LMFDB - Embedded newform 4650.2.d.s.3349.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4650,2,Mod(3349,4650)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4650, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4650.3349");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4650.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$37.1304369399$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3349.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	4650.3349
Dual form			4650.2.d.s.3349.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -5.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -5.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -1.00000 q^{11} -1.00000i q^{12} -5.00000i q^{13} -5.00000 q^{14} +1.00000 q^{16} -4.00000i q^{17} +1.00000i q^{18} +2.00000 q^{19} +5.00000 q^{21} +1.00000i q^{22} -4.00000i q^{23} -1.00000 q^{24} -5.00000 q^{26} -1.00000i q^{27} +5.00000i q^{28} -6.00000 q^{29} -1.00000 q^{31} -1.00000i q^{32} -1.00000i q^{33} -4.00000 q^{34} +1.00000 q^{36} +1.00000i q^{37} -2.00000i q^{38} +5.00000 q^{39} +7.00000 q^{41} -5.00000i q^{42} +11.0000i q^{43} +1.00000 q^{44} -4.00000 q^{46} -11.0000i q^{47} +1.00000i q^{48} -18.0000 q^{49} +4.00000 q^{51} +5.00000i q^{52} -3.00000i q^{53} -1.00000 q^{54} +5.00000 q^{56} +2.00000i q^{57} +6.00000i q^{58} -6.00000 q^{59} +9.00000 q^{61} +1.00000i q^{62} +5.00000i q^{63} -1.00000 q^{64} -1.00000 q^{66} +10.0000i q^{67} +4.00000i q^{68} +4.00000 q^{69} -1.00000 q^{71} -1.00000i q^{72} +2.00000i q^{73} +1.00000 q^{74} -2.00000 q^{76} +5.00000i q^{77} -5.00000i q^{78} +1.00000 q^{81} -7.00000i q^{82} -15.0000i q^{83} -5.00000 q^{84} +11.0000 q^{86} -6.00000i q^{87} -1.00000i q^{88} +6.00000 q^{89} -25.0000 q^{91} +4.00000i q^{92} -1.00000i q^{93} -11.0000 q^{94} +1.00000 q^{96} +10.0000i q^{97} +18.0000i q^{98} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9	$ 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 2 q^{11} - 10 q^{14} + 2 q^{16} + 4 q^{19} + 10 q^{21} - 2 q^{24} - 10 q^{26} - 12 q^{29} - 2 q^{31} - 8 q^{34} + 2 q^{36} + 10 q^{39} + 14 q^{41} + 2 q^{44} - 8 q^{46} - 36 q^{49} + 8 q^{51} - 2 q^{54} + 10 q^{56} - 12 q^{59} + 18 q^{61} - 2 q^{64} - 2 q^{66} + 8 q^{69} - 2 q^{71} + 2 q^{74} - 4 q^{76} + 2 q^{81} - 10 q^{84} + 22 q^{86} + 12 q^{89} - 50 q^{91} - 22 q^{94} + 2 q^{96} + 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 - 2 * q^11 - 10 * q^14 + 2 * q^16 + 4 * q^19 + 10 * q^21 - 2 * q^24 - 10 * q^26 - 12 * q^29 - 2 * q^31 - 8 * q^34 + 2 * q^36 + 10 * q^39 + 14 * q^41 + 2 * q^44 - 8 * q^46 - 36 * q^49 + 8 * q^51 - 2 * q^54 + 10 * q^56 - 12 * q^59 + 18 * q^61 - 2 * q^64 - 2 * q^66 + 8 * q^69 - 2 * q^71 + 2 * q^74 - 4 * q^76 + 2 * q^81 - 10 * q^84 + 22 * q^86 + 12 * q^89 - 50 * q^91 - 22 * q^94 + 2 * q^96 + 2 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4650\mathbb{Z}\right)^\times$.

$n$	$1801$	$2977$	$3101$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	− 5.00000i	− 1.88982i	−0.327327	−	0.944911i	$-0.606148\pi$
$7$	− 5.00000i	− 1.88982i	0.327327	−	0.944911i	$-0.393852\pi$
$8$	1.00000i	0.353553i
$9$	−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	$0.548171\pi$
$12$	− 1.00000i	− 0.288675i
$13$	− 5.00000i	− 1.38675i	−0.720577	−	0.693375i	$-0.756123\pi$
$13$	− 5.00000i	− 1.38675i	0.720577	−	0.693375i	$-0.243877\pi$
$14$	−5.00000	−1.33631
$15$	0	0
$16$	1.00000	0.250000
$17$	− 4.00000i	− 0.970143i	−0.874475	−	0.485071i	$-0.838794\pi$
$17$	− 4.00000i	− 0.970143i	0.874475	−	0.485071i	$-0.161206\pi$
$18$	1.00000i	0.235702i
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	5.00000	1.09109
$22$	1.00000i	0.213201i
$23$	− 4.00000i	− 0.834058i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$	− 4.00000i	− 0.834058i	0.908893	−	0.417029i	$-0.136929\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	−5.00000	−0.980581
$27$	− 1.00000i	− 0.192450i
$28$	5.00000i	0.944911i
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	− 1.00000i	− 0.176777i
$33$	− 1.00000i	− 0.174078i
$34$	−4.00000	−0.685994
$35$	0	0
$36$	1.00000	0.166667
$37$	1.00000i	0.164399i	0.996616	+	0.0821995i	$0.0261945\pi$
$37$	1.00000i	0.164399i	−0.996616	+	0.0821995i	$0.973806\pi$
$38$	− 2.00000i	− 0.324443i
$39$	5.00000	0.800641
$40$	0	0
$41$	7.00000	1.09322	0.546608	−	0.837389i	$-0.315919\pi$
$41$	7.00000	1.09322	0.546608	+	0.837389i	$0.315919\pi$
$42$	− 5.00000i	− 0.771517i
$43$	11.0000i	1.67748i	0.544529	+	0.838742i	$0.316708\pi$
$43$	11.0000i	1.67748i	−0.544529	+	0.838742i	$0.683292\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	−4.00000	−0.589768
$47$	− 11.0000i	− 1.60451i	−0.596978	−	0.802257i	$-0.703632\pi$
$47$	− 11.0000i	− 1.60451i	0.596978	−	0.802257i	$-0.296368\pi$
$48$	1.00000i	0.144338i
$49$	−18.0000	−2.57143
$50$	0	0
$51$	4.00000	0.560112
$52$	5.00000i	0.693375i
$53$	− 3.00000i	− 0.412082i	−0.978543	−	0.206041i	$-0.933942\pi$
$53$	− 3.00000i	− 0.412082i	0.978543	−	0.206041i	$-0.0660580\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	5.00000	0.668153
$57$	2.00000i	0.264906i
$58$	6.00000i	0.787839i
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	9.00000	1.15233	0.576166	−	0.817333i	$-0.304548\pi$
$61$	9.00000	1.15233	0.576166	+	0.817333i	$0.304548\pi$
$62$	1.00000i	0.127000i
$63$	5.00000i	0.629941i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−1.00000	−0.123091
$67$	10.0000i	1.22169i	0.791748	+	0.610847i	$0.209171\pi$
$67$	10.0000i	1.22169i	−0.791748	+	0.610847i	$0.790829\pi$
$68$	4.00000i	0.485071i
$69$	4.00000	0.481543
$70$	0	0
$71$	−1.00000	−0.118678	−0.0593391	−	0.998238i	$-0.518899\pi$
$71$	−1.00000	−0.118678	−0.0593391	+	0.998238i	$0.518899\pi$
$72$	− 1.00000i	− 0.117851i
$73$	2.00000i	0.234082i	0.993127	+	0.117041i	$0.0373409\pi$
$73$	2.00000i	0.234082i	−0.993127	+	0.117041i	$0.962659\pi$
$74$	1.00000	0.116248
$75$	0	0
$76$	−2.00000	−0.229416
$77$	5.00000i	0.569803i
$78$	− 5.00000i	− 0.566139i
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 7.00000i	− 0.773021i
$83$	− 15.0000i	− 1.64646i	−0.567705	−	0.823232i	$-0.692169\pi$
$83$	− 15.0000i	− 1.64646i	0.567705	−	0.823232i	$-0.307831\pi$
$84$	−5.00000	−0.545545
$85$	0	0
$86$	11.0000	1.18616
$87$	− 6.00000i	− 0.643268i
$88$	− 1.00000i	− 0.106600i
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	−25.0000	−2.62071
$92$	4.00000i	0.417029i
$93$	− 1.00000i	− 0.103695i
$94$	−11.0000	−1.13456
$95$	0	0
$96$	1.00000	0.102062
$97$	10.0000i	1.01535i	0.861550	+	0.507673i	$0.169494\pi$
$97$	10.0000i	1.01535i	−0.861550	+	0.507673i	$0.830506\pi$
$98$	18.0000i	1.81827i
$99$	1.00000	0.100504
$100$	0	0
$101$	−8.00000	−0.796030	−0.398015	−	0.917379i	$-0.630301\pi$
$101$	−8.00000	−0.796030	−0.398015	+	0.917379i	$0.630301\pi$
$102$	− 4.00000i	− 0.396059i
$103$	13.0000i	1.28093i	0.767988	+	0.640464i	$0.221258\pi$
$103$	13.0000i	1.28093i	−0.767988	+	0.640464i	$0.778742\pi$
$104$	5.00000	0.490290
$105$	0	0
$106$	−3.00000	−0.291386
$107$	4.00000i	0.386695i	0.981130	+	0.193347i	$0.0619344\pi$
$107$	4.00000i	0.386695i	−0.981130	+	0.193347i	$0.938066\pi$
$108$	1.00000i	0.0962250i
$109$	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
$111$	−1.00000	−0.0949158
$112$	− 5.00000i	− 0.472456i
$113$	14.0000i	1.31701i	0.752577	+	0.658505i	$0.228811\pi$
$113$	14.0000i	1.31701i	−0.752577	+	0.658505i	$0.771189\pi$
$114$	2.00000	0.187317
$115$	0	0
$116$	6.00000	0.557086
$117$	5.00000i	0.462250i
$118$	6.00000i	0.552345i
$119$	−20.0000	−1.83340
$120$	0	0
$121$	−10.0000	−0.909091
$122$	− 9.00000i	− 0.814822i
$123$	7.00000i	0.631169i
$124$	1.00000	0.0898027
$125$	0	0
$126$	5.00000	0.445435
$127$	4.00000i	0.354943i	0.984126	+	0.177471i	$0.0567917\pi$
$127$	4.00000i	0.354943i	−0.984126	+	0.177471i	$0.943208\pi$
$128$	1.00000i	0.0883883i
$129$	−11.0000	−0.968496
$130$	0	0
$131$	−2.00000	−0.174741	−0.0873704	−	0.996176i	$-0.527846\pi$
$131$	−2.00000	−0.174741	−0.0873704	+	0.996176i	$0.527846\pi$
$132$	1.00000i	0.0870388i
$133$	− 10.0000i	− 0.867110i
$134$	10.0000	0.863868
$135$	0	0
$136$	4.00000	0.342997
$137$	4.00000i	0.341743i	0.985293	+	0.170872i	$0.0546583\pi$
$137$	4.00000i	0.341743i	−0.985293	+	0.170872i	$0.945342\pi$
$138$	− 4.00000i	− 0.340503i
$139$	11.0000	0.933008	0.466504	−	0.884519i	$-0.345513\pi$
$139$	11.0000	0.933008	0.466504	+	0.884519i	$0.345513\pi$
$140$	0	0
$141$	11.0000	0.926367
$142$	1.00000i	0.0839181i
$143$	5.00000i	0.418121i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	2.00000	0.165521
$147$	− 18.0000i	− 1.48461i
$148$	− 1.00000i	− 0.0821995i
$149$	−16.0000	−1.31077	−0.655386	−	0.755295i	$-0.727494\pi$
$149$	−16.0000	−1.31077	−0.655386	+	0.755295i	$0.727494\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	2.00000i	0.162221i
$153$	4.00000i	0.323381i
$154$	5.00000	0.402911
$155$	0	0
$156$	−5.00000	−0.400320
$157$	− 14.0000i	− 1.11732i	−0.829396	−	0.558661i	$-0.811315\pi$
$157$	− 14.0000i	− 1.11732i	0.829396	−	0.558661i	$-0.188685\pi$
$158$	0	0
$159$	3.00000	0.237915
$160$	0	0
$161$	−20.0000	−1.57622
$162$	− 1.00000i	− 0.0785674i
$163$	− 2.00000i	− 0.156652i	−0.996928	−	0.0783260i	$-0.975042\pi$
$163$	− 2.00000i	− 0.156652i	0.996928	−	0.0783260i	$-0.0249575\pi$
$164$	−7.00000	−0.546608
$165$	0	0
$166$	−15.0000	−1.16423
$167$	− 4.00000i	− 0.309529i	−0.987951	−	0.154765i	$-0.950538\pi$
$167$	− 4.00000i	− 0.309529i	0.987951	−	0.154765i	$-0.0494619\pi$
$168$	5.00000i	0.385758i
$169$	−12.0000	−0.923077
$170$	0	0
$171$	−2.00000	−0.152944
$172$	− 11.0000i	− 0.838742i
$173$	− 18.0000i	− 1.36851i	−0.729241	−	0.684257i	$-0.760127\pi$
$173$	− 18.0000i	− 1.36851i	0.729241	−	0.684257i	$-0.239873\pi$
$174$	−6.00000	−0.454859
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	− 6.00000i	− 0.450988i
$178$	− 6.00000i	− 0.449719i
$179$	3.00000	0.224231	0.112115	−	0.993695i	$-0.464237\pi$
$179$	3.00000	0.224231	0.112115	+	0.993695i	$0.464237\pi$
$180$	0	0
$181$	−19.0000	−1.41226	−0.706129	−	0.708083i	$-0.749560\pi$
$181$	−19.0000	−1.41226	−0.706129	+	0.708083i	$0.749560\pi$
$182$	25.0000i	1.85312i
$183$	9.00000i	0.665299i
$184$	4.00000	0.294884
$185$	0	0
$186$	−1.00000	−0.0733236
$187$	4.00000i	0.292509i
$188$	11.0000i	0.802257i
$189$	−5.00000	−0.363696
$190$	0	0
$191$	4.00000	0.289430	0.144715	−	0.989473i	$-0.453773\pi$
$191$	4.00000	0.289430	0.144715	+	0.989473i	$0.453773\pi$
$192$	− 1.00000i	− 0.0721688i
$193$	− 5.00000i	− 0.359908i	−0.983675	−	0.179954i	$-0.942405\pi$
$193$	− 5.00000i	− 0.359908i	0.983675	−	0.179954i	$-0.0575949\pi$
$194$	10.0000	0.717958
$195$	0	0
$196$	18.0000	1.28571
$197$	− 17.0000i	− 1.21120i	−0.795769	−	0.605600i	$-0.792933\pi$
$197$	− 17.0000i	− 1.21120i	0.795769	−	0.605600i	$-0.207067\pi$
$198$	− 1.00000i	− 0.0710669i
$199$	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
$201$	−10.0000	−0.705346
$202$	8.00000i	0.562878i
$203$	30.0000i	2.10559i
$204$	−4.00000	−0.280056
$205$	0	0
$206$	13.0000	0.905753
$207$	4.00000i	0.278019i
$208$	− 5.00000i	− 0.346688i
$209$	−2.00000	−0.138343
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	3.00000i	0.206041i
$213$	− 1.00000i	− 0.0685189i
$214$	4.00000	0.273434
$215$	0	0
$216$	1.00000	0.0680414
$217$	5.00000i	0.339422i
$218$	− 10.0000i	− 0.677285i
$219$	−2.00000	−0.135147
$220$	0	0
$221$	−20.0000	−1.34535
$222$	1.00000i	0.0671156i
$223$	− 18.0000i	− 1.20537i	−0.797980	−	0.602685i	$-0.794098\pi$
$223$	− 18.0000i	− 1.20537i	0.797980	−	0.602685i	$-0.205902\pi$
$224$	−5.00000	−0.334077
$225$	0	0
$226$	14.0000	0.931266
$227$	4.00000i	0.265489i	0.991150	+	0.132745i	$0.0423790\pi$
$227$	4.00000i	0.265489i	−0.991150	+	0.132745i	$0.957621\pi$
$228$	− 2.00000i	− 0.132453i
$229$	22.0000	1.45380	0.726900	−	0.686743i	$-0.240960\pi$
$229$	22.0000	1.45380	0.726900	+	0.686743i	$0.240960\pi$
$230$	0	0
$231$	−5.00000	−0.328976
$232$	− 6.00000i	− 0.393919i
$233$	7.00000i	0.458585i	0.973358	+	0.229293i	$0.0736413\pi$
$233$	7.00000i	0.458585i	−0.973358	+	0.229293i	$0.926359\pi$
$234$	5.00000	0.326860
$235$	0	0
$236$	6.00000	0.390567
$237$	0	0
$238$	20.0000i	1.29641i
$239$	−18.0000	−1.16432	−0.582162	−	0.813073i	$-0.697793\pi$
$239$	−18.0000	−1.16432	−0.582162	+	0.813073i	$0.697793\pi$
$240$	0	0
$241$	−26.0000	−1.67481	−0.837404	−	0.546585i	$-0.815928\pi$
$241$	−26.0000	−1.67481	−0.837404	+	0.546585i	$0.815928\pi$
$242$	10.0000i	0.642824i
$243$	1.00000i	0.0641500i
$244$	−9.00000	−0.576166
$245$	0	0
$246$	7.00000	0.446304
$247$	− 10.0000i	− 0.636285i
$248$	− 1.00000i	− 0.0635001i
$249$	15.0000	0.950586
$250$	0	0
$251$	23.0000	1.45175	0.725874	−	0.687828i	$-0.241436\pi$
$251$	23.0000	1.45175	0.725874	+	0.687828i	$0.241436\pi$
$252$	− 5.00000i	− 0.314970i
$253$	4.00000i	0.251478i
$254$	4.00000	0.250982
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 27.0000i	− 1.68421i	−0.539311	−	0.842107i	$-0.681315\pi$
$257$	− 27.0000i	− 1.68421i	0.539311	−	0.842107i	$-0.318685\pi$
$258$	11.0000i	0.684830i
$259$	5.00000	0.310685
$260$	0	0
$261$	6.00000	0.371391
$262$	2.00000i	0.123560i
$263$	22.0000i	1.35658i	0.734795	+	0.678289i	$0.237278\pi$
$263$	22.0000i	1.35658i	−0.734795	+	0.678289i	$0.762722\pi$
$264$	1.00000	0.0615457
$265$	0	0
$266$	−10.0000	−0.613139
$267$	6.00000i	0.367194i
$268$	− 10.0000i	− 0.610847i
$269$	−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
$271$	−12.0000	−0.728948	−0.364474	−	0.931214i	$-0.618751\pi$
$271$	−12.0000	−0.728948	−0.364474	+	0.931214i	$0.618751\pi$
$272$	− 4.00000i	− 0.242536i
$273$	− 25.0000i	− 1.51307i
$274$	4.00000	0.241649
$275$	0	0
$276$	−4.00000	−0.240772
$277$	6.00000i	0.360505i	0.983620	+	0.180253i	$0.0576915\pi$
$277$	6.00000i	0.360505i	−0.983620	+	0.180253i	$0.942309\pi$
$278$	− 11.0000i	− 0.659736i
$279$	1.00000	0.0598684
$280$	0	0
$281$	−25.0000	−1.49137	−0.745687	−	0.666296i	$-0.767879\pi$
$281$	−25.0000	−1.49137	−0.745687	+	0.666296i	$0.767879\pi$
$282$	− 11.0000i	− 0.655040i
$283$	− 12.0000i	− 0.713326i	−0.934233	−	0.356663i	$-0.883914\pi$
$283$	− 12.0000i	− 0.713326i	0.934233	−	0.356663i	$-0.116086\pi$
$284$	1.00000	0.0593391
$285$	0	0
$286$	5.00000	0.295656
$287$	− 35.0000i	− 2.06598i
$288$	1.00000i	0.0589256i
$289$	1.00000	0.0588235
$290$	0	0
$291$	−10.0000	−0.586210
$292$	− 2.00000i	− 0.117041i
$293$	− 20.0000i	− 1.16841i	−0.811605	−	0.584206i	$-0.801406\pi$
$293$	− 20.0000i	− 1.16841i	0.811605	−	0.584206i	$-0.198594\pi$
$294$	−18.0000	−1.04978
$295$	0	0
$296$	−1.00000	−0.0581238
$297$	1.00000i	0.0580259i
$298$	16.0000i	0.926855i
$299$	−20.0000	−1.15663
$300$	0	0
$301$	55.0000	3.17015
$302$	8.00000i	0.460348i
$303$	− 8.00000i	− 0.459588i
$304$	2.00000	0.114708
$305$	0	0
$306$	4.00000	0.228665
$307$	− 8.00000i	− 0.456584i	−0.973593	−	0.228292i	$-0.926686\pi$
$307$	− 8.00000i	− 0.456584i	0.973593	−	0.228292i	$-0.0733141\pi$
$308$	− 5.00000i	− 0.284901i
$309$	−13.0000	−0.739544
$310$	0	0
$311$	−25.0000	−1.41762	−0.708810	−	0.705399i	$-0.750768\pi$
$311$	−25.0000	−1.41762	−0.708810	+	0.705399i	$0.750768\pi$
$312$	5.00000i	0.283069i
$313$	34.0000i	1.92179i	0.276907	+	0.960897i	$0.410691\pi$
$313$	34.0000i	1.92179i	−0.276907	+	0.960897i	$0.589309\pi$
$314$	−14.0000	−0.790066
$315$	0	0
$316$	0	0
$317$	28.0000i	1.57264i	0.617822	+	0.786318i	$0.288015\pi$
$317$	28.0000i	1.57264i	−0.617822	+	0.786318i	$0.711985\pi$
$318$	− 3.00000i	− 0.168232i
$319$	6.00000	0.335936
$320$	0	0
$321$	−4.00000	−0.223258
$322$	20.0000i	1.11456i
$323$	− 8.00000i	− 0.445132i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−2.00000	−0.110770
$327$	10.0000i	0.553001i
$328$	7.00000i	0.386510i
$329$	−55.0000	−3.03225
$330$	0	0
$331$	−17.0000	−0.934405	−0.467202	−	0.884150i	$-0.654738\pi$
$331$	−17.0000	−0.934405	−0.467202	+	0.884150i	$0.654738\pi$
$332$	15.0000i	0.823232i
$333$	− 1.00000i	− 0.0547997i
$334$	−4.00000	−0.218870
$335$	0	0
$336$	5.00000	0.272772
$337$	30.0000i	1.63420i	0.576493	+	0.817102i	$0.304421\pi$
$337$	30.0000i	1.63420i	−0.576493	+	0.817102i	$0.695579\pi$
$338$	12.0000i	0.652714i
$339$	−14.0000	−0.760376
$340$	0	0
$341$	1.00000	0.0541530
$342$	2.00000i	0.108148i
$343$	55.0000i	2.96972i
$344$	−11.0000	−0.593080
$345$	0	0
$346$	−18.0000	−0.967686
$347$	− 5.00000i	− 0.268414i	−0.990953	−	0.134207i	$-0.957151\pi$
$347$	− 5.00000i	− 0.268414i	0.990953	−	0.134207i	$-0.0428487\pi$
$348$	6.00000i	0.321634i
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	0	0
$351$	−5.00000	−0.266880
$352$	1.00000i	0.0533002i
$353$	− 32.0000i	− 1.70319i	−0.524202	−	0.851594i	$-0.675636\pi$
$353$	− 32.0000i	− 1.70319i	0.524202	−	0.851594i	$-0.324364\pi$
$354$	−6.00000	−0.318896
$355$	0	0
$356$	−6.00000	−0.317999
$357$	− 20.0000i	− 1.05851i
$358$	− 3.00000i	− 0.158555i
$359$	−20.0000	−1.05556	−0.527780	−	0.849381i	$-0.676975\pi$
$359$	−20.0000	−1.05556	−0.527780	+	0.849381i	$0.676975\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	19.0000i	0.998618i
$363$	− 10.0000i	− 0.524864i
$364$	25.0000	1.31036
$365$	0	0
$366$	9.00000	0.470438
$367$	− 24.0000i	− 1.25279i	−0.779506	−	0.626395i	$-0.784530\pi$
$367$	− 24.0000i	− 1.25279i	0.779506	−	0.626395i	$-0.215470\pi$
$368$	− 4.00000i	− 0.208514i
$369$	−7.00000	−0.364405
$370$	0	0
$371$	−15.0000	−0.778761
$372$	1.00000i	0.0518476i
$373$	8.00000i	0.414224i	0.978317	+	0.207112i	$0.0664065\pi$
$373$	8.00000i	0.414224i	−0.978317	+	0.207112i	$0.933593\pi$
$374$	4.00000	0.206835
$375$	0	0
$376$	11.0000	0.567282
$377$	30.0000i	1.54508i
$378$	5.00000i	0.257172i
$379$	34.0000	1.74646	0.873231	−	0.487306i	$-0.162020\pi$
$379$	34.0000	1.74646	0.873231	+	0.487306i	$0.162020\pi$
$380$	0	0
$381$	−4.00000	−0.204926
$382$	− 4.00000i	− 0.204658i
$383$	20.0000i	1.02195i	0.859595	+	0.510976i	$0.170716\pi$
$383$	20.0000i	1.02195i	−0.859595	+	0.510976i	$0.829284\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−5.00000	−0.254493
$387$	− 11.0000i	− 0.559161i
$388$	− 10.0000i	− 0.507673i
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	−16.0000	−0.809155
$392$	− 18.0000i	− 0.909137i
$393$	− 2.00000i	− 0.100887i
$394$	−17.0000	−0.856448
$395$	0	0
$396$	−1.00000	−0.0502519
$397$	− 38.0000i	− 1.90717i	−0.301131	−	0.953583i	$-0.597364\pi$
$397$	− 38.0000i	− 1.90717i	0.301131	−	0.953583i	$-0.402636\pi$
$398$	− 14.0000i	− 0.701757i
$399$	10.0000	0.500626
$400$	0	0
$401$	−24.0000	−1.19850	−0.599251	−	0.800561i	$-0.704535\pi$
$401$	−24.0000	−1.19850	−0.599251	+	0.800561i	$0.704535\pi$
$402$	10.0000i	0.498755i
$403$	5.00000i	0.249068i
$404$	8.00000	0.398015
$405$	0	0
$406$	30.0000	1.48888
$407$	− 1.00000i	− 0.0495682i
$408$	4.00000i	0.198030i
$409$	−34.0000	−1.68119	−0.840596	−	0.541663i	$-0.817795\pi$
$409$	−34.0000	−1.68119	−0.840596	+	0.541663i	$0.817795\pi$
$410$	0	0
$411$	−4.00000	−0.197305
$412$	− 13.0000i	− 0.640464i
$413$	30.0000i	1.47620i
$414$	4.00000	0.196589
$415$	0	0
$416$	−5.00000	−0.245145
$417$	11.0000i	0.538672i
$418$	2.00000i	0.0978232i
$419$	−36.0000	−1.75872	−0.879358	−	0.476162i	$-0.842028\pi$
$419$	−36.0000	−1.75872	−0.879358	+	0.476162i	$0.842028\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	− 4.00000i	− 0.194717i
$423$	11.0000i	0.534838i
$424$	3.00000	0.145693
$425$	0	0
$426$	−1.00000	−0.0484502
$427$	− 45.0000i	− 2.17770i
$428$	− 4.00000i	− 0.193347i
$429$	−5.00000	−0.241402
$430$	0	0
$431$	23.0000	1.10787	0.553936	−	0.832560i	$-0.313125\pi$
$431$	23.0000	1.10787	0.553936	+	0.832560i	$0.313125\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	− 16.0000i	− 0.768911i	−0.923144	−	0.384455i	$-0.874389\pi$
$433$	− 16.0000i	− 0.768911i	0.923144	−	0.384455i	$-0.125611\pi$
$434$	5.00000	0.240008
$435$	0	0
$436$	−10.0000	−0.478913
$437$	− 8.00000i	− 0.382692i
$438$	2.00000i	0.0955637i
$439$	24.0000	1.14546	0.572729	−	0.819745i	$-0.305885\pi$
$439$	24.0000	1.14546	0.572729	+	0.819745i	$0.305885\pi$
$440$	0	0
$441$	18.0000	0.857143
$442$	20.0000i	0.951303i
$443$	10.0000i	0.475114i	0.971374	+	0.237557i	$0.0763467\pi$
$443$	10.0000i	0.475114i	−0.971374	+	0.237557i	$0.923653\pi$
$444$	1.00000	0.0474579
$445$	0	0
$446$	−18.0000	−0.852325
$447$	− 16.0000i	− 0.756774i
$448$	5.00000i	0.236228i
$449$	34.0000	1.60456	0.802280	−	0.596948i	$-0.203620\pi$
$449$	34.0000	1.60456	0.802280	+	0.596948i	$0.203620\pi$
$450$	0	0
$451$	−7.00000	−0.329617
$452$	− 14.0000i	− 0.658505i
$453$	− 8.00000i	− 0.375873i
$454$	4.00000	0.187729
$455$	0	0
$456$	−2.00000	−0.0936586
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	− 22.0000i	− 1.02799i
$459$	−4.00000	−0.186704
$460$	0	0
$461$	−21.0000	−0.978068	−0.489034	−	0.872265i	$-0.662651\pi$
$461$	−21.0000	−0.978068	−0.489034	+	0.872265i	$0.662651\pi$
$462$	5.00000i	0.232621i
$463$	− 4.00000i	− 0.185896i	−0.995671	−	0.0929479i	$-0.970371\pi$
$463$	− 4.00000i	− 0.185896i	0.995671	−	0.0929479i	$-0.0296290\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	7.00000	0.324269
$467$	18.0000i	0.832941i	0.909149	+	0.416470i	$0.136733\pi$
$467$	18.0000i	0.832941i	−0.909149	+	0.416470i	$0.863267\pi$
$468$	− 5.00000i	− 0.231125i
$469$	50.0000	2.30879
$470$	0	0
$471$	14.0000	0.645086
$472$	− 6.00000i	− 0.276172i
$473$	− 11.0000i	− 0.505781i
$474$	0	0
$475$	0	0
$476$	20.0000	0.916698
$477$	3.00000i	0.137361i
$478$	18.0000i	0.823301i
$479$	16.0000	0.731059	0.365529	−	0.930800i	$-0.380888\pi$
$479$	16.0000	0.731059	0.365529	+	0.930800i	$0.380888\pi$
$480$	0	0
$481$	5.00000	0.227980
$482$	26.0000i	1.18427i
$483$	− 20.0000i	− 0.910032i
$484$	10.0000	0.454545
$485$	0	0
$486$	1.00000	0.0453609
$487$	− 4.00000i	− 0.181257i	−0.995885	−	0.0906287i	$-0.971112\pi$
$487$	− 4.00000i	− 0.181257i	0.995885	−	0.0906287i	$-0.0288876\pi$
$488$	9.00000i	0.407411i
$489$	2.00000	0.0904431
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	− 7.00000i	− 0.315584i
$493$	24.0000i	1.08091i
$494$	−10.0000	−0.449921
$495$	0	0
$496$	−1.00000	−0.0449013
$497$	5.00000i	0.224281i
$498$	− 15.0000i	− 0.672166i
$499$	32.0000	1.43252	0.716258	−	0.697835i	$-0.245853\pi$
$499$	32.0000	1.43252	0.716258	+	0.697835i	$0.245853\pi$
$500$	0	0
$501$	4.00000	0.178707
$502$	− 23.0000i	− 1.02654i
$503$	3.00000i	0.133763i	0.997761	+	0.0668817i	$0.0213050\pi$
$503$	3.00000i	0.133763i	−0.997761	+	0.0668817i	$0.978695\pi$
$504$	−5.00000	−0.222718
$505$	0	0
$506$	4.00000	0.177822
$507$	− 12.0000i	− 0.532939i
$508$	− 4.00000i	− 0.177471i
$509$	21.0000	0.930809	0.465404	−	0.885098i	$-0.345909\pi$
$509$	21.0000	0.930809	0.465404	+	0.885098i	$0.345909\pi$
$510$	0	0
$511$	10.0000	0.442374
$512$	− 1.00000i	− 0.0441942i
$513$	− 2.00000i	− 0.0883022i
$514$	−27.0000	−1.19092
$515$	0	0
$516$	11.0000	0.484248
$517$	11.0000i	0.483779i
$518$	− 5.00000i	− 0.219687i
$519$	18.0000	0.790112
$520$	0	0
$521$	35.0000	1.53338	0.766689	−	0.642019i	$-0.221903\pi$
$521$	35.0000	1.53338	0.766689	+	0.642019i	$0.221903\pi$
$522$	− 6.00000i	− 0.262613i
$523$	36.0000i	1.57417i	0.616844	+	0.787085i	$0.288411\pi$
$523$	36.0000i	1.57417i	−0.616844	+	0.787085i	$0.711589\pi$
$524$	2.00000	0.0873704
$525$	0	0
$526$	22.0000	0.959246
$527$	4.00000i	0.174243i
$528$	− 1.00000i	− 0.0435194i
$529$	7.00000	0.304348
$530$	0	0
$531$	6.00000	0.260378
$532$	10.0000i	0.433555i
$533$	− 35.0000i	− 1.51602i
$534$	6.00000	0.259645
$535$	0	0
$536$	−10.0000	−0.431934
$537$	3.00000i	0.129460i
$538$	10.0000i	0.431131i
$539$	18.0000	0.775315
$540$	0	0
$541$	14.0000	0.601907	0.300954	−	0.953639i	$-0.402695\pi$
$541$	14.0000	0.601907	0.300954	+	0.953639i	$0.402695\pi$
$542$	12.0000i	0.515444i
$543$	− 19.0000i	− 0.815368i
$544$	−4.00000	−0.171499
$545$	0	0
$546$	−25.0000	−1.06990
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	− 4.00000i	− 0.170872i
$549$	−9.00000	−0.384111
$550$	0	0
$551$	−12.0000	−0.511217
$552$	4.00000i	0.170251i
$553$	0	0
$554$	6.00000	0.254916
$555$	0	0
$556$	−11.0000	−0.466504
$557$	2.00000i	0.0847427i	0.999102	+	0.0423714i	$0.0134913\pi$
$557$	2.00000i	0.0847427i	−0.999102	+	0.0423714i	$0.986509\pi$
$558$	− 1.00000i	− 0.0423334i
$559$	55.0000	2.32625
$560$	0	0
$561$	−4.00000	−0.168880
$562$	25.0000i	1.05456i
$563$	16.0000i	0.674320i	0.941447	+	0.337160i	$0.109466\pi$
$563$	16.0000i	0.674320i	−0.941447	+	0.337160i	$0.890534\pi$
$564$	−11.0000	−0.463184
$565$	0	0
$566$	−12.0000	−0.504398
$567$	− 5.00000i	− 0.209980i
$568$	− 1.00000i	− 0.0419591i
$569$	36.0000	1.50920	0.754599	−	0.656186i	$-0.227831\pi$
$569$	36.0000	1.50920	0.754599	+	0.656186i	$0.227831\pi$
$570$	0	0
$571$	−21.0000	−0.878823	−0.439411	−	0.898286i	$-0.644813\pi$
$571$	−21.0000	−0.878823	−0.439411	+	0.898286i	$0.644813\pi$
$572$	− 5.00000i	− 0.209061i
$573$	4.00000i	0.167102i
$574$	−35.0000	−1.46087
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 30.0000i	− 1.24892i	−0.781058	−	0.624458i	$-0.785320\pi$
$577$	− 30.0000i	− 1.24892i	0.781058	−	0.624458i	$-0.214680\pi$
$578$	− 1.00000i	− 0.0415945i
$579$	5.00000	0.207793
$580$	0	0
$581$	−75.0000	−3.11152
$582$	10.0000i	0.414513i
$583$	3.00000i	0.124247i
$584$	−2.00000	−0.0827606
$585$	0	0
$586$	−20.0000	−0.826192
$587$	45.0000i	1.85735i	0.370896	+	0.928674i	$0.379051\pi$
$587$	45.0000i	1.85735i	−0.370896	+	0.928674i	$0.620949\pi$
$588$	18.0000i	0.742307i
$589$	−2.00000	−0.0824086
$590$	0	0
$591$	17.0000	0.699287
$592$	1.00000i	0.0410997i
$593$	1.00000i	0.0410651i	0.999789	+	0.0205325i	$0.00653617\pi$
$593$	1.00000i	0.0410651i	−0.999789	+	0.0205325i	$0.993464\pi$
$594$	1.00000	0.0410305
$595$	0	0
$596$	16.0000	0.655386
$597$	14.0000i	0.572982i
$598$	20.0000i	0.817861i
$599$	15.0000	0.612883	0.306442	−	0.951889i	$-0.400862\pi$
$599$	15.0000	0.612883	0.306442	+	0.951889i	$0.400862\pi$
$600$	0	0
$601$	−18.0000	−0.734235	−0.367118	−	0.930175i	$-0.619655\pi$
$601$	−18.0000	−0.734235	−0.367118	+	0.930175i	$0.619655\pi$
$602$	− 55.0000i	− 2.24163i
$603$	− 10.0000i	− 0.407231i
$604$	8.00000	0.325515
$605$	0	0
$606$	−8.00000	−0.324978
$607$	27.0000i	1.09590i	0.836512	+	0.547948i	$0.184591\pi$
$607$	27.0000i	1.09590i	−0.836512	+	0.547948i	$0.815409\pi$
$608$	− 2.00000i	− 0.0811107i
$609$	−30.0000	−1.21566
$610$	0	0
$611$	−55.0000	−2.22506
$612$	− 4.00000i	− 0.161690i
$613$	6.00000i	0.242338i	0.992632	+	0.121169i	$0.0386643\pi$
$613$	6.00000i	0.242338i	−0.992632	+	0.121169i	$0.961336\pi$
$614$	−8.00000	−0.322854
$615$	0	0
$616$	−5.00000	−0.201456
$617$	− 33.0000i	− 1.32853i	−0.747497	−	0.664265i	$-0.768745\pi$
$617$	− 33.0000i	− 1.32853i	0.747497	−	0.664265i	$-0.231255\pi$
$618$	13.0000i	0.522937i
$619$	43.0000	1.72832	0.864158	−	0.503221i	$-0.167852\pi$
$619$	43.0000	1.72832	0.864158	+	0.503221i	$0.167852\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	25.0000i	1.00241i
$623$	− 30.0000i	− 1.20192i
$624$	5.00000	0.200160
$625$	0	0
$626$	34.0000	1.35891
$627$	− 2.00000i	− 0.0798723i
$628$	14.0000i	0.558661i
$629$	4.00000	0.159490
$630$	0	0
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	0	0
$633$	4.00000i	0.158986i
$634$	28.0000	1.11202
$635$	0	0
$636$	−3.00000	−0.118958
$637$	90.0000i	3.56593i
$638$	− 6.00000i	− 0.237542i
$639$	1.00000	0.0395594
$640$	0	0
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	4.00000i	0.157867i
$643$	− 7.00000i	− 0.276053i	−0.990429	−	0.138027i	$-0.955924\pi$
$643$	− 7.00000i	− 0.276053i	0.990429	−	0.138027i	$-0.0440759\pi$
$644$	20.0000	0.788110
$645$	0	0
$646$	−8.00000	−0.314756
$647$	− 12.0000i	− 0.471769i	−0.971781	−	0.235884i	$-0.924201\pi$
$647$	− 12.0000i	− 0.471769i	0.971781	−	0.235884i	$-0.0757987\pi$
$648$	1.00000i	0.0392837i
$649$	6.00000	0.235521
$650$	0	0
$651$	−5.00000	−0.195965
$652$	2.00000i	0.0783260i
$653$	− 42.0000i	− 1.64359i	−0.569785	−	0.821794i	$-0.692974\pi$
$653$	− 42.0000i	− 1.64359i	0.569785	−	0.821794i	$-0.307026\pi$
$654$	10.0000	0.391031
$655$	0	0
$656$	7.00000	0.273304
$657$	− 2.00000i	− 0.0780274i
$658$	55.0000i	2.14412i
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	0	0		−	1.00000i	$-0.5\pi$
$661$	0	0			1.00000i	$0.5\pi$
$662$	17.0000i	0.660724i
$663$	− 20.0000i	− 0.776736i
$664$	15.0000	0.582113
$665$	0	0
$666$	−1.00000	−0.0387492
$667$	24.0000i	0.929284i
$668$	4.00000i	0.154765i
$669$	18.0000	0.695920
$670$	0	0
$671$	−9.00000	−0.347441
$672$	− 5.00000i	− 0.192879i
$673$	6.00000i	0.231283i	0.993291	+	0.115642i	$0.0368924\pi$
$673$	6.00000i	0.231283i	−0.993291	+	0.115642i	$0.963108\pi$
$674$	30.0000	1.15556
$675$	0	0
$676$	12.0000	0.461538
$677$	− 49.0000i	− 1.88322i	−0.336701	−	0.941611i	$-0.609311\pi$
$677$	− 49.0000i	− 1.88322i	0.336701	−	0.941611i	$-0.390689\pi$
$678$	14.0000i	0.537667i
$679$	50.0000	1.91882
$680$	0	0
$681$	−4.00000	−0.153280
$682$	− 1.00000i	− 0.0382920i
$683$	− 24.0000i	− 0.918334i	−0.888350	−	0.459167i	$-0.848148\pi$
$683$	− 24.0000i	− 0.918334i	0.888350	−	0.459167i	$-0.151852\pi$
$684$	2.00000	0.0764719
$685$	0	0
$686$	55.0000	2.09991
$687$	22.0000i	0.839352i
$688$	11.0000i	0.419371i
$689$	−15.0000	−0.571454
$690$	0	0
$691$	40.0000	1.52167	0.760836	−	0.648944i	$-0.224789\pi$
$691$	40.0000	1.52167	0.760836	+	0.648944i	$0.224789\pi$
$692$	18.0000i	0.684257i
$693$	− 5.00000i	− 0.189934i
$694$	−5.00000	−0.189797
$695$	0	0
$696$	6.00000	0.227429
$697$	− 28.0000i	− 1.06058i
$698$	− 2.00000i	− 0.0757011i
$699$	−7.00000	−0.264764
$700$	0	0
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	5.00000i	0.188713i
$703$	2.00000i	0.0754314i
$704$	1.00000	0.0376889
$705$	0	0
$706$	−32.0000	−1.20434
$707$	40.0000i	1.50435i
$708$	6.00000i	0.225494i
$709$	−6.00000	−0.225335	−0.112667	−	0.993633i	$-0.535939\pi$
$709$	−6.00000	−0.225335	−0.112667	+	0.993633i	$0.535939\pi$
$710$	0	0
$711$	0	0
$712$	6.00000i	0.224860i
$713$	4.00000i	0.149801i
$714$	−20.0000	−0.748481
$715$	0	0
$716$	−3.00000	−0.112115
$717$	− 18.0000i	− 0.672222i
$718$	20.0000i	0.746393i
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	65.0000	2.42073
$722$	15.0000i	0.558242i
$723$	− 26.0000i	− 0.966950i
$724$	19.0000	0.706129
$725$	0	0
$726$	−10.0000	−0.371135
$727$	− 27.0000i	− 1.00137i	−0.865628	−	0.500687i	$-0.833081\pi$
$727$	− 27.0000i	− 1.00137i	0.865628	−	0.500687i	$-0.166919\pi$
$728$	− 25.0000i	− 0.926562i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	44.0000	1.62740
$732$	− 9.00000i	− 0.332650i
$733$	− 4.00000i	− 0.147743i	−0.997268	−	0.0738717i	$-0.976464\pi$
$733$	− 4.00000i	− 0.147743i	0.997268	−	0.0738717i	$-0.0235355\pi$
$734$	−24.0000	−0.885856
$735$	0	0
$736$	−4.00000	−0.147442
$737$	− 10.0000i	− 0.368355i
$738$	7.00000i	0.257674i
$739$	−36.0000	−1.32428	−0.662141	−	0.749380i	$-0.730352\pi$
$739$	−36.0000	−1.32428	−0.662141	+	0.749380i	$0.730352\pi$
$740$	0	0
$741$	10.0000	0.367359
$742$	15.0000i	0.550667i
$743$	44.0000i	1.61420i	0.590412	+	0.807102i	$0.298965\pi$
$743$	44.0000i	1.61420i	−0.590412	+	0.807102i	$0.701035\pi$
$744$	1.00000	0.0366618
$745$	0	0
$746$	8.00000	0.292901
$747$	15.0000i	0.548821i
$748$	− 4.00000i	− 0.146254i
$749$	20.0000	0.730784
$750$	0	0
$751$	−33.0000	−1.20419	−0.602094	−	0.798426i	$-0.705667\pi$
$751$	−33.0000	−1.20419	−0.602094	+	0.798426i	$0.705667\pi$
$752$	− 11.0000i	− 0.401129i
$753$	23.0000i	0.838167i
$754$	30.0000	1.09254
$755$	0	0
$756$	5.00000	0.181848
$757$	11.0000i	0.399802i	0.979816	+	0.199901i	$0.0640620\pi$
$757$	11.0000i	0.399802i	−0.979816	+	0.199901i	$0.935938\pi$
$758$	− 34.0000i	− 1.23494i
$759$	−4.00000	−0.145191
$760$	0	0
$761$	14.0000	0.507500	0.253750	−	0.967270i	$-0.418336\pi$
$761$	14.0000	0.507500	0.253750	+	0.967270i	$0.418336\pi$
$762$	4.00000i	0.144905i
$763$	− 50.0000i	− 1.81012i
$764$	−4.00000	−0.144715
$765$	0	0
$766$	20.0000	0.722629
$767$	30.0000i	1.08324i
$768$	1.00000i	0.0360844i
$769$	−43.0000	−1.55062	−0.775310	−	0.631581i	$-0.782406\pi$
$769$	−43.0000	−1.55062	−0.775310	+	0.631581i	$0.782406\pi$
$770$	0	0
$771$	27.0000	0.972381
$772$	5.00000i	0.179954i
$773$	6.00000i	0.215805i	0.994161	+	0.107903i	$0.0344134\pi$
$773$	6.00000i	0.215805i	−0.994161	+	0.107903i	$0.965587\pi$
$774$	−11.0000	−0.395387
$775$	0	0
$776$	−10.0000	−0.358979
$777$	5.00000i	0.179374i
$778$	− 6.00000i	− 0.215110i
$779$	14.0000	0.501602
$780$	0	0
$781$	1.00000	0.0357828
$782$	16.0000i	0.572159i
$783$	6.00000i	0.214423i
$784$	−18.0000	−0.642857
$785$	0	0
$786$	−2.00000	−0.0713376
$787$	35.0000i	1.24762i	0.781578	+	0.623808i	$0.214415\pi$
$787$	35.0000i	1.24762i	−0.781578	+	0.623808i	$0.785585\pi$
$788$	17.0000i	0.605600i
$789$	−22.0000	−0.783221
$790$	0	0
$791$	70.0000	2.48891
$792$	1.00000i	0.0355335i
$793$	− 45.0000i	− 1.59800i
$794$	−38.0000	−1.34857
$795$	0	0
$796$	−14.0000	−0.496217
$797$	− 34.0000i	− 1.20434i	−0.798367	−	0.602171i	$-0.794303\pi$
$797$	− 34.0000i	− 1.20434i	0.798367	−	0.602171i	$-0.205697\pi$
$798$	− 10.0000i	− 0.353996i
$799$	−44.0000	−1.55661
$800$	0	0
$801$	−6.00000	−0.212000
$802$	24.0000i	0.847469i
$803$	− 2.00000i	− 0.0705785i
$804$	10.0000	0.352673
$805$	0	0
$806$	5.00000	0.176117
$807$	− 10.0000i	− 0.352017i
$808$	− 8.00000i	− 0.281439i
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	0	0
$811$	18.0000	0.632065	0.316033	−	0.948748i	$-0.397649\pi$
$811$	18.0000	0.632065	0.316033	+	0.948748i	$0.397649\pi$
$812$	− 30.0000i	− 1.05279i
$813$	− 12.0000i	− 0.420858i
$814$	−1.00000	−0.0350500
$815$	0	0
$816$	4.00000	0.140028
$817$	22.0000i	0.769683i
$818$	34.0000i	1.18878i
$819$	25.0000	0.873571
$820$	0	0
$821$	45.0000	1.57051	0.785255	−	0.619172i	$-0.212532\pi$
$821$	45.0000	1.57051	0.785255	+	0.619172i	$0.212532\pi$
$822$	4.00000i	0.139516i
$823$	− 8.00000i	− 0.278862i	−0.990232	−	0.139431i	$-0.955473\pi$
$823$	− 8.00000i	− 0.278862i	0.990232	−	0.139431i	$-0.0445274\pi$
$824$	−13.0000	−0.452876
$825$	0	0
$826$	30.0000	1.04383
$827$	36.0000i	1.25184i	0.779886	+	0.625921i	$0.215277\pi$
$827$	36.0000i	1.25184i	−0.779886	+	0.625921i	$0.784723\pi$
$828$	− 4.00000i	− 0.139010i
$829$	31.0000	1.07667	0.538337	−	0.842729i	$-0.319053\pi$
$829$	31.0000	1.07667	0.538337	+	0.842729i	$0.319053\pi$
$830$	0	0
$831$	−6.00000	−0.208138
$832$	5.00000i	0.173344i
$833$	72.0000i	2.49465i
$834$	11.0000	0.380899
$835$	0	0
$836$	2.00000	0.0691714
$837$	1.00000i	0.0345651i
$838$	36.0000i	1.24360i
$839$	5.00000	0.172619	0.0863096	−	0.996268i	$-0.472493\pi$
$839$	5.00000	0.172619	0.0863096	+	0.996268i	$0.472493\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	− 2.00000i	− 0.0689246i
$843$	− 25.0000i	− 0.861046i
$844$	−4.00000	−0.137686
$845$	0	0
$846$	11.0000	0.378188
$847$	50.0000i	1.71802i
$848$	− 3.00000i	− 0.103020i
$849$	12.0000	0.411839
$850$	0	0
$851$	4.00000	0.137118
$852$	1.00000i	0.0342594i
$853$	− 54.0000i	− 1.84892i	−0.381273	−	0.924462i	$-0.624514\pi$
$853$	− 54.0000i	− 1.84892i	0.381273	−	0.924462i	$-0.375486\pi$
$854$	−45.0000	−1.53987
$855$	0	0
$856$	−4.00000	−0.136717
$857$	− 53.0000i	− 1.81045i	−0.424937	−	0.905223i	$-0.639704\pi$
$857$	− 53.0000i	− 1.81045i	0.424937	−	0.905223i	$-0.360296\pi$
$858$	5.00000i	0.170697i
$859$	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
$861$	35.0000	1.19280
$862$	− 23.0000i	− 0.783383i
$863$	− 12.0000i	− 0.408485i	−0.978920	−	0.204242i	$-0.934527\pi$
$863$	− 12.0000i	− 0.408485i	0.978920	−	0.204242i	$-0.0654731\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	−16.0000	−0.543702
$867$	1.00000i	0.0339618i
$868$	− 5.00000i	− 0.169711i
$869$	0	0
$870$	0	0
$871$	50.0000	1.69419
$872$	10.0000i	0.338643i
$873$	− 10.0000i	− 0.338449i
$874$	−8.00000	−0.270604
$875$	0	0
$876$	2.00000	0.0675737
$877$	4.00000i	0.135070i	0.997717	+	0.0675352i	$0.0215135\pi$
$877$	4.00000i	0.135070i	−0.997717	+	0.0675352i	$0.978487\pi$
$878$	− 24.0000i	− 0.809961i
$879$	20.0000	0.674583
$880$	0	0
$881$	10.0000	0.336909	0.168454	−	0.985709i	$-0.446122\pi$
$881$	10.0000	0.336909	0.168454	+	0.985709i	$0.446122\pi$
$882$	− 18.0000i	− 0.606092i
$883$	− 31.0000i	− 1.04323i	−0.853180	−	0.521617i	$-0.825329\pi$
$883$	− 31.0000i	− 1.04323i	0.853180	−	0.521617i	$-0.174671\pi$
$884$	20.0000	0.672673
$885$	0	0
$886$	10.0000	0.335957
$887$	13.0000i	0.436497i	0.975893	+	0.218249i	$0.0700344\pi$
$887$	13.0000i	0.436497i	−0.975893	+	0.218249i	$0.929966\pi$
$888$	− 1.00000i	− 0.0335578i
$889$	20.0000	0.670778
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	18.0000i	0.602685i
$893$	− 22.0000i	− 0.736202i
$894$	−16.0000	−0.535120
$895$	0	0
$896$	5.00000	0.167038
$897$	− 20.0000i	− 0.667781i
$898$	− 34.0000i	− 1.13459i
$899$	6.00000	0.200111
$900$	0	0
$901$	−12.0000	−0.399778
$902$	7.00000i	0.233075i
$903$	55.0000i	1.83029i
$904$	−14.0000	−0.465633
$905$	0	0
$906$	−8.00000	−0.265782
$907$	24.0000i	0.796907i	0.917189	+	0.398453i	$0.130453\pi$
$907$	24.0000i	0.796907i	−0.917189	+	0.398453i	$0.869547\pi$
$908$	− 4.00000i	− 0.132745i
$909$	8.00000	0.265343
$910$	0	0
$911$	−2.00000	−0.0662630	−0.0331315	−	0.999451i	$-0.510548\pi$
$911$	−2.00000	−0.0662630	−0.0331315	+	0.999451i	$0.510548\pi$
$912$	2.00000i	0.0662266i
$913$	15.0000i	0.496428i
$914$	0	0
$915$	0	0
$916$	−22.0000	−0.726900
$917$	10.0000i	0.330229i
$918$	4.00000i	0.132020i
$919$	−29.0000	−0.956622	−0.478311	−	0.878191i	$-0.658751\pi$
$919$	−29.0000	−0.956622	−0.478311	+	0.878191i	$0.658751\pi$
$920$	0	0
$921$	8.00000	0.263609
$922$	21.0000i	0.691598i
$923$	5.00000i	0.164577i
$924$	5.00000	0.164488
$925$	0	0
$926$	−4.00000	−0.131448
$927$	− 13.0000i	− 0.426976i
$928$	6.00000i	0.196960i
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	0	0
$931$	−36.0000	−1.17985
$932$	− 7.00000i	− 0.229293i
$933$	− 25.0000i	− 0.818463i
$934$	18.0000	0.588978
$935$	0	0
$936$	−5.00000	−0.163430
$937$	− 38.0000i	− 1.24141i	−0.784046	−	0.620703i	$-0.786847\pi$
$937$	− 38.0000i	− 1.24141i	0.784046	−	0.620703i	$-0.213153\pi$
$938$	− 50.0000i	− 1.63256i
$939$	−34.0000	−1.10955
$940$	0	0
$941$	18.0000	0.586783	0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000	0.586783	0.293392	+	0.955992i	$0.405216\pi$
$942$	− 14.0000i	− 0.456145i
$943$	− 28.0000i	− 0.911805i
$944$	−6.00000	−0.195283
$945$	0	0
$946$	−11.0000	−0.357641
$947$	− 27.0000i	− 0.877382i	−0.898638	−	0.438691i	$-0.855442\pi$
$947$	− 27.0000i	− 0.877382i	0.898638	−	0.438691i	$-0.144558\pi$
$948$	0	0
$949$	10.0000	0.324614
$950$	0	0
$951$	−28.0000	−0.907962
$952$	− 20.0000i	− 0.648204i
$953$	16.0000i	0.518291i	0.965838	+	0.259145i	$0.0834409\pi$
$953$	16.0000i	0.518291i	−0.965838	+	0.259145i	$0.916559\pi$
$954$	3.00000	0.0971286
$955$	0	0
$956$	18.0000	0.582162
$957$	6.00000i	0.193952i
$958$	− 16.0000i	− 0.516937i
$959$	20.0000	0.645834
$960$	0	0
$961$	1.00000	0.0322581
$962$	− 5.00000i	− 0.161206i
$963$	− 4.00000i	− 0.128898i
$964$	26.0000	0.837404
$965$	0	0
$966$	−20.0000	−0.643489
$967$	58.0000i	1.86515i	0.360971	+	0.932577i	$0.382445\pi$
$967$	58.0000i	1.86515i	−0.360971	+	0.932577i	$0.617555\pi$
$968$	− 10.0000i	− 0.321412i
$969$	8.00000	0.256997
$970$	0	0
$971$	−36.0000	−1.15529	−0.577647	−	0.816286i	$-0.696029\pi$
$971$	−36.0000	−1.15529	−0.577647	+	0.816286i	$0.696029\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	− 55.0000i	− 1.76322i
$974$	−4.00000	−0.128168
$975$	0	0
$976$	9.00000	0.288083
$977$	− 35.0000i	− 1.11975i	−0.828577	−	0.559875i	$-0.810849\pi$
$977$	− 35.0000i	− 1.11975i	0.828577	−	0.559875i	$-0.189151\pi$
$978$	− 2.00000i	− 0.0639529i
$979$	−6.00000	−0.191761
$980$	0	0
$981$	−10.0000	−0.319275
$982$	− 12.0000i	− 0.382935i
$983$	− 58.0000i	− 1.84991i	−0.380073	−	0.924956i	$-0.624101\pi$
$983$	− 58.0000i	− 1.84991i	0.380073	−	0.924956i	$-0.375899\pi$
$984$	−7.00000	−0.223152
$985$	0	0
$986$	24.0000	0.764316
$987$	− 55.0000i	− 1.75067i
$988$	10.0000i	0.318142i
$989$	44.0000	1.39912
$990$	0	0
$991$	−2.00000	−0.0635321	−0.0317660	−	0.999495i	$-0.510113\pi$
$991$	−2.00000	−0.0635321	−0.0317660	+	0.999495i	$0.510113\pi$
$992$	1.00000i	0.0317500i
$993$	− 17.0000i	− 0.539479i
$994$	5.00000	0.158590
$995$	0	0
$996$	−15.0000	−0.475293
$997$	− 24.0000i	− 0.760088i	−0.924968	−	0.380044i	$-0.875909\pi$
$997$	− 24.0000i	− 0.760088i	0.924968	−	0.380044i	$-0.124091\pi$
$998$	− 32.0000i	− 1.01294i
$999$	1.00000	0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4650.2.d.s.3349.1		2
5.2	odd	4		4650.2.a.bx.1.1	yes	1
5.3	odd	4		4650.2.a.a.1.1	✓	1
5.4	even	2	inner	4650.2.d.s.3349.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4650.2.a.a.1.1	✓	1	5.3	odd	4
4650.2.a.bx.1.1	yes	1	5.2	odd	4
4650.2.d.s.3349.1		2	1.1	even	1	trivial
4650.2.d.s.3349.2		2	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4650.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -5.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -5.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -1.00000 q^{11} -1.00000i q^{12} -5.00000i q^{13} -5.00000 q^{14} +1.00000 q^{16} -4.00000i q^{17} +1.00000i q^{18} +2.00000 q^{19} +5.00000 q^{21} +1.00000i q^{22} -4.00000i q^{23} -1.00000 q^{24} -5.00000 q^{26} -1.00000i q^{27} +5.00000i q^{28} -6.00000 q^{29} -1.00000 q^{31} -1.00000i q^{32} -1.00000i q^{33} -4.00000 q^{34} +1.00000 q^{36} +1.00000i q^{37} -2.00000i q^{38} +5.00000 q^{39} +7.00000 q^{41} -5.00000i q^{42} +11.0000i q^{43} +1.00000 q^{44} -4.00000 q^{46} -11.0000i q^{47} +1.00000i q^{48} -18.0000 q^{49} +4.00000 q^{51} +5.00000i q^{52} -3.00000i q^{53} -1.00000 q^{54} +5.00000 q^{56} +2.00000i q^{57} +6.00000i q^{58} -6.00000 q^{59} +9.00000 q^{61} +1.00000i q^{62} +5.00000i q^{63} -1.00000 q^{64} -1.00000 q^{66} +10.0000i q^{67} +4.00000i q^{68} +4.00000 q^{69} -1.00000 q^{71} -1.00000i q^{72} +2.00000i q^{73} +1.00000 q^{74} -2.00000 q^{76} +5.00000i q^{77} -5.00000i q^{78} +1.00000 q^{81} -7.00000i q^{82} -15.0000i q^{83} -5.00000 q^{84} +11.0000 q^{86} -6.00000i q^{87} -1.00000i q^{88} +6.00000 q^{89} -25.0000 q^{91} +4.00000i q^{92} -1.00000i q^{93} -11.0000 q^{94} +1.00000 q^{96} +10.0000i q^{97} +18.0000i q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 2 q^{11} - 10 q^{14} + 2 q^{16} + 4 q^{19} + 10 q^{21} - 2 q^{24} - 10 q^{26} - 12 q^{29} - 2 q^{31} - 8 q^{34} + 2 q^{36} + 10 q^{39} + 14 q^{41} + 2 q^{44} - 8 q^{46} - 36 q^{49} + 8 q^{51} - 2 q^{54} + 10 q^{56} - 12 q^{59} + 18 q^{61} - 2 q^{64} - 2 q^{66} + 8 q^{69} - 2 q^{71} + 2 q^{74} - 4 q^{76} + 2 q^{81} - 10 q^{84} + 22 q^{86} + 12 q^{89} - 50 q^{91} - 22 q^{94} + 2 q^{96} + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 - 2 * q^11 - 10 * q^14 + 2 * q^16 + 4 * q^19 + 10 * q^21 - 2 * q^24 - 10 * q^26 - 12 * q^29 - 2 * q^31 - 8 * q^34 + 2 * q^36 + 10 * q^39 + 14 * q^41 + 2 * q^44 - 8 * q^46 - 36 * q^49 + 8 * q^51 - 2 * q^54 + 10 * q^56 - 12 * q^59 + 18 * q^61 - 2 * q^64 - 2 * q^66 + 8 * q^69 - 2 * q^71 + 2 * q^74 - 4 * q^76 + 2 * q^81 - 10 * q^84 + 22 * q^86 + 12 * q^89 - 50 * q^91 - 22 * q^94 + 2 * q^96 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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