LMFDB - Embedded newform 4600.2.e.l.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:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4049.1
Root			$-1.61803i$ of defining polynomial
Character	$\chi$	$=$	4600.4049
Dual form			4600.2.e.l.4049.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.23607i q^{3} -4.23607i q^{7} -7.47214 q^{9} +1.00000 q^{11} +2.23607i q^{13} +6.47214i q^{17} -1.00000 q^{19} -13.7082 q^{21} +1.00000i q^{23} +14.4721i q^{27} +1.76393 q^{29} +0.472136 q^{31} -3.23607i q^{33} +11.2361i q^{37} +7.23607 q^{39} -5.94427 q^{41} -2.52786i q^{43} +11.7082i q^{47} -10.9443 q^{49} +20.9443 q^{51} -1.23607i q^{53} +3.23607i q^{57} -3.23607 q^{59} -7.23607 q^{61} +31.6525i q^{63} +12.9443i q^{67} +3.23607 q^{69} +10.0000 q^{71} +9.47214i q^{73} -4.23607i q^{77} -15.1803 q^{79} +24.4164 q^{81} -9.00000i q^{83} -5.70820i q^{87} -2.00000 q^{89} +9.47214 q^{91} -1.52786i q^{93} +6.18034i q^{97} -7.47214 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{9} + 4 q^{11} - 4 q^{19} - 28 q^{21} + 16 q^{29} - 16 q^{31} + 20 q^{39} + 12 q^{41} - 8 q^{49} + 48 q^{51} - 4 q^{59} - 20 q^{61} + 4 q^{69} + 40 q^{71} - 16 q^{79} + 44 q^{81} - 8 q^{89} + 20 q^{91}+ \cdots - 12 q^{99}+O(q^{100}) $ 4 * q - 12 * q^9 + 4 * q^11 - 4 * q^19 - 28 * q^21 + 16 * q^29 - 16 * q^31 + 20 * q^39 + 12 * q^41 - 8 * q^49 + 48 * q^51 - 4 * q^59 - 20 * q^61 + 4 * q^69 + 40 * q^71 - 16 * q^79 + 44 * q^81 - 8 * q^89 + 20 * q^91 - 12 * q^99

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
$3$	− 3.23607i	− 1.86834i	−0.356822	−	0.934172i	$-0.616140\pi$
$3$	− 3.23607i	− 1.86834i	0.356822	−	0.934172i	$-0.383860\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 4.23607i	− 1.60108i	−0.599277	−	0.800542i	$-0.704545\pi$
$7$	− 4.23607i	− 1.60108i	0.599277	−	0.800542i	$-0.295455\pi$
$8$	0	0
$9$	−7.47214	−2.49071
$10$	0	0
$11$	1.00000	0.301511	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	0	0
$13$	2.23607i	0.620174i	0.950708	+	0.310087i	$0.100358\pi$
$13$	2.23607i	0.620174i	−0.950708	+	0.310087i	$0.899642\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.47214i	1.56972i	0.619671	+	0.784862i	$0.287266\pi$
$17$	6.47214i	1.56972i	−0.619671	+	0.784862i	$0.712734\pi$
$18$	0	0
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	0	0
$21$	−13.7082	−2.99138
$22$	0	0
$23$	1.00000i	0.208514i
$23$	1.00000i	0.208514i
$24$	0	0
$25$	0	0
$26$	0	0
$27$	14.4721i	2.78516i
$28$	0	0
$29$	1.76393	0.327554	0.163777	−	0.986497i	$-0.447632\pi$
$29$	1.76393	0.327554	0.163777	+	0.986497i	$0.447632\pi$
$30$	0	0
$31$	0.472136	0.0847981	0.0423991	−	0.999101i	$-0.486500\pi$
$31$	0.472136	0.0847981	0.0423991	+	0.999101i	$0.486500\pi$
$32$	0	0
$33$	− 3.23607i	− 0.563327i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	11.2361i	1.84720i	0.383360	+	0.923599i	$0.374767\pi$
$37$	11.2361i	1.84720i	−0.383360	+	0.923599i	$0.625233\pi$
$38$	0	0
$39$	7.23607	1.15870
$40$	0	0
$41$	−5.94427	−0.928339	−0.464170	−	0.885746i	$-0.653647\pi$
$41$	−5.94427	−0.928339	−0.464170	+	0.885746i	$0.653647\pi$
$42$	0	0
$43$	− 2.52786i	− 0.385496i	−0.981248	−	0.192748i	$-0.938260\pi$
$43$	− 2.52786i	− 0.385496i	0.981248	−	0.192748i	$-0.0617399\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	11.7082i	1.70782i	0.520423	+	0.853909i	$0.325774\pi$
$47$	11.7082i	1.70782i	−0.520423	+	0.853909i	$0.674226\pi$
$48$	0	0
$49$	−10.9443	−1.56347
$50$	0	0
$51$	20.9443	2.93278
$52$	0	0
$53$	− 1.23607i	− 0.169787i	−0.996390	−	0.0848935i	$-0.972945\pi$
$53$	− 1.23607i	− 0.169787i	0.996390	−	0.0848935i	$-0.0270550\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	3.23607i	0.428628i
$58$	0	0
$59$	−3.23607	−0.421300	−0.210650	−	0.977562i	$-0.567558\pi$
$59$	−3.23607	−0.421300	−0.210650	+	0.977562i	$0.567558\pi$
$60$	0	0
$61$	−7.23607	−0.926484	−0.463242	−	0.886232i	$-0.653314\pi$
$61$	−7.23607	−0.926484	−0.463242	+	0.886232i	$0.653314\pi$
$62$	0	0
$63$	31.6525i	3.98784i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	12.9443i	1.58139i	0.612207	+	0.790697i	$0.290282\pi$
$67$	12.9443i	1.58139i	−0.612207	+	0.790697i	$0.709718\pi$
$68$	0	0
$69$	3.23607	0.389577
$70$	0	0
$71$	10.0000	1.18678	0.593391	−	0.804914i	$-0.297789\pi$
$71$	10.0000	1.18678	0.593391	+	0.804914i	$0.297789\pi$
$72$	0	0
$73$	9.47214i	1.10863i	0.832307	+	0.554315i	$0.187020\pi$
$73$	9.47214i	1.10863i	−0.832307	+	0.554315i	$0.812980\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 4.23607i	− 0.482745i
$78$	0	0
$79$	−15.1803	−1.70792	−0.853961	−	0.520337i	$-0.825806\pi$
$79$	−15.1803	−1.70792	−0.853961	+	0.520337i	$0.825806\pi$
$80$	0	0
$81$	24.4164	2.71293
$82$	0	0
$83$	− 9.00000i	− 0.987878i	−0.869496	−	0.493939i	$-0.835557\pi$
$83$	− 9.00000i	− 0.987878i	0.869496	−	0.493939i	$-0.164443\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 5.70820i	− 0.611984i
$88$	0	0
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	9.47214	0.992950
$92$	0	0
$93$	− 1.52786i	− 0.158432i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	6.18034i	0.627518i	0.949503	+	0.313759i	$0.101588\pi$
$97$	6.18034i	0.627518i	−0.949503	+	0.313759i	$0.898412\pi$
$98$	0	0
$99$	−7.47214	−0.750978
$100$	0	0
$101$	−18.9443	−1.88503	−0.942513	−	0.334170i	$-0.891544\pi$
$101$	−18.9443	−1.88503	−0.942513	+	0.334170i	$0.891544\pi$
$102$	0	0
$103$	2.23607i	0.220326i	0.993914	+	0.110163i	$0.0351373\pi$
$103$	2.23607i	0.220326i	−0.993914	+	0.110163i	$0.964863\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 19.4164i	− 1.87705i	−0.345204	−	0.938527i	$-0.612190\pi$
$107$	− 19.4164i	− 1.87705i	0.345204	−	0.938527i	$-0.387810\pi$
$108$	0	0
$109$	12.7639	1.22256	0.611281	−	0.791413i	$-0.290654\pi$
$109$	12.7639	1.22256	0.611281	+	0.791413i	$0.290654\pi$
$110$	0	0
$111$	36.3607	3.45120
$112$	0	0
$113$	− 6.76393i	− 0.636297i	−0.948041	−	0.318149i	$-0.896939\pi$
$113$	− 6.76393i	− 0.636297i	0.948041	−	0.318149i	$-0.103061\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 16.7082i	− 1.54467i
$118$	0	0
$119$	27.4164	2.51326
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	19.2361i	1.73446i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 4.18034i	− 0.370945i	−0.982649	−	0.185473i	$-0.940618\pi$
$127$	− 4.18034i	− 0.370945i	0.982649	−	0.185473i	$-0.0593816\pi$
$128$	0	0
$129$	−8.18034	−0.720239
$130$	0	0
$131$	−14.9443	−1.30569	−0.652844	−	0.757493i	$-0.726424\pi$
$131$	−14.9443	−1.30569	−0.652844	+	0.757493i	$0.726424\pi$
$132$	0	0
$133$	4.23607i	0.367314i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	11.5279i	0.984892i	0.870343	+	0.492446i	$0.163897\pi$
$137$	11.5279i	0.984892i	−0.870343	+	0.492446i	$0.836103\pi$
$138$	0	0
$139$	−20.0000	−1.69638	−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000	−1.69638	−0.848189	+	0.529694i	$0.822307\pi$
$140$	0	0
$141$	37.8885	3.19079
$142$	0	0
$143$	2.23607i	0.186989i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	35.4164i	2.92110i
$148$	0	0
$149$	−10.9443	−0.896590	−0.448295	−	0.893886i	$-0.647969\pi$
$149$	−10.9443	−0.896590	−0.448295	+	0.893886i	$0.647969\pi$
$150$	0	0
$151$	11.5279	0.938124	0.469062	−	0.883165i	$-0.344592\pi$
$151$	11.5279	0.938124	0.469062	+	0.883165i	$0.344592\pi$
$152$	0	0
$153$	− 48.3607i	− 3.90973i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0.291796i	0.0232879i	0.999932	+	0.0116439i	$0.00370646\pi$
$157$	0.291796i	0.0232879i	−0.999932	+	0.0116439i	$0.996294\pi$
$158$	0	0
$159$	−4.00000	−0.317221
$160$	0	0
$161$	4.23607	0.333849
$162$	0	0
$163$	11.7082i	0.917057i	0.888680	+	0.458529i	$0.151623\pi$
$163$	11.7082i	0.917057i	−0.888680	+	0.458529i	$0.848377\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 5.41641i	− 0.419134i	−0.977794	−	0.209567i	$-0.932795\pi$
$167$	− 5.41641i	− 0.419134i	0.977794	−	0.209567i	$-0.0672055\pi$
$168$	0	0
$169$	8.00000	0.615385
$170$	0	0
$171$	7.47214	0.571409
$172$	0	0
$173$	− 14.2361i	− 1.08235i	−0.840911	−	0.541174i	$-0.817980\pi$
$173$	− 14.2361i	− 1.08235i	0.840911	−	0.541174i	$-0.182020\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	10.4721i	0.787134i
$178$	0	0
$179$	15.7082	1.17409	0.587043	−	0.809556i	$-0.300292\pi$
$179$	15.7082	1.17409	0.587043	+	0.809556i	$0.300292\pi$
$180$	0	0
$181$	−0.944272	−0.0701872	−0.0350936	−	0.999384i	$-0.511173\pi$
$181$	−0.944272	−0.0701872	−0.0350936	+	0.999384i	$0.511173\pi$
$182$	0	0
$183$	23.4164i	1.73099i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	6.47214i	0.473289i
$188$	0	0
$189$	61.3050	4.45928
$190$	0	0
$191$	−19.1803	−1.38784	−0.693920	−	0.720052i	$-0.744118\pi$
$191$	−19.1803	−1.38784	−0.693920	+	0.720052i	$0.744118\pi$
$192$	0	0
$193$	− 2.94427i	− 0.211933i	−0.994370	−	0.105967i	$-0.966206\pi$
$193$	− 2.94427i	− 0.211933i	0.994370	−	0.105967i	$-0.0337937\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 8.23607i	− 0.586796i	−0.955990	−	0.293398i	$-0.905214\pi$
$197$	− 8.23607i	− 0.586796i	0.955990	−	0.293398i	$-0.0947860\pi$
$198$	0	0
$199$	−12.7082	−0.900861	−0.450430	−	0.892812i	$-0.648729\pi$
$199$	−12.7082	−0.900861	−0.450430	+	0.892812i	$0.648729\pi$
$200$	0	0
$201$	41.8885	2.95459
$202$	0	0
$203$	− 7.47214i	− 0.524441i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 7.47214i	− 0.519349i
$208$	0	0
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	−14.9443	−1.02881	−0.514403	−	0.857549i	$-0.671986\pi$
$211$	−14.9443	−1.02881	−0.514403	+	0.857549i	$0.671986\pi$
$212$	0	0
$213$	− 32.3607i	− 2.21732i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 2.00000i	− 0.135769i
$218$	0	0
$219$	30.6525	2.07130
$220$	0	0
$221$	−14.4721	−0.973501
$222$	0	0
$223$	− 13.2361i	− 0.886353i	−0.896434	−	0.443176i	$-0.853852\pi$
$223$	− 13.2361i	− 0.886353i	0.896434	−	0.443176i	$-0.146148\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	12.9443i	0.859142i	0.903033	+	0.429571i	$0.141335\pi$
$227$	12.9443i	0.859142i	−0.903033	+	0.429571i	$0.858665\pi$
$228$	0	0
$229$	−6.94427	−0.458890	−0.229445	−	0.973322i	$-0.573691\pi$
$229$	−6.94427	−0.458890	−0.229445	+	0.973322i	$0.573691\pi$
$230$	0	0
$231$	−13.7082	−0.901934
$232$	0	0
$233$	− 4.52786i	− 0.296630i	−0.988940	−	0.148315i	$-0.952615\pi$
$233$	− 4.52786i	− 0.296630i	0.988940	−	0.148315i	$-0.0473850\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	49.1246i	3.19099i
$238$	0	0
$239$	−6.76393	−0.437522	−0.218761	−	0.975778i	$-0.570202\pi$
$239$	−6.76393	−0.437522	−0.218761	+	0.975778i	$0.570202\pi$
$240$	0	0
$241$	21.2361	1.36794	0.683968	−	0.729512i	$-0.260253\pi$
$241$	21.2361	1.36794	0.683968	+	0.729512i	$0.260253\pi$
$242$	0	0
$243$	− 35.5967i	− 2.28353i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 2.23607i	− 0.142278i
$248$	0	0
$249$	−29.1246	−1.84570
$250$	0	0
$251$	−5.52786	−0.348916	−0.174458	−	0.984665i	$-0.555817\pi$
$251$	−5.52786	−0.348916	−0.174458	+	0.984665i	$0.555817\pi$
$252$	0	0
$253$	1.00000i	0.0628695i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 14.0000i	− 0.873296i	−0.899632	−	0.436648i	$-0.856166\pi$
$257$	− 14.0000i	− 0.873296i	0.899632	−	0.436648i	$-0.143834\pi$
$258$	0	0
$259$	47.5967	2.95752
$260$	0	0
$261$	−13.1803	−0.815843
$262$	0	0
$263$	− 8.00000i	− 0.493301i	−0.969104	−	0.246651i	$-0.920670\pi$
$263$	− 8.00000i	− 0.493301i	0.969104	−	0.246651i	$-0.0793300\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	6.47214i	0.396088i
$268$	0	0
$269$	25.1803	1.53527	0.767636	−	0.640886i	$-0.221433\pi$
$269$	25.1803	1.53527	0.767636	+	0.640886i	$0.221433\pi$
$270$	0	0
$271$	−32.0689	−1.94805	−0.974023	−	0.226449i	$-0.927288\pi$
$271$	−32.0689	−1.94805	−0.974023	+	0.226449i	$0.927288\pi$
$272$	0	0
$273$	− 30.6525i	− 1.85517i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	14.2361i	0.855362i	0.903930	+	0.427681i	$0.140669\pi$
$277$	14.2361i	0.855362i	−0.903930	+	0.427681i	$0.859331\pi$
$278$	0	0
$279$	−3.52786	−0.211208
$280$	0	0
$281$	−22.1803	−1.32317	−0.661584	−	0.749871i	$-0.730116\pi$
$281$	−22.1803	−1.32317	−0.661584	+	0.749871i	$0.730116\pi$
$282$	0	0
$283$	− 28.3607i	− 1.68587i	−0.538017	−	0.842934i	$-0.680827\pi$
$283$	− 28.3607i	− 1.68587i	0.538017	−	0.842934i	$-0.319173\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	25.1803i	1.48635i
$288$	0	0
$289$	−24.8885	−1.46403
$290$	0	0
$291$	20.0000	1.17242
$292$	0	0
$293$	14.4721i	0.845471i	0.906253	+	0.422736i	$0.138930\pi$
$293$	14.4721i	0.845471i	−0.906253	+	0.422736i	$0.861070\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	14.4721i	0.839759i
$298$	0	0
$299$	−2.23607	−0.129315
$300$	0	0
$301$	−10.7082	−0.617211
$302$	0	0
$303$	61.3050i	3.52188i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	24.0000i	1.36975i	0.728659	+	0.684876i	$0.240144\pi$
$307$	24.0000i	1.36975i	−0.728659	+	0.684876i	$0.759856\pi$
$308$	0	0
$309$	7.23607	0.411646
$310$	0	0
$311$	−3.23607	−0.183501	−0.0917503	−	0.995782i	$-0.529246\pi$
$311$	−3.23607	−0.183501	−0.0917503	+	0.995782i	$0.529246\pi$
$312$	0	0
$313$	10.6525i	0.602114i	0.953606	+	0.301057i	$0.0973394\pi$
$313$	10.6525i	0.602114i	−0.953606	+	0.301057i	$0.902661\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 22.1246i	− 1.24264i	−0.783556	−	0.621321i	$-0.786596\pi$
$317$	− 22.1246i	− 1.24264i	0.783556	−	0.621321i	$-0.213404\pi$
$318$	0	0
$319$	1.76393	0.0987612
$320$	0	0
$321$	−62.8328	−3.50699
$322$	0	0
$323$	− 6.47214i	− 0.360119i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 41.3050i	− 2.28417i
$328$	0	0
$329$	49.5967	2.73436
$330$	0	0
$331$	30.9443	1.70085	0.850426	−	0.526095i	$-0.176345\pi$
$331$	30.9443	1.70085	0.850426	+	0.526095i	$0.176345\pi$
$332$	0	0
$333$	− 83.9574i	− 4.60084i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 24.6525i	− 1.34291i	−0.741047	−	0.671453i	$-0.765671\pi$
$337$	− 24.6525i	− 1.34291i	0.741047	−	0.671453i	$-0.234329\pi$
$338$	0	0
$339$	−21.8885	−1.18882
$340$	0	0
$341$	0.472136	0.0255676
$342$	0	0
$343$	16.7082i	0.902158i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	3.05573i	0.164040i	0.996631	+	0.0820200i	$0.0261372\pi$
$347$	3.05573i	0.164040i	−0.996631	+	0.0820200i	$0.973863\pi$
$348$	0	0
$349$	5.18034	0.277297	0.138649	−	0.990342i	$-0.455724\pi$
$349$	5.18034	0.277297	0.138649	+	0.990342i	$0.455724\pi$
$350$	0	0
$351$	−32.3607	−1.72729
$352$	0	0
$353$	4.88854i	0.260191i	0.991501	+	0.130095i	$0.0415284\pi$
$353$	4.88854i	0.260191i	−0.991501	+	0.130095i	$0.958472\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 88.7214i	− 4.69563i
$358$	0	0
$359$	−19.7639	−1.04310	−0.521550	−	0.853221i	$-0.674646\pi$
$359$	−19.7639	−1.04310	−0.521550	+	0.853221i	$0.674646\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	32.3607i	1.69850i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 23.6525i	− 1.23465i	−0.786709	−	0.617325i	$-0.788217\pi$
$367$	− 23.6525i	− 1.23465i	0.786709	−	0.617325i	$-0.211783\pi$
$368$	0	0
$369$	44.4164	2.31223
$370$	0	0
$371$	−5.23607	−0.271843
$372$	0	0
$373$	12.0000i	0.621336i	0.950518	+	0.310668i	$0.100553\pi$
$373$	12.0000i	0.621336i	−0.950518	+	0.310668i	$0.899447\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.94427i	0.203140i
$378$	0	0
$379$	−3.05573	−0.156962	−0.0784811	−	0.996916i	$-0.525007\pi$
$379$	−3.05573	−0.156962	−0.0784811	+	0.996916i	$0.525007\pi$
$380$	0	0
$381$	−13.5279	−0.693053
$382$	0	0
$383$	0.236068i	0.0120625i	0.999982	+	0.00603126i	$0.00191982\pi$
$383$	0.236068i	0.0120625i	−0.999982	+	0.00603126i	$0.998080\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	18.8885i	0.960159i
$388$	0	0
$389$	−6.18034	−0.313356	−0.156678	−	0.987650i	$-0.550078\pi$
$389$	−6.18034	−0.313356	−0.156678	+	0.987650i	$0.550078\pi$
$390$	0	0
$391$	−6.47214	−0.327310
$392$	0	0
$393$	48.3607i	2.43947i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	22.3607i	1.12225i	0.827731	+	0.561125i	$0.189631\pi$
$397$	22.3607i	1.12225i	−0.827731	+	0.561125i	$0.810369\pi$
$398$	0	0
$399$	13.7082	0.686269
$400$	0	0
$401$	−13.1246	−0.655412	−0.327706	−	0.944780i	$-0.606276\pi$
$401$	−13.1246	−0.655412	−0.327706	+	0.944780i	$0.606276\pi$
$402$	0	0
$403$	1.05573i	0.0525896i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	11.2361i	0.556951i
$408$	0	0
$409$	26.4164	1.30621	0.653104	−	0.757269i	$-0.273467\pi$
$409$	26.4164	1.30621	0.653104	+	0.757269i	$0.273467\pi$
$410$	0	0
$411$	37.3050	1.84012
$412$	0	0
$413$	13.7082i	0.674537i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	64.7214i	3.16942i
$418$	0	0
$419$	36.7771	1.79668	0.898339	−	0.439303i	$-0.144774\pi$
$419$	36.7771	1.79668	0.898339	+	0.439303i	$0.144774\pi$
$420$	0	0
$421$	15.8885	0.774360	0.387180	−	0.922004i	$-0.373449\pi$
$421$	15.8885	0.774360	0.387180	+	0.922004i	$0.373449\pi$
$422$	0	0
$423$	− 87.4853i	− 4.25368i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	30.6525i	1.48338i
$428$	0	0
$429$	7.23607	0.349361
$430$	0	0
$431$	12.9443	0.623504	0.311752	−	0.950164i	$-0.399084\pi$
$431$	12.9443	0.623504	0.311752	+	0.950164i	$0.399084\pi$
$432$	0	0
$433$	29.1246i	1.39964i	0.714319	+	0.699820i	$0.246736\pi$
$433$	29.1246i	1.39964i	−0.714319	+	0.699820i	$0.753264\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 1.00000i	− 0.0478365i
$438$	0	0
$439$	−12.3607	−0.589943	−0.294972	−	0.955506i	$-0.595310\pi$
$439$	−12.3607	−0.589943	−0.294972	+	0.955506i	$0.595310\pi$
$440$	0	0
$441$	81.7771	3.89415
$442$	0	0
$443$	10.0000i	0.475114i	0.971374	+	0.237557i	$0.0763467\pi$
$443$	10.0000i	0.475114i	−0.971374	+	0.237557i	$0.923653\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	35.4164i	1.67514i
$448$	0	0
$449$	36.8328	1.73825	0.869124	−	0.494594i	$-0.164683\pi$
$449$	36.8328	1.73825	0.869124	+	0.494594i	$0.164683\pi$
$450$	0	0
$451$	−5.94427	−0.279905
$452$	0	0
$453$	− 37.3050i	− 1.75274i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 20.6525i	− 0.966082i	−0.875598	−	0.483041i	$-0.839532\pi$
$457$	− 20.6525i	− 0.966082i	0.875598	−	0.483041i	$-0.160468\pi$
$458$	0	0
$459$	−93.6656	−4.37194
$460$	0	0
$461$	33.6525	1.56735	0.783676	−	0.621170i	$-0.213342\pi$
$461$	33.6525	1.56735	0.783676	+	0.621170i	$0.213342\pi$
$462$	0	0
$463$	26.4721i	1.23026i	0.788424	+	0.615132i	$0.210897\pi$
$463$	26.4721i	1.23026i	−0.788424	+	0.615132i	$0.789103\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 17.0000i	− 0.786666i	−0.919396	−	0.393333i	$-0.871322\pi$
$467$	− 17.0000i	− 0.786666i	0.919396	−	0.393333i	$-0.128678\pi$
$468$	0	0
$469$	54.8328	2.53194
$470$	0	0
$471$	0.944272	0.0435098
$472$	0	0
$473$	− 2.52786i	− 0.116231i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	9.23607i	0.422891i
$478$	0	0
$479$	12.1246	0.553988	0.276994	−	0.960872i	$-0.410662\pi$
$479$	12.1246	0.553988	0.276994	+	0.960872i	$0.410662\pi$
$480$	0	0
$481$	−25.1246	−1.14558
$482$	0	0
$483$	− 13.7082i	− 0.623745i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 39.1246i	− 1.77291i	−0.462819	−	0.886453i	$-0.653162\pi$
$487$	− 39.1246i	− 1.77291i	0.462819	−	0.886453i	$-0.346838\pi$
$488$	0	0
$489$	37.8885	1.71338
$490$	0	0
$491$	−3.23607	−0.146042	−0.0730209	−	0.997330i	$-0.523264\pi$
$491$	−3.23607	−0.146042	−0.0730209	+	0.997330i	$0.523264\pi$
$492$	0	0
$493$	11.4164i	0.514169i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 42.3607i	− 1.90014i
$498$	0	0
$499$	42.5410	1.90440	0.952199	−	0.305479i	$-0.0988166\pi$
$499$	42.5410	1.90440	0.952199	+	0.305479i	$0.0988166\pi$
$500$	0	0
$501$	−17.5279	−0.783087
$502$	0	0
$503$	− 16.5967i	− 0.740012i	−0.929029	−	0.370006i	$-0.879356\pi$
$503$	− 16.5967i	− 0.740012i	0.929029	−	0.370006i	$-0.120644\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 25.8885i	− 1.14975i
$508$	0	0
$509$	−30.3607	−1.34571	−0.672857	−	0.739773i	$-0.734933\pi$
$509$	−30.3607	−1.34571	−0.672857	+	0.739773i	$0.734933\pi$
$510$	0	0
$511$	40.1246	1.77501
$512$	0	0
$513$	− 14.4721i	− 0.638960i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	11.7082i	0.514926i
$518$	0	0
$519$	−46.0689	−2.02220
$520$	0	0
$521$	9.34752	0.409522	0.204761	−	0.978812i	$-0.434358\pi$
$521$	9.34752	0.409522	0.204761	+	0.978812i	$0.434358\pi$
$522$	0	0
$523$	18.8885i	0.825938i	0.910745	+	0.412969i	$0.135508\pi$
$523$	18.8885i	0.825938i	−0.910745	+	0.412969i	$0.864492\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.05573i	0.133110i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	24.1803	1.04934
$532$	0	0
$533$	− 13.2918i	− 0.575732i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 50.8328i	− 2.19360i
$538$	0	0
$539$	−10.9443	−0.471403
$540$	0	0
$541$	8.70820	0.374395	0.187197	−	0.982322i	$-0.440060\pi$
$541$	8.70820	0.374395	0.187197	+	0.982322i	$0.440060\pi$
$542$	0	0
$543$	3.05573i	0.131134i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 15.7082i	− 0.671634i	−0.941927	−	0.335817i	$-0.890988\pi$
$547$	− 15.7082i	− 0.671634i	0.941927	−	0.335817i	$-0.109012\pi$
$548$	0	0
$549$	54.0689	2.30760
$550$	0	0
$551$	−1.76393	−0.0751460
$552$	0	0
$553$	64.3050i	2.73452i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	36.1803i	1.53301i	0.642238	+	0.766505i	$0.278006\pi$
$557$	36.1803i	1.53301i	−0.642238	+	0.766505i	$0.721994\pi$
$558$	0	0
$559$	5.65248	0.239074
$560$	0	0
$561$	20.9443	0.884268
$562$	0	0
$563$	39.3607i	1.65885i	0.558614	+	0.829427i	$0.311333\pi$
$563$	39.3607i	1.65885i	−0.558614	+	0.829427i	$0.688667\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 103.430i	− 4.34363i
$568$	0	0
$569$	−12.5836	−0.527532	−0.263766	−	0.964587i	$-0.584965\pi$
$569$	−12.5836	−0.527532	−0.263766	+	0.964587i	$0.584965\pi$
$570$	0	0
$571$	24.0000	1.00437	0.502184	−	0.864761i	$-0.332530\pi$
$571$	24.0000	1.00437	0.502184	+	0.864761i	$0.332530\pi$
$572$	0	0
$573$	62.0689i	2.59296i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	9.00000i	0.374675i	0.982296	+	0.187337i	$0.0599858\pi$
$577$	9.00000i	0.374675i	−0.982296	+	0.187337i	$0.940014\pi$
$578$	0	0
$579$	−9.52786	−0.395965
$580$	0	0
$581$	−38.1246	−1.58168
$582$	0	0
$583$	− 1.23607i	− 0.0511927i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	27.5967i	1.13904i	0.821978	+	0.569520i	$0.192871\pi$
$587$	27.5967i	1.13904i	−0.821978	+	0.569520i	$0.807129\pi$
$588$	0	0
$589$	−0.472136	−0.0194540
$590$	0	0
$591$	−26.6525	−1.09634
$592$	0	0
$593$	− 44.8885i	− 1.84335i	−0.387961	−	0.921676i	$-0.626820\pi$
$593$	− 44.8885i	− 1.84335i	0.387961	−	0.921676i	$-0.373180\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	41.1246i	1.68312i
$598$	0	0
$599$	−16.8328	−0.687770	−0.343885	−	0.939012i	$-0.611743\pi$
$599$	−16.8328	−0.687770	−0.343885	+	0.939012i	$0.611743\pi$
$600$	0	0
$601$	−14.9443	−0.609590	−0.304795	−	0.952418i	$-0.598588\pi$
$601$	−14.9443	−0.609590	−0.304795	+	0.952418i	$0.598588\pi$
$602$	0	0
$603$	− 96.7214i	− 3.93880i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	23.1246i	0.938599i	0.883039	+	0.469300i	$0.155494\pi$
$607$	23.1246i	0.938599i	−0.883039	+	0.469300i	$0.844506\pi$
$608$	0	0
$609$	−24.1803	−0.979837
$610$	0	0
$611$	−26.1803	−1.05914
$612$	0	0
$613$	18.0689i	0.729795i	0.931048	+	0.364898i	$0.118896\pi$
$613$	18.0689i	0.729795i	−0.931048	+	0.364898i	$0.881104\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	25.3050i	1.01874i	0.860548	+	0.509369i	$0.170121\pi$
$617$	25.3050i	1.01874i	−0.860548	+	0.509369i	$0.829879\pi$
$618$	0	0
$619$	9.52786	0.382957	0.191479	−	0.981497i	$-0.438672\pi$
$619$	9.52786	0.382957	0.191479	+	0.981497i	$0.438672\pi$
$620$	0	0
$621$	−14.4721	−0.580747
$622$	0	0
$623$	8.47214i	0.339429i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	3.23607i	0.129236i
$628$	0	0
$629$	−72.7214	−2.89959
$630$	0	0
$631$	−24.1246	−0.960386	−0.480193	−	0.877163i	$-0.659433\pi$
$631$	−24.1246	−0.960386	−0.480193	+	0.877163i	$0.659433\pi$
$632$	0	0
$633$	48.3607i	1.92216i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 24.4721i	− 0.969621i
$638$	0	0
$639$	−74.7214	−2.95593
$640$	0	0
$641$	27.7771	1.09713	0.548564	−	0.836108i	$-0.315175\pi$
$641$	27.7771	1.09713	0.548564	+	0.836108i	$0.315175\pi$
$642$	0	0
$643$	− 1.94427i	− 0.0766746i	−0.999265	−	0.0383373i	$-0.987794\pi$
$643$	− 1.94427i	− 0.0766746i	0.999265	−	0.0383373i	$-0.0122061\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	10.1803i	0.400230i	0.979772	+	0.200115i	$0.0641317\pi$
$647$	10.1803i	0.400230i	−0.979772	+	0.200115i	$0.935868\pi$
$648$	0	0
$649$	−3.23607	−0.127027
$650$	0	0
$651$	−6.47214	−0.253663
$652$	0	0
$653$	39.5410i	1.54736i	0.633577	+	0.773680i	$0.281586\pi$
$653$	39.5410i	1.54736i	−0.633577	+	0.773680i	$0.718414\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 70.7771i	− 2.76128i
$658$	0	0
$659$	−13.4721	−0.524800	−0.262400	−	0.964959i	$-0.584514\pi$
$659$	−13.4721	−0.524800	−0.262400	+	0.964959i	$0.584514\pi$
$660$	0	0
$661$	−21.8197	−0.848686	−0.424343	−	0.905501i	$-0.639495\pi$
$661$	−21.8197	−0.848686	−0.424343	+	0.905501i	$0.639495\pi$
$662$	0	0
$663$	46.8328i	1.81884i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.76393i	0.0682997i
$668$	0	0
$669$	−42.8328	−1.65601
$670$	0	0
$671$	−7.23607	−0.279345
$672$	0	0
$673$	− 48.3050i	− 1.86202i	−0.364995	−	0.931010i	$-0.618929\pi$
$673$	− 48.3050i	− 1.86202i	0.364995	−	0.931010i	$-0.381071\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 17.7082i	− 0.680582i	−0.940320	−	0.340291i	$-0.889474\pi$
$677$	− 17.7082i	− 0.680582i	0.940320	−	0.340291i	$-0.110526\pi$
$678$	0	0
$679$	26.1803	1.00471
$680$	0	0
$681$	41.8885	1.60517
$682$	0	0
$683$	− 41.7771i	− 1.59856i	−0.600962	−	0.799278i	$-0.705216\pi$
$683$	− 41.7771i	− 1.59856i	0.600962	−	0.799278i	$-0.294784\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	22.4721i	0.857365i
$688$	0	0
$689$	2.76393	0.105297
$690$	0	0
$691$	−42.4721	−1.61572	−0.807858	−	0.589377i	$-0.799373\pi$
$691$	−42.4721	−1.61572	−0.807858	+	0.589377i	$0.799373\pi$
$692$	0	0
$693$	31.6525i	1.20238i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 38.4721i	− 1.45724i
$698$	0	0
$699$	−14.6525	−0.554208
$700$	0	0
$701$	−37.5279	−1.41741	−0.708704	−	0.705506i	$-0.750720\pi$
$701$	−37.5279	−1.41741	−0.708704	+	0.705506i	$0.750720\pi$
$702$	0	0
$703$	− 11.2361i	− 0.423776i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	80.2492i	3.01808i
$708$	0	0
$709$	−9.41641	−0.353641	−0.176820	−	0.984243i	$-0.556581\pi$
$709$	−9.41641	−0.353641	−0.176820	+	0.984243i	$0.556581\pi$
$710$	0	0
$711$	113.430	4.25394
$712$	0	0
$713$	0.472136i	0.0176816i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	21.8885i	0.817443i
$718$	0	0
$719$	−37.7082	−1.40628	−0.703139	−	0.711052i	$-0.748219\pi$
$719$	−37.7082	−1.40628	−0.703139	+	0.711052i	$0.748219\pi$
$720$	0	0
$721$	9.47214	0.352761
$722$	0	0
$723$	− 68.7214i	− 2.55577i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	44.7214i	1.65862i	0.558786	+	0.829312i	$0.311267\pi$
$727$	44.7214i	1.65862i	−0.558786	+	0.829312i	$0.688733\pi$
$728$	0	0
$729$	−41.9443	−1.55349
$730$	0	0
$731$	16.3607	0.605122
$732$	0	0
$733$	39.5279i	1.45999i	0.683450	+	0.729997i	$0.260479\pi$
$733$	39.5279i	1.45999i	−0.683450	+	0.729997i	$0.739521\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	12.9443i	0.476808i
$738$	0	0
$739$	−9.05573	−0.333120	−0.166560	−	0.986031i	$-0.553266\pi$
$739$	−9.05573	−0.333120	−0.166560	+	0.986031i	$0.553266\pi$
$740$	0	0
$741$	−7.23607	−0.265824
$742$	0	0
$743$	− 39.1803i	− 1.43739i	−0.695327	−	0.718694i	$-0.744740\pi$
$743$	− 39.1803i	− 1.43739i	0.695327	−	0.718694i	$-0.255260\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	67.2492i	2.46052i
$748$	0	0
$749$	−82.2492	−3.00532
$750$	0	0
$751$	−22.5967	−0.824567	−0.412284	−	0.911056i	$-0.635269\pi$
$751$	−22.5967	−0.824567	−0.412284	+	0.911056i	$0.635269\pi$
$752$	0	0
$753$	17.8885i	0.651895i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 16.7639i	− 0.609295i	−0.952465	−	0.304648i	$-0.901461\pi$
$757$	− 16.7639i	− 0.609295i	0.952465	−	0.304648i	$-0.0985387\pi$
$758$	0	0
$759$	3.23607	0.117462
$760$	0	0
$761$	−52.8885	−1.91721	−0.958604	−	0.284742i	$-0.908092\pi$
$761$	−52.8885	−1.91721	−0.958604	+	0.284742i	$0.908092\pi$
$762$	0	0
$763$	− 54.0689i	− 1.95743i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 7.23607i	− 0.261279i
$768$	0	0
$769$	46.0689	1.66129	0.830643	−	0.556805i	$-0.187973\pi$
$769$	46.0689	1.66129	0.830643	+	0.556805i	$0.187973\pi$
$770$	0	0
$771$	−45.3050	−1.63162
$772$	0	0
$773$	4.76393i	0.171347i	0.996323	+	0.0856734i	$0.0273041\pi$
$773$	4.76393i	0.171347i	−0.996323	+	0.0856734i	$0.972696\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 154.026i	− 5.52566i
$778$	0	0
$779$	5.94427	0.212976
$780$	0	0
$781$	10.0000	0.357828
$782$	0	0
$783$	25.5279i	0.912291i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	39.7214i	1.41591i	0.706256	+	0.707957i	$0.250383\pi$
$787$	39.7214i	1.41591i	−0.706256	+	0.707957i	$0.749617\pi$
$788$	0	0
$789$	−25.8885	−0.921657
$790$	0	0
$791$	−28.6525	−1.01876
$792$	0	0
$793$	− 16.1803i	− 0.574581i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	15.2361i	0.539689i	0.962904	+	0.269845i	$0.0869723\pi$
$797$	15.2361i	0.539689i	−0.962904	+	0.269845i	$0.913028\pi$
$798$	0	0
$799$	−75.7771	−2.68080
$800$	0	0
$801$	14.9443	0.528030
$802$	0	0
$803$	9.47214i	0.334264i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 81.4853i	− 2.86842i
$808$	0	0
$809$	9.11146	0.320342	0.160171	−	0.987089i	$-0.448795\pi$
$809$	9.11146	0.320342	0.160171	+	0.987089i	$0.448795\pi$
$810$	0	0
$811$	15.4164	0.541343	0.270672	−	0.962672i	$-0.412754\pi$
$811$	15.4164	0.541343	0.270672	+	0.962672i	$0.412754\pi$
$812$	0	0
$813$	103.777i	3.63962i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	2.52786i	0.0884388i
$818$	0	0
$819$	−70.7771	−2.47315
$820$	0	0
$821$	−22.2361	−0.776044	−0.388022	−	0.921650i	$-0.626842\pi$
$821$	−22.2361	−0.776044	−0.388022	+	0.921650i	$0.626842\pi$
$822$	0	0
$823$	− 8.94427i	− 0.311778i	−0.987775	−	0.155889i	$-0.950176\pi$
$823$	− 8.94427i	− 0.311778i	0.987775	−	0.155889i	$-0.0498242\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3.00000i	0.104320i	0.998639	+	0.0521601i	$0.0166106\pi$
$827$	3.00000i	0.104320i	−0.998639	+	0.0521601i	$0.983389\pi$
$828$	0	0
$829$	4.23607	0.147125	0.0735624	−	0.997291i	$-0.476563\pi$
$829$	4.23607	0.147125	0.0735624	+	0.997291i	$0.476563\pi$
$830$	0	0
$831$	46.0689	1.59811
$832$	0	0
$833$	− 70.8328i	− 2.45421i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	6.83282i	0.236177i
$838$	0	0
$839$	−8.59675	−0.296793	−0.148396	−	0.988928i	$-0.547411\pi$
$839$	−8.59675	−0.296793	−0.148396	+	0.988928i	$0.547411\pi$
$840$	0	0
$841$	−25.8885	−0.892708
$842$	0	0
$843$	71.7771i	2.47213i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	42.3607i	1.45553i
$848$	0	0
$849$	−91.7771	−3.14978
$850$	0	0
$851$	−11.2361	−0.385167
$852$	0	0
$853$	− 28.2361i	− 0.966785i	−0.875404	−	0.483392i	$-0.839404\pi$
$853$	− 28.2361i	− 0.966785i	0.875404	−	0.483392i	$-0.160596\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 35.3050i	− 1.20599i	−0.797743	−	0.602997i	$-0.793973\pi$
$857$	− 35.3050i	− 1.20599i	0.797743	−	0.602997i	$-0.206027\pi$
$858$	0	0
$859$	21.7082	0.740674	0.370337	−	0.928897i	$-0.379242\pi$
$859$	21.7082	0.740674	0.370337	+	0.928897i	$0.379242\pi$
$860$	0	0
$861$	81.4853	2.77701
$862$	0	0
$863$	34.5410i	1.17579i	0.808937	+	0.587895i	$0.200043\pi$
$863$	34.5410i	1.17579i	−0.808937	+	0.587895i	$0.799957\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	80.5410i	2.73532i
$868$	0	0
$869$	−15.1803	−0.514958
$870$	0	0
$871$	−28.9443	−0.980739
$872$	0	0
$873$	− 46.1803i	− 1.56297i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 2.58359i	− 0.0872417i	−0.999048	−	0.0436209i	$-0.986111\pi$
$877$	− 2.58359i	− 0.0872417i	0.999048	−	0.0436209i	$-0.0138894\pi$
$878$	0	0
$879$	46.8328	1.57963
$880$	0	0
$881$	37.7082	1.27042	0.635211	−	0.772339i	$-0.280913\pi$
$881$	37.7082	1.27042	0.635211	+	0.772339i	$0.280913\pi$
$882$	0	0
$883$	3.05573i	0.102833i	0.998677	+	0.0514167i	$0.0163737\pi$
$883$	3.05573i	0.102833i	−0.998677	+	0.0514167i	$0.983626\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 12.3607i	− 0.415031i	−0.978232	−	0.207516i	$-0.933462\pi$
$887$	− 12.3607i	− 0.415031i	0.978232	−	0.207516i	$-0.0665377\pi$
$888$	0	0
$889$	−17.7082	−0.593914
$890$	0	0
$891$	24.4164	0.817980
$892$	0	0
$893$	− 11.7082i	− 0.391800i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	7.23607i	0.241605i
$898$	0	0
$899$	0.832816	0.0277760
$900$	0	0
$901$	8.00000	0.266519
$902$	0	0
$903$	34.6525i	1.15316i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 5.58359i	− 0.185400i	−0.995694	−	0.0927001i	$-0.970450\pi$
$907$	− 5.58359i	− 0.185400i	0.995694	−	0.0927001i	$-0.0295498\pi$
$908$	0	0
$909$	141.554	4.69506
$910$	0	0
$911$	−31.0689	−1.02936	−0.514679	−	0.857383i	$-0.672089\pi$
$911$	−31.0689	−1.02936	−0.514679	+	0.857383i	$0.672089\pi$
$912$	0	0
$913$	− 9.00000i	− 0.297857i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	63.3050i	2.09051i
$918$	0	0
$919$	−43.7771	−1.44407	−0.722036	−	0.691855i	$-0.756794\pi$
$919$	−43.7771	−1.44407	−0.722036	+	0.691855i	$0.756794\pi$
$920$	0	0
$921$	77.6656	2.55917
$922$	0	0
$923$	22.3607i	0.736011i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 16.7082i	− 0.548769i
$928$	0	0
$929$	−31.0000	−1.01708	−0.508539	−	0.861039i	$-0.669814\pi$
$929$	−31.0000	−1.01708	−0.508539	+	0.861039i	$0.669814\pi$
$930$	0	0
$931$	10.9443	0.358684
$932$	0	0
$933$	10.4721i	0.342842i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	7.12461i	0.232751i	0.993205	+	0.116375i	$0.0371276\pi$
$937$	7.12461i	0.232751i	−0.993205	+	0.116375i	$0.962872\pi$
$938$	0	0
$939$	34.4721	1.12496
$940$	0	0
$941$	−13.2361	−0.431483	−0.215742	−	0.976450i	$-0.569217\pi$
$941$	−13.2361	−0.431483	−0.215742	+	0.976450i	$0.569217\pi$
$942$	0	0
$943$	− 5.94427i	− 0.193572i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	45.1935i	1.46859i	0.678830	+	0.734296i	$0.262487\pi$
$947$	45.1935i	1.46859i	−0.678830	+	0.734296i	$0.737513\pi$
$948$	0	0
$949$	−21.1803	−0.687543
$950$	0	0
$951$	−71.5967	−2.32168
$952$	0	0
$953$	− 5.34752i	− 0.173223i	−0.996242	−	0.0866116i	$-0.972396\pi$
$953$	− 5.34752i	− 0.173223i	0.996242	−	0.0866116i	$-0.0276039\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 5.70820i	− 0.184520i
$958$	0	0
$959$	48.8328	1.57689
$960$	0	0
$961$	−30.7771	−0.992809
$962$	0	0
$963$	145.082i	4.67520i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	5.88854i	0.189363i	0.995508	+	0.0946814i	$0.0301832\pi$
$967$	5.88854i	0.189363i	−0.995508	+	0.0946814i	$0.969817\pi$
$968$	0	0
$969$	−20.9443	−0.672827
$970$	0	0
$971$	37.2492	1.19538	0.597692	−	0.801726i	$-0.296084\pi$
$971$	37.2492	1.19538	0.597692	+	0.801726i	$0.296084\pi$
$972$	0	0
$973$	84.7214i	2.71604i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	22.5410i	0.721151i	0.932730	+	0.360576i	$0.117420\pi$
$977$	22.5410i	0.721151i	−0.932730	+	0.360576i	$0.882580\pi$
$978$	0	0
$979$	−2.00000	−0.0639203
$980$	0	0
$981$	−95.3738	−3.04505
$982$	0	0
$983$	− 27.6525i	− 0.881977i	−0.897513	−	0.440989i	$-0.854628\pi$
$983$	− 27.6525i	− 0.881977i	0.897513	−	0.440989i	$-0.145372\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 160.498i	− 5.10872i
$988$	0	0
$989$	2.52786	0.0803814
$990$	0	0
$991$	−41.5967	−1.32136	−0.660682	−	0.750666i	$-0.729733\pi$
$991$	−41.5967	−1.32136	−0.660682	+	0.750666i	$0.729733\pi$
$992$	0	0
$993$	− 100.138i	− 3.17778i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 2.12461i	− 0.0672871i	−0.999434	−	0.0336436i	$-0.989289\pi$
$997$	− 2.12461i	− 0.0672871i	0.999434	−	0.0336436i	$-0.0107111\pi$
$998$	0	0
$999$	−162.610	−5.14475

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4600.2.e.l.4049.1		4
5.2	odd	4		4600.2.a.q.1.1	✓	2
5.3	odd	4		4600.2.a.u.1.2	yes	2
5.4	even	2	inner	4600.2.e.l.4049.4		4
20.3	even	4		9200.2.a.bn.1.1		2
20.7	even	4		9200.2.a.bz.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4600.2.a.q.1.1	✓	2	5.2	odd	4
4600.2.a.u.1.2	yes	2	5.3	odd	4
4600.2.e.l.4049.1		4	1.1	even	1	trivial
4600.2.e.l.4049.4		4	5.4	even	2	inner
9200.2.a.bn.1.1		2	20.3	even	4
9200.2.a.bz.1.2		2	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)

\(f(q)\)	\(=\)	\(q-3.23607i q^{3} -4.23607i q^{7} -7.47214 q^{9} +1.00000 q^{11} +2.23607i q^{13} +6.47214i q^{17} -1.00000 q^{19} -13.7082 q^{21} +1.00000i q^{23} +14.4721i q^{27} +1.76393 q^{29} +0.472136 q^{31} -3.23607i q^{33} +11.2361i q^{37} +7.23607 q^{39} -5.94427 q^{41} -2.52786i q^{43} +11.7082i q^{47} -10.9443 q^{49} +20.9443 q^{51} -1.23607i q^{53} +3.23607i q^{57} -3.23607 q^{59} -7.23607 q^{61} +31.6525i q^{63} +12.9443i q^{67} +3.23607 q^{69} +10.0000 q^{71} +9.47214i q^{73} -4.23607i q^{77} -15.1803 q^{79} +24.4164 q^{81} -9.00000i q^{83} -5.70820i q^{87} -2.00000 q^{89} +9.47214 q^{91} -1.52786i q^{93} +6.18034i q^{97} -7.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{9} + 4 q^{11} - 4 q^{19} - 28 q^{21} + 16 q^{29} - 16 q^{31} + 20 q^{39} + 12 q^{41} - 8 q^{49} + 48 q^{51} - 4 q^{59} - 20 q^{61} + 4 q^{69} + 40 q^{71} - 16 q^{79} + 44 q^{81} - 8 q^{89} + 20 q^{91}+ \cdots - 12 q^{99}+O(q^{100}) \) 4 * q - 12 * q^9 + 4 * q^11 - 4 * q^19 - 28 * q^21 + 16 * q^29 - 16 * q^31 + 20 * q^39 + 12 * q^41 - 8 * q^49 + 48 * q^51 - 4 * q^59 - 20 * q^61 + 4 * q^69 + 40 * q^71 - 16 * q^79 + 44 * q^81 - 8 * q^89 + 20 * q^91 - 12 * q^99

Introduction

L-functions

Modular forms

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Database

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