i	− 0.353827i	0.984226	−	0.176913i	$-0.0566112\pi$
$648$	0	0
$649$	−2.00000	−0.0785069
$650$	0	0
$651$	12.0000	0.470317
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 18.0000i	− 0.702247i
$658$	0	0
$659$	−19.0000	−0.740135	−0.370067	−	0.929005i	$-0.620665\pi$
$659$	−19.0000	−0.740135	−0.370067	+	0.929005i	$0.620665\pi$
$660$	0	0
$661$	−17.0000	−0.661223	−0.330612	−	0.943767i	$-0.607255\pi$
$661$	−17.0000	−0.661223	−0.330612	+	0.943767i	$0.607255\pi$
$662$	0	0
$663$	5.00000i	0.194184i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	3.00000i	0.116160i
$668$	0	0
$669$	14.0000	0.541271
$670$	0	0
$671$	28.0000	1.08093
$672$	0	0
$673$	12.0000i	0.462566i	0.972887	+	0.231283i	$0.0742923\pi$
$673$	12.0000i	0.462566i	−0.972887	+	0.231283i	$0.925708\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	27.0000i	1.03769i	0.854867	+	0.518847i	$0.173639\pi$
$677$	27.0000i	1.03769i	−0.854867	+	0.518847i	$0.826361\pi$
$678$	0	0
$679$	−42.0000	−1.61181
$680$	0	0
$681$	9.00000	0.344881
$682$	0	0
$683$	12.0000i	0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$683$	12.0000i	0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	2.00000i	0.0763048i
$688$	0	0
$689$	−9.00000	−0.342873
$690$	0	0
$691$	−42.0000	−1.59776	−0.798878	−	0.601494i	$-0.794573\pi$
$691$	−42.0000	−1.59776	−0.798878	+	0.601494i	$0.794573\pi$
$692$	0	0
$693$	12.0000i	0.455842i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	40.0000i	1.51511i
$698$	0	0
$699$	−26.0000	−0.983410
$700$	0	0
$701$	44.0000	1.66186	0.830929	−	0.556379i	$-0.187810\pi$
$701$	44.0000	1.66186	0.830929	+	0.556379i	$0.187810\pi$
$702$	0	0
$703$	− 2.00000i	− 0.0754314i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 42.0000i	− 1.57957i
$708$	0	0
$709$	−46.0000	−1.72757	−0.863783	−	0.503864i	$-0.831911\pi$
$709$	−46.0000	−1.72757	−0.863783	+	0.503864i	$0.831911\pi$
$710$	0	0
$711$	20.0000	0.750059
$712$	0	0
$713$	4.00000i	0.149801i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 9.00000i	− 0.336111i
$718$	0	0
$719$	−27.0000	−1.00693	−0.503465	−	0.864016i	$-0.667942\pi$
$719$	−27.0000	−1.00693	−0.503465	+	0.864016i	$0.667942\pi$
$720$	0	0
$721$	18.0000	0.670355
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	7.00000i	0.259616i	0.991539	+	0.129808i	$0.0414360\pi$
$727$	7.00000i	0.259616i	−0.991539	+	0.129808i	$0.958564\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	40.0000	1.47945
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	26.0000i	0.957722i
$738$	0	0
$739$	−16.0000	−0.588570	−0.294285	−	0.955718i	$-0.595081\pi$
$739$	−16.0000	−0.588570	−0.294285	+	0.955718i	$0.595081\pi$
$740$	0	0
$741$	1.00000	0.0367359
$742$	0	0
$743$	28.0000i	1.02722i	0.858024	+	0.513610i	$0.171692\pi$
$743$	28.0000i	1.02722i	−0.858024	+	0.513610i	$0.828308\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 20.0000i	− 0.731762i
$748$	0	0
$749$	−45.0000	−1.64426
$750$	0	0
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	0	0
$753$	18.0000i	0.655956i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 22.0000i	− 0.799604i	−0.916602	−	0.399802i	$-0.869079\pi$
$757$	− 22.0000i	− 0.799604i	0.916602	−	0.399802i	$-0.130921\pi$
$758$	0	0
$759$	2.00000	0.0725954
$760$	0	0
$761$	−37.0000	−1.34125	−0.670624	−	0.741797i	$-0.733974\pi$
$761$	−37.0000	−1.34125	−0.670624	+	0.741797i	$0.733974\pi$
$762$	0	0
$763$	− 21.0000i	− 0.760251i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.00000i	0.0361079i
$768$	0	0
$769$	−5.00000	−0.180305	−0.0901523	−	0.995928i	$-0.528735\pi$
$769$	−5.00000	−0.180305	−0.0901523	+	0.995928i	$0.528735\pi$
$770$	0	0
$771$	−4.00000	−0.144056
$772$	0	0
$773$	49.0000i	1.76241i	0.472737	+	0.881204i	$0.343266\pi$
$773$	49.0000i	1.76241i	−0.472737	+	0.881204i	$0.656734\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	6.00000i	0.215249i
$778$	0	0
$779$	8.00000	0.286630
$780$	0	0
$781$	20.0000	0.715656
$782$	0	0
$783$	− 15.0000i	− 0.536056i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 7.00000i	− 0.249523i	−0.992187	−	0.124762i	$-0.960183\pi$
$787$	− 7.00000i	− 0.249523i	0.992187	−	0.124762i	$-0.0398166\pi$
$788$	0	0
$789$	−32.0000	−1.13923
$790$	0	0
$791$	−54.0000	−1.92002
$792$	0	0
$793$	− 14.0000i	− 0.497155i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 3.00000i	− 0.106265i	−0.998587	−	0.0531327i	$-0.983079\pi$
$797$	− 3.00000i	− 0.106265i	0.998587	−	0.0531327i	$-0.0169206\pi$
$798$	0	0
$799$	−40.0000	−1.41510
$800$	0	0
$801$	24.0000	0.847998
$802$	0	0
$803$	− 18.0000i	− 0.635206i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	10.0000i	0.352017i
$808$	0	0
$809$	15.0000	0.527372	0.263686	−	0.964609i	$-0.415062\pi$
$809$	15.0000	0.527372	0.263686	+	0.964609i	$0.415062\pi$
$810$	0	0
$811$	27.0000	0.948098	0.474049	−	0.880498i	$-0.342792\pi$
$811$	27.0000	0.948098	0.474049	+	0.880498i	$0.342792\pi$
$812$	0	0
$813$	25.0000i	0.876788i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 8.00000i	− 0.279885i
$818$	0	0
$819$	6.00000	0.209657
$820$	0	0
$821$	8.00000	0.279202	0.139601	−	0.990208i	$-0.455418\pi$
$821$	8.00000	0.279202	0.139601	+	0.990208i	$0.455418\pi$
$822$	0	0
$823$	11.0000i	0.383436i	0.981450	+	0.191718i	$0.0614059\pi$
$823$	11.0000i	0.383436i	−0.981450	+	0.191718i	$0.938594\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	33.0000i	1.14752i	0.819023	+	0.573761i	$0.194516\pi$
$827$	33.0000i	1.14752i	−0.819023	+	0.573761i	$0.805484\pi$
$828$	0	0
$829$	−39.0000	−1.35453	−0.677263	−	0.735741i	$-0.736834\pi$
$829$	−39.0000	−1.35453	−0.677263	+	0.735741i	$0.736834\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	10.0000i	0.346479i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 20.0000i	− 0.691301i
$838$	0	0
$839$	−16.0000	−0.552381	−0.276191	−	0.961103i	$-0.589072\pi$
$839$	−16.0000	−0.552381	−0.276191	+	0.961103i	$0.589072\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 21.0000i	− 0.721569i
$848$	0	0
$849$	6.00000	0.205919
$850$	0	0
$851$	−2.00000	−0.0685591
$852$	0	0
$853$	6.00000i	0.205436i	0.994711	+	0.102718i	$0.0327539\pi$
$853$	6.00000i	0.205436i	−0.994711	+	0.102718i	$0.967246\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 44.0000i	− 1.50301i	−0.659727	−	0.751506i	$-0.729328\pi$
$857$	− 44.0000i	− 1.50301i	0.659727	−	0.751506i	$-0.270672\pi$
$858$	0	0
$859$	−14.0000	−0.477674	−0.238837	−	0.971060i	$-0.576766\pi$
$859$	−14.0000	−0.477674	−0.238837	+	0.971060i	$0.576766\pi$
$860$	0	0
$861$	−24.0000	−0.817918
$862$	0	0
$863$	− 54.0000i	− 1.83818i	−0.394046	−	0.919091i	$-0.628925\pi$
$863$	− 54.0000i	− 1.83818i	0.394046	−	0.919091i	$-0.371075\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	8.00000i	0.271694i
$868$	0	0
$869$	20.0000	0.678454
$870$	0	0
$871$	13.0000	0.440488
$872$	0	0
$873$	28.0000i	0.947656i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 37.0000i	− 1.24940i	−0.780864	−	0.624701i	$-0.785221\pi$
$877$	− 37.0000i	− 1.24940i	0.780864	−	0.624701i	$-0.214779\pi$
$878$	0	0
$879$	−7.00000	−0.236104
$880$	0	0
$881$	14.0000	0.471672	0.235836	−	0.971793i	$-0.424217\pi$
$881$	14.0000	0.471672	0.235836	+	0.971793i	$0.424217\pi$
$882$	0	0
$883$	− 46.0000i	− 1.54802i	−0.633171	−	0.774012i	$-0.718247\pi$
$883$	− 46.0000i	− 1.54802i	0.633171	−	0.774012i	$-0.281753\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 18.0000i	− 0.604381i	−0.953248	−	0.302190i	$-0.902282\pi$
$887$	− 18.0000i	− 0.604381i	0.953248	−	0.302190i	$-0.0977178\pi$
$888$	0	0
$889$	18.0000	0.603701
$890$	0	0
$891$	2.00000	0.0670025
$892$	0	0
$893$	8.00000i	0.267710i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 1.00000i	− 0.0333890i
$898$	0	0
$899$	12.0000	0.400222
$900$	0	0
$901$	−45.0000	−1.49917
$902$	0	0
$903$	24.0000i	0.798670i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 5.00000i	− 0.166022i	−0.996549	−	0.0830111i	$-0.973546\pi$
$907$	− 5.00000i	− 0.166022i	0.996549	−	0.0830111i	$-0.0264537\pi$
$908$	0	0
$909$	−28.0000	−0.928701
$910$	0	0
$911$	−32.0000	−1.06021	−0.530104	−	0.847933i	$-0.677847\pi$
$911$	−32.0000	−1.06021	−0.530104	+	0.847933i	$0.677847\pi$
$912$	0	0
$913$	− 20.0000i	− 0.661903i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	43.0000	1.41844	0.709220	−	0.704988i	$-0.249047\pi$
$919$	43.0000	1.41844	0.709220	+	0.704988i	$0.249047\pi$
$920$	0	0
$921$	−12.0000	−0.395413
$922$	0	0
$923$	− 10.0000i	− 0.329154i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 12.0000i	− 0.394132i
$928$	0	0
$929$	−1.00000	−0.0328089	−0.0164045	−	0.999865i	$-0.505222\pi$
$929$	−1.00000	−0.0328089	−0.0164045	+	0.999865i	$0.505222\pi$
$930$	0	0
$931$	2.00000	0.0655474
$932$	0	0
$933$	9.00000i	0.294647i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	49.0000i	1.60076i	0.599493	+	0.800380i	$0.295369\pi$
$937$	49.0000i	1.60076i	−0.599493	+	0.800380i	$0.704631\pi$
$938$	0	0
$939$	−13.0000	−0.424239
$940$	0	0
$941$	−15.0000	−0.488986	−0.244493	−	0.969651i	$-0.578622\pi$
$941$	−15.0000	−0.488986	−0.244493	+	0.969651i	$0.578622\pi$
$942$	0	0
$943$	− 8.00000i	− 0.260516i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 32.0000i	− 1.03986i	−0.854209	−	0.519930i	$-0.825958\pi$
$947$	− 32.0000i	− 1.03986i	0.854209	−	0.519930i	$-0.174042\pi$
$948$	0	0
$949$	−9.00000	−0.292152
$950$	0	0
$951$	3.00000	0.0972817
$952$	0	0
$953$	18.0000i	0.583077i	0.956559	+	0.291539i	$0.0941672\pi$
$953$	18.0000i	0.583077i	−0.956559	+	0.291539i	$0.905833\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 6.00000i	− 0.193952i
$958$	0	0
$959$	−21.0000	−0.678125
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	30.0000i	0.966736i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	40.0000i	1.28631i	0.765735	+	0.643157i	$0.222376\pi$
$967$	40.0000i	1.28631i	−0.765735	+	0.643157i	$0.777624\pi$
$968$	0	0
$969$	5.00000	0.160623
$970$	0	0
$971$	60.0000	1.92549	0.962746	−	0.270408i	$-0.0871586\pi$
$971$	60.0000	1.92549	0.962746	+	0.270408i	$0.0871586\pi$
$972$	0	0
$973$	36.0000i	1.15411i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	12.0000i	0.383914i	0.981403	+	0.191957i	$0.0614834\pi$
$977$	12.0000i	0.383914i	−0.981403	+	0.191957i	$0.938517\pi$
$978$	0	0
$979$	24.0000	0.767043
$980$	0	0
$981$	−14.0000	−0.446986
$982$	0	0
$983$	− 10.0000i	− 0.318950i	−0.987202	−	0.159475i	$-0.949020\pi$
$983$	− 10.0000i	− 0.318950i	0.987202	−	0.159475i	$-0.0509802\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 24.0000i	− 0.763928i
$988$	0	0
$989$	−8.00000	−0.254385
$990$	0	0
$991$	−28.0000	−0.889449	−0.444725	−	0.895667i	$-0.646698\pi$
$991$	−28.0000	−0.889449	−0.444725	+	0.895667i	$0.646698\pi$
$992$	0	0
$993$	25.0000i	0.793351i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 16.0000i	− 0.506725i	−0.967371	−	0.253363i	$-0.918463\pi$
$997$	− 16.0000i	− 0.506725i	0.967371	−	0.253363i	$-0.0815366\pi$
$998$	0	0
$999$	10.0000	0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3800.2.d.f.3649.1		2
5.2	odd	4		3800.2.a.d.1.1		1
5.3	odd	4		152.2.a.b.1.1	✓	1
5.4	even	2	inner	3800.2.d.f.3649.2		2
15.8	even	4		1368.2.a.g.1.1		1
20.3	even	4		304.2.a.b.1.1		1
20.7	even	4		7600.2.a.o.1.1		1
35.13	even	4		7448.2.a.g.1.1		1
40.3	even	4		1216.2.a.l.1.1		1
40.13	odd	4		1216.2.a.f.1.1		1
60.23	odd	4		2736.2.a.k.1.1		1
95.18	even	4		2888.2.a.b.1.1		1
380.303	odd	4		5776.2.a.l.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.2.a.b.1.1	✓	1	5.3	odd	4
304.2.a.b.1.1		1	20.3	even	4
1216.2.a.f.1.1		1	40.13	odd	4
1216.2.a.l.1.1		1	40.3	even	4
1368.2.a.g.1.1		1	15.8	even	4
2736.2.a.k.1.1		1	60.23	odd	4
2888.2.a.b.1.1		1	95.18	even	4
3800.2.a.d.1.1		1	5.2	odd	4
3800.2.d.f.3649.1		2	1.1	even	1	trivial
3800.2.d.f.3649.2		2	5.4	even	2	inner
5776.2.a.l.1.1		1	380.303	odd	4
7448.2.a.g.1.1		1	35.13	even	4
7600.2.a.o.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:	\( N \)	\(=\)	\( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3800.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +3.00000i q^{7} +2.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +3.00000i q^{7} +2.00000 q^{9} +2.00000 q^{11} -1.00000i q^{13} -5.00000i q^{17} -1.00000 q^{19} +3.00000 q^{21} +1.00000i q^{23} -5.00000i q^{27} +3.00000 q^{29} +4.00000 q^{31} -2.00000i q^{33} +2.00000i q^{37} -1.00000 q^{39} -8.00000 q^{41} +8.00000i q^{43} -8.00000i q^{47} -2.00000 q^{49} -5.00000 q^{51} -9.00000i q^{53} +1.00000i q^{57} -1.00000 q^{59} +14.0000 q^{61} +6.00000i q^{63} +13.0000i q^{67} +1.00000 q^{69} +10.0000 q^{71} -9.00000i q^{73} +6.00000i q^{77} +10.0000 q^{79} +1.00000 q^{81} -10.0000i q^{83} -3.00000i q^{87} +12.0000 q^{89} +3.00000 q^{91} -4.00000i q^{93} +14.0000i q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{9}+O(q^{10}) \) 2 * q + 4 * q^9	\( 2 q + 4 q^{9} + 4 q^{11} - 2 q^{19} + 6 q^{21} + 6 q^{29} + 8 q^{31} - 2 q^{39} - 16 q^{41} - 4 q^{49} - 10 q^{51} - 2 q^{59} + 28 q^{61} + 2 q^{69} + 20 q^{71} + 20 q^{79} + 2 q^{81} + 24 q^{89} + 6 q^{91} + 8 q^{99}+O(q^{100}) \) 2 * q + 4 * q^9 + 4 * q^11 - 2 * q^19 + 6 * q^21 + 6 * q^29 + 8 * q^31 - 2 * q^39 - 16 * q^41 - 4 * q^49 - 10 * q^51 - 2 * q^59 + 28 * q^61 + 2 * q^69 + 20 * q^71 + 20 * q^79 + 2 * q^81 + 24 * q^89 + 6 * q^91 + 8 * q^99

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