LMFDB - Embedded newform 4200.2.t.h.1849.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4200,2,Mod(1849,4200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4200, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4200.1849");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4200 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4200.t (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:	$33.5371688489$
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1849.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	4200.1849
Dual form			4200.2.t.h.1849.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -1.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} -1.00000i q^{7} -1.00000 q^{9} -1.00000 q^{11} -4.00000i q^{13} +2.00000i q^{17} +4.00000 q^{19} +1.00000 q^{21} +3.00000i q^{23} -1.00000i q^{27} +1.00000 q^{29} -6.00000 q^{31} -1.00000i q^{33} -3.00000i q^{37} +4.00000 q^{39} -5.00000i q^{43} +4.00000i q^{47} -1.00000 q^{49} -2.00000 q^{51} -6.00000i q^{53} +4.00000i q^{57} +4.00000 q^{59} -4.00000 q^{61} +1.00000i q^{63} -5.00000i q^{67} -3.00000 q^{69} -1.00000 q^{71} -10.0000i q^{73} +1.00000i q^{77} +1.00000 q^{79} +1.00000 q^{81} -14.0000i q^{83} +1.00000i q^{87} +8.00000 q^{89} -4.00000 q^{91} -6.00000i q^{93} -14.0000i q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^9	$ 2 q - 2 q^{9} - 2 q^{11} + 8 q^{19} + 2 q^{21} + 2 q^{29} - 12 q^{31} + 8 q^{39} - 2 q^{49} - 4 q^{51} + 8 q^{59} - 8 q^{61} - 6 q^{69} - 2 q^{71} + 2 q^{79} + 2 q^{81} + 16 q^{89} - 8 q^{91} + 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^9 - 2 * q^11 + 8 * q^19 + 2 * q^21 + 2 * q^29 - 12 * q^31 + 8 * q^39 - 2 * q^49 - 4 * q^51 + 8 * q^59 - 8 * q^61 - 6 * q^69 - 2 * q^71 + 2 * q^79 + 2 * q^81 + 16 * q^89 - 8 * q^91 + 2 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4200\mathbb{Z}\right)^\times$.

$n$	$1177$	$2101$	$2801$	$3151$	$3601$
$\chi(n)$	$-1$	$1$	$1$	$1$	$1$

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
$3$	1.00000i	0.577350i
$4$	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
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	0	0
$13$	− 4.00000i	− 1.10940i	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	− 4.00000i	− 1.10940i	0.832050	−	0.554700i	$-0.187167\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.00000i	0.485071i	0.970143	+	0.242536i	$0.0779791\pi$
$17$	2.00000i	0.485071i	−0.970143	+	0.242536i	$0.922021\pi$
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	3.00000i	0.625543i	0.949828	+	0.312772i	$0.101257\pi$
$23$	3.00000i	0.625543i	−0.949828	+	0.312772i	$0.898743\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	1.00000	0.185695	0.0928477	−	0.995680i	$-0.470403\pi$
$29$	1.00000	0.185695	0.0928477	+	0.995680i	$0.470403\pi$
$30$	0	0
$31$	−6.00000	−1.07763	−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000	−1.07763	−0.538816	+	0.842424i	$0.681128\pi$
$32$	0	0
$33$	− 1.00000i	− 0.174078i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 3.00000i	− 0.493197i	−0.969118	−	0.246598i	$-0.920687\pi$
$37$	− 3.00000i	− 0.493197i	0.969118	−	0.246598i	$-0.0793129\pi$
$38$	0	0
$39$	4.00000	0.640513
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	− 5.00000i	− 0.762493i	−0.924473	−	0.381246i	$-0.875495\pi$
$43$	− 5.00000i	− 0.762493i	0.924473	−	0.381246i	$-0.124505\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.00000i	0.583460i	0.956501	+	0.291730i	$0.0942309\pi$
$47$	4.00000i	0.583460i	−0.956501	+	0.291730i	$0.905769\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	−2.00000	−0.280056
$52$	0	0
$53$	− 6.00000i	− 0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$	− 6.00000i	− 0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	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	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	−4.00000	−0.512148	−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000	−0.512148	−0.256074	+	0.966657i	$0.582429\pi$
$62$	0	0
$63$	1.00000i	0.125988i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 5.00000i	− 0.610847i	−0.952217	−	0.305424i	$-0.901202\pi$
$67$	− 5.00000i	− 0.610847i	0.952217	−	0.305424i	$-0.0987981\pi$
$68$	0	0
$69$	−3.00000	−0.361158
$70$	0	0
$71$	−1.00000	−0.118678	−0.0593391	−	0.998238i	$-0.518899\pi$
$71$	−1.00000	−0.118678	−0.0593391	+	0.998238i	$0.518899\pi$
$72$	0	0
$73$	− 10.0000i	− 1.17041i	−0.810885	−	0.585206i	$-0.801014\pi$
$73$	− 10.0000i	− 1.17041i	0.810885	−	0.585206i	$-0.198986\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.00000i	0.113961i
$78$	0	0
$79$	1.00000	0.112509	0.0562544	−	0.998416i	$-0.482084\pi$
$79$	1.00000	0.112509	0.0562544	+	0.998416i	$0.482084\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 14.0000i	− 1.53670i	−0.640030	−	0.768350i	$-0.721078\pi$
$83$	− 14.0000i	− 1.53670i	0.640030	−	0.768350i	$-0.278922\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1.00000i	0.107211i
$88$	0	0
$89$	8.00000	0.847998	0.423999	−	0.905663i	$-0.360626\pi$
$89$	8.00000	0.847998	0.423999	+	0.905663i	$0.360626\pi$
$90$	0	0
$91$	−4.00000	−0.419314
$92$	0	0
$93$	− 6.00000i	− 0.622171i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 14.0000i	− 1.42148i	−0.703452	−	0.710742i	$-0.748359\pi$
$97$	− 14.0000i	− 1.42148i	0.703452	−	0.710742i	$-0.251641\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	0	0
$103$	4.00000i	0.394132i	0.980390	+	0.197066i	$0.0631413\pi$
$103$	4.00000i	0.394132i	−0.980390	+	0.197066i	$0.936859\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 4.00000i	− 0.386695i	−0.981130	−	0.193347i	$-0.938066\pi$
$107$	− 4.00000i	− 0.386695i	0.981130	−	0.193347i	$-0.0619344\pi$
$108$	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
$111$	3.00000	0.284747
$112$	0	0
$113$	− 9.00000i	− 0.846649i	−0.905978	−	0.423324i	$-0.860863\pi$
$113$	− 9.00000i	− 0.846649i	0.905978	−	0.423324i	$-0.139137\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	4.00000i	0.369800i
$118$	0	0
$119$	2.00000	0.183340
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 17.0000i	− 1.50851i	−0.656584	−	0.754253i	$-0.727999\pi$
$127$	− 17.0000i	− 1.50851i	0.656584	−	0.754253i	$-0.272001\pi$
$128$	0	0
$129$	5.00000	0.440225
$130$	0	0
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	0	0
$133$	− 4.00000i	− 0.346844i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 18.0000i	− 1.53784i	−0.639343	−	0.768922i	$-0.720793\pi$
$137$	− 18.0000i	− 1.53784i	0.639343	−	0.768922i	$-0.279207\pi$
$138$	0	0
$139$	14.0000	1.18746	0.593732	−	0.804663i	$-0.297654\pi$
$139$	14.0000	1.18746	0.593732	+	0.804663i	$0.297654\pi$
$140$	0	0
$141$	−4.00000	−0.336861
$142$	0	0
$143$	4.00000i	0.334497i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 1.00000i	− 0.0824786i
$148$	0	0
$149$	11.0000	0.901155	0.450578	−	0.892737i	$-0.351218\pi$
$149$	11.0000	0.901155	0.450578	+	0.892737i	$0.351218\pi$
$150$	0	0
$151$	13.0000	1.05792	0.528962	−	0.848645i	$-0.322581\pi$
$151$	13.0000	1.05792	0.528962	+	0.848645i	$0.322581\pi$
$152$	0	0
$153$	− 2.00000i	− 0.161690i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 4.00000i	− 0.319235i	−0.987179	−	0.159617i	$-0.948974\pi$
$157$	− 4.00000i	− 0.319235i	0.987179	−	0.159617i	$-0.0510260\pi$
$158$	0	0
$159$	6.00000	0.475831
$160$	0	0
$161$	3.00000	0.236433
$162$	0	0
$163$	20.0000i	1.56652i	0.621694	+	0.783260i	$0.286445\pi$
$163$	20.0000i	1.56652i	−0.621694	+	0.783260i	$0.713555\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	−3.00000	−0.230769
$170$	0	0
$171$	−4.00000	−0.305888
$172$	0	0
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	4.00000i	0.300658i
$178$	0	0
$179$	20.0000	1.49487	0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000	1.49487	0.747435	+	0.664335i	$0.231285\pi$
$180$	0	0
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	− 4.00000i	− 0.295689i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 2.00000i	− 0.146254i
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	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
$193$	1.00000i	0.0719816i	0.999352	+	0.0359908i	$0.0114587\pi$
$193$	1.00000i	0.0719816i	−0.999352	+	0.0359908i	$0.988541\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	9.00000i	0.641223i	0.947211	+	0.320612i	$0.103888\pi$
$197$	9.00000i	0.641223i	−0.947211	+	0.320612i	$0.896112\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	5.00000	0.352673
$202$	0	0
$203$	− 1.00000i	− 0.0701862i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 3.00000i	− 0.208514i
$208$	0	0
$209$	−4.00000	−0.276686
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	0	0
$213$	− 1.00000i	− 0.0685189i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	6.00000i	0.407307i
$218$	0	0
$219$	10.0000	0.675737
$220$	0	0
$221$	8.00000	0.538138
$222$	0	0
$223$	10.0000i	0.669650i	0.942280	+	0.334825i	$0.108677\pi$
$223$	10.0000i	0.669650i	−0.942280	+	0.334825i	$0.891323\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	8.00000i	0.530979i	0.964114	+	0.265489i	$0.0855335\pi$
$227$	8.00000i	0.530979i	−0.964114	+	0.265489i	$0.914466\pi$
$228$	0	0
$229$	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
$231$	−1.00000	−0.0657952
$232$	0	0
$233$	− 15.0000i	− 0.982683i	−0.870967	−	0.491341i	$-0.836507\pi$
$233$	− 15.0000i	− 0.982683i	0.870967	−	0.491341i	$-0.163493\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1.00000i	0.0649570i
$238$	0	0
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	0	0
$241$	−8.00000	−0.515325	−0.257663	−	0.966235i	$-0.582952\pi$
$241$	−8.00000	−0.515325	−0.257663	+	0.966235i	$0.582952\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 16.0000i	− 1.01806i
$248$	0	0
$249$	14.0000	0.887214
$250$	0	0
$251$	10.0000	0.631194	0.315597	−	0.948893i	$-0.397795\pi$
$251$	10.0000	0.631194	0.315597	+	0.948893i	$0.397795\pi$
$252$	0	0
$253$	− 3.00000i	− 0.188608i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 22.0000i	− 1.37232i	−0.727450	−	0.686161i	$-0.759294\pi$
$257$	− 22.0000i	− 1.37232i	0.727450	−	0.686161i	$-0.240706\pi$
$258$	0	0
$259$	−3.00000	−0.186411
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	0	0
$263$	21.0000i	1.29492i	0.762101	+	0.647458i	$0.224168\pi$
$263$	21.0000i	1.29492i	−0.762101	+	0.647458i	$0.775832\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	8.00000i	0.489592i
$268$	0	0
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	0	0
$271$	−24.0000	−1.45790	−0.728948	−	0.684569i	$-0.759990\pi$
$271$	−24.0000	−1.45790	−0.728948	+	0.684569i	$0.759990\pi$
$272$	0	0
$273$	− 4.00000i	− 0.242091i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	2.00000i	0.120168i	0.998193	+	0.0600842i	$0.0191369\pi$
$277$	2.00000i	0.120168i	−0.998193	+	0.0600842i	$0.980863\pi$
$278$	0	0
$279$	6.00000	0.359211
$280$	0	0
$281$	−25.0000	−1.49137	−0.745687	−	0.666296i	$-0.767879\pi$
$281$	−25.0000	−1.49137	−0.745687	+	0.666296i	$0.767879\pi$
$282$	0	0
$283$	− 14.0000i	− 0.832214i	−0.909316	−	0.416107i	$-0.863394\pi$
$283$	− 14.0000i	− 0.832214i	0.909316	−	0.416107i	$-0.136606\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	13.0000	0.764706
$290$	0	0
$291$	14.0000	0.820695
$292$	0	0
$293$	− 24.0000i	− 1.40209i	−0.713115	−	0.701047i	$-0.752716\pi$
$293$	− 24.0000i	− 1.40209i	0.713115	−	0.701047i	$-0.247284\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.00000i	0.0580259i
$298$	0	0
$299$	12.0000	0.693978
$300$	0	0
$301$	−5.00000	−0.288195
$302$	0	0
$303$	− 10.0000i	− 0.574485i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 10.0000i	− 0.570730i	−0.958419	−	0.285365i	$-0.907885\pi$
$307$	− 10.0000i	− 0.570730i	0.958419	−	0.285365i	$-0.0921148\pi$
$308$	0	0
$309$	−4.00000	−0.227552
$310$	0	0
$311$	30.0000	1.70114	0.850572	−	0.525859i	$-0.176256\pi$
$311$	30.0000	1.70114	0.850572	+	0.525859i	$0.176256\pi$
$312$	0	0
$313$	− 12.0000i	− 0.678280i	−0.940736	−	0.339140i	$-0.889864\pi$
$313$	− 12.0000i	− 0.678280i	0.940736	−	0.339140i	$-0.110136\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	17.0000i	0.954815i	0.878682	+	0.477408i	$0.158423\pi$
$317$	17.0000i	0.954815i	−0.878682	+	0.477408i	$0.841577\pi$
$318$	0	0
$319$	−1.00000	−0.0559893
$320$	0	0
$321$	4.00000	0.223258
$322$	0	0
$323$	8.00000i	0.445132i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 5.00000i	− 0.276501i
$328$	0	0
$329$	4.00000	0.220527
$330$	0	0
$331$	3.00000	0.164895	0.0824475	−	0.996595i	$-0.473726\pi$
$331$	3.00000	0.164895	0.0824475	+	0.996595i	$0.473726\pi$
$332$	0	0
$333$	3.00000i	0.164399i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 6.00000i	− 0.326841i	−0.986557	−	0.163420i	$-0.947747\pi$
$337$	− 6.00000i	− 0.326841i	0.986557	−	0.163420i	$-0.0522527\pi$
$338$	0	0
$339$	9.00000	0.488813
$340$	0	0
$341$	6.00000	0.324918
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	3.00000i	0.161048i	0.996753	+	0.0805242i	$0.0256594\pi$
$347$	3.00000i	0.161048i	−0.996753	+	0.0805242i	$0.974341\pi$
$348$	0	0
$349$	−28.0000	−1.49881	−0.749403	−	0.662114i	$-0.769659\pi$
$349$	−28.0000	−1.49881	−0.749403	+	0.662114i	$0.769659\pi$
$350$	0	0
$351$	−4.00000	−0.213504
$352$	0	0
$353$	− 34.0000i	− 1.80964i	−0.425797	−	0.904819i	$-0.640006\pi$
$353$	− 34.0000i	− 1.80964i	0.425797	−	0.904819i	$-0.359994\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	2.00000i	0.105851i
$358$	0	0
$359$	1.00000	0.0527780	0.0263890	−	0.999652i	$-0.491599\pi$
$359$	1.00000	0.0527780	0.0263890	+	0.999652i	$0.491599\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	− 10.0000i	− 0.524864i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 28.0000i	− 1.46159i	−0.682598	−	0.730794i	$-0.739150\pi$
$367$	− 28.0000i	− 1.46159i	0.682598	−	0.730794i	$-0.260850\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−6.00000	−0.311504
$372$	0	0
$373$	25.0000i	1.29445i	0.762299	+	0.647225i	$0.224071\pi$
$373$	25.0000i	1.29445i	−0.762299	+	0.647225i	$0.775929\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 4.00000i	− 0.206010i
$378$	0	0
$379$	13.0000	0.667765	0.333883	−	0.942615i	$-0.391641\pi$
$379$	13.0000	0.667765	0.333883	+	0.942615i	$0.391641\pi$
$380$	0	0
$381$	17.0000	0.870936
$382$	0	0
$383$	36.0000i	1.83951i	0.392488	+	0.919757i	$0.371614\pi$
$383$	36.0000i	1.83951i	−0.392488	+	0.919757i	$0.628386\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	5.00000i	0.254164i
$388$	0	0
$389$	21.0000	1.06474	0.532371	−	0.846511i	$-0.321301\pi$
$389$	21.0000	1.06474	0.532371	+	0.846511i	$0.321301\pi$
$390$	0	0
$391$	−6.00000	−0.303433
$392$	0	0
$393$	6.00000i	0.302660i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 30.0000i	− 1.50566i	−0.658217	−	0.752828i	$-0.728689\pi$
$397$	− 30.0000i	− 1.50566i	0.658217	−	0.752828i	$-0.271311\pi$
$398$	0	0
$399$	4.00000	0.200250
$400$	0	0
$401$	3.00000	0.149813	0.0749064	−	0.997191i	$-0.476134\pi$
$401$	3.00000	0.149813	0.0749064	+	0.997191i	$0.476134\pi$
$402$	0	0
$403$	24.0000i	1.19553i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.00000i	0.148704i
$408$	0	0
$409$	−32.0000	−1.58230	−0.791149	−	0.611623i	$-0.790517\pi$
$409$	−32.0000	−1.58230	−0.791149	+	0.611623i	$0.790517\pi$
$410$	0	0
$411$	18.0000	0.887875
$412$	0	0
$413$	− 4.00000i	− 0.196827i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	14.0000i	0.685583i
$418$	0	0
$419$	−10.0000	−0.488532	−0.244266	−	0.969708i	$-0.578547\pi$
$419$	−10.0000	−0.488532	−0.244266	+	0.969708i	$0.578547\pi$
$420$	0	0
$421$	−29.0000	−1.41337	−0.706687	−	0.707527i	$-0.749811\pi$
$421$	−29.0000	−1.41337	−0.706687	+	0.707527i	$0.749811\pi$
$422$	0	0
$423$	− 4.00000i	− 0.194487i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	4.00000i	0.193574i
$428$	0	0
$429$	−4.00000	−0.193122
$430$	0	0
$431$	16.0000	0.770693	0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000	0.770693	0.385346	+	0.922772i	$0.374082\pi$
$432$	0	0
$433$	− 16.0000i	− 0.768911i	−0.923144	−	0.384455i	$-0.874389\pi$
$433$	− 16.0000i	− 0.768911i	0.923144	−	0.384455i	$-0.125611\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	12.0000i	0.574038i
$438$	0	0
$439$	20.0000	0.954548	0.477274	−	0.878755i	$-0.341625\pi$
$439$	20.0000	0.954548	0.477274	+	0.878755i	$0.341625\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	20.0000i	0.950229i	0.879924	+	0.475114i	$0.157593\pi$
$443$	20.0000i	0.950229i	−0.879924	+	0.475114i	$0.842407\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	11.0000i	0.520282i
$448$	0	0
$449$	23.0000	1.08544	0.542719	−	0.839915i	$-0.317395\pi$
$449$	23.0000	1.08544	0.542719	+	0.839915i	$0.317395\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	13.0000i	0.610793i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	3.00000i	0.140334i	0.997535	+	0.0701670i	$0.0223532\pi$
$457$	3.00000i	0.140334i	−0.997535	+	0.0701670i	$0.977647\pi$
$458$	0	0
$459$	2.00000	0.0933520
$460$	0	0
$461$	−8.00000	−0.372597	−0.186299	−	0.982493i	$-0.559649\pi$
$461$	−8.00000	−0.372597	−0.186299	+	0.982493i	$0.559649\pi$
$462$	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
$467$	22.0000i	1.01804i	0.860755	+	0.509019i	$0.169992\pi$
$467$	22.0000i	1.01804i	−0.860755	+	0.509019i	$0.830008\pi$
$468$	0	0
$469$	−5.00000	−0.230879
$470$	0	0
$471$	4.00000	0.184310
$472$	0	0
$473$	5.00000i	0.229900i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	6.00000i	0.274721i
$478$	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
$481$	−12.0000	−0.547153
$482$	0	0
$483$	3.00000i	0.136505i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 43.0000i	− 1.94852i	−0.225436	−	0.974258i	$-0.572381\pi$
$487$	− 43.0000i	− 1.94852i	0.225436	−	0.974258i	$-0.427619\pi$
$488$	0	0
$489$	−20.0000	−0.904431
$490$	0	0
$491$	35.0000	1.57953	0.789764	−	0.613411i	$-0.210203\pi$
$491$	35.0000	1.57953	0.789764	+	0.613411i	$0.210203\pi$
$492$	0	0
$493$	2.00000i	0.0900755i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.00000i	0.0448561i
$498$	0	0
$499$	12.0000	0.537194	0.268597	−	0.963253i	$-0.413440\pi$
$499$	12.0000	0.537194	0.268597	+	0.963253i	$0.413440\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 6.00000i	− 0.267527i	−0.991013	−	0.133763i	$-0.957294\pi$
$503$	− 6.00000i	− 0.267527i	0.991013	−	0.133763i	$-0.0427062\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 3.00000i	− 0.133235i
$508$	0	0
$509$	−20.0000	−0.886484	−0.443242	−	0.896402i	$-0.646172\pi$
$509$	−20.0000	−0.886484	−0.443242	+	0.896402i	$0.646172\pi$
$510$	0	0
$511$	−10.0000	−0.442374
$512$	0	0
$513$	− 4.00000i	− 0.176604i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 4.00000i	− 0.175920i
$518$	0	0
$519$	0	0
$520$	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
$523$	8.00000i	0.349816i	0.984585	+	0.174908i	$0.0559627\pi$
$523$	8.00000i	0.349816i	−0.984585	+	0.174908i	$0.944037\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 12.0000i	− 0.522728i
$528$	0	0
$529$	14.0000	0.608696
$530$	0	0
$531$	−4.00000	−0.173585
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	20.0000i	0.863064i
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	19.0000	0.816874	0.408437	−	0.912787i	$-0.366074\pi$
$541$	19.0000	0.816874	0.408437	+	0.912787i	$0.366074\pi$
$542$	0	0
$543$	− 2.00000i	− 0.0858282i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 1.00000i	− 0.0427569i	−0.999771	−	0.0213785i	$-0.993195\pi$
$547$	− 1.00000i	− 0.0427569i	0.999771	−	0.0213785i	$-0.00680549\pi$
$548$	0	0
$549$	4.00000	0.170716
$550$	0	0
$551$	4.00000	0.170406
$552$	0	0
$553$	− 1.00000i	− 0.0425243i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	9.00000i	0.381342i	0.981654	+	0.190671i	$0.0610664\pi$
$557$	9.00000i	0.381342i	−0.981654	+	0.190671i	$0.938934\pi$
$558$	0	0
$559$	−20.0000	−0.845910
$560$	0	0
$561$	2.00000	0.0844401
$562$	0	0
$563$	− 14.0000i	− 0.590030i	−0.955493	−	0.295015i	$-0.904675\pi$
$563$	− 14.0000i	− 0.590030i	0.955493	−	0.295015i	$-0.0953246\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 1.00000i	− 0.0419961i
$568$	0	0
$569$	1.00000	0.0419222	0.0209611	−	0.999780i	$-0.493327\pi$
$569$	1.00000	0.0419222	0.0209611	+	0.999780i	$0.493327\pi$
$570$	0	0
$571$	11.0000	0.460336	0.230168	−	0.973151i	$-0.426072\pi$
$571$	11.0000	0.460336	0.230168	+	0.973151i	$0.426072\pi$
$572$	0	0
$573$	− 12.0000i	− 0.501307i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	16.0000i	0.666089i	0.942911	+	0.333044i	$0.108076\pi$
$577$	16.0000i	0.666089i	−0.942911	+	0.333044i	$0.891924\pi$
$578$	0	0
$579$	−1.00000	−0.0415586
$580$	0	0
$581$	−14.0000	−0.580818
$582$	0	0
$583$	6.00000i	0.248495i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	30.0000i	1.23823i	0.785299	+	0.619116i	$0.212509\pi$
$587$	30.0000i	1.23823i	−0.785299	+	0.619116i	$0.787491\pi$
$588$	0	0
$589$	−24.0000	−0.988903
$590$	0	0
$591$	−9.00000	−0.370211
$592$	0	0
$593$	− 4.00000i	− 0.164260i	−0.996622	−	0.0821302i	$-0.973828\pi$
$593$	− 4.00000i	− 0.164260i	0.996622	−	0.0821302i	$-0.0261723\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−17.0000	−0.694601	−0.347301	−	0.937754i	$-0.612902\pi$
$599$	−17.0000	−0.694601	−0.347301	+	0.937754i	$0.612902\pi$
$600$	0	0
$601$	−20.0000	−0.815817	−0.407909	−	0.913023i	$-0.633742\pi$
$601$	−20.0000	−0.815817	−0.407909	+	0.913023i	$0.633742\pi$
$602$	0	0
$603$	5.00000i	0.203616i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 16.0000i	− 0.649420i	−0.945814	−	0.324710i	$-0.894733\pi$
$607$	− 16.0000i	− 0.649420i	0.945814	−	0.324710i	$-0.105267\pi$
$608$	0	0
$609$	1.00000	0.0405220
$610$	0	0
$611$	16.0000	0.647291
$612$	0	0
$613$	39.0000i	1.57520i	0.616190	+	0.787598i	$0.288675\pi$
$613$	39.0000i	1.57520i	−0.616190	+	0.787598i	$0.711325\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	27.0000i	1.08698i	0.839416	+	0.543490i	$0.182897\pi$
$617$	27.0000i	1.08698i	−0.839416	+	0.543490i	$0.817103\pi$
$618$	0	0
$619$	−32.0000	−1.28619	−0.643094	−	0.765787i	$-0.722350\pi$
$619$	−32.0000	−1.28619	−0.643094	+	0.765787i	$0.722350\pi$
$620$	0	0
$621$	3.00000	0.120386
$622$	0	0
$623$	− 8.00000i	− 0.320513i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 4.00000i	− 0.159745i
$628$	0	0
$629$	6.00000	0.239236
$630$	0	0
$631$	−13.0000	−0.517522	−0.258761	−	0.965941i	$-0.583314\pi$
$631$	−13.0000	−0.517522	−0.258761	+	0.965941i	$0.583314\pi$
$632$	0	0
$633$	− 4.00000i	− 0.158986i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	4.00000i	0.158486i
$638$	0	0
$639$	1.00000	0.0395594
$640$	0	0
$641$	27.0000	1.06644	0.533218	−	0.845978i	$-0.320983\pi$
$641$	27.0000	1.06644	0.533218	+	0.845978i	$0.320983\pi$
$642$	0	0
$643$	− 20.0000i	− 0.788723i	−0.918955	−	0.394362i	$-0.870966\pi$
$643$	− 20.0000i	− 0.788723i	0.918955	−	0.394362i	$-0.129034\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 22.0000i	− 0.864909i	−0.901656	−	0.432455i	$-0.857648\pi$
$647$	− 22.0000i	− 0.864909i	0.901656	−	0.432455i	$-0.142352\pi$
$648$	0	0
$649$	−4.00000	−0.157014
$650$	0	0
$651$	−6.00000	−0.235159
$652$	0	0
$653$	− 26.0000i	− 1.01746i	−0.860927	−	0.508729i	$-0.830115\pi$
$653$	− 26.0000i	− 1.01746i	0.860927	−	0.508729i	$-0.169885\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	10.0000i	0.390137i
$658$	0	0
$659$	−44.0000	−1.71400	−0.856998	−	0.515319i	$-0.827673\pi$
$659$	−44.0000	−1.71400	−0.856998	+	0.515319i	$0.827673\pi$
$660$	0	0
$661$	−8.00000	−0.311164	−0.155582	−	0.987823i	$-0.549725\pi$
$661$	−8.00000	−0.311164	−0.155582	+	0.987823i	$0.549725\pi$
$662$	0	0
$663$	8.00000i	0.310694i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	3.00000i	0.116160i
$668$	0	0
$669$	−10.0000	−0.386622
$670$	0	0
$671$	4.00000	0.154418
$672$	0	0
$673$	30.0000i	1.15642i	0.815890	+	0.578208i	$0.196248\pi$
$673$	30.0000i	1.15642i	−0.815890	+	0.578208i	$0.803752\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	18.0000i	0.691796i	0.938272	+	0.345898i	$0.112426\pi$
$677$	18.0000i	0.691796i	−0.938272	+	0.345898i	$0.887574\pi$
$678$	0	0
$679$	−14.0000	−0.537271
$680$	0	0
$681$	−8.00000	−0.306561
$682$	0	0
$683$	29.0000i	1.10965i	0.831966	+	0.554827i	$0.187216\pi$
$683$	29.0000i	1.10965i	−0.831966	+	0.554827i	$0.812784\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	2.00000i	0.0763048i
$688$	0	0
$689$	−24.0000	−0.914327
$690$	0	0
$691$	12.0000	0.456502	0.228251	−	0.973602i	$-0.426699\pi$
$691$	12.0000	0.456502	0.228251	+	0.973602i	$0.426699\pi$
$692$	0	0
$693$	− 1.00000i	− 0.0379869i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	15.0000	0.567352
$700$	0	0
$701$	−46.0000	−1.73740	−0.868698	−	0.495342i	$-0.835043\pi$
$701$	−46.0000	−1.73740	−0.868698	+	0.495342i	$0.835043\pi$
$702$	0	0
$703$	− 12.0000i	− 0.452589i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	10.0000i	0.376089i
$708$	0	0
$709$	−38.0000	−1.42712	−0.713560	−	0.700594i	$-0.752918\pi$
$709$	−38.0000	−1.42712	−0.713560	+	0.700594i	$0.752918\pi$
$710$	0	0
$711$	−1.00000	−0.0375029
$712$	0	0
$713$	− 18.0000i	− 0.674105i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	8.00000i	0.298765i
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	4.00000	0.148968
$722$	0	0
$723$	− 8.00000i	− 0.297523i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 6.00000i	− 0.222528i	−0.993791	−	0.111264i	$-0.964510\pi$
$727$	− 6.00000i	− 0.222528i	0.993791	−	0.111264i	$-0.0354899\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	10.0000	0.369863
$732$	0	0
$733$	4.00000i	0.147743i	0.997268	+	0.0738717i	$0.0235355\pi$
$733$	4.00000i	0.147743i	−0.997268	+	0.0738717i	$0.976464\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	5.00000i	0.184177i
$738$	0	0
$739$	−5.00000	−0.183928	−0.0919640	−	0.995762i	$-0.529314\pi$
$739$	−5.00000	−0.183928	−0.0919640	+	0.995762i	$0.529314\pi$
$740$	0	0
$741$	16.0000	0.587775
$742$	0	0
$743$	− 32.0000i	− 1.17397i	−0.809599	−	0.586983i	$-0.800316\pi$
$743$	− 32.0000i	− 1.17397i	0.809599	−	0.586983i	$-0.199684\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	14.0000i	0.512233i
$748$	0	0
$749$	−4.00000	−0.146157
$750$	0	0
$751$	−8.00000	−0.291924	−0.145962	−	0.989290i	$-0.546628\pi$
$751$	−8.00000	−0.291924	−0.145962	+	0.989290i	$0.546628\pi$
$752$	0	0
$753$	10.0000i	0.364420i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	13.0000i	0.472493i	0.971693	+	0.236247i	$0.0759173\pi$
$757$	13.0000i	0.472493i	−0.971693	+	0.236247i	$0.924083\pi$
$758$	0	0
$759$	3.00000	0.108893
$760$	0	0
$761$	12.0000	0.435000	0.217500	−	0.976060i	$-0.430210\pi$
$761$	12.0000	0.435000	0.217500	+	0.976060i	$0.430210\pi$
$762$	0	0
$763$	5.00000i	0.181012i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 16.0000i	− 0.577727i
$768$	0	0
$769$	−32.0000	−1.15395	−0.576975	−	0.816762i	$-0.695767\pi$
$769$	−32.0000	−1.15395	−0.576975	+	0.816762i	$0.695767\pi$
$770$	0	0
$771$	22.0000	0.792311
$772$	0	0
$773$	18.0000i	0.647415i	0.946157	+	0.323708i	$0.104929\pi$
$773$	18.0000i	0.647415i	−0.946157	+	0.323708i	$0.895071\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 3.00000i	− 0.107624i
$778$	0	0
$779$	0	0
$780$	0	0
$781$	1.00000	0.0357828
$782$	0	0
$783$	− 1.00000i	− 0.0357371i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	14.0000i	0.499046i	0.968369	+	0.249523i	$0.0802738\pi$
$787$	14.0000i	0.499046i	−0.968369	+	0.249523i	$0.919726\pi$
$788$	0	0
$789$	−21.0000	−0.747620
$790$	0	0
$791$	−9.00000	−0.320003
$792$	0	0
$793$	16.0000i	0.568177i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	18.0000i	0.637593i	0.947823	+	0.318796i	$0.103279\pi$
$797$	18.0000i	0.637593i	−0.947823	+	0.318796i	$0.896721\pi$
$798$	0	0
$799$	−8.00000	−0.283020
$800$	0	0
$801$	−8.00000	−0.282666
$802$	0	0
$803$	10.0000i	0.352892i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	6.00000i	0.211210i
$808$	0	0
$809$	51.0000	1.79306	0.896532	−	0.442978i	$-0.146078\pi$
$809$	51.0000	1.79306	0.896532	+	0.442978i	$0.146078\pi$
$810$	0	0
$811$	−2.00000	−0.0702295	−0.0351147	−	0.999383i	$-0.511180\pi$
$811$	−2.00000	−0.0702295	−0.0351147	+	0.999383i	$0.511180\pi$
$812$	0	0
$813$	− 24.0000i	− 0.841717i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 20.0000i	− 0.699711i
$818$	0	0
$819$	4.00000	0.139771
$820$	0	0
$821$	10.0000	0.349002	0.174501	−	0.984657i	$-0.444169\pi$
$821$	10.0000	0.349002	0.174501	+	0.984657i	$0.444169\pi$
$822$	0	0
$823$	45.0000i	1.56860i	0.620381	+	0.784301i	$0.286978\pi$
$823$	45.0000i	1.56860i	−0.620381	+	0.784301i	$0.713022\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	23.0000i	0.799788i	0.916561	+	0.399894i	$0.130953\pi$
$827$	23.0000i	0.799788i	−0.916561	+	0.399894i	$0.869047\pi$
$828$	0	0
$829$	−30.0000	−1.04194	−0.520972	−	0.853574i	$-0.674430\pi$
$829$	−30.0000	−1.04194	−0.520972	+	0.853574i	$0.674430\pi$
$830$	0	0
$831$	−2.00000	−0.0693792
$832$	0	0
$833$	− 2.00000i	− 0.0692959i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	6.00000i	0.207390i
$838$	0	0
$839$	−34.0000	−1.17381	−0.586905	−	0.809656i	$-0.699654\pi$
$839$	−34.0000	−1.17381	−0.586905	+	0.809656i	$0.699654\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	0	0
$843$	− 25.0000i	− 0.861046i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	10.0000i	0.343604i
$848$	0	0
$849$	14.0000	0.480479
$850$	0	0
$851$	9.00000	0.308516
$852$	0	0
$853$	− 44.0000i	− 1.50653i	−0.657716	−	0.753266i	$-0.728477\pi$
$853$	− 44.0000i	− 1.50653i	0.657716	−	0.753266i	$-0.271523\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	20.0000i	0.683187i	0.939848	+	0.341593i	$0.110967\pi$
$857$	20.0000i	0.683187i	−0.939848	+	0.341593i	$0.889033\pi$
$858$	0	0
$859$	38.0000	1.29654	0.648272	−	0.761409i	$-0.275492\pi$
$859$	38.0000	1.29654	0.648272	+	0.761409i	$0.275492\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	29.0000i	0.987171i	0.869697	+	0.493586i	$0.164314\pi$
$863$	29.0000i	0.987171i	−0.869697	+	0.493586i	$0.835686\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	13.0000i	0.441503i
$868$	0	0
$869$	−1.00000	−0.0339227
$870$	0	0
$871$	−20.0000	−0.677674
$872$	0	0
$873$	14.0000i	0.473828i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 18.0000i	− 0.607817i	−0.952701	−	0.303908i	$-0.901708\pi$
$877$	− 18.0000i	− 0.607817i	0.952701	−	0.303908i	$-0.0982917\pi$
$878$	0	0
$879$	24.0000	0.809500
$880$	0	0
$881$	26.0000	0.875962	0.437981	−	0.898984i	$-0.355694\pi$
$881$	26.0000	0.875962	0.437981	+	0.898984i	$0.355694\pi$
$882$	0	0
$883$	− 11.0000i	− 0.370179i	−0.982722	−	0.185090i	$-0.940742\pi$
$883$	− 11.0000i	− 0.370179i	0.982722	−	0.185090i	$-0.0592576\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	52.0000i	1.74599i	0.487730	+	0.872995i	$0.337825\pi$
$887$	52.0000i	1.74599i	−0.487730	+	0.872995i	$0.662175\pi$
$888$	0	0
$889$	−17.0000	−0.570162
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	16.0000i	0.535420i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	12.0000i	0.400668i
$898$	0	0
$899$	−6.00000	−0.200111
$900$	0	0
$901$	12.0000	0.399778
$902$	0	0
$903$	− 5.00000i	− 0.166390i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 16.0000i	− 0.531271i	−0.964073	−	0.265636i	$-0.914418\pi$
$907$	− 16.0000i	− 0.531271i	0.964073	−	0.265636i	$-0.0855818\pi$
$908$	0	0
$909$	10.0000	0.331679
$910$	0	0
$911$	13.0000	0.430709	0.215355	−	0.976536i	$-0.430909\pi$
$911$	13.0000	0.430709	0.215355	+	0.976536i	$0.430909\pi$
$912$	0	0
$913$	14.0000i	0.463332i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 6.00000i	− 0.198137i
$918$	0	0
$919$	−43.0000	−1.41844	−0.709220	−	0.704988i	$-0.750953\pi$
$919$	−43.0000	−1.41844	−0.709220	+	0.704988i	$0.750953\pi$
$920$	0	0
$921$	10.0000	0.329511
$922$	0	0
$923$	4.00000i	0.131662i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 4.00000i	− 0.131377i
$928$	0	0
$929$	−14.0000	−0.459325	−0.229663	−	0.973270i	$-0.573762\pi$
$929$	−14.0000	−0.459325	−0.229663	+	0.973270i	$0.573762\pi$
$930$	0	0
$931$	−4.00000	−0.131095
$932$	0	0
$933$	30.0000i	0.982156i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 4.00000i	− 0.130674i	−0.997863	−	0.0653372i	$-0.979188\pi$
$937$	− 4.00000i	− 0.130674i	0.997863	−	0.0653372i	$-0.0208123\pi$
$938$	0	0
$939$	12.0000	0.391605
$940$	0	0
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	60.0000i	1.94974i	0.222779	+	0.974869i	$0.428487\pi$
$947$	60.0000i	1.94974i	−0.222779	+	0.974869i	$0.571513\pi$
$948$	0	0
$949$	−40.0000	−1.29845
$950$	0	0
$951$	−17.0000	−0.551263
$952$	0	0
$953$	15.0000i	0.485898i	0.970039	+	0.242949i	$0.0781147\pi$
$953$	15.0000i	0.485898i	−0.970039	+	0.242949i	$0.921885\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 1.00000i	− 0.0323254i
$958$	0	0
$959$	−18.0000	−0.581250
$960$	0	0
$961$	5.00000	0.161290
$962$	0	0
$963$	4.00000i	0.128898i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	8.00000i	0.257263i	0.991692	+	0.128631i	$0.0410584\pi$
$967$	8.00000i	0.257263i	−0.991692	+	0.128631i	$0.958942\pi$
$968$	0	0
$969$	−8.00000	−0.256997
$970$	0	0
$971$	32.0000	1.02693	0.513464	−	0.858111i	$-0.328362\pi$
$971$	32.0000	1.02693	0.513464	+	0.858111i	$0.328362\pi$
$972$	0	0
$973$	− 14.0000i	− 0.448819i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 45.0000i	− 1.43968i	−0.694141	−	0.719839i	$-0.744216\pi$
$977$	− 45.0000i	− 1.43968i	0.694141	−	0.719839i	$-0.255784\pi$
$978$	0	0
$979$	−8.00000	−0.255681
$980$	0	0
$981$	5.00000	0.159638
$982$	0	0
$983$	6.00000i	0.191370i	0.995412	+	0.0956851i	$0.0305042\pi$
$983$	6.00000i	0.191370i	−0.995412	+	0.0956851i	$0.969496\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	4.00000i	0.127321i
$988$	0	0
$989$	15.0000	0.476972
$990$	0	0
$991$	35.0000	1.11181	0.555906	−	0.831245i	$-0.312372\pi$
$991$	35.0000	1.11181	0.555906	+	0.831245i	$0.312372\pi$
$992$	0	0
$993$	3.00000i	0.0952021i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	28.0000i	0.886769i	0.896332	+	0.443384i	$0.146222\pi$
$997$	28.0000i	0.886769i	−0.896332	+	0.443384i	$0.853778\pi$
$998$	0	0
$999$	−3.00000	−0.0949158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4200.2.t.h.1849.2		2
5.2	odd	4		4200.2.a.bd.1.1	yes	1
5.3	odd	4		4200.2.a.b.1.1	✓	1
5.4	even	2	inner	4200.2.t.h.1849.1		2
20.3	even	4		8400.2.a.cq.1.1		1
20.7	even	4		8400.2.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4200.2.a.b.1.1	✓	1	5.3	odd	4
4200.2.a.bd.1.1	yes	1	5.2	odd	4
4200.2.t.h.1849.1		2	5.4	even	2	inner
4200.2.t.h.1849.2		2	1.1	even	1	trivial
8400.2.a.g.1.1		1	20.7	even	4
8400.2.a.cq.1.1		1	20.3	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:	\( N \)	\(=\)	\( 4200 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4200.t (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -1.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} -1.00000i q^{7} -1.00000 q^{9} -1.00000 q^{11} -4.00000i q^{13} +2.00000i q^{17} +4.00000 q^{19} +1.00000 q^{21} +3.00000i q^{23} -1.00000i q^{27} +1.00000 q^{29} -6.00000 q^{31} -1.00000i q^{33} -3.00000i q^{37} +4.00000 q^{39} -5.00000i q^{43} +4.00000i q^{47} -1.00000 q^{49} -2.00000 q^{51} -6.00000i q^{53} +4.00000i q^{57} +4.00000 q^{59} -4.00000 q^{61} +1.00000i q^{63} -5.00000i q^{67} -3.00000 q^{69} -1.00000 q^{71} -10.0000i q^{73} +1.00000i q^{77} +1.00000 q^{79} +1.00000 q^{81} -14.0000i q^{83} +1.00000i q^{87} +8.00000 q^{89} -4.00000 q^{91} -6.00000i q^{93} -14.0000i q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^9	\( 2 q - 2 q^{9} - 2 q^{11} + 8 q^{19} + 2 q^{21} + 2 q^{29} - 12 q^{31} + 8 q^{39} - 2 q^{49} - 4 q^{51} + 8 q^{59} - 8 q^{61} - 6 q^{69} - 2 q^{71} + 2 q^{79} + 2 q^{81} + 16 q^{89} - 8 q^{91} + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^9 - 2 * q^11 + 8 * q^19 + 2 * q^21 + 2 * q^29 - 12 * q^31 + 8 * q^39 - 2 * q^49 - 4 * q^51 + 8 * q^59 - 8 * q^61 - 6 * q^69 - 2 * q^71 + 2 * q^79 + 2 * q^81 + 16 * q^89 - 8 * q^91 + 2 * q^99

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