LMFDB - Embedded newform 4600.2.e.i.4049.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4600,2,Mod(4049,4600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4600, 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("4600.4049");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4600 = 2^{3} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4600.e (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:	$36.7311849298$
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			4049.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	4600.4049
Dual form			4600.2.e.i.4049.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +4.00000i q^{7} +2.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +4.00000i q^{7} +2.00000 q^{9} +3.00000 q^{11} +2.00000i q^{13} -1.00000i q^{17} +1.00000 q^{19} +4.00000 q^{21} -1.00000i q^{23} -5.00000i q^{27} +8.00000 q^{31} -3.00000i q^{33} -2.00000i q^{37} +2.00000 q^{39} +1.00000 q^{41} -12.0000i q^{43} +6.00000i q^{47} -9.00000 q^{49} -1.00000 q^{51} +4.00000i q^{53} -1.00000i q^{57} -12.0000 q^{59} +8.00000i q^{63} +13.0000i q^{67} -1.00000 q^{69} +12.0000 q^{71} +17.0000i q^{73} +12.0000i q^{77} +14.0000 q^{79} +1.00000 q^{81} -5.00000i q^{83} -1.00000 q^{89} -8.00000 q^{91} -8.00000i q^{93} -2.00000i q^{97} +6.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{9}+O(q^{10}) $ 2 * q + 4 * q^9	$ 2 q + 4 q^{9} + 6 q^{11} + 2 q^{19} + 8 q^{21} + 16 q^{31} + 4 q^{39} + 2 q^{41} - 18 q^{49} - 2 q^{51} - 24 q^{59} - 2 q^{69} + 24 q^{71} + 28 q^{79} + 2 q^{81} - 2 q^{89} - 16 q^{91} + 12 q^{99}+O(q^{100}) $ 2 * q + 4 * q^9 + 6 * q^11 + 2 * q^19 + 8 * q^21 + 16 * q^31 + 4 * q^39 + 2 * q^41 - 18 * q^49 - 2 * q^51 - 24 * q^59 - 2 * q^69 + 24 * q^71 + 28 * q^79 + 2 * q^81 - 2 * q^89 - 16 * q^91 + 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times$.

$n$	$1151$	$1201$	$2301$	$2577$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	− 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
$7$	4.00000i	1.51186i	0.654654	+	0.755929i	$0.272814\pi$
$7$	4.00000i	1.51186i	−0.654654	+	0.755929i	$0.727186\pi$
$8$	0	0
$9$	2.00000	0.666667
$10$	0	0
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	0	0
$13$	2.00000i	0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$	2.00000i	0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 1.00000i	− 0.242536i	−0.992620	−	0.121268i	$-0.961304\pi$
$17$	− 1.00000i	− 0.242536i	0.992620	−	0.121268i	$-0.0386960\pi$
$18$	0	0
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	0	0
$21$	4.00000	0.872872
$22$	0	0
$23$	− 1.00000i	− 0.208514i
$23$	− 1.00000i	− 0.208514i
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 5.00000i	− 0.962250i
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	0	0
$33$	− 3.00000i	− 0.522233i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 2.00000i	− 0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	− 2.00000i	− 0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	0	0
$41$	1.00000	0.156174	0.0780869	−	0.996947i	$-0.475119\pi$
$41$	1.00000	0.156174	0.0780869	+	0.996947i	$0.475119\pi$
$42$	0	0
$43$	− 12.0000i	− 1.82998i	−0.403473	−	0.914991i	$-0.632197\pi$
$43$	− 12.0000i	− 1.82998i	0.403473	−	0.914991i	$-0.367803\pi$
$44$	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
$49$	−9.00000	−1.28571
$50$	0	0
$51$	−1.00000	−0.140028
$52$	0	0
$53$	4.00000i	0.549442i	0.961524	+	0.274721i	$0.0885855\pi$
$53$	4.00000i	0.549442i	−0.961524	+	0.274721i	$0.911414\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 1.00000i	− 0.132453i
$58$	0	0
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	0	0
$63$	8.00000i	1.00791i
$64$	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
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	0	0
$73$	17.0000i	1.98970i	0.101361	+	0.994850i	$0.467680\pi$
$73$	17.0000i	1.98970i	−0.101361	+	0.994850i	$0.532320\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	12.0000i	1.36753i
$78$	0	0
$79$	14.0000	1.57512	0.787562	−	0.616236i	$-0.211343\pi$
$79$	14.0000	1.57512	0.787562	+	0.616236i	$0.211343\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 5.00000i	− 0.548821i	−0.961613	−	0.274411i	$-0.911517\pi$
$83$	− 5.00000i	− 0.548821i	0.961613	−	0.274411i	$-0.0884828\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1.00000	−0.106000	−0.0529999	−	0.998595i	$-0.516878\pi$
$89$	−1.00000	−0.106000	−0.0529999	+	0.998595i	$0.516878\pi$
$90$	0	0
$91$	−8.00000	−0.838628
$92$	0	0
$93$	− 8.00000i	− 0.829561i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 2.00000i	− 0.203069i	−0.994832	−	0.101535i	$-0.967625\pi$
$97$	− 2.00000i	− 0.203069i	0.994832	−	0.101535i	$-0.0323753\pi$
$98$	0	0
$99$	6.00000	0.603023
$100$	0	0
$101$	−4.00000	−0.398015	−0.199007	−	0.979998i	$-0.563772\pi$
$101$	−4.00000	−0.398015	−0.199007	+	0.979998i	$0.563772\pi$
$102$	0	0
$103$	− 14.0000i	− 1.37946i	−0.724066	−	0.689730i	$-0.757729\pi$
$103$	− 14.0000i	− 1.37946i	0.724066	−	0.689730i	$-0.242271\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	19.0000i	1.83680i	0.395654	+	0.918400i	$0.370518\pi$
$107$	19.0000i	1.83680i	−0.395654	+	0.918400i	$0.629482\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	0	0
$113$	3.00000i	0.282216i	0.989994	+	0.141108i	$0.0450665\pi$
$113$	3.00000i	0.282216i	−0.989994	+	0.141108i	$0.954933\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	4.00000i	0.369800i
$118$	0	0
$119$	4.00000	0.366679
$120$	0	0
$121$	−2.00000	−0.181818
$122$	0	0
$123$	− 1.00000i	− 0.0901670i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 8.00000i	− 0.709885i	−0.934888	−	0.354943i	$-0.884500\pi$
$127$	− 8.00000i	− 0.709885i	0.934888	−	0.354943i	$-0.115500\pi$
$128$	0	0
$129$	−12.0000	−1.05654
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	0	0
$133$	4.00000i	0.346844i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	15.0000i	1.28154i	0.767734	+	0.640768i	$0.221384\pi$
$137$	15.0000i	1.28154i	−0.767734	+	0.640768i	$0.778616\pi$
$138$	0	0
$139$	−1.00000	−0.0848189	−0.0424094	−	0.999100i	$-0.513503\pi$
$139$	−1.00000	−0.0848189	−0.0424094	+	0.999100i	$0.513503\pi$
$140$	0	0
$141$	6.00000	0.505291
$142$	0	0
$143$	6.00000i	0.501745i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	9.00000i	0.742307i
$148$	0	0
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	0	0
$153$	− 2.00000i	− 0.161690i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	10.0000i	0.798087i	0.916932	+	0.399043i	$0.130658\pi$
$157$	10.0000i	0.798087i	−0.916932	+	0.399043i	$0.869342\pi$
$158$	0	0
$159$	4.00000	0.317221
$160$	0	0
$161$	4.00000	0.315244
$162$	0	0
$163$	1.00000i	0.0783260i	0.999233	+	0.0391630i	$0.0124692\pi$
$163$	1.00000i	0.0783260i	−0.999233	+	0.0391630i	$0.987531\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 6.00000i	− 0.464294i	−0.972681	−	0.232147i	$-0.925425\pi$
$167$	− 6.00000i	− 0.464294i	0.972681	−	0.232147i	$-0.0745750\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	2.00000	0.152944
$172$	0	0
$173$	4.00000i	0.304114i	0.988372	+	0.152057i	$0.0485898\pi$
$173$	4.00000i	0.304114i	−0.988372	+	0.152057i	$0.951410\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	12.0000i	0.901975i
$178$	0	0
$179$	15.0000	1.12115	0.560576	−	0.828103i	$-0.310580\pi$
$179$	15.0000	1.12115	0.560576	+	0.828103i	$0.310580\pi$
$180$	0	0
$181$	18.0000	1.33793	0.668965	−	0.743294i	$-0.266738\pi$
$181$	18.0000	1.33793	0.668965	+	0.743294i	$0.266738\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 3.00000i	− 0.219382i
$188$	0	0
$189$	20.0000	1.45479
$190$	0	0
$191$	−4.00000	−0.289430	−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000	−0.289430	−0.144715	+	0.989473i	$0.546227\pi$
$192$	0	0
$193$	− 9.00000i	− 0.647834i	−0.946085	−	0.323917i	$-0.895000\pi$
$193$	− 9.00000i	− 0.647834i	0.946085	−	0.323917i	$-0.105000\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	8.00000i	0.569976i	0.958531	+	0.284988i	$0.0919897\pi$
$197$	8.00000i	0.569976i	−0.958531	+	0.284988i	$0.908010\pi$
$198$	0	0
$199$	14.0000	0.992434	0.496217	−	0.868199i	$-0.334722\pi$
$199$	14.0000	0.992434	0.496217	+	0.868199i	$0.334722\pi$
$200$	0	0
$201$	13.0000	0.916949
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 2.00000i	− 0.139010i
$208$	0	0
$209$	3.00000	0.207514
$210$	0	0
$211$	−5.00000	−0.344214	−0.172107	−	0.985078i	$-0.555058\pi$
$211$	−5.00000	−0.344214	−0.172107	+	0.985078i	$0.555058\pi$
$212$	0	0
$213$	− 12.0000i	− 0.822226i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	32.0000i	2.17230i
$218$	0	0
$219$	17.0000	1.14875
$220$	0	0
$221$	2.00000	0.134535
$222$	0	0
$223$	− 16.0000i	− 1.07144i	−0.844396	−	0.535720i	$-0.820040\pi$
$223$	− 16.0000i	− 1.07144i	0.844396	−	0.535720i	$-0.179960\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 8.00000i	− 0.530979i	−0.964114	−	0.265489i	$-0.914466\pi$
$227$	− 8.00000i	− 0.530979i	0.964114	−	0.265489i	$-0.0855335\pi$
$228$	0	0
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	0	0
$231$	12.0000	0.789542
$232$	0	0
$233$	18.0000i	1.17922i	0.807688	+	0.589610i	$0.200718\pi$
$233$	18.0000i	1.17922i	−0.807688	+	0.589610i	$0.799282\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 14.0000i	− 0.909398i
$238$	0	0
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	0	0
$241$	−5.00000	−0.322078	−0.161039	−	0.986948i	$-0.551485\pi$
$241$	−5.00000	−0.322078	−0.161039	+	0.986948i	$0.551485\pi$
$242$	0	0
$243$	− 16.0000i	− 1.02640i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.00000i	0.127257i
$248$	0	0
$249$	−5.00000	−0.316862
$250$	0	0
$251$	−15.0000	−0.946792	−0.473396	−	0.880850i	$-0.656972\pi$
$251$	−15.0000	−0.946792	−0.473396	+	0.880850i	$0.656972\pi$
$252$	0	0
$253$	− 3.00000i	− 0.188608i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	30.0000i	1.87135i	0.352865	+	0.935674i	$0.385208\pi$
$257$	30.0000i	1.87135i	−0.352865	+	0.935674i	$0.614792\pi$
$258$	0	0
$259$	8.00000	0.497096
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 4.00000i	− 0.246651i	−0.992366	−	0.123325i	$-0.960644\pi$
$263$	− 4.00000i	− 0.246651i	0.992366	−	0.123325i	$-0.0393559\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	1.00000i	0.0611990i
$268$	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
$271$	2.00000	0.121491	0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000	0.121491	0.0607457	+	0.998153i	$0.480652\pi$
$272$	0	0
$273$	8.00000i	0.484182i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 12.0000i	− 0.721010i	−0.932757	−	0.360505i	$-0.882604\pi$
$277$	− 12.0000i	− 0.721010i	0.932757	−	0.360505i	$-0.117396\pi$
$278$	0	0
$279$	16.0000	0.957895
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	0	0
$283$	− 17.0000i	− 1.01055i	−0.862960	−	0.505273i	$-0.831392\pi$
$283$	− 17.0000i	− 1.01055i	0.862960	−	0.505273i	$-0.168608\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.00000i	0.236113i
$288$	0	0
$289$	16.0000	0.941176
$290$	0	0
$291$	−2.00000	−0.117242
$292$	0	0
$293$	16.0000i	0.934730i	0.884064	+	0.467365i	$0.154797\pi$
$293$	16.0000i	0.934730i	−0.884064	+	0.467365i	$0.845203\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 15.0000i	− 0.870388i
$298$	0	0
$299$	2.00000	0.115663
$300$	0	0
$301$	48.0000	2.76667
$302$	0	0
$303$	4.00000i	0.229794i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 19.0000i	− 1.08439i	−0.840254	−	0.542194i	$-0.817594\pi$
$307$	− 19.0000i	− 1.08439i	0.840254	−	0.542194i	$-0.182406\pi$
$308$	0	0
$309$	−14.0000	−0.796432
$310$	0	0
$311$	18.0000	1.02069	0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000	1.02069	0.510343	+	0.859971i	$0.329518\pi$
$312$	0	0
$313$	− 26.0000i	− 1.46961i	−0.678280	−	0.734803i	$-0.737274\pi$
$313$	− 26.0000i	− 1.46961i	0.678280	−	0.734803i	$-0.262726\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	20.0000i	1.12331i	0.827371	+	0.561656i	$0.189836\pi$
$317$	20.0000i	1.12331i	−0.827371	+	0.561656i	$0.810164\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	19.0000	1.06048
$322$	0	0
$323$	− 1.00000i	− 0.0556415i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 10.0000i	− 0.553001i
$328$	0	0
$329$	−24.0000	−1.32316
$330$	0	0
$331$	−19.0000	−1.04433	−0.522167	−	0.852843i	$-0.674876\pi$
$331$	−19.0000	−1.04433	−0.522167	+	0.852843i	$0.674876\pi$
$332$	0	0
$333$	− 4.00000i	− 0.219199i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 7.00000i	− 0.381314i	−0.981657	−	0.190657i	$-0.938938\pi$
$337$	− 7.00000i	− 0.381314i	0.981657	−	0.190657i	$-0.0610619\pi$
$338$	0	0
$339$	3.00000	0.162938
$340$	0	0
$341$	24.0000	1.29967
$342$	0	0
$343$	− 8.00000i	− 0.431959i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 11.0000i	− 0.590511i	−0.955418	−	0.295255i	$-0.904595\pi$
$347$	− 11.0000i	− 0.590511i	0.955418	−	0.295255i	$-0.0954048\pi$
$348$	0	0
$349$	30.0000	1.60586	0.802932	−	0.596071i	$-0.203272\pi$
$349$	30.0000	1.60586	0.802932	+	0.596071i	$0.203272\pi$
$350$	0	0
$351$	10.0000	0.533761
$352$	0	0
$353$	18.0000i	0.958043i	0.877803	+	0.479022i	$0.159008\pi$
$353$	18.0000i	0.958043i	−0.877803	+	0.479022i	$0.840992\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 4.00000i	− 0.211702i
$358$	0	0
$359$	−4.00000	−0.211112	−0.105556	−	0.994413i	$-0.533662\pi$
$359$	−4.00000	−0.211112	−0.105556	+	0.994413i	$0.533662\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	2.00000i	0.104973i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	30.0000i	1.56599i	0.622030	+	0.782994i	$0.286308\pi$
$367$	30.0000i	1.56599i	−0.622030	+	0.782994i	$0.713692\pi$
$368$	0	0
$369$	2.00000	0.104116
$370$	0	0
$371$	−16.0000	−0.830679
$372$	0	0
$373$	− 12.0000i	− 0.621336i	−0.950518	−	0.310668i	$-0.899447\pi$
$373$	− 12.0000i	− 0.621336i	0.950518	−	0.310668i	$-0.100553\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−29.0000	−1.48963	−0.744815	−	0.667271i	$-0.767462\pi$
$379$	−29.0000	−1.48963	−0.744815	+	0.667271i	$0.767462\pi$
$380$	0	0
$381$	−8.00000	−0.409852
$382$	0	0
$383$	− 14.0000i	− 0.715367i	−0.933843	−	0.357683i	$-0.883567\pi$
$383$	− 14.0000i	− 0.715367i	0.933843	−	0.357683i	$-0.116433\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 24.0000i	− 1.21999i
$388$	0	0
$389$	−12.0000	−0.608424	−0.304212	−	0.952604i	$-0.598393\pi$
$389$	−12.0000	−0.608424	−0.304212	+	0.952604i	$0.598393\pi$
$390$	0	0
$391$	−1.00000	−0.0505722
$392$	0	0
$393$	12.0000i	0.605320i
$394$	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
$399$	4.00000	0.200250
$400$	0	0
$401$	−25.0000	−1.24844	−0.624220	−	0.781248i	$-0.714583\pi$
$401$	−25.0000	−1.24844	−0.624220	+	0.781248i	$0.714583\pi$
$402$	0	0
$403$	16.0000i	0.797017i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 6.00000i	− 0.297409i
$408$	0	0
$409$	19.0000	0.939490	0.469745	−	0.882802i	$-0.344346\pi$
$409$	19.0000	0.939490	0.469745	+	0.882802i	$0.344346\pi$
$410$	0	0
$411$	15.0000	0.739895
$412$	0	0
$413$	− 48.0000i	− 2.36193i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	1.00000i	0.0489702i
$418$	0	0
$419$	−3.00000	−0.146560	−0.0732798	−	0.997311i	$-0.523347\pi$
$419$	−3.00000	−0.146560	−0.0732798	+	0.997311i	$0.523347\pi$
$420$	0	0
$421$	−6.00000	−0.292422	−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000	−0.292422	−0.146211	+	0.989253i	$0.546708\pi$
$422$	0	0
$423$	12.0000i	0.583460i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	6.00000	0.289683
$430$	0	0
$431$	36.0000	1.73406	0.867029	−	0.498257i	$-0.166026\pi$
$431$	36.0000	1.73406	0.867029	+	0.498257i	$0.166026\pi$
$432$	0	0
$433$	25.0000i	1.20142i	0.799466	+	0.600712i	$0.205116\pi$
$433$	25.0000i	1.20142i	−0.799466	+	0.600712i	$0.794884\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 1.00000i	− 0.0478365i
$438$	0	0
$439$	22.0000	1.05000	0.525001	−	0.851101i	$-0.324065\pi$
$439$	22.0000	1.05000	0.525001	+	0.851101i	$0.324065\pi$
$440$	0	0
$441$	−18.0000	−0.857143
$442$	0	0
$443$	− 35.0000i	− 1.66290i	−0.555599	−	0.831450i	$-0.687511\pi$
$443$	− 35.0000i	− 1.66290i	0.555599	−	0.831450i	$-0.312489\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	6.00000i	0.283790i
$448$	0	0
$449$	27.0000	1.27421	0.637104	−	0.770778i	$-0.280132\pi$
$449$	27.0000	1.27421	0.637104	+	0.770778i	$0.280132\pi$
$450$	0	0
$451$	3.00000	0.141264
$452$	0	0
$453$	4.00000i	0.187936i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 13.0000i	− 0.608114i	−0.952654	−	0.304057i	$-0.901659\pi$
$457$	− 13.0000i	− 0.608114i	0.952654	−	0.304057i	$-0.0983414\pi$
$458$	0	0
$459$	−5.00000	−0.233380
$460$	0	0
$461$	−24.0000	−1.11779	−0.558896	−	0.829238i	$-0.688775\pi$
$461$	−24.0000	−1.11779	−0.558896	+	0.829238i	$0.688775\pi$
$462$	0	0
$463$	2.00000i	0.0929479i	0.998920	+	0.0464739i	$0.0147984\pi$
$463$	2.00000i	0.0929479i	−0.998920	+	0.0464739i	$0.985202\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 12.0000i	− 0.555294i	−0.960683	−	0.277647i	$-0.910445\pi$
$467$	− 12.0000i	− 0.555294i	0.960683	−	0.277647i	$-0.0895545\pi$
$468$	0	0
$469$	−52.0000	−2.40114
$470$	0	0
$471$	10.0000	0.460776
$472$	0	0
$473$	− 36.0000i	− 1.65528i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	8.00000i	0.366295i
$478$	0	0
$479$	4.00000	0.182765	0.0913823	−	0.995816i	$-0.470871\pi$
$479$	4.00000	0.182765	0.0913823	+	0.995816i	$0.470871\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	0	0
$483$	− 4.00000i	− 0.182006i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 26.0000i	− 1.17817i	−0.808070	−	0.589086i	$-0.799488\pi$
$487$	− 26.0000i	− 1.17817i	0.808070	−	0.589086i	$-0.200512\pi$
$488$	0	0
$489$	1.00000	0.0452216
$490$	0	0
$491$	24.0000	1.08310	0.541552	−	0.840667i	$-0.317837\pi$
$491$	24.0000	1.08310	0.541552	+	0.840667i	$0.317837\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	48.0000i	2.15309i
$498$	0	0
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	0	0
$501$	−6.00000	−0.268060
$502$	0	0
$503$	24.0000i	1.07011i	0.844818	+	0.535054i	$0.179709\pi$
$503$	24.0000i	1.07011i	−0.844818	+	0.535054i	$0.820291\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 9.00000i	− 0.399704i
$508$	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
$511$	−68.0000	−3.00814
$512$	0	0
$513$	− 5.00000i	− 0.220755i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	18.0000i	0.791639i
$518$	0	0
$519$	4.00000	0.175581
$520$	0	0
$521$	−37.0000	−1.62100	−0.810500	−	0.585739i	$-0.800804\pi$
$521$	−37.0000	−1.62100	−0.810500	+	0.585739i	$0.800804\pi$
$522$	0	0
$523$	29.0000i	1.26808i	0.773300	+	0.634041i	$0.218605\pi$
$523$	29.0000i	1.26808i	−0.773300	+	0.634041i	$0.781395\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 8.00000i	− 0.348485i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	−24.0000	−1.04151
$532$	0	0
$533$	2.00000i	0.0866296i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 15.0000i	− 0.647298i
$538$	0	0
$539$	−27.0000	−1.16297
$540$	0	0
$541$	32.0000	1.37579	0.687894	−	0.725811i	$-0.258536\pi$
$541$	32.0000	1.37579	0.687894	+	0.725811i	$0.258536\pi$
$542$	0	0
$543$	− 18.0000i	− 0.772454i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	29.0000i	1.23995i	0.784621	+	0.619975i	$0.212857\pi$
$547$	29.0000i	1.23995i	−0.784621	+	0.619975i	$0.787143\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	56.0000i	2.38136i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 2.00000i	− 0.0847427i	−0.999102	−	0.0423714i	$-0.986509\pi$
$557$	− 2.00000i	− 0.0847427i	0.999102	−	0.0423714i	$-0.0134913\pi$
$558$	0	0
$559$	24.0000	1.01509
$560$	0	0
$561$	−3.00000	−0.126660
$562$	0	0
$563$	− 36.0000i	− 1.51722i	−0.651546	−	0.758610i	$-0.725879\pi$
$563$	− 36.0000i	− 1.51722i	0.651546	−	0.758610i	$-0.274121\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	4.00000i	0.167984i
$568$	0	0
$569$	−19.0000	−0.796521	−0.398261	−	0.917272i	$-0.630386\pi$
$569$	−19.0000	−0.796521	−0.398261	+	0.917272i	$0.630386\pi$
$570$	0	0
$571$	−12.0000	−0.502184	−0.251092	−	0.967963i	$-0.580790\pi$
$571$	−12.0000	−0.502184	−0.251092	+	0.967963i	$0.580790\pi$
$572$	0	0
$573$	4.00000i	0.167102i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 1.00000i	− 0.0416305i	−0.999783	−	0.0208153i	$-0.993374\pi$
$577$	− 1.00000i	− 0.0416305i	0.999783	−	0.0208153i	$-0.00662619\pi$
$578$	0	0
$579$	−9.00000	−0.374027
$580$	0	0
$581$	20.0000	0.829740
$582$	0	0
$583$	12.0000i	0.496989i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 13.0000i	− 0.536567i	−0.963340	−	0.268284i	$-0.913544\pi$
$587$	− 13.0000i	− 0.536567i	0.963340	−	0.268284i	$-0.0864565\pi$
$588$	0	0
$589$	8.00000	0.329634
$590$	0	0
$591$	8.00000	0.329076
$592$	0	0
$593$	− 11.0000i	− 0.451716i	−0.974160	−	0.225858i	$-0.927481\pi$
$593$	− 11.0000i	− 0.451716i	0.974160	−	0.225858i	$-0.0725185\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 14.0000i	− 0.572982i
$598$	0	0
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	0	0
$601$	−9.00000	−0.367118	−0.183559	−	0.983009i	$-0.558762\pi$
$601$	−9.00000	−0.367118	−0.183559	+	0.983009i	$0.558762\pi$
$602$	0	0
$603$	26.0000i	1.05880i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	14.0000i	0.568242i	0.958788	+	0.284121i	$0.0917018\pi$
$607$	14.0000i	0.568242i	−0.958788	+	0.284121i	$0.908298\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−12.0000	−0.485468
$612$	0	0
$613$	40.0000i	1.61558i	0.589467	+	0.807792i	$0.299338\pi$
$613$	40.0000i	1.61558i	−0.589467	+	0.807792i	$0.700662\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	2.00000i	0.0805170i	0.999189	+	0.0402585i	$0.0128181\pi$
$617$	2.00000i	0.0805170i	−0.999189	+	0.0402585i	$0.987182\pi$
$618$	0	0
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	0	0
$621$	−5.00000	−0.200643
$622$	0	0
$623$	− 4.00000i	− 0.160257i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 3.00000i	− 0.119808i
$628$	0	0
$629$	−2.00000	−0.0797452
$630$	0	0
$631$	−34.0000	−1.35352	−0.676759	−	0.736204i	$-0.736616\pi$
$631$	−34.0000	−1.35352	−0.676759	+	0.736204i	$0.736616\pi$
$632$	0	0
$633$	5.00000i	0.198732i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 18.0000i	− 0.713186i
$638$	0	0
$639$	24.0000	0.949425
$640$	0	0
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	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
$647$	36.0000i	1.41531i	0.706560	+	0.707653i	$0.250246\pi$
$647$	36.0000i	1.41531i	−0.706560	+	0.707653i	$0.749754\pi$
$648$	0	0
$649$	−36.0000	−1.41312
$650$	0	0
$651$	32.0000	1.25418
$652$	0	0
$653$	− 10.0000i	− 0.391330i	−0.980671	−	0.195665i	$-0.937313\pi$
$653$	− 10.0000i	− 0.391330i	0.980671	−	0.195665i	$-0.0626866\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	34.0000i	1.32647i
$658$	0	0
$659$	27.0000	1.05177	0.525885	−	0.850555i	$-0.323734\pi$
$659$	27.0000	1.05177	0.525885	+	0.850555i	$0.323734\pi$
$660$	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
$663$	− 2.00000i	− 0.0776736i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−16.0000	−0.618596
$670$	0	0
$671$	0	0
$672$	0	0
$673$	34.0000i	1.31060i	0.755367	+	0.655302i	$0.227459\pi$
$673$	34.0000i	1.31060i	−0.755367	+	0.655302i	$0.772541\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 34.0000i	− 1.30673i	−0.757045	−	0.653363i	$-0.773358\pi$
$677$	− 34.0000i	− 1.30673i	0.757045	−	0.653363i	$-0.226642\pi$
$678$	0	0
$679$	8.00000	0.307012
$680$	0	0
$681$	−8.00000	−0.306561
$682$	0	0
$683$	− 33.0000i	− 1.26271i	−0.775494	−	0.631355i	$-0.782499\pi$
$683$	− 33.0000i	− 1.26271i	0.775494	−	0.631355i	$-0.217501\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	10.0000i	0.381524i
$688$	0	0
$689$	−8.00000	−0.304776
$690$	0	0
$691$	41.0000	1.55971	0.779857	−	0.625958i	$-0.215292\pi$
$691$	41.0000	1.55971	0.779857	+	0.625958i	$0.215292\pi$
$692$	0	0
$693$	24.0000i	0.911685i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 1.00000i	− 0.0378777i
$698$	0	0
$699$	18.0000	0.680823
$700$	0	0
$701$	−48.0000	−1.81293	−0.906467	−	0.422276i	$-0.861231\pi$
$701$	−48.0000	−1.81293	−0.906467	+	0.422276i	$0.861231\pi$
$702$	0	0
$703$	− 2.00000i	− 0.0754314i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 16.0000i	− 0.601742i
$708$	0	0
$709$	−16.0000	−0.600893	−0.300446	−	0.953799i	$-0.597136\pi$
$709$	−16.0000	−0.600893	−0.300446	+	0.953799i	$0.597136\pi$
$710$	0	0
$711$	28.0000	1.05008
$712$	0	0
$713$	− 8.00000i	− 0.299602i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 6.00000i	− 0.224074i
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	56.0000	2.08555
$722$	0	0
$723$	5.00000i	0.185952i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	4.00000i	0.148352i	0.997245	+	0.0741759i	$0.0236326\pi$
$727$	4.00000i	0.148352i	−0.997245	+	0.0741759i	$0.976367\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	−12.0000	−0.443836
$732$	0	0
$733$	− 26.0000i	− 0.960332i	−0.877178	−	0.480166i	$-0.840576\pi$
$733$	− 26.0000i	− 0.960332i	0.877178	−	0.480166i	$-0.159424\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	39.0000i	1.43658i
$738$	0	0
$739$	−4.00000	−0.147142	−0.0735712	−	0.997290i	$-0.523440\pi$
$739$	−4.00000	−0.147142	−0.0735712	+	0.997290i	$0.523440\pi$
$740$	0	0
$741$	2.00000	0.0734718
$742$	0	0
$743$	26.0000i	0.953847i	0.878945	+	0.476924i	$0.158248\pi$
$743$	26.0000i	0.953847i	−0.878945	+	0.476924i	$0.841752\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 10.0000i	− 0.365881i
$748$	0	0
$749$	−76.0000	−2.77698
$750$	0	0
$751$	−16.0000	−0.583848	−0.291924	−	0.956441i	$-0.594295\pi$
$751$	−16.0000	−0.583848	−0.291924	+	0.956441i	$0.594295\pi$
$752$	0	0
$753$	15.0000i	0.546630i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 16.0000i	− 0.581530i	−0.956795	−	0.290765i	$-0.906090\pi$
$757$	− 16.0000i	− 0.581530i	0.956795	−	0.290765i	$-0.0939098\pi$
$758$	0	0
$759$	−3.00000	−0.108893
$760$	0	0
$761$	−55.0000	−1.99375	−0.996874	−	0.0790050i	$-0.974826\pi$
$761$	−55.0000	−1.99375	−0.996874	+	0.0790050i	$0.974826\pi$
$762$	0	0
$763$	40.0000i	1.44810i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 24.0000i	− 0.866590i
$768$	0	0
$769$	1.00000	0.0360609	0.0180305	−	0.999837i	$-0.494260\pi$
$769$	1.00000	0.0360609	0.0180305	+	0.999837i	$0.494260\pi$
$770$	0	0
$771$	30.0000	1.08042
$772$	0	0
$773$	12.0000i	0.431610i	0.976436	+	0.215805i	$0.0692376\pi$
$773$	12.0000i	0.431610i	−0.976436	+	0.215805i	$0.930762\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 8.00000i	− 0.286998i
$778$	0	0
$779$	1.00000	0.0358287
$780$	0	0
$781$	36.0000	1.28818
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 28.0000i	− 0.998092i	−0.866575	−	0.499046i	$-0.833684\pi$
$787$	− 28.0000i	− 0.998092i	0.866575	−	0.499046i	$-0.166316\pi$
$788$	0	0
$789$	−4.00000	−0.142404
$790$	0	0
$791$	−12.0000	−0.426671
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	42.0000i	1.48772i	0.668338	+	0.743858i	$0.267006\pi$
$797$	42.0000i	1.48772i	−0.668338	+	0.743858i	$0.732994\pi$
$798$	0	0
$799$	6.00000	0.212265
$800$	0	0
$801$	−2.00000	−0.0706665
$802$	0	0
$803$	51.0000i	1.79975i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	10.0000i	0.352017i
$808$	0	0
$809$	−26.0000	−0.914111	−0.457056	−	0.889438i	$-0.651096\pi$
$809$	−26.0000	−0.914111	−0.457056	+	0.889438i	$0.651096\pi$
$810$	0	0
$811$	12.0000	0.421377	0.210688	−	0.977553i	$-0.432429\pi$
$811$	12.0000	0.421377	0.210688	+	0.977553i	$0.432429\pi$
$812$	0	0
$813$	− 2.00000i	− 0.0701431i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 12.0000i	− 0.419827i
$818$	0	0
$819$	−16.0000	−0.559085
$820$	0	0
$821$	−32.0000	−1.11681	−0.558404	−	0.829569i	$-0.688586\pi$
$821$	−32.0000	−1.11681	−0.558404	+	0.829569i	$0.688586\pi$
$822$	0	0
$823$	30.0000i	1.04573i	0.852414	+	0.522867i	$0.175138\pi$
$823$	30.0000i	1.04573i	−0.852414	+	0.522867i	$0.824862\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 23.0000i	− 0.799788i	−0.916561	−	0.399894i	$-0.869047\pi$
$827$	− 23.0000i	− 0.799788i	0.916561	−	0.399894i	$-0.130953\pi$
$828$	0	0
$829$	−32.0000	−1.11141	−0.555703	−	0.831381i	$-0.687551\pi$
$829$	−32.0000	−1.11141	−0.555703	+	0.831381i	$0.687551\pi$
$830$	0	0
$831$	−12.0000	−0.416275
$832$	0	0
$833$	9.00000i	0.311832i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 40.0000i	− 1.38260i
$838$	0	0
$839$	−42.0000	−1.45000	−0.725001	−	0.688748i	$-0.758161\pi$
$839$	−42.0000	−1.45000	−0.725001	+	0.688748i	$0.758161\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	− 18.0000i	− 0.619953i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 8.00000i	− 0.274883i
$848$	0	0
$849$	−17.0000	−0.583438
$850$	0	0
$851$	−2.00000	−0.0685591
$852$	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
$857$	− 43.0000i	− 1.46885i	−0.678689	−	0.734426i	$-0.737451\pi$
$857$	− 43.0000i	− 1.46885i	0.678689	−	0.734426i	$-0.262549\pi$
$858$	0	0
$859$	15.0000	0.511793	0.255897	−	0.966704i	$-0.417629\pi$
$859$	15.0000	0.511793	0.255897	+	0.966704i	$0.417629\pi$
$860$	0	0
$861$	4.00000	0.136320
$862$	0	0
$863$	16.0000i	0.544646i	0.962206	+	0.272323i	$0.0877920\pi$
$863$	16.0000i	0.544646i	−0.962206	+	0.272323i	$0.912208\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 16.0000i	− 0.543388i
$868$	0	0
$869$	42.0000	1.42475
$870$	0	0
$871$	−26.0000	−0.880976
$872$	0	0
$873$	− 4.00000i	− 0.135379i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 22.0000i	− 0.742887i	−0.928456	−	0.371444i	$-0.878863\pi$
$877$	− 22.0000i	− 0.742887i	0.928456	−	0.371444i	$-0.121137\pi$
$878$	0	0
$879$	16.0000	0.539667
$880$	0	0
$881$	2.00000	0.0673817	0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000	0.0673817	0.0336909	+	0.999432i	$0.489274\pi$
$882$	0	0
$883$	− 1.00000i	− 0.0336527i	−0.999858	−	0.0168263i	$-0.994644\pi$
$883$	− 1.00000i	− 0.0336527i	0.999858	−	0.0168263i	$-0.00535624\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	36.0000i	1.20876i	0.796696	+	0.604381i	$0.206579\pi$
$887$	36.0000i	1.20876i	−0.796696	+	0.604381i	$0.793421\pi$
$888$	0	0
$889$	32.0000	1.07325
$890$	0	0
$891$	3.00000	0.100504
$892$	0	0
$893$	6.00000i	0.200782i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 2.00000i	− 0.0667781i
$898$	0	0
$899$	0	0
$900$	0	0
$901$	4.00000	0.133259
$902$	0	0
$903$	− 48.0000i	− 1.59734i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 4.00000i	− 0.132818i	−0.997792	−	0.0664089i	$-0.978846\pi$
$907$	− 4.00000i	− 0.132818i	0.997792	−	0.0664089i	$-0.0211542\pi$
$908$	0	0
$909$	−8.00000	−0.265343
$910$	0	0
$911$	−58.0000	−1.92163	−0.960813	−	0.277198i	$-0.910594\pi$
$911$	−58.0000	−1.92163	−0.960813	+	0.277198i	$0.910594\pi$
$912$	0	0
$913$	− 15.0000i	− 0.496428i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 48.0000i	− 1.58510i
$918$	0	0
$919$	−52.0000	−1.71532	−0.857661	−	0.514216i	$-0.828083\pi$
$919$	−52.0000	−1.71532	−0.857661	+	0.514216i	$0.828083\pi$
$920$	0	0
$921$	−19.0000	−0.626071
$922$	0	0
$923$	24.0000i	0.789970i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 28.0000i	− 0.919641i
$928$	0	0
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	0	0
$931$	−9.00000	−0.294963
$932$	0	0
$933$	− 18.0000i	− 0.589294i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 51.0000i	− 1.66610i	−0.553200	−	0.833049i	$-0.686593\pi$
$937$	− 51.0000i	− 1.66610i	0.553200	−	0.833049i	$-0.313407\pi$
$938$	0	0
$939$	−26.0000	−0.848478
$940$	0	0
$941$	60.0000	1.95594	0.977972	−	0.208736i	$-0.0669349\pi$
$941$	60.0000	1.95594	0.977972	+	0.208736i	$0.0669349\pi$
$942$	0	0
$943$	− 1.00000i	− 0.0325645i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 36.0000i	− 1.16984i	−0.811090	−	0.584921i	$-0.801125\pi$
$947$	− 36.0000i	− 1.16984i	0.811090	−	0.584921i	$-0.198875\pi$
$948$	0	0
$949$	−34.0000	−1.10369
$950$	0	0
$951$	20.0000	0.648544
$952$	0	0
$953$	− 39.0000i	− 1.26333i	−0.775240	−	0.631667i	$-0.782371\pi$
$953$	− 39.0000i	− 1.26333i	0.775240	−	0.631667i	$-0.217629\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−60.0000	−1.93750
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	38.0000i	1.22453i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 52.0000i	− 1.67221i	−0.548572	−	0.836104i	$-0.684828\pi$
$967$	− 52.0000i	− 1.67221i	0.548572	−	0.836104i	$-0.315172\pi$
$968$	0	0
$969$	−1.00000	−0.0321246
$970$	0	0
$971$	−49.0000	−1.57248	−0.786242	−	0.617918i	$-0.787976\pi$
$971$	−49.0000	−1.57248	−0.786242	+	0.617918i	$0.787976\pi$
$972$	0	0
$973$	− 4.00000i	− 0.128234i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 21.0000i	− 0.671850i	−0.941889	−	0.335925i	$-0.890951\pi$
$977$	− 21.0000i	− 0.671850i	0.941889	−	0.335925i	$-0.109049\pi$
$978$	0	0
$979$	−3.00000	−0.0958804
$980$	0	0
$981$	20.0000	0.638551
$982$	0	0
$983$	− 30.0000i	− 0.956851i	−0.878128	−	0.478426i	$-0.841208\pi$
$983$	− 30.0000i	− 0.956851i	0.878128	−	0.478426i	$-0.158792\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	24.0000i	0.763928i
$988$	0	0
$989$	−12.0000	−0.381578
$990$	0	0
$991$	50.0000	1.58830	0.794151	−	0.607720i	$-0.207916\pi$
$991$	50.0000	1.58830	0.794151	+	0.607720i	$0.207916\pi$
$992$	0	0
$993$	19.0000i	0.602947i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	10.0000i	0.316703i	0.987383	+	0.158352i	$0.0506179\pi$
$997$	10.0000i	0.316703i	−0.987383	+	0.158352i	$0.949382\pi$
$998$	0	0
$999$	−10.0000	−0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4600.2.e.i.4049.1		2
5.2	odd	4		4600.2.a.e.1.1	✓	1
5.3	odd	4		4600.2.a.l.1.1	yes	1
5.4	even	2	inner	4600.2.e.i.4049.2		2
20.3	even	4		9200.2.a.k.1.1		1
20.7	even	4		9200.2.a.bb.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4600.2.a.e.1.1	✓	1	5.2	odd	4
4600.2.a.l.1.1	yes	1	5.3	odd	4
4600.2.e.i.4049.1		2	1.1	even	1	trivial
4600.2.e.i.4049.2		2	5.4	even	2	inner
9200.2.a.k.1.1		1	20.3	even	4
9200.2.a.bb.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:	\( N \)	\(=\)	\( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4600.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +4.00000i q^{7} +2.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +4.00000i q^{7} +2.00000 q^{9} +3.00000 q^{11} +2.00000i q^{13} -1.00000i q^{17} +1.00000 q^{19} +4.00000 q^{21} -1.00000i q^{23} -5.00000i q^{27} +8.00000 q^{31} -3.00000i q^{33} -2.00000i q^{37} +2.00000 q^{39} +1.00000 q^{41} -12.0000i q^{43} +6.00000i q^{47} -9.00000 q^{49} -1.00000 q^{51} +4.00000i q^{53} -1.00000i q^{57} -12.0000 q^{59} +8.00000i q^{63} +13.0000i q^{67} -1.00000 q^{69} +12.0000 q^{71} +17.0000i q^{73} +12.0000i q^{77} +14.0000 q^{79} +1.00000 q^{81} -5.00000i q^{83} -1.00000 q^{89} -8.00000 q^{91} -8.00000i q^{93} -2.00000i q^{97} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{9}+O(q^{10}) \) 2 * q + 4 * q^9	\( 2 q + 4 q^{9} + 6 q^{11} + 2 q^{19} + 8 q^{21} + 16 q^{31} + 4 q^{39} + 2 q^{41} - 18 q^{49} - 2 q^{51} - 24 q^{59} - 2 q^{69} + 24 q^{71} + 28 q^{79} + 2 q^{81} - 2 q^{89} - 16 q^{91} + 12 q^{99}+O(q^{100}) \) 2 * q + 4 * q^9 + 6 * q^11 + 2 * q^19 + 8 * q^21 + 16 * q^31 + 4 * q^39 + 2 * q^41 - 18 * q^49 - 2 * q^51 - 24 * q^59 - 2 * q^69 + 24 * q^71 + 28 * q^79 + 2 * q^81 - 2 * q^89 - 16 * q^91 + 12 * q^99

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