LMFDB - Embedded newform 2800.2.g.d.449.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2800,2,Mod(449,2800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2800, 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("2800.449");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2800 = 2^{4} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2800.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$22.3581125660$
Analytic rank:	$1$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1400)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	2800.449
Dual form			2800.2.g.d.449.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{3} +1.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-2.00000i q^{3} +1.00000i q^{7} -1.00000 q^{9} -5.00000 q^{11} -8.00000i q^{17} -2.00000 q^{19} +2.00000 q^{21} +7.00000i q^{23} -4.00000i q^{27} +3.00000 q^{29} -4.00000 q^{31} +10.0000i q^{33} +1.00000i q^{37} -2.00000 q^{41} -3.00000i q^{43} +6.00000i q^{47} -1.00000 q^{49} -16.0000 q^{51} +10.0000i q^{53} +4.00000i q^{57} -4.00000 q^{59} -6.00000 q^{61} -1.00000i q^{63} +13.0000i q^{67} +14.0000 q^{69} -5.00000 q^{71} +6.00000i q^{73} -5.00000i q^{77} -13.0000 q^{79} -11.0000 q^{81} -16.0000i q^{83} -6.00000i q^{87} +8.00000i q^{93} +12.0000i q^{97} +5.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^9	$ 2 q - 2 q^{9} - 10 q^{11} - 4 q^{19} + 4 q^{21} + 6 q^{29} - 8 q^{31} - 4 q^{41} - 2 q^{49} - 32 q^{51} - 8 q^{59} - 12 q^{61} + 28 q^{69} - 10 q^{71} - 26 q^{79} - 22 q^{81} + 10 q^{99}+O(q^{100}) $ 2 * q - 2 * q^9 - 10 * q^11 - 4 * q^19 + 4 * q^21 + 6 * q^29 - 8 * q^31 - 4 * q^41 - 2 * q^49 - 32 * q^51 - 8 * q^59 - 12 * q^61 + 28 * q^69 - 10 * q^71 - 26 * q^79 - 22 * q^81 + 10 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2800\mathbb{Z}\right)^\times$.

$n$	$351$	$801$	$2101$	$2577$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	− 2.00000i	− 1.15470i	−0.816497	−	0.577350i	$-0.804087\pi$
$3$	− 2.00000i	− 1.15470i	0.816497	−	0.577350i	$-0.195913\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−5.00000	−1.50756	−0.753778	−	0.657129i	$-0.771771\pi$
$11$	−5.00000	−1.50756	−0.753778	+	0.657129i	$0.771771\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 8.00000i	− 1.94029i	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	− 8.00000i	− 1.94029i	0.242536	−	0.970143i	$-0.422021\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	0	0
$23$	7.00000i	1.45960i	0.683660	+	0.729800i	$0.260387\pi$
$23$	7.00000i	1.45960i	−0.683660	+	0.729800i	$0.739613\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 4.00000i	− 0.769800i
$28$	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	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	0	0
$33$	10.0000i	1.74078i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.00000i	0.164399i	0.996616	+	0.0821995i	$0.0261945\pi$
$37$	1.00000i	0.164399i	−0.996616	+	0.0821995i	$0.973806\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	− 3.00000i	− 0.457496i	−0.973486	−	0.228748i	$-0.926537\pi$
$43$	− 3.00000i	− 0.457496i	0.973486	−	0.228748i	$-0.0734631\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.00000i	0.875190i	0.899172	+	0.437595i	$0.144170\pi$
$47$	6.00000i	0.875190i	−0.899172	+	0.437595i	$0.855830\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	−16.0000	−2.24045
$52$	0	0
$53$	10.0000i	1.37361i	0.726844	+	0.686803i	$0.240986\pi$
$53$	10.0000i	1.37361i	−0.726844	+	0.686803i	$0.759014\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	4.00000i	0.529813i
$58$	0	0
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	0	0
$63$	− 1.00000i	− 0.125988i
$64$	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
$69$	14.0000	1.68540
$70$	0	0
$71$	−5.00000	−0.593391	−0.296695	−	0.954972i	$-0.595885\pi$
$71$	−5.00000	−0.593391	−0.296695	+	0.954972i	$0.595885\pi$
$72$	0	0
$73$	6.00000i	0.702247i	0.936329	+	0.351123i	$0.114200\pi$
$73$	6.00000i	0.702247i	−0.936329	+	0.351123i	$0.885800\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 5.00000i	− 0.569803i
$78$	0	0
$79$	−13.0000	−1.46261	−0.731307	−	0.682048i	$-0.761089\pi$
$79$	−13.0000	−1.46261	−0.731307	+	0.682048i	$0.761089\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	− 16.0000i	− 1.75623i	−0.478451	−	0.878114i	$-0.658802\pi$
$83$	− 16.0000i	− 1.75623i	0.478451	−	0.878114i	$-0.341198\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 6.00000i	− 0.643268i
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	8.00000i	0.829561i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	12.0000i	1.21842i	0.793011	+	0.609208i	$0.208512\pi$
$97$	12.0000i	1.21842i	−0.793011	+	0.609208i	$0.791488\pi$
$98$	0	0
$99$	5.00000	0.502519
$100$	0	0
$101$	−18.0000	−1.79107	−0.895533	−	0.444994i	$-0.853206\pi$
$101$	−18.0000	−1.79107	−0.895533	+	0.444994i	$0.853206\pi$
$102$	0	0
$103$	2.00000i	0.197066i	0.995134	+	0.0985329i	$0.0314150\pi$
$103$	2.00000i	0.197066i	−0.995134	+	0.0985329i	$0.968585\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 12.0000i	− 1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$	− 12.0000i	− 1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$	0	0
$109$	−5.00000	−0.478913	−0.239457	−	0.970907i	$-0.576969\pi$
$109$	−5.00000	−0.478913	−0.239457	+	0.970907i	$0.576969\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	0	0
$113$	1.00000i	0.0940721i	0.998893	+	0.0470360i	$0.0149776\pi$
$113$	1.00000i	0.0940721i	−0.998893	+	0.0470360i	$0.985022\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	8.00000	0.733359
$120$	0	0
$121$	14.0000	1.27273
$122$	0	0
$123$	4.00000i	0.360668i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	13.0000i	1.15356i	0.816898	+	0.576782i	$0.195692\pi$
$127$	13.0000i	1.15356i	−0.816898	+	0.576782i	$0.804308\pi$
$128$	0	0
$129$	−6.00000	−0.528271
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	0	0
$133$	− 2.00000i	− 0.173422i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	18.0000i	1.53784i	0.639343	+	0.768922i	$0.279207\pi$
$137$	18.0000i	1.53784i	−0.639343	+	0.768922i	$0.720793\pi$
$138$	0	0
$139$	−2.00000	−0.169638	−0.0848189	−	0.996396i	$-0.527031\pi$
$139$	−2.00000	−0.169638	−0.0848189	+	0.996396i	$0.527031\pi$
$140$	0	0
$141$	12.0000	1.01058
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	2.00000i	0.164957i
$148$	0	0
$149$	17.0000	1.39269	0.696347	−	0.717705i	$-0.254807\pi$
$149$	17.0000	1.39269	0.696347	+	0.717705i	$0.254807\pi$
$150$	0	0
$151$	19.0000	1.54620	0.773099	−	0.634285i	$-0.218706\pi$
$151$	19.0000	1.54620	0.773099	+	0.634285i	$0.218706\pi$
$152$	0	0
$153$	8.00000i	0.646762i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 10.0000i	− 0.798087i	−0.916932	−	0.399043i	$-0.869342\pi$
$157$	− 10.0000i	− 0.798087i	0.916932	−	0.399043i	$-0.130658\pi$
$158$	0	0
$159$	20.0000	1.58610
$160$	0	0
$161$	−7.00000	−0.551677
$162$	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
$167$	− 8.00000i	− 0.619059i	−0.950890	−	0.309529i	$-0.899829\pi$
$167$	− 8.00000i	− 0.619059i	0.950890	−	0.309529i	$-0.100171\pi$
$168$	0	0
$169$	13.0000	1.00000
$170$	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
$177$	8.00000i	0.601317i
$178$	0	0
$179$	−24.0000	−1.79384	−0.896922	−	0.442189i	$-0.854202\pi$
$179$	−24.0000	−1.79384	−0.896922	+	0.442189i	$0.854202\pi$
$180$	0	0
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	0	0
$183$	12.0000i	0.887066i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	40.0000i	2.92509i
$188$	0	0
$189$	4.00000	0.290957
$190$	0	0
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	0	0
$193$	17.0000i	1.22369i	0.790979	+	0.611843i	$0.209572\pi$
$193$	17.0000i	1.22369i	−0.790979	+	0.611843i	$0.790428\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3.00000i	0.213741i	0.994273	+	0.106871i	$0.0340831\pi$
$197$	3.00000i	0.213741i	−0.994273	+	0.106871i	$0.965917\pi$
$198$	0	0
$199$	26.0000	1.84309	0.921546	−	0.388270i	$-0.126927\pi$
$199$	26.0000	1.84309	0.921546	+	0.388270i	$0.126927\pi$
$200$	0	0
$201$	26.0000	1.83390
$202$	0	0
$203$	3.00000i	0.210559i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 7.00000i	− 0.486534i
$208$	0	0
$209$	10.0000	0.691714
$210$	0	0
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	0	0
$213$	10.0000i	0.685189i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 4.00000i	− 0.271538i
$218$	0	0
$219$	12.0000	0.810885
$220$	0	0
$221$	0	0
$222$	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
$227$	14.0000i	0.929213i	0.885517	+	0.464606i	$0.153804\pi$
$227$	14.0000i	0.929213i	−0.885517	+	0.464606i	$0.846196\pi$
$228$	0	0
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	−10.0000	−0.657952
$232$	0	0
$233$	− 29.0000i	− 1.89985i	−0.312473	−	0.949927i	$-0.601157\pi$
$233$	− 29.0000i	− 1.89985i	0.312473	−	0.949927i	$-0.398843\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	26.0000i	1.68888i
$238$	0	0
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	0	0
$243$	10.0000i	0.641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−32.0000	−2.02792
$250$	0	0
$251$	4.00000	0.252478	0.126239	−	0.992000i	$-0.459709\pi$
$251$	4.00000	0.252478	0.126239	+	0.992000i	$0.459709\pi$
$252$	0	0
$253$	− 35.0000i	− 2.20043i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	12.0000i	0.748539i	0.927320	+	0.374270i	$0.122107\pi$
$257$	12.0000i	0.748539i	−0.927320	+	0.374270i	$0.877893\pi$
$258$	0	0
$259$	−1.00000	−0.0621370
$260$	0	0
$261$	−3.00000	−0.185695
$262$	0	0
$263$	5.00000i	0.308313i	0.988046	+	0.154157i	$0.0492660\pi$
$263$	5.00000i	0.308313i	−0.988046	+	0.154157i	$0.950734\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	24.0000	1.46331	0.731653	−	0.681677i	$-0.238749\pi$
$269$	24.0000	1.46331	0.731653	+	0.681677i	$0.238749\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 22.0000i	− 1.32185i	−0.750451	−	0.660926i	$-0.770164\pi$
$277$	− 22.0000i	− 1.32185i	0.750451	−	0.660926i	$-0.229836\pi$
$278$	0	0
$279$	4.00000	0.239474
$280$	0	0
$281$	9.00000	0.536895	0.268447	−	0.963294i	$-0.413489\pi$
$281$	9.00000	0.536895	0.268447	+	0.963294i	$0.413489\pi$
$282$	0	0
$283$	− 10.0000i	− 0.594438i	−0.954809	−	0.297219i	$-0.903941\pi$
$283$	− 10.0000i	− 0.594438i	0.954809	−	0.297219i	$-0.0960592\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 2.00000i	− 0.118056i
$288$	0	0
$289$	−47.0000	−2.76471
$290$	0	0
$291$	24.0000	1.40690
$292$	0	0
$293$	− 12.0000i	− 0.701047i	−0.936554	−	0.350524i	$-0.886004\pi$
$293$	− 12.0000i	− 0.701047i	0.936554	−	0.350524i	$-0.113996\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	20.0000i	1.16052i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	3.00000	0.172917
$302$	0	0
$303$	36.0000i	2.06815i
$304$	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
$309$	4.00000	0.227552
$310$	0	0
$311$	4.00000	0.226819	0.113410	−	0.993548i	$-0.463823\pi$
$311$	4.00000	0.226819	0.113410	+	0.993548i	$0.463823\pi$
$312$	0	0
$313$	14.0000i	0.791327i	0.918396	+	0.395663i	$0.129485\pi$
$313$	14.0000i	0.791327i	−0.918396	+	0.395663i	$0.870515\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 1.00000i	− 0.0561656i	−0.999606	−	0.0280828i	$-0.991060\pi$
$317$	− 1.00000i	− 0.0561656i	0.999606	−	0.0280828i	$-0.00894021\pi$
$318$	0	0
$319$	−15.0000	−0.839839
$320$	0	0
$321$	−24.0000	−1.33955
$322$	0	0
$323$	16.0000i	0.890264i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	10.0000i	0.553001i
$328$	0	0
$329$	−6.00000	−0.330791
$330$	0	0
$331$	−3.00000	−0.164895	−0.0824475	−	0.996595i	$-0.526274\pi$
$331$	−3.00000	−0.164895	−0.0824475	+	0.996595i	$0.526274\pi$
$332$	0	0
$333$	− 1.00000i	− 0.0547997i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 14.0000i	− 0.762629i	−0.924445	−	0.381314i	$-0.875472\pi$
$337$	− 14.0000i	− 0.762629i	0.924445	−	0.381314i	$-0.124528\pi$
$338$	0	0
$339$	2.00000	0.108625
$340$	0	0
$341$	20.0000	1.08306
$342$	0	0
$343$	− 1.00000i	− 0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 29.0000i	− 1.55680i	−0.627768	−	0.778401i	$-0.716031\pi$
$347$	− 29.0000i	− 1.55680i	0.627768	−	0.778401i	$-0.283969\pi$
$348$	0	0
$349$	22.0000	1.17763	0.588817	−	0.808267i	$-0.299594\pi$
$349$	22.0000	1.17763	0.588817	+	0.808267i	$0.299594\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 20.0000i	− 1.06449i	−0.846590	−	0.532246i	$-0.821348\pi$
$353$	− 20.0000i	− 1.06449i	0.846590	−	0.532246i	$-0.178652\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 16.0000i	− 0.846810i
$358$	0	0
$359$	−15.0000	−0.791670	−0.395835	−	0.918322i	$-0.629545\pi$
$359$	−15.0000	−0.791670	−0.395835	+	0.918322i	$0.629545\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	− 28.0000i	− 1.46962i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 22.0000i	− 1.14839i	−0.818718	−	0.574195i	$-0.805315\pi$
$367$	− 22.0000i	− 1.14839i	0.818718	−	0.574195i	$-0.194685\pi$
$368$	0	0
$369$	2.00000	0.104116
$370$	0	0
$371$	−10.0000	−0.519174
$372$	0	0
$373$	1.00000i	0.0517780i	0.999665	+	0.0258890i	$0.00824165\pi$
$373$	1.00000i	0.0517780i	−0.999665	+	0.0258890i	$0.991758\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	15.0000	0.770498	0.385249	−	0.922813i	$-0.374116\pi$
$379$	15.0000	0.770498	0.385249	+	0.922813i	$0.374116\pi$
$380$	0	0
$381$	26.0000	1.33202
$382$	0	0
$383$	− 12.0000i	− 0.613171i	−0.951843	−	0.306586i	$-0.900813\pi$
$383$	− 12.0000i	− 0.613171i	0.951843	−	0.306586i	$-0.0991866\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3.00000i	0.152499i
$388$	0	0
$389$	−5.00000	−0.253510	−0.126755	−	0.991934i	$-0.540456\pi$
$389$	−5.00000	−0.253510	−0.126755	+	0.991934i	$0.540456\pi$
$390$	0	0
$391$	56.0000	2.83204
$392$	0	0
$393$	8.00000i	0.403547i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	18.0000i	0.903394i	0.892171	+	0.451697i	$0.149181\pi$
$397$	18.0000i	0.903394i	−0.892171	+	0.451697i	$0.850819\pi$
$398$	0	0
$399$	−4.00000	−0.200250
$400$	0	0
$401$	25.0000	1.24844	0.624220	−	0.781248i	$-0.285417\pi$
$401$	25.0000	1.24844	0.624220	+	0.781248i	$0.285417\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 5.00000i	− 0.247841i
$408$	0	0
$409$	−6.00000	−0.296681	−0.148340	−	0.988936i	$-0.547393\pi$
$409$	−6.00000	−0.296681	−0.148340	+	0.988936i	$0.547393\pi$
$410$	0	0
$411$	36.0000	1.77575
$412$	0	0
$413$	− 4.00000i	− 0.196827i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	4.00000i	0.195881i
$418$	0	0
$419$	16.0000	0.781651	0.390826	−	0.920465i	$-0.372190\pi$
$419$	16.0000	0.781651	0.390826	+	0.920465i	$0.372190\pi$
$420$	0	0
$421$	−33.0000	−1.60832	−0.804161	−	0.594412i	$-0.797385\pi$
$421$	−33.0000	−1.60832	−0.804161	+	0.594412i	$0.797385\pi$
$422$	0	0
$423$	− 6.00000i	− 0.291730i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 6.00000i	− 0.290360i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−20.0000	−0.963366	−0.481683	−	0.876346i	$-0.659974\pi$
$431$	−20.0000	−0.963366	−0.481683	+	0.876346i	$0.659974\pi$
$432$	0	0
$433$	24.0000i	1.15337i	0.816968	+	0.576683i	$0.195653\pi$
$433$	24.0000i	1.15337i	−0.816968	+	0.576683i	$0.804347\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 14.0000i	− 0.669711i
$438$	0	0
$439$	−16.0000	−0.763638	−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000	−0.763638	−0.381819	+	0.924237i	$0.624702\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	4.00000i	0.190046i	0.995475	+	0.0950229i	$0.0302924\pi$
$443$	4.00000i	0.190046i	−0.995475	+	0.0950229i	$0.969708\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 34.0000i	− 1.60814i
$448$	0	0
$449$	−31.0000	−1.46298	−0.731490	−	0.681852i	$-0.761175\pi$
$449$	−31.0000	−1.46298	−0.731490	+	0.681852i	$0.761175\pi$
$450$	0	0
$451$	10.0000	0.470882
$452$	0	0
$453$	− 38.0000i	− 1.78540i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 37.0000i	− 1.73079i	−0.501093	−	0.865393i	$-0.667069\pi$
$457$	− 37.0000i	− 1.73079i	0.501093	−	0.865393i	$-0.332931\pi$
$458$	0	0
$459$	−32.0000	−1.49363
$460$	0	0
$461$	−40.0000	−1.86299	−0.931493	−	0.363760i	$-0.881493\pi$
$461$	−40.0000	−1.86299	−0.931493	+	0.363760i	$0.881493\pi$
$462$	0	0
$463$	− 24.0000i	− 1.11537i	−0.830051	−	0.557687i	$-0.811689\pi$
$463$	− 24.0000i	− 1.11537i	0.830051	−	0.557687i	$-0.188311\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 14.0000i	− 0.647843i	−0.946084	−	0.323921i	$-0.894999\pi$
$467$	− 14.0000i	− 0.647843i	0.946084	−	0.323921i	$-0.105001\pi$
$468$	0	0
$469$	−13.0000	−0.600284
$470$	0	0
$471$	−20.0000	−0.921551
$472$	0	0
$473$	15.0000i	0.689701i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 10.0000i	− 0.457869i
$478$	0	0
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	14.0000i	0.637022i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 13.0000i	− 0.589086i	−0.955638	−	0.294543i	$-0.904833\pi$
$487$	− 13.0000i	− 0.589086i	0.955638	−	0.294543i	$-0.0951675\pi$
$488$	0	0
$489$	8.00000	0.361773
$490$	0	0
$491$	11.0000	0.496423	0.248212	−	0.968706i	$-0.420157\pi$
$491$	11.0000	0.496423	0.248212	+	0.968706i	$0.420157\pi$
$492$	0	0
$493$	− 24.0000i	− 1.08091i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 5.00000i	− 0.224281i
$498$	0	0
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	0	0
$501$	−16.0000	−0.714827
$502$	0	0
$503$	− 38.0000i	− 1.69434i	−0.531325	−	0.847168i	$-0.678306\pi$
$503$	− 38.0000i	− 1.69434i	0.531325	−	0.847168i	$-0.321694\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 26.0000i	− 1.15470i
$508$	0	0
$509$	24.0000	1.06378	0.531891	−	0.846813i	$-0.321482\pi$
$509$	24.0000	1.06378	0.531891	+	0.846813i	$0.321482\pi$
$510$	0	0
$511$	−6.00000	−0.265424
$512$	0	0
$513$	8.00000i	0.353209i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 30.0000i	− 1.31940i
$518$	0	0
$519$	−44.0000	−1.93139
$520$	0	0
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	0	0
$523$	4.00000i	0.174908i	0.996169	+	0.0874539i	$0.0278730\pi$
$523$	4.00000i	0.174908i	−0.996169	+	0.0874539i	$0.972127\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	32.0000i	1.39394i
$528$	0	0
$529$	−26.0000	−1.13043
$530$	0	0
$531$	4.00000	0.173585
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	48.0000i	2.07135i
$538$	0	0
$539$	5.00000	0.215365
$540$	0	0
$541$	−13.0000	−0.558914	−0.279457	−	0.960158i	$-0.590154\pi$
$541$	−13.0000	−0.558914	−0.279457	+	0.960158i	$0.590154\pi$
$542$	0	0
$543$	44.0000i	1.88822i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	17.0000i	0.726868i	0.931620	+	0.363434i	$0.118396\pi$
$547$	17.0000i	0.726868i	−0.931620	+	0.363434i	$0.881604\pi$
$548$	0	0
$549$	6.00000	0.256074
$550$	0	0
$551$	−6.00000	−0.255609
$552$	0	0
$553$	− 13.0000i	− 0.552816i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 9.00000i	− 0.381342i	−0.981654	−	0.190671i	$-0.938934\pi$
$557$	− 9.00000i	− 0.381342i	0.981654	−	0.190671i	$-0.0610664\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	80.0000	3.37760
$562$	0	0
$563$	6.00000i	0.252870i	0.991975	+	0.126435i	$0.0403535\pi$
$563$	6.00000i	0.252870i	−0.991975	+	0.126435i	$0.959647\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 11.0000i	− 0.461957i
$568$	0	0
$569$	−13.0000	−0.544988	−0.272494	−	0.962157i	$-0.587849\pi$
$569$	−13.0000	−0.544988	−0.272494	+	0.962157i	$0.587849\pi$
$570$	0	0
$571$	−19.0000	−0.795125	−0.397563	−	0.917575i	$-0.630144\pi$
$571$	−19.0000	−0.795125	−0.397563	+	0.917575i	$0.630144\pi$
$572$	0	0
$573$	24.0000i	1.00261i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	2.00000i	0.0832611i	0.999133	+	0.0416305i	$0.0132552\pi$
$577$	2.00000i	0.0832611i	−0.999133	+	0.0416305i	$0.986745\pi$
$578$	0	0
$579$	34.0000	1.41299
$580$	0	0
$581$	16.0000	0.663792
$582$	0	0
$583$	− 50.0000i	− 2.07079i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 30.0000i	− 1.23823i	−0.785299	−	0.619116i	$-0.787491\pi$
$587$	− 30.0000i	− 1.23823i	0.785299	−	0.619116i	$-0.212509\pi$
$588$	0	0
$589$	8.00000	0.329634
$590$	0	0
$591$	6.00000	0.246807
$592$	0	0
$593$	12.0000i	0.492781i	0.969171	+	0.246390i	$0.0792446\pi$
$593$	12.0000i	0.492781i	−0.969171	+	0.246390i	$0.920755\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 52.0000i	− 2.12822i
$598$	0	0
$599$	−5.00000	−0.204294	−0.102147	−	0.994769i	$-0.532571\pi$
$599$	−5.00000	−0.204294	−0.102147	+	0.994769i	$0.532571\pi$
$600$	0	0
$601$	40.0000	1.63163	0.815817	−	0.578310i	$-0.196288\pi$
$601$	40.0000	1.63163	0.815817	+	0.578310i	$0.196288\pi$
$602$	0	0
$603$	− 13.0000i	− 0.529401i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	48.0000i	1.94826i	0.225989	+	0.974130i	$0.427439\pi$
$607$	48.0000i	1.94826i	−0.225989	+	0.974130i	$0.572561\pi$
$608$	0	0
$609$	6.00000	0.243132
$610$	0	0
$611$	0	0
$612$	0	0
$613$	35.0000i	1.41364i	0.707395	+	0.706818i	$0.249870\pi$
$613$	35.0000i	1.41364i	−0.707395	+	0.706818i	$0.750130\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	9.00000i	0.362326i	0.983453	+	0.181163i	$0.0579862\pi$
$617$	9.00000i	0.362326i	−0.983453	+	0.181163i	$0.942014\pi$
$618$	0	0
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	0	0
$621$	28.0000	1.12360
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 20.0000i	− 0.798723i
$628$	0	0
$629$	8.00000	0.318981
$630$	0	0
$631$	−7.00000	−0.278666	−0.139333	−	0.990246i	$-0.544496\pi$
$631$	−7.00000	−0.278666	−0.139333	+	0.990246i	$0.544496\pi$
$632$	0	0
$633$	40.0000i	1.58986i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	5.00000	0.197797
$640$	0	0
$641$	−7.00000	−0.276483	−0.138242	−	0.990399i	$-0.544145\pi$
$641$	−7.00000	−0.276483	−0.138242	+	0.990399i	$0.544145\pi$
$642$	0	0
$643$	− 16.0000i	− 0.630978i	−0.948929	−	0.315489i	$-0.897831\pi$
$643$	− 16.0000i	− 0.630978i	0.948929	−	0.315489i	$-0.102169\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 26.0000i	− 1.02217i	−0.859532	−	0.511083i	$-0.829245\pi$
$647$	− 26.0000i	− 1.02217i	0.859532	−	0.511083i	$-0.170755\pi$
$648$	0	0
$649$	20.0000	0.785069
$650$	0	0
$651$	−8.00000	−0.313545
$652$	0	0
$653$	18.0000i	0.704394i	0.935926	+	0.352197i	$0.114565\pi$
$653$	18.0000i	0.704394i	−0.935926	+	0.352197i	$0.885435\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 6.00000i	− 0.234082i
$658$	0	0
$659$	20.0000	0.779089	0.389545	−	0.921008i	$-0.372632\pi$
$659$	20.0000	0.779089	0.389545	+	0.921008i	$0.372632\pi$
$660$	0	0
$661$	30.0000	1.16686	0.583432	−	0.812162i	$-0.301709\pi$
$661$	30.0000	1.16686	0.583432	+	0.812162i	$0.301709\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	21.0000i	0.813123i
$668$	0	0
$669$	28.0000	1.08254
$670$	0	0
$671$	30.0000	1.15814
$672$	0	0
$673$	− 30.0000i	− 1.15642i	−0.815890	−	0.578208i	$-0.803752\pi$
$673$	− 30.0000i	− 1.15642i	0.815890	−	0.578208i	$-0.196248\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 6.00000i	− 0.230599i	−0.993331	−	0.115299i	$-0.963217\pi$
$677$	− 6.00000i	− 0.230599i	0.993331	−	0.115299i	$-0.0367827\pi$
$678$	0	0
$679$	−12.0000	−0.460518
$680$	0	0
$681$	28.0000	1.07296
$682$	0	0
$683$	− 27.0000i	− 1.03313i	−0.856249	−	0.516563i	$-0.827211\pi$
$683$	− 27.0000i	− 1.03313i	0.856249	−	0.516563i	$-0.172789\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	28.0000i	1.06827i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−14.0000	−0.532585	−0.266293	−	0.963892i	$-0.585799\pi$
$691$	−14.0000	−0.532585	−0.266293	+	0.963892i	$0.585799\pi$
$692$	0	0
$693$	5.00000i	0.189934i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	16.0000i	0.606043i
$698$	0	0
$699$	−58.0000	−2.19376
$700$	0	0
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	0	0
$703$	− 2.00000i	− 0.0754314i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 18.0000i	− 0.676960i
$708$	0	0
$709$	−30.0000	−1.12667	−0.563337	−	0.826227i	$-0.690483\pi$
$709$	−30.0000	−1.12667	−0.563337	+	0.826227i	$0.690483\pi$
$710$	0	0
$711$	13.0000	0.487538
$712$	0	0
$713$	− 28.0000i	− 1.04861i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 48.0000i	− 1.79259i
$718$	0	0
$719$	6.00000	0.223762	0.111881	−	0.993722i	$-0.464312\pi$
$719$	6.00000	0.223762	0.111881	+	0.993722i	$0.464312\pi$
$720$	0	0
$721$	−2.00000	−0.0744839
$722$	0	0
$723$	44.0000i	1.63638i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	40.0000i	1.48352i	0.670667	+	0.741759i	$0.266008\pi$
$727$	40.0000i	1.48352i	−0.670667	+	0.741759i	$0.733992\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	−24.0000	−0.887672
$732$	0	0
$733$	− 28.0000i	− 1.03420i	−0.855924	−	0.517102i	$-0.827011\pi$
$733$	− 28.0000i	− 1.03420i	0.855924	−	0.517102i	$-0.172989\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 65.0000i	− 2.39431i
$738$	0	0
$739$	1.00000	0.0367856	0.0183928	−	0.999831i	$-0.494145\pi$
$739$	1.00000	0.0367856	0.0183928	+	0.999831i	$0.494145\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 24.0000i	− 0.880475i	−0.897881	−	0.440237i	$-0.854894\pi$
$743$	− 24.0000i	− 0.880475i	0.897881	−	0.440237i	$-0.145106\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	16.0000i	0.585409i
$748$	0	0
$749$	12.0000	0.438470
$750$	0	0
$751$	−12.0000	−0.437886	−0.218943	−	0.975738i	$-0.570261\pi$
$751$	−12.0000	−0.437886	−0.218943	+	0.975738i	$0.570261\pi$
$752$	0	0
$753$	− 8.00000i	− 0.291536i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 15.0000i	− 0.545184i	−0.962130	−	0.272592i	$-0.912119\pi$
$757$	− 15.0000i	− 0.545184i	0.962130	−	0.272592i	$-0.0878810\pi$
$758$	0	0
$759$	−70.0000	−2.54084
$760$	0	0
$761$	48.0000	1.74000	0.869999	−	0.493053i	$-0.164119\pi$
$761$	48.0000	1.74000	0.869999	+	0.493053i	$0.164119\pi$
$762$	0	0
$763$	− 5.00000i	− 0.181012i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−34.0000	−1.22607	−0.613036	−	0.790055i	$-0.710052\pi$
$769$	−34.0000	−1.22607	−0.613036	+	0.790055i	$0.710052\pi$
$770$	0	0
$771$	24.0000	0.864339
$772$	0	0
$773$	− 36.0000i	− 1.29483i	−0.762138	−	0.647415i	$-0.775850\pi$
$773$	− 36.0000i	− 1.29483i	0.762138	−	0.647415i	$-0.224150\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	2.00000i	0.0717496i
$778$	0	0
$779$	4.00000	0.143315
$780$	0	0
$781$	25.0000	0.894570
$782$	0	0
$783$	− 12.0000i	− 0.428845i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 18.0000i	− 0.641631i	−0.947142	−	0.320815i	$-0.896043\pi$
$787$	− 18.0000i	− 0.641631i	0.947142	−	0.320815i	$-0.103957\pi$
$788$	0	0
$789$	10.0000	0.356009
$790$	0	0
$791$	−1.00000	−0.0355559
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	44.0000i	1.55856i	0.626676	+	0.779280i	$0.284415\pi$
$797$	44.0000i	1.55856i	−0.626676	+	0.779280i	$0.715585\pi$
$798$	0	0
$799$	48.0000	1.69812
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 30.0000i	− 1.05868i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 48.0000i	− 1.68968i
$808$	0	0
$809$	29.0000	1.01959	0.509793	−	0.860297i	$-0.329722\pi$
$809$	29.0000	1.01959	0.509793	+	0.860297i	$0.329722\pi$
$810$	0	0
$811$	50.0000	1.75574	0.877869	−	0.478901i	$-0.158965\pi$
$811$	50.0000	1.75574	0.877869	+	0.478901i	$0.158965\pi$
$812$	0	0
$813$	− 16.0000i	− 0.561144i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	6.00000i	0.209913i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−54.0000	−1.88461	−0.942306	−	0.334751i	$-0.891348\pi$
$821$	−54.0000	−1.88461	−0.942306	+	0.334751i	$0.891348\pi$
$822$	0	0
$823$	− 53.0000i	− 1.84746i	−0.383040	−	0.923732i	$-0.625123\pi$
$823$	− 53.0000i	− 1.84746i	0.383040	−	0.923732i	$-0.374877\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	11.0000i	0.382507i	0.981541	+	0.191254i	$0.0612553\pi$
$827$	11.0000i	0.382507i	−0.981541	+	0.191254i	$0.938745\pi$
$828$	0	0
$829$	−20.0000	−0.694629	−0.347314	−	0.937749i	$-0.612906\pi$
$829$	−20.0000	−0.694629	−0.347314	+	0.937749i	$0.612906\pi$
$830$	0	0
$831$	−44.0000	−1.52634
$832$	0	0
$833$	8.00000i	0.277184i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	16.0000i	0.553041i
$838$	0	0
$839$	46.0000	1.58810	0.794048	−	0.607855i	$-0.207970\pi$
$839$	46.0000	1.58810	0.794048	+	0.607855i	$0.207970\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	− 18.0000i	− 0.619953i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	14.0000i	0.481046i
$848$	0	0
$849$	−20.0000	−0.686398
$850$	0	0
$851$	−7.00000	−0.239957
$852$	0	0
$853$	28.0000i	0.958702i	0.877623	+	0.479351i	$0.159128\pi$
$853$	28.0000i	0.958702i	−0.877623	+	0.479351i	$0.840872\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	54.0000i	1.84460i	0.386469	+	0.922302i	$0.373695\pi$
$857$	54.0000i	1.84460i	−0.386469	+	0.922302i	$0.626305\pi$
$858$	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
$861$	−4.00000	−0.136320
$862$	0	0
$863$	− 43.0000i	− 1.46374i	−0.681446	−	0.731869i	$-0.738649\pi$
$863$	− 43.0000i	− 1.46374i	0.681446	−	0.731869i	$-0.261351\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	94.0000i	3.19241i
$868$	0	0
$869$	65.0000	2.20497
$870$	0	0
$871$	0	0
$872$	0	0
$873$	− 12.0000i	− 0.406138i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	18.0000i	0.607817i	0.952701	+	0.303908i	$0.0982917\pi$
$877$	18.0000i	0.607817i	−0.952701	+	0.303908i	$0.901708\pi$
$878$	0	0
$879$	−24.0000	−0.809500
$880$	0	0
$881$	20.0000	0.673817	0.336909	−	0.941537i	$-0.390619\pi$
$881$	20.0000	0.673817	0.336909	+	0.941537i	$0.390619\pi$
$882$	0	0
$883$	35.0000i	1.17784i	0.808190	+	0.588922i	$0.200447\pi$
$883$	35.0000i	1.17784i	−0.808190	+	0.588922i	$0.799553\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	6.00000i	0.201460i	0.994914	+	0.100730i	$0.0321179\pi$
$887$	6.00000i	0.201460i	−0.994914	+	0.100730i	$0.967882\pi$
$888$	0	0
$889$	−13.0000	−0.436006
$890$	0	0
$891$	55.0000	1.84257
$892$	0	0
$893$	− 12.0000i	− 0.401565i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−12.0000	−0.400222
$900$	0	0
$901$	80.0000	2.66519
$902$	0	0
$903$	− 6.00000i	− 0.199667i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 28.0000i	− 0.929725i	−0.885383	−	0.464862i	$-0.846104\pi$
$907$	− 28.0000i	− 0.929725i	0.885383	−	0.464862i	$-0.153896\pi$
$908$	0	0
$909$	18.0000	0.597022
$910$	0	0
$911$	−27.0000	−0.894550	−0.447275	−	0.894397i	$-0.647605\pi$
$911$	−27.0000	−0.894550	−0.447275	+	0.894397i	$0.647605\pi$
$912$	0	0
$913$	80.0000i	2.64761i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 4.00000i	− 0.132092i
$918$	0	0
$919$	15.0000	0.494804	0.247402	−	0.968913i	$-0.420423\pi$
$919$	15.0000	0.494804	0.247402	+	0.968913i	$0.420423\pi$
$920$	0	0
$921$	−24.0000	−0.790827
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 2.00000i	− 0.0656886i
$928$	0	0
$929$	22.0000	0.721797	0.360898	−	0.932605i	$-0.382470\pi$
$929$	22.0000	0.721797	0.360898	+	0.932605i	$0.382470\pi$
$930$	0	0
$931$	2.00000	0.0655474
$932$	0	0
$933$	− 8.00000i	− 0.261908i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	22.0000i	0.718709i	0.933201	+	0.359354i	$0.117003\pi$
$937$	22.0000i	0.718709i	−0.933201	+	0.359354i	$0.882997\pi$
$938$	0	0
$939$	28.0000	0.913745
$940$	0	0
$941$	−12.0000	−0.391189	−0.195594	−	0.980685i	$-0.562664\pi$
$941$	−12.0000	−0.391189	−0.195594	+	0.980685i	$0.562664\pi$
$942$	0	0
$943$	− 14.0000i	− 0.455903i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 20.0000i	− 0.649913i	−0.945729	−	0.324956i	$-0.894650\pi$
$947$	− 20.0000i	− 0.649913i	0.945729	−	0.324956i	$-0.105350\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−2.00000	−0.0648544
$952$	0	0
$953$	5.00000i	0.161966i	0.996715	+	0.0809829i	$0.0258059\pi$
$953$	5.00000i	0.161966i	−0.996715	+	0.0809829i	$0.974194\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	30.0000i	0.969762i
$958$	0	0
$959$	−18.0000	−0.581250
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	12.0000i	0.386695i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	56.0000i	1.80084i	0.435023	+	0.900419i	$0.356740\pi$
$967$	56.0000i	1.80084i	−0.435023	+	0.900419i	$0.643260\pi$
$968$	0	0
$969$	32.0000	1.02799
$970$	0	0
$971$	26.0000	0.834380	0.417190	−	0.908819i	$-0.363015\pi$
$971$	26.0000	0.834380	0.417190	+	0.908819i	$0.363015\pi$
$972$	0	0
$973$	− 2.00000i	− 0.0641171i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 51.0000i	− 1.63163i	−0.578310	−	0.815817i	$-0.696287\pi$
$977$	− 51.0000i	− 1.63163i	0.578310	−	0.815817i	$-0.303713\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	5.00000	0.159638
$982$	0	0
$983$	12.0000i	0.382741i	0.981518	+	0.191370i	$0.0612931\pi$
$983$	12.0000i	0.382741i	−0.981518	+	0.191370i	$0.938707\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	12.0000i	0.381964i
$988$	0	0
$989$	21.0000	0.667761
$990$	0	0
$991$	13.0000	0.412959	0.206479	−	0.978451i	$-0.433799\pi$
$991$	13.0000	0.412959	0.206479	+	0.978451i	$0.433799\pi$
$992$	0	0
$993$	6.00000i	0.190404i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 2.00000i	− 0.0633406i	−0.999498	−	0.0316703i	$-0.989917\pi$
$997$	− 2.00000i	− 0.0633406i	0.999498	−	0.0316703i	$-0.0100827\pi$
$998$	0	0
$999$	4.00000	0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2800.2.g.d.449.1		2
4.3	odd	2		1400.2.g.d.449.2		2
5.2	odd	4		2800.2.a.d.1.1		1
5.3	odd	4		2800.2.a.bc.1.1		1
5.4	even	2	inner	2800.2.g.d.449.2		2
20.3	even	4		1400.2.a.b.1.1	✓	1
20.7	even	4		1400.2.a.m.1.1	yes	1
20.19	odd	2		1400.2.g.d.449.1		2
140.27	odd	4		9800.2.a.l.1.1		1
140.83	odd	4		9800.2.a.bo.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1400.2.a.b.1.1	✓	1	20.3	even	4
1400.2.a.m.1.1	yes	1	20.7	even	4
1400.2.g.d.449.1		2	20.19	odd	2
1400.2.g.d.449.2		2	4.3	odd	2
2800.2.a.d.1.1		1	5.2	odd	4
2800.2.a.bc.1.1		1	5.3	odd	4
2800.2.g.d.449.1		2	1.1	even	1	trivial
2800.2.g.d.449.2		2	5.4	even	2	inner
9800.2.a.l.1.1		1	140.27	odd	4
9800.2.a.bo.1.1		1	140.83	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:	\( N \)	\(=\)	\( 2800 = 2^{4} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2800.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{3} +1.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-2.00000i q^{3} +1.00000i q^{7} -1.00000 q^{9} -5.00000 q^{11} -8.00000i q^{17} -2.00000 q^{19} +2.00000 q^{21} +7.00000i q^{23} -4.00000i q^{27} +3.00000 q^{29} -4.00000 q^{31} +10.0000i q^{33} +1.00000i q^{37} -2.00000 q^{41} -3.00000i q^{43} +6.00000i q^{47} -1.00000 q^{49} -16.0000 q^{51} +10.0000i q^{53} +4.00000i q^{57} -4.00000 q^{59} -6.00000 q^{61} -1.00000i q^{63} +13.0000i q^{67} +14.0000 q^{69} -5.00000 q^{71} +6.00000i q^{73} -5.00000i q^{77} -13.0000 q^{79} -11.0000 q^{81} -16.0000i q^{83} -6.00000i q^{87} +8.00000i q^{93} +12.0000i q^{97} +5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^9	\( 2 q - 2 q^{9} - 10 q^{11} - 4 q^{19} + 4 q^{21} + 6 q^{29} - 8 q^{31} - 4 q^{41} - 2 q^{49} - 32 q^{51} - 8 q^{59} - 12 q^{61} + 28 q^{69} - 10 q^{71} - 26 q^{79} - 22 q^{81} + 10 q^{99}+O(q^{100}) \) 2 * q - 2 * q^9 - 10 * q^11 - 4 * q^19 + 4 * q^21 + 6 * q^29 - 8 * q^31 - 4 * q^41 - 2 * q^49 - 32 * q^51 - 8 * q^59 - 12 * q^61 + 28 * q^69 - 10 * q^71 - 26 * q^79 - 22 * q^81 + 10 * q^99

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