LMFDB - Embedded newform 4650.2.d.bg.3349.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4650,2,Mod(3349,4650)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [4,0,0,-4,0,4,0,0,-4,0,10,0,0,2,0,4,0,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$37.1304369399$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{65})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 33x^{2} + 256 $ x^4 + 33*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 930)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3349.3
Root			$-4.53113i$ of defining polynomial
Character	$\chi$	$=$	4650.3349
Dual form			4650.2.d.bg.3349.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -4.53113i q^{7} -1.00000i q^{8} -1.00000 q^{9} +6.53113 q^{11} +1.00000i q^{12} -6.00000i q^{13} +4.53113 q^{14} +1.00000 q^{16} -4.00000i q^{17} -1.00000i q^{18} -4.53113 q^{19} -4.53113 q^{21} +6.53113i q^{22} -6.53113i q^{23} -1.00000 q^{24} +6.00000 q^{26} +1.00000i q^{27} +4.53113i q^{28} +1.00000 q^{31} +1.00000i q^{32} -6.53113i q^{33} +4.00000 q^{34} +1.00000 q^{36} -7.06226i q^{37} -4.53113i q^{38} -6.00000 q^{39} +7.06226 q^{41} -4.53113i q^{42} +8.53113i q^{43} -6.53113 q^{44} +6.53113 q^{46} -5.06226i q^{47} -1.00000i q^{48} -13.5311 q^{49} -4.00000 q^{51} +6.00000i q^{52} +2.53113i q^{53} -1.00000 q^{54} -4.53113 q^{56} +4.53113i q^{57} +9.06226 q^{59} +5.06226 q^{61} +1.00000i q^{62} +4.53113i q^{63} -1.00000 q^{64} +6.53113 q^{66} +5.06226i q^{67} +4.00000i q^{68} -6.53113 q^{69} -12.5311 q^{71} +1.00000i q^{72} +8.53113i q^{73} +7.06226 q^{74} +4.53113 q^{76} -29.5934i q^{77} -6.00000i q^{78} +8.53113 q^{79} +1.00000 q^{81} +7.06226i q^{82} +8.00000i q^{83} +4.53113 q^{84} -8.53113 q^{86} -6.53113i q^{88} +6.53113 q^{89} -27.1868 q^{91} +6.53113i q^{92} -1.00000i q^{93} +5.06226 q^{94} +1.00000 q^{96} +16.1245i q^{97} -13.5311i q^{98} -6.53113 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} + 4 q^{6} - 4 q^{9} + 10 q^{11} + 2 q^{14} + 4 q^{16} - 2 q^{19} - 2 q^{21} - 4 q^{24} + 24 q^{26} + 4 q^{31} + 16 q^{34} + 4 q^{36} - 24 q^{39} - 4 q^{41} - 10 q^{44} + 10 q^{46} - 38 q^{49}+ \cdots - 10 q^{99}+O(q^{100}) $ 4 * q - 4 * q^4 + 4 * q^6 - 4 * q^9 + 10 * q^11 + 2 * q^14 + 4 * q^16 - 2 * q^19 - 2 * q^21 - 4 * q^24 + 24 * q^26 + 4 * q^31 + 16 * q^34 + 4 * q^36 - 24 * q^39 - 4 * q^41 - 10 * q^44 + 10 * q^46 - 38 * q^49 - 16 * q^51 - 4 * q^54 - 2 * q^56 + 4 * q^59 - 12 * q^61 - 4 * q^64 + 10 * q^66 - 10 * q^69 - 34 * q^71 - 4 * q^74 + 2 * q^76 + 18 * q^79 + 4 * q^81 + 2 * q^84 - 18 * q^86 + 10 * q^89 - 12 * q^91 - 12 * q^94 + 4 * q^96 - 10 * q^99

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
$7$	− 4.53113i	− 1.71261i	−0.516474	−	0.856303i	$-0.672756\pi$
$7$	− 4.53113i	− 1.71261i	0.516474	−	0.856303i	$-0.327244\pi$
$8$	− 1.00000i	− 0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	6.53113	1.96921	0.984605	−	0.174796i	$-0.0559265\pi$
$11$	6.53113	1.96921	0.984605	+	0.174796i	$0.0559265\pi$
$12$	1.00000i	0.288675i
$13$	− 6.00000i	− 1.66410i	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	− 6.00000i	− 1.66410i	0.554700	−	0.832050i	$-0.312833\pi$
$14$	4.53113	1.21100
$15$	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
$19$	−4.53113	−1.03951	−0.519756	−	0.854315i	$-0.673977\pi$
$19$	−4.53113	−1.03951	−0.519756	+	0.854315i	$0.673977\pi$
$20$	0	0
$21$	−4.53113	−0.988773
$22$	6.53113i	1.39244i
$23$	− 6.53113i	− 1.36183i	−0.732360	−	0.680917i	$-0.761581\pi$
$23$	− 6.53113i	− 1.36183i	0.732360	−	0.680917i	$-0.238419\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	6.00000	1.17670
$27$	1.00000i	0.192450i
$28$	4.53113i	0.856303i
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	1.00000i	0.176777i
$33$	− 6.53113i	− 1.13692i
$34$	4.00000	0.685994
$35$	0	0
$36$	1.00000	0.166667
$37$	− 7.06226i	− 1.16103i	−0.814250	−	0.580514i	$-0.802852\pi$
$37$	− 7.06226i	− 1.16103i	0.814250	−	0.580514i	$-0.197148\pi$
$38$	− 4.53113i	− 0.735046i
$39$	−6.00000	−0.960769
$40$	0	0
$41$	7.06226	1.10294	0.551470	−	0.834195i	$-0.314067\pi$
$41$	7.06226	1.10294	0.551470	+	0.834195i	$0.314067\pi$
$42$	− 4.53113i	− 0.699168i
$43$	8.53113i	1.30098i	0.759513	+	0.650492i	$0.225437\pi$
$43$	8.53113i	1.30098i	−0.759513	+	0.650492i	$0.774563\pi$
$44$	−6.53113	−0.984605
$45$	0	0
$46$	6.53113	0.962962
$47$	− 5.06226i	− 0.738406i	−0.929349	−	0.369203i	$-0.879631\pi$
$47$	− 5.06226i	− 0.738406i	0.929349	−	0.369203i	$-0.120369\pi$
$48$	− 1.00000i	− 0.144338i
$49$	−13.5311	−1.93302
$50$	0	0
$51$	−4.00000	−0.560112
$52$	6.00000i	0.832050i
$53$	2.53113i	0.347677i	0.984774	+	0.173839i	$0.0556171\pi$
$53$	2.53113i	0.347677i	−0.984774	+	0.173839i	$0.944383\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	−4.53113	−0.605498
$57$	4.53113i	0.600163i
$58$	0	0
$59$	9.06226	1.17981	0.589903	−	0.807474i	$-0.299166\pi$
$59$	9.06226	1.17981	0.589903	+	0.807474i	$0.299166\pi$
$60$	0	0
$61$	5.06226	0.648156	0.324078	−	0.946030i	$-0.394946\pi$
$61$	5.06226	0.648156	0.324078	+	0.946030i	$0.394946\pi$
$62$	1.00000i	0.127000i
$63$	4.53113i	0.570869i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	6.53113	0.803926
$67$	5.06226i	0.618453i	0.950988	+	0.309227i	$0.100070\pi$
$67$	5.06226i	0.618453i	−0.950988	+	0.309227i	$0.899930\pi$
$68$	4.00000i	0.485071i
$69$	−6.53113	−0.786256
$70$	0	0
$71$	−12.5311	−1.48717	−0.743586	−	0.668641i	$-0.766876\pi$
$71$	−12.5311	−1.48717	−0.743586	+	0.668641i	$0.766876\pi$
$72$	1.00000i	0.117851i
$73$	8.53113i	0.998493i	0.866460	+	0.499247i	$0.166390\pi$
$73$	8.53113i	0.998493i	−0.866460	+	0.499247i	$0.833610\pi$
$74$	7.06226	0.820971
$75$	0	0
$76$	4.53113	0.519756
$77$	− 29.5934i	− 3.37248i
$78$	− 6.00000i	− 0.679366i
$79$	8.53113	0.959827	0.479913	−	0.877316i	$-0.340668\pi$
$79$	8.53113	0.959827	0.479913	+	0.877316i	$0.340668\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	7.06226i	0.779896i
$83$	8.00000i	0.878114i	0.898459	+	0.439057i	$0.144687\pi$
$83$	8.00000i	0.878114i	−0.898459	+	0.439057i	$0.855313\pi$
$84$	4.53113	0.494387
$85$	0	0
$86$	−8.53113	−0.919935
$87$	0	0
$88$	− 6.53113i	− 0.696221i
$89$	6.53113	0.692298	0.346149	−	0.938180i	$-0.387489\pi$
$89$	6.53113	0.692298	0.346149	+	0.938180i	$0.387489\pi$
$90$	0	0
$91$	−27.1868	−2.84995
$92$	6.53113i	0.680917i
$93$	− 1.00000i	− 0.103695i
$94$	5.06226	0.522132
$95$	0	0
$96$	1.00000	0.102062
$97$	16.1245i	1.63720i	0.574367	+	0.818598i	$0.305248\pi$
$97$	16.1245i	1.63720i	−0.574367	+	0.818598i	$0.694752\pi$
$98$	− 13.5311i	− 1.36685i
$99$	−6.53113	−0.656403
$100$	0	0
$101$	−9.46887	−0.942188	−0.471094	−	0.882083i	$-0.656141\pi$
$101$	−9.46887	−0.942188	−0.471094	+	0.882083i	$0.656141\pi$
$102$	− 4.00000i	− 0.396059i
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	−6.00000	−0.588348
$105$	0	0
$106$	−2.53113	−0.245845
$107$	9.59339i	0.927428i	0.885985	+	0.463714i	$0.153483\pi$
$107$	9.59339i	0.927428i	−0.885985	+	0.463714i	$0.846517\pi$
$108$	− 1.00000i	− 0.0962250i
$109$	−15.0623	−1.44270	−0.721351	−	0.692569i	$-0.756479\pi$
$109$	−15.0623	−1.44270	−0.721351	+	0.692569i	$0.756479\pi$
$110$	0	0
$111$	−7.06226	−0.670320
$112$	− 4.53113i	− 0.428151i
$113$	7.59339i	0.714326i	0.934042	+	0.357163i	$0.116256\pi$
$113$	7.59339i	0.714326i	−0.934042	+	0.357163i	$0.883744\pi$
$114$	−4.53113	−0.424379
$115$	0	0
$116$	0	0
$117$	6.00000i	0.554700i
$118$	9.06226i	0.834248i
$119$	−18.1245	−1.66147
$120$	0	0
$121$	31.6556	2.87779
$122$	5.06226i	0.458315i
$123$	− 7.06226i	− 0.636782i
$124$	−1.00000	−0.0898027
$125$	0	0
$126$	−4.53113	−0.403665
$127$	7.06226i	0.626674i	0.949642	+	0.313337i	$0.101447\pi$
$127$	7.06226i	0.626674i	−0.949642	+	0.313337i	$0.898553\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	8.53113	0.751124
$130$	0	0
$131$	−9.06226	−0.791773	−0.395887	−	0.918299i	$-0.629563\pi$
$131$	−9.06226	−0.791773	−0.395887	+	0.918299i	$0.629563\pi$
$132$	6.53113i	0.568462i
$133$	20.5311i	1.78027i
$134$	−5.06226	−0.437312
$135$	0	0
$136$	−4.00000	−0.342997
$137$	17.0623i	1.45773i	0.684659	+	0.728864i	$0.259951\pi$
$137$	17.0623i	1.45773i	−0.684659	+	0.728864i	$0.740049\pi$
$138$	− 6.53113i	− 0.555967i
$139$	−6.00000	−0.508913	−0.254457	−	0.967084i	$-0.581897\pi$
$139$	−6.00000	−0.508913	−0.254457	+	0.967084i	$0.581897\pi$
$140$	0	0
$141$	−5.06226	−0.426319
$142$	− 12.5311i	− 1.05159i
$143$	− 39.1868i	− 3.27696i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−8.53113	−0.706041
$147$	13.5311i	1.11603i
$148$	7.06226i	0.580514i
$149$	18.5311	1.51813	0.759065	−	0.651015i	$-0.225657\pi$
$149$	18.5311	1.51813	0.759065	+	0.651015i	$0.225657\pi$
$150$	0	0
$151$	14.1245	1.14944	0.574718	−	0.818351i	$-0.305112\pi$
$151$	14.1245	1.14944	0.574718	+	0.818351i	$0.305112\pi$
$152$	4.53113i	0.367523i
$153$	4.00000i	0.323381i
$154$	29.5934	2.38470
$155$	0	0
$156$	6.00000	0.480384
$157$	− 22.5311i	− 1.79818i	−0.437764	−	0.899090i	$-0.644229\pi$
$157$	− 22.5311i	− 1.79818i	0.437764	−	0.899090i	$-0.355771\pi$
$158$	8.53113i	0.678700i
$159$	2.53113	0.200732
$160$	0	0
$161$	−29.5934	−2.33229
$162$	1.00000i	0.0785674i
$163$	− 5.06226i	− 0.396507i	−0.980151	−	0.198253i	$-0.936473\pi$
$163$	− 5.06226i	− 0.396507i	0.980151	−	0.198253i	$-0.0635269\pi$
$164$	−7.06226	−0.551470
$165$	0	0
$166$	−8.00000	−0.620920
$167$	− 23.5934i	− 1.82571i	−0.408283	−	0.912856i	$-0.633872\pi$
$167$	− 23.5934i	− 1.82571i	0.408283	−	0.912856i	$-0.366128\pi$
$168$	4.53113i	0.349584i
$169$	−23.0000	−1.76923
$170$	0	0
$171$	4.53113	0.346504
$172$	− 8.53113i	− 0.650492i
$173$	− 14.0000i	− 1.06440i	−0.846619	−	0.532200i	$-0.821365\pi$
$173$	− 14.0000i	− 1.06440i	0.846619	−	0.532200i	$-0.178635\pi$
$174$	0	0
$175$	0	0
$176$	6.53113	0.492302
$177$	− 9.06226i	− 0.681161i
$178$	6.53113i	0.489529i
$179$	3.06226	0.228884	0.114442	−	0.993430i	$-0.463492\pi$
$179$	3.06226	0.228884	0.114442	+	0.993430i	$0.463492\pi$
$180$	0	0
$181$	2.40661	0.178882	0.0894411	−	0.995992i	$-0.471492\pi$
$181$	2.40661	0.178882	0.0894411	+	0.995992i	$0.471492\pi$
$182$	− 27.1868i	− 2.01522i
$183$	− 5.06226i	− 0.374213i
$184$	−6.53113	−0.481481
$185$	0	0
$186$	1.00000	0.0733236
$187$	− 26.1245i	− 1.91041i
$188$	5.06226i	0.369203i
$189$	4.53113	0.329591
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	1.00000i	0.0721688i
$193$	− 14.0000i	− 1.00774i	−0.863779	−	0.503871i	$-0.831909\pi$
$193$	− 14.0000i	− 1.00774i	0.863779	−	0.503871i	$-0.168091\pi$
$194$	−16.1245	−1.15767
$195$	0	0
$196$	13.5311	0.966509
$197$	6.00000i	0.427482i	0.976890	+	0.213741i	$0.0685649\pi$
$197$	6.00000i	0.427482i	−0.976890	+	0.213741i	$0.931435\pi$
$198$	− 6.53113i	− 0.464147i
$199$	8.53113	0.604756	0.302378	−	0.953188i	$-0.402220\pi$
$199$	8.53113	0.604756	0.302378	+	0.953188i	$0.402220\pi$
$200$	0	0
$201$	5.06226	0.357064
$202$	− 9.46887i	− 0.666227i
$203$	0	0
$204$	4.00000	0.280056
$205$	0	0
$206$	0	0
$207$	6.53113i	0.453945i
$208$	− 6.00000i	− 0.416025i
$209$	−29.5934	−2.04702
$210$	0	0
$211$	−1.59339	−0.109693	−0.0548466	−	0.998495i	$-0.517467\pi$
$211$	−1.59339	−0.109693	−0.0548466	+	0.998495i	$0.517467\pi$
$212$	− 2.53113i	− 0.173839i
$213$	12.5311i	0.858619i
$214$	−9.59339	−0.655790
$215$	0	0
$216$	1.00000	0.0680414
$217$	− 4.53113i	− 0.307593i
$218$	− 15.0623i	− 1.02014i
$219$	8.53113	0.576480
$220$	0	0
$221$	−24.0000	−1.61441
$222$	− 7.06226i	− 0.473988i
$223$	− 2.00000i	− 0.133930i	−0.997755	−	0.0669650i	$-0.978668\pi$
$223$	− 2.00000i	− 0.133930i	0.997755	−	0.0669650i	$-0.0213316\pi$
$224$	4.53113	0.302749
$225$	0	0
$226$	−7.59339	−0.505105
$227$	− 24.5311i	− 1.62819i	−0.580733	−	0.814094i	$-0.697234\pi$
$227$	− 24.5311i	− 1.62819i	0.580733	−	0.814094i	$-0.302766\pi$
$228$	− 4.53113i	− 0.300081i
$229$	5.59339	0.369621	0.184811	−	0.982774i	$-0.440833\pi$
$229$	5.59339	0.369621	0.184811	+	0.982774i	$0.440833\pi$
$230$	0	0
$231$	−29.5934	−1.94710
$232$	0	0
$233$	− 1.46887i	− 0.0962289i	−0.998842	−	0.0481145i	$-0.984679\pi$
$233$	− 1.46887i	− 0.0962289i	0.998842	−	0.0481145i	$-0.0153212\pi$
$234$	−6.00000	−0.392232
$235$	0	0
$236$	−9.06226	−0.589903
$237$	− 8.53113i	− 0.554156i
$238$	− 18.1245i	− 1.17484i
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	31.6556i	2.03490i
$243$	− 1.00000i	− 0.0641500i
$244$	−5.06226	−0.324078
$245$	0	0
$246$	7.06226	0.450273
$247$	27.1868i	1.72985i
$248$	− 1.00000i	− 0.0635001i
$249$	8.00000	0.506979
$250$	0	0
$251$	3.06226	0.193288	0.0966440	−	0.995319i	$-0.469189\pi$
$251$	3.06226	0.193288	0.0966440	+	0.995319i	$0.469189\pi$
$252$	− 4.53113i	− 0.285434i
$253$	− 42.6556i	− 2.68174i
$254$	−7.06226	−0.443125
$255$	0	0
$256$	1.00000	0.0625000
$257$	12.6556i	0.789437i	0.918802	+	0.394719i	$0.129158\pi$
$257$	12.6556i	0.789437i	−0.918802	+	0.394719i	$0.870842\pi$
$258$	8.53113i	0.531125i
$259$	−32.0000	−1.98838
$260$	0	0
$261$	0	0
$262$	− 9.06226i	− 0.559868i
$263$	23.0623i	1.42208i	0.703152	+	0.711040i	$0.251775\pi$
$263$	23.0623i	1.42208i	−0.703152	+	0.711040i	$0.748225\pi$
$264$	−6.53113	−0.401963
$265$	0	0
$266$	−20.5311	−1.25884
$267$	− 6.53113i	− 0.399699i
$268$	− 5.06226i	− 0.309227i
$269$	19.1868	1.16984	0.584919	−	0.811092i	$-0.301126\pi$
$269$	19.1868	1.16984	0.584919	+	0.811092i	$0.301126\pi$
$270$	0	0
$271$	−11.4689	−0.696684	−0.348342	−	0.937367i	$-0.613255\pi$
$271$	−11.4689	−0.696684	−0.348342	+	0.937367i	$0.613255\pi$
$272$	− 4.00000i	− 0.242536i
$273$	27.1868i	1.64542i
$274$	−17.0623	−1.03077
$275$	0	0
$276$	6.53113	0.393128
$277$	− 15.0623i	− 0.905003i	−0.891764	−	0.452502i	$-0.850532\pi$
$277$	− 15.0623i	− 0.905003i	0.891764	−	0.452502i	$-0.149468\pi$
$278$	− 6.00000i	− 0.359856i
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	15.0623	0.898539	0.449269	−	0.893396i	$-0.351684\pi$
$281$	15.0623	0.898539	0.449269	+	0.893396i	$0.351684\pi$
$282$	− 5.06226i	− 0.301453i
$283$	12.0000i	0.713326i	0.934233	+	0.356663i	$0.116086\pi$
$283$	12.0000i	0.713326i	−0.934233	+	0.356663i	$0.883914\pi$
$284$	12.5311	0.743586
$285$	0	0
$286$	39.1868	2.31716
$287$	− 32.0000i	− 1.88890i
$288$	− 1.00000i	− 0.0589256i
$289$	1.00000	0.0588235
$290$	0	0
$291$	16.1245	0.945236
$292$	− 8.53113i	− 0.499247i
$293$	14.0000i	0.817889i	0.912559	+	0.408944i	$0.134103\pi$
$293$	14.0000i	0.817889i	−0.912559	+	0.408944i	$0.865897\pi$
$294$	−13.5311	−0.789151
$295$	0	0
$296$	−7.06226	−0.410485
$297$	6.53113i	0.378975i
$298$	18.5311i	1.07348i
$299$	−39.1868	−2.26623
$300$	0	0
$301$	38.6556	2.22807
$302$	14.1245i	0.812775i
$303$	9.46887i	0.543972i
$304$	−4.53113	−0.259878
$305$	0	0
$306$	−4.00000	−0.228665
$307$	− 30.1245i	− 1.71930i	−0.510886	−	0.859648i	$-0.670683\pi$
$307$	− 30.1245i	− 1.71930i	0.510886	−	0.859648i	$-0.329317\pi$
$308$	29.5934i	1.68624i
$309$	0	0
$310$	0	0
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	6.00000i	0.339683i
$313$	− 10.9377i	− 0.618238i	−0.951023	−	0.309119i	$-0.899966\pi$
$313$	− 10.9377i	− 0.618238i	0.951023	−	0.309119i	$-0.100034\pi$
$314$	22.5311	1.27151
$315$	0	0
$316$	−8.53113	−0.479913
$317$	12.9377i	0.726656i	0.931661	+	0.363328i	$0.118360\pi$
$317$	12.9377i	0.726656i	−0.931661	+	0.363328i	$0.881640\pi$
$318$	2.53113i	0.141939i
$319$	0	0
$320$	0	0
$321$	9.59339	0.535451
$322$	− 29.5934i	− 1.64917i
$323$	18.1245i	1.00848i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	5.06226	0.280373
$327$	15.0623i	0.832945i
$328$	− 7.06226i	− 0.389948i
$329$	−22.9377	−1.26460
$330$	0	0
$331$	12.9377	0.711123	0.355561	−	0.934653i	$-0.384290\pi$
$331$	12.9377	0.711123	0.355561	+	0.934653i	$0.384290\pi$
$332$	− 8.00000i	− 0.439057i
$333$	7.06226i	0.387009i
$334$	23.5934	1.29097
$335$	0	0
$336$	−4.53113	−0.247193
$337$	− 19.1868i	− 1.04517i	−0.852587	−	0.522585i	$-0.824968\pi$
$337$	− 19.1868i	− 1.04517i	0.852587	−	0.522585i	$-0.175032\pi$
$338$	− 23.0000i	− 1.25104i
$339$	7.59339	0.412416
$340$	0	0
$341$	6.53113	0.353680
$342$	4.53113i	0.245015i
$343$	29.5934i	1.59789i
$344$	8.53113	0.459968
$345$	0	0
$346$	14.0000	0.752645
$347$	10.1245i	0.543512i	0.962366	+	0.271756i	$0.0876044\pi$
$347$	10.1245i	0.543512i	−0.962366	+	0.271756i	$0.912396\pi$
$348$	0	0
$349$	−8.93774	−0.478426	−0.239213	−	0.970967i	$-0.576889\pi$
$349$	−8.93774	−0.478426	−0.239213	+	0.970967i	$0.576889\pi$
$350$	0	0
$351$	6.00000	0.320256
$352$	6.53113i	0.348110i
$353$	6.93774i	0.369259i	0.982808	+	0.184629i	$0.0591085\pi$
$353$	6.93774i	0.369259i	−0.982808	+	0.184629i	$0.940892\pi$
$354$	9.06226	0.481654
$355$	0	0
$356$	−6.53113	−0.346149
$357$	18.1245i	0.959251i
$358$	3.06226i	0.161845i
$359$	−3.46887	−0.183080	−0.0915400	−	0.995801i	$-0.529179\pi$
$359$	−3.46887	−0.183080	−0.0915400	+	0.995801i	$0.529179\pi$
$360$	0	0
$361$	1.53113	0.0805857
$362$	2.40661i	0.126489i
$363$	− 31.6556i	− 1.66149i
$364$	27.1868	1.42497
$365$	0	0
$366$	5.06226	0.264608
$367$	− 10.0000i	− 0.521996i	−0.965339	−	0.260998i	$-0.915948\pi$
$367$	− 10.0000i	− 0.521996i	0.965339	−	0.260998i	$-0.0840516\pi$
$368$	− 6.53113i	− 0.340459i
$369$	−7.06226	−0.367646
$370$	0	0
$371$	11.4689	0.595434
$372$	1.00000i	0.0518476i
$373$	18.5311i	0.959505i	0.877404	+	0.479753i	$0.159274\pi$
$373$	18.5311i	0.959505i	−0.877404	+	0.479753i	$0.840726\pi$
$374$	26.1245	1.35087
$375$	0	0
$376$	−5.06226	−0.261066
$377$	0	0
$378$	4.53113i	0.233056i
$379$	−21.5934	−1.10918	−0.554589	−	0.832124i	$-0.687124\pi$
$379$	−21.5934	−1.10918	−0.554589	+	0.832124i	$0.687124\pi$
$380$	0	0
$381$	7.06226	0.361810
$382$	− 8.00000i	− 0.409316i
$383$	3.06226i	0.156474i	0.996935	+	0.0782370i	$0.0249291\pi$
$383$	3.06226i	0.156474i	−0.996935	+	0.0782370i	$0.975071\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	14.0000	0.712581
$387$	− 8.53113i	− 0.433662i
$388$	− 16.1245i	− 0.818598i
$389$	−10.9377	−0.554566	−0.277283	−	0.960788i	$-0.589434\pi$
$389$	−10.9377	−0.554566	−0.277283	+	0.960788i	$0.589434\pi$
$390$	0	0
$391$	−26.1245	−1.32117
$392$	13.5311i	0.683425i
$393$	9.06226i	0.457130i
$394$	−6.00000	−0.302276
$395$	0	0
$396$	6.53113	0.328202
$397$	37.7179i	1.89301i	0.322693	+	0.946504i	$0.395412\pi$
$397$	37.7179i	1.89301i	−0.322693	+	0.946504i	$0.604588\pi$
$398$	8.53113i	0.427627i
$399$	20.5311	1.02784
$400$	0	0
$401$	21.7179	1.08454	0.542270	−	0.840204i	$-0.317565\pi$
$401$	21.7179	1.08454	0.542270	+	0.840204i	$0.317565\pi$
$402$	5.06226i	0.252482i
$403$	− 6.00000i	− 0.298881i
$404$	9.46887	0.471094
$405$	0	0
$406$	0	0
$407$	− 46.1245i	− 2.28631i
$408$	4.00000i	0.198030i
$409$	−7.06226	−0.349206	−0.174603	−	0.984639i	$-0.555864\pi$
$409$	−7.06226	−0.349206	−0.174603	+	0.984639i	$0.555864\pi$
$410$	0	0
$411$	17.0623	0.841619
$412$	0	0
$413$	− 41.0623i	− 2.02054i
$414$	−6.53113	−0.320987
$415$	0	0
$416$	6.00000	0.294174
$417$	6.00000i	0.293821i
$418$	− 29.5934i	− 1.44746i
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	−26.2490	−1.27930	−0.639650	−	0.768667i	$-0.720921\pi$
$421$	−26.2490	−1.27930	−0.639650	+	0.768667i	$0.720921\pi$
$422$	− 1.59339i	− 0.0775648i
$423$	5.06226i	0.246135i
$424$	2.53113	0.122922
$425$	0	0
$426$	−12.5311	−0.607135
$427$	− 22.9377i	− 1.11004i
$428$	− 9.59339i	− 0.463714i
$429$	−39.1868	−1.89196
$430$	0	0
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	1.00000i	0.0481125i
$433$	25.5934i	1.22994i	0.788551	+	0.614970i	$0.210832\pi$
$433$	25.5934i	1.22994i	−0.788551	+	0.614970i	$0.789168\pi$
$434$	4.53113	0.217501
$435$	0	0
$436$	15.0623	0.721351
$437$	29.5934i	1.41564i
$438$	8.53113i	0.407633i
$439$	25.0623	1.19616	0.598078	−	0.801438i	$-0.295931\pi$
$439$	25.0623	1.19616	0.598078	+	0.801438i	$0.295931\pi$
$440$	0	0
$441$	13.5311	0.644339
$442$	− 24.0000i	− 1.14156i
$443$	− 14.4066i	− 0.684479i	−0.939613	−	0.342239i	$-0.888815\pi$
$443$	− 14.4066i	− 0.684479i	0.939613	−	0.342239i	$-0.111185\pi$
$444$	7.06226	0.335160
$445$	0	0
$446$	2.00000	0.0947027
$447$	− 18.5311i	− 0.876492i
$448$	4.53113i	0.214076i
$449$	4.12452	0.194648	0.0973240	−	0.995253i	$-0.468972\pi$
$449$	4.12452	0.194648	0.0973240	+	0.995253i	$0.468972\pi$
$450$	0	0
$451$	46.1245	2.17192
$452$	− 7.59339i	− 0.357163i
$453$	− 14.1245i	− 0.663628i
$454$	24.5311	1.15130
$455$	0	0
$456$	4.53113	0.212190
$457$	14.9377i	0.698758i	0.936981	+	0.349379i	$0.113607\pi$
$457$	14.9377i	0.698758i	−0.936981	+	0.349379i	$0.886393\pi$
$458$	5.59339i	0.261362i
$459$	4.00000	0.186704
$460$	0	0
$461$	24.0000	1.11779	0.558896	−	0.829238i	$-0.311225\pi$
$461$	24.0000	1.11779	0.558896	+	0.829238i	$0.311225\pi$
$462$	− 29.5934i	− 1.37681i
$463$	− 13.1868i	− 0.612841i	−0.951896	−	0.306421i	$-0.900869\pi$
$463$	− 13.1868i	− 0.612841i	0.951896	−	0.306421i	$-0.0991314\pi$
$464$	0	0
$465$	0	0
$466$	1.46887	0.0680441
$467$	− 22.1245i	− 1.02380i	−0.859045	−	0.511900i	$-0.828942\pi$
$467$	− 22.1245i	− 1.02380i	0.859045	−	0.511900i	$-0.171058\pi$
$468$	− 6.00000i	− 0.277350i
$469$	22.9377	1.05917
$470$	0	0
$471$	−22.5311	−1.03818
$472$	− 9.06226i	− 0.417124i
$473$	55.7179i	2.56191i
$474$	8.53113	0.391848
$475$	0	0
$476$	18.1245	0.830736
$477$	− 2.53113i	− 0.115892i
$478$	8.00000i	0.365911i
$479$	−1.34436	−0.0614252	−0.0307126	−	0.999528i	$-0.509778\pi$
$479$	−1.34436	−0.0614252	−0.0307126	+	0.999528i	$0.509778\pi$
$480$	0	0
$481$	−42.3735	−1.93207
$482$	2.00000i	0.0910975i
$483$	29.5934i	1.34655i
$484$	−31.6556	−1.43889
$485$	0	0
$486$	1.00000	0.0453609
$487$	23.0623i	1.04505i	0.852624	+	0.522525i	$0.175010\pi$
$487$	23.0623i	1.04505i	−0.852624	+	0.522525i	$0.824990\pi$
$488$	− 5.06226i	− 0.229158i
$489$	−5.06226	−0.228923
$490$	0	0
$491$	−7.59339	−0.342685	−0.171342	−	0.985212i	$-0.554810\pi$
$491$	−7.59339	−0.342685	−0.171342	+	0.985212i	$0.554810\pi$
$492$	7.06226i	0.318391i
$493$	0	0
$494$	−27.1868	−1.22319
$495$	0	0
$496$	1.00000	0.0449013
$497$	56.7802i	2.54694i
$498$	8.00000i	0.358489i
$499$	7.06226	0.316150	0.158075	−	0.987427i	$-0.449471\pi$
$499$	7.06226	0.316150	0.158075	+	0.987427i	$0.449471\pi$
$500$	0	0
$501$	−23.5934	−1.05407
$502$	3.06226i	0.136675i
$503$	− 16.0000i	− 0.713405i	−0.934218	−	0.356702i	$-0.883901\pi$
$503$	− 16.0000i	− 0.713405i	0.934218	−	0.356702i	$-0.116099\pi$
$504$	4.53113	0.201833
$505$	0	0
$506$	42.6556	1.89627
$507$	23.0000i	1.02147i
$508$	− 7.06226i	− 0.313337i
$509$	−38.1245	−1.68984	−0.844920	−	0.534893i	$-0.820352\pi$
$509$	−38.1245	−1.68984	−0.844920	+	0.534893i	$0.820352\pi$
$510$	0	0
$511$	38.6556	1.71003
$512$	1.00000i	0.0441942i
$513$	− 4.53113i	− 0.200054i
$514$	−12.6556	−0.558217
$515$	0	0
$516$	−8.53113	−0.375562
$517$	− 33.0623i	− 1.45408i
$518$	− 32.0000i	− 1.40600i
$519$	−14.0000	−0.614532
$520$	0	0
$521$	−12.1245	−0.531185	−0.265592	−	0.964085i	$-0.585568\pi$
$521$	−12.1245	−0.531185	−0.265592	+	0.964085i	$0.585568\pi$
$522$	0	0
$523$	− 9.59339i	− 0.419490i	−0.977756	−	0.209745i	$-0.932737\pi$
$523$	− 9.59339i	− 0.419490i	0.977756	−	0.209745i	$-0.0672633\pi$
$524$	9.06226	0.395887
$525$	0	0
$526$	−23.0623	−1.00556
$527$	− 4.00000i	− 0.174243i
$528$	− 6.53113i	− 0.284231i
$529$	−19.6556	−0.854593
$530$	0	0
$531$	−9.06226	−0.393268
$532$	− 20.5311i	− 0.890137i
$533$	− 42.3735i	− 1.83540i
$534$	6.53113	0.282630
$535$	0	0
$536$	5.06226	0.218656
$537$	− 3.06226i	− 0.132146i
$538$	19.1868i	0.827201i
$539$	−88.3735	−3.80652
$540$	0	0
$541$	−4.12452	−0.177327	−0.0886634	−	0.996062i	$-0.528260\pi$
$541$	−4.12452	−0.177327	−0.0886634	+	0.996062i	$0.528260\pi$
$542$	− 11.4689i	− 0.492630i
$543$	− 2.40661i	− 0.103278i
$544$	4.00000	0.171499
$545$	0	0
$546$	−27.1868	−1.16349
$547$	− 7.18677i	− 0.307284i	−0.988127	−	0.153642i	$-0.950900\pi$
$547$	− 7.18677i	− 0.307284i	0.988127	−	0.153642i	$-0.0491003\pi$
$548$	− 17.0623i	− 0.728864i
$549$	−5.06226	−0.216052
$550$	0	0
$551$	0	0
$552$	6.53113i	0.277983i
$553$	− 38.6556i	− 1.64381i
$554$	15.0623	0.639934
$555$	0	0
$556$	6.00000	0.254457
$557$	− 26.7802i	− 1.13471i	−0.823473	−	0.567356i	$-0.807966\pi$
$557$	− 26.7802i	− 1.13471i	0.823473	−	0.567356i	$-0.192034\pi$
$558$	− 1.00000i	− 0.0423334i
$559$	51.1868	2.16497
$560$	0	0
$561$	−26.1245	−1.10298
$562$	15.0623i	0.635363i
$563$	4.00000i	0.168580i	0.996441	+	0.0842900i	$0.0268622\pi$
$563$	4.00000i	0.168580i	−0.996441	+	0.0842900i	$0.973138\pi$
$564$	5.06226	0.213160
$565$	0	0
$566$	−12.0000	−0.504398
$567$	− 4.53113i	− 0.190290i
$568$	12.5311i	0.525794i
$569$	−9.46887	−0.396956	−0.198478	−	0.980105i	$-0.563600\pi$
$569$	−9.46887	−0.396956	−0.198478	+	0.980105i	$0.563600\pi$
$570$	0	0
$571$	36.1245	1.51176	0.755882	−	0.654708i	$-0.227208\pi$
$571$	36.1245	1.51176	0.755882	+	0.654708i	$0.227208\pi$
$572$	39.1868i	1.63848i
$573$	8.00000i	0.334205i
$574$	32.0000	1.33565
$575$	0	0
$576$	1.00000	0.0416667
$577$	37.1868i	1.54811i	0.633121	+	0.774053i	$0.281774\pi$
$577$	37.1868i	1.54811i	−0.633121	+	0.774053i	$0.718226\pi$
$578$	1.00000i	0.0415945i
$579$	−14.0000	−0.581820
$580$	0	0
$581$	36.2490	1.50386
$582$	16.1245i	0.668383i
$583$	16.5311i	0.684649i
$584$	8.53113	0.353021
$585$	0	0
$586$	−14.0000	−0.578335
$587$	− 44.2490i	− 1.82635i	−0.407564	−	0.913176i	$-0.633622\pi$
$587$	− 44.2490i	− 1.82635i	0.407564	−	0.913176i	$-0.366378\pi$
$588$	− 13.5311i	− 0.558014i
$589$	−4.53113	−0.186702
$590$	0	0
$591$	6.00000	0.246807
$592$	− 7.06226i	− 0.290257i
$593$	14.0000i	0.574911i	0.957794	+	0.287456i	$0.0928094\pi$
$593$	14.0000i	0.574911i	−0.957794	+	0.287456i	$0.907191\pi$
$594$	−6.53113	−0.267975
$595$	0	0
$596$	−18.5311	−0.759065
$597$	− 8.53113i	− 0.349156i
$598$	− 39.1868i	− 1.60247i
$599$	13.5934	0.555411	0.277705	−	0.960666i	$-0.410426\pi$
$599$	13.5934	0.555411	0.277705	+	0.960666i	$0.410426\pi$
$600$	0	0
$601$	30.0000	1.22373	0.611863	−	0.790964i	$-0.290420\pi$
$601$	30.0000	1.22373	0.611863	+	0.790964i	$0.290420\pi$
$602$	38.6556i	1.57549i
$603$	− 5.06226i	− 0.206151i
$604$	−14.1245	−0.574718
$605$	0	0
$606$	−9.46887	−0.384647
$607$	− 14.4066i	− 0.584746i	−0.956304	−	0.292373i	$-0.905555\pi$
$607$	− 14.4066i	− 0.584746i	0.956304	−	0.292373i	$-0.0944449\pi$
$608$	− 4.53113i	− 0.183762i
$609$	0	0
$610$	0	0
$611$	−30.3735	−1.22878
$612$	− 4.00000i	− 0.161690i
$613$	30.0000i	1.21169i	0.795583	+	0.605844i	$0.207165\pi$
$613$	30.0000i	1.21169i	−0.795583	+	0.605844i	$0.792835\pi$
$614$	30.1245	1.21573
$615$	0	0
$616$	−29.5934	−1.19235
$617$	− 3.59339i	− 0.144664i	−0.997381	−	0.0723321i	$-0.976956\pi$
$617$	− 3.59339i	− 0.144664i	0.997381	−	0.0723321i	$-0.0230442\pi$
$618$	0	0
$619$	34.0000	1.36658	0.683288	−	0.730149i	$-0.260549\pi$
$619$	34.0000	1.36658	0.683288	+	0.730149i	$0.260549\pi$
$620$	0	0
$621$	6.53113	0.262085
$622$	− 8.00000i	− 0.320771i
$623$	− 29.5934i	− 1.18563i
$624$	−6.00000	−0.240192
$625$	0	0
$626$	10.9377	0.437160
$627$	29.5934i	1.18185i
$628$	22.5311i	0.899090i
$629$	−28.2490	−1.12636
$630$	0	0
$631$	30.6556	1.22038	0.610191	−	0.792254i	$-0.291093\pi$
$631$	30.6556	1.22038	0.610191	+	0.792254i	$0.291093\pi$
$632$	− 8.53113i	− 0.339350i
$633$	1.59339i	0.0633314i
$634$	−12.9377	−0.513823
$635$	0	0
$636$	−2.53113	−0.100366
$637$	81.1868i	3.21674i
$638$	0	0
$639$	12.5311	0.495724
$640$	0	0
$641$	−22.0000	−0.868948	−0.434474	−	0.900684i	$-0.643066\pi$
$641$	−22.0000	−0.868948	−0.434474	+	0.900684i	$0.643066\pi$
$642$	9.59339i	0.378621i
$643$	17.5934i	0.693815i	0.937899	+	0.346908i	$0.112768\pi$
$643$	17.5934i	0.693815i	−0.937899	+	0.346908i	$0.887232\pi$
$644$	29.5934	1.16614
$645$	0	0
$646$	−18.1245	−0.713100
$647$	− 6.53113i	− 0.256765i	−0.991725	−	0.128383i	$-0.959021\pi$
$647$	− 6.53113i	− 0.256765i	0.991725	−	0.128383i	$-0.0409785\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	59.1868	2.32328
$650$	0	0
$651$	−4.53113	−0.177589
$652$	5.06226i	0.198253i
$653$	− 38.2490i	− 1.49680i	−0.663248	−	0.748400i	$-0.730822\pi$
$653$	− 38.2490i	− 1.49680i	0.663248	−	0.748400i	$-0.269178\pi$
$654$	−15.0623	−0.588981
$655$	0	0
$656$	7.06226	0.275735
$657$	− 8.53113i	− 0.332831i
$658$	− 22.9377i	− 0.894206i
$659$	−16.0000	−0.623272	−0.311636	−	0.950202i	$-0.600877\pi$
$659$	−16.0000	−0.623272	−0.311636	+	0.950202i	$0.600877\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	12.9377i	0.502840i
$663$	24.0000i	0.932083i
$664$	8.00000	0.310460
$665$	0	0
$666$	−7.06226	−0.273657
$667$	0	0
$668$	23.5934i	0.912856i
$669$	−2.00000	−0.0773245
$670$	0	0
$671$	33.0623	1.27635
$672$	− 4.53113i	− 0.174792i
$673$	9.06226i	0.349324i	0.984628	+	0.174662i	$0.0558833\pi$
$673$	9.06226i	0.349324i	−0.984628	+	0.174662i	$0.944117\pi$
$674$	19.1868	0.739047
$675$	0	0
$676$	23.0000	0.884615
$677$	27.5934i	1.06050i	0.847841	+	0.530250i	$0.177902\pi$
$677$	27.5934i	1.06050i	−0.847841	+	0.530250i	$0.822098\pi$
$678$	7.59339i	0.291622i
$679$	73.0623	2.80387
$680$	0	0
$681$	−24.5311	−0.940035
$682$	6.53113i	0.250090i
$683$	− 7.46887i	− 0.285788i	−0.989738	−	0.142894i	$-0.954359\pi$
$683$	− 7.46887i	− 0.285788i	0.989738	−	0.142894i	$-0.0456409\pi$
$684$	−4.53113	−0.173252
$685$	0	0
$686$	−29.5934	−1.12988
$687$	− 5.59339i	− 0.213401i
$688$	8.53113i	0.325246i
$689$	15.1868	0.578570
$690$	0	0
$691$	0.531129	0.0202051	0.0101025	−	0.999949i	$-0.496784\pi$
$691$	0.531129	0.0202051	0.0101025	+	0.999949i	$0.496784\pi$
$692$	14.0000i	0.532200i
$693$	29.5934i	1.12416i
$694$	−10.1245	−0.384321
$695$	0	0
$696$	0	0
$697$	− 28.2490i	− 1.07001i
$698$	− 8.93774i	− 0.338299i
$699$	−1.46887	−0.0555578
$700$	0	0
$701$	−13.4689	−0.508712	−0.254356	−	0.967111i	$-0.581864\pi$
$701$	−13.4689	−0.508712	−0.254356	+	0.967111i	$0.581864\pi$
$702$	6.00000i	0.226455i
$703$	32.0000i	1.20690i
$704$	−6.53113	−0.246151
$705$	0	0
$706$	−6.93774	−0.261105
$707$	42.9047i	1.61360i
$708$	9.06226i	0.340581i
$709$	−49.5934	−1.86252	−0.931259	−	0.364357i	$-0.881289\pi$
$709$	−49.5934	−1.86252	−0.931259	+	0.364357i	$0.881289\pi$
$710$	0	0
$711$	−8.53113	−0.319942
$712$	− 6.53113i	− 0.244764i
$713$	− 6.53113i	− 0.244593i
$714$	−18.1245	−0.678293
$715$	0	0
$716$	−3.06226	−0.114442
$717$	− 8.00000i	− 0.298765i
$718$	− 3.46887i	− 0.129457i
$719$	−15.1868	−0.566371	−0.283186	−	0.959065i	$-0.591391\pi$
$719$	−15.1868	−0.566371	−0.283186	+	0.959065i	$0.591391\pi$
$720$	0	0
$721$	0	0
$722$	1.53113i	0.0569827i
$723$	− 2.00000i	− 0.0743808i
$724$	−2.40661	−0.0894411
$725$	0	0
$726$	31.6556	1.17485
$727$	37.8424i	1.40350i	0.712424	+	0.701749i	$0.247597\pi$
$727$	37.8424i	1.40350i	−0.712424	+	0.701749i	$0.752403\pi$
$728$	27.1868i	1.00761i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	34.1245	1.26214
$732$	5.06226i	0.187106i
$733$	28.1245i	1.03880i	0.854530	+	0.519401i	$0.173845\pi$
$733$	28.1245i	1.03880i	−0.854530	+	0.519401i	$0.826155\pi$
$734$	10.0000	0.369107
$735$	0	0
$736$	6.53113	0.240741
$737$	33.0623i	1.21786i
$738$	− 7.06226i	− 0.259965i
$739$	21.1868	0.779368	0.389684	−	0.920949i	$-0.372584\pi$
$739$	21.1868	0.779368	0.389684	+	0.920949i	$0.372584\pi$
$740$	0	0
$741$	27.1868	0.998731
$742$	11.4689i	0.421036i
$743$	28.6556i	1.05127i	0.850709	+	0.525637i	$0.176173\pi$
$743$	28.6556i	1.05127i	−0.850709	+	0.525637i	$0.823827\pi$
$744$	−1.00000	−0.0366618
$745$	0	0
$746$	−18.5311	−0.678473
$747$	− 8.00000i	− 0.292705i
$748$	26.1245i	0.955207i
$749$	43.4689	1.58832
$750$	0	0
$751$	30.9377	1.12893	0.564467	−	0.825456i	$-0.309082\pi$
$751$	30.9377	1.12893	0.564467	+	0.825456i	$0.309082\pi$
$752$	− 5.06226i	− 0.184602i
$753$	− 3.06226i	− 0.111595i
$754$	0	0
$755$	0	0
$756$	−4.53113	−0.164796
$757$	26.0000i	0.944986i	0.881334	+	0.472493i	$0.156646\pi$
$757$	26.0000i	0.944986i	−0.881334	+	0.472493i	$0.843354\pi$
$758$	− 21.5934i	− 0.784307i
$759$	−42.6556	−1.54830
$760$	0	0
$761$	11.5934	0.420260	0.210130	−	0.977673i	$-0.432611\pi$
$761$	11.5934	0.420260	0.210130	+	0.977673i	$0.432611\pi$
$762$	7.06226i	0.255839i
$763$	68.2490i	2.47078i
$764$	8.00000	0.289430
$765$	0	0
$766$	−3.06226	−0.110644
$767$	− 54.3735i	− 1.96331i
$768$	− 1.00000i	− 0.0360844i
$769$	4.40661	0.158907	0.0794533	−	0.996839i	$-0.474683\pi$
$769$	4.40661	0.158907	0.0794533	+	0.996839i	$0.474683\pi$
$770$	0	0
$771$	12.6556	0.455782
$772$	14.0000i	0.503871i
$773$	− 22.5311i	− 0.810388i	−0.914231	−	0.405194i	$-0.867204\pi$
$773$	− 22.5311i	− 0.810388i	0.914231	−	0.405194i	$-0.132796\pi$
$774$	8.53113	0.306645
$775$	0	0
$776$	16.1245	0.578836
$777$	32.0000i	1.14799i
$778$	− 10.9377i	− 0.392137i
$779$	−32.0000	−1.14652
$780$	0	0
$781$	−81.8424	−2.92855
$782$	− 26.1245i	− 0.934211i
$783$	0	0
$784$	−13.5311	−0.483255
$785$	0	0
$786$	−9.06226	−0.323240
$787$	− 36.5311i	− 1.30219i	−0.758994	−	0.651097i	$-0.774309\pi$
$787$	− 36.5311i	− 1.30219i	0.758994	−	0.651097i	$-0.225691\pi$
$788$	− 6.00000i	− 0.213741i
$789$	23.0623	0.821038
$790$	0	0
$791$	34.4066	1.22336
$792$	6.53113i	0.232074i
$793$	− 30.3735i	− 1.07860i
$794$	−37.7179	−1.33856
$795$	0	0
$796$	−8.53113	−0.302378
$797$	12.1245i	0.429472i	0.976672	+	0.214736i	$0.0688892\pi$
$797$	12.1245i	0.429472i	−0.976672	+	0.214736i	$0.931111\pi$
$798$	20.5311i	0.726794i
$799$	−20.2490	−0.716359
$800$	0	0
$801$	−6.53113	−0.230766
$802$	21.7179i	0.766886i
$803$	55.7179i	1.96624i
$804$	−5.06226	−0.178532
$805$	0	0
$806$	6.00000	0.211341
$807$	− 19.1868i	− 0.675406i
$808$	9.46887i	0.333114i
$809$	−15.5934	−0.548234	−0.274117	−	0.961696i	$-0.588386\pi$
$809$	−15.5934	−0.548234	−0.274117	+	0.961696i	$0.588386\pi$
$810$	0	0
$811$	32.7802	1.15107	0.575534	−	0.817778i	$-0.304794\pi$
$811$	32.7802	1.15107	0.575534	+	0.817778i	$0.304794\pi$
$812$	0	0
$813$	11.4689i	0.402231i
$814$	46.1245	1.61666
$815$	0	0
$816$	−4.00000	−0.140028
$817$	− 38.6556i	− 1.35239i
$818$	− 7.06226i	− 0.246926i
$819$	27.1868	0.949983
$820$	0	0
$821$	42.1245	1.47016	0.735078	−	0.677983i	$-0.237146\pi$
$821$	42.1245	1.47016	0.735078	+	0.677983i	$0.237146\pi$
$822$	17.0623i	0.595115i
$823$	− 40.1245i	− 1.39865i	−0.714803	−	0.699326i	$-0.753483\pi$
$823$	− 40.1245i	− 1.39865i	0.714803	−	0.699326i	$-0.246517\pi$
$824$	0	0
$825$	0	0
$826$	41.0623	1.42874
$827$	8.00000i	0.278187i	0.990279	+	0.139094i	$0.0444189\pi$
$827$	8.00000i	0.278187i	−0.990279	+	0.139094i	$0.955581\pi$
$828$	− 6.53113i	− 0.226972i
$829$	−30.6556	−1.06471	−0.532357	−	0.846520i	$-0.678694\pi$
$829$	−30.6556	−1.06471	−0.532357	+	0.846520i	$0.678694\pi$
$830$	0	0
$831$	−15.0623	−0.522504
$832$	6.00000i	0.208013i
$833$	54.1245i	1.87530i
$834$	−6.00000	−0.207763
$835$	0	0
$836$	29.5934	1.02351
$837$	1.00000i	0.0345651i
$838$	0	0
$839$	−33.8424	−1.16837	−0.584185	−	0.811621i	$-0.698586\pi$
$839$	−33.8424	−1.16837	−0.584185	+	0.811621i	$0.698586\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	− 26.2490i	− 0.904601i
$843$	− 15.0623i	− 0.518772i
$844$	1.59339	0.0548466
$845$	0	0
$846$	−5.06226	−0.174044
$847$	− 143.436i	− 4.92851i
$848$	2.53113i	0.0869193i
$849$	12.0000	0.411839
$850$	0	0
$851$	−46.1245	−1.58113
$852$	− 12.5311i	− 0.429309i
$853$	− 57.7179i	− 1.97622i	−0.153738	−	0.988112i	$-0.549131\pi$
$853$	− 57.7179i	− 1.97622i	0.153738	−	0.988112i	$-0.450869\pi$
$854$	22.9377	0.784913
$855$	0	0
$856$	9.59339	0.327895
$857$	14.0000i	0.478231i	0.970991	+	0.239115i	$0.0768574\pi$
$857$	14.0000i	0.478231i	−0.970991	+	0.239115i	$0.923143\pi$
$858$	− 39.1868i	− 1.33781i
$859$	23.0623	0.786874	0.393437	−	0.919352i	$-0.371286\pi$
$859$	23.0623	0.786874	0.393437	+	0.919352i	$0.371286\pi$
$860$	0	0
$861$	−32.0000	−1.09056
$862$	− 24.0000i	− 0.817443i
$863$	− 33.4689i	− 1.13929i	−0.821890	−	0.569647i	$-0.807080\pi$
$863$	− 33.4689i	− 1.13929i	0.821890	−	0.569647i	$-0.192920\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	−25.5934	−0.869699
$867$	− 1.00000i	− 0.0339618i
$868$	4.53113i	0.153797i
$869$	55.7179	1.89010
$870$	0	0
$871$	30.3735	1.02917
$872$	15.0623i	0.510072i
$873$	− 16.1245i	− 0.545732i
$874$	−29.5934	−1.00101
$875$	0	0
$876$	−8.53113	−0.288240
$877$	11.8755i	0.401007i	0.979693	+	0.200503i	$0.0642577\pi$
$877$	11.8755i	0.401007i	−0.979693	+	0.200503i	$0.935742\pi$
$878$	25.0623i	0.845810i
$879$	14.0000	0.472208
$880$	0	0
$881$	3.87548	0.130568	0.0652842	−	0.997867i	$-0.479205\pi$
$881$	3.87548	0.130568	0.0652842	+	0.997867i	$0.479205\pi$
$882$	13.5311i	0.455617i
$883$	13.3444i	0.449073i	0.974466	+	0.224537i	$0.0720869\pi$
$883$	13.3444i	0.449073i	−0.974466	+	0.224537i	$0.927913\pi$
$884$	24.0000	0.807207
$885$	0	0
$886$	14.4066	0.484000
$887$	23.1868i	0.778536i	0.921125	+	0.389268i	$0.127272\pi$
$887$	23.1868i	0.778536i	−0.921125	+	0.389268i	$0.872728\pi$
$888$	7.06226i	0.236994i
$889$	32.0000	1.07325
$890$	0	0
$891$	6.53113	0.218801
$892$	2.00000i	0.0669650i
$893$	22.9377i	0.767582i
$894$	18.5311	0.619774
$895$	0	0
$896$	−4.53113	−0.151374
$897$	39.1868i	1.30841i
$898$	4.12452i	0.137637i
$899$	0	0
$900$	0	0
$901$	10.1245	0.337297
$902$	46.1245i	1.53578i
$903$	− 38.6556i	− 1.28638i
$904$	7.59339	0.252552
$905$	0	0
$906$	14.1245	0.469256
$907$	32.2490i	1.07081i	0.844595	+	0.535406i	$0.179841\pi$
$907$	32.2490i	1.07081i	−0.844595	+	0.535406i	$0.820159\pi$
$908$	24.5311i	0.814094i
$909$	9.46887	0.314063
$910$	0	0
$911$	−16.0000	−0.530104	−0.265052	−	0.964234i	$-0.585389\pi$
$911$	−16.0000	−0.530104	−0.265052	+	0.964234i	$0.585389\pi$
$912$	4.53113i	0.150041i
$913$	52.2490i	1.72919i
$914$	−14.9377	−0.494097
$915$	0	0
$916$	−5.59339	−0.184811
$917$	41.0623i	1.35600i
$918$	4.00000i	0.132020i
$919$	25.0623	0.826728	0.413364	−	0.910566i	$-0.364354\pi$
$919$	25.0623	0.826728	0.413364	+	0.910566i	$0.364354\pi$
$920$	0	0
$921$	−30.1245	−0.992637
$922$	24.0000i	0.790398i
$923$	75.1868i	2.47480i
$924$	29.5934	0.973551
$925$	0	0
$926$	13.1868	0.433344
$927$	0	0
$928$	0	0
$929$	51.8424	1.70089	0.850447	−	0.526060i	$-0.176331\pi$
$929$	51.8424	1.70089	0.850447	+	0.526060i	$0.176331\pi$
$930$	0	0
$931$	61.3113	2.00940
$932$	1.46887i	0.0481145i
$933$	8.00000i	0.261908i
$934$	22.1245	0.723936
$935$	0	0
$936$	6.00000	0.196116
$937$	− 8.93774i	− 0.291983i	−0.989286	−	0.145992i	$-0.953363\pi$
$937$	− 8.93774i	− 0.291983i	0.989286	−	0.145992i	$-0.0466373\pi$
$938$	22.9377i	0.748944i
$939$	−10.9377	−0.356940
$940$	0	0
$941$	−54.1245	−1.76441	−0.882204	−	0.470867i	$-0.843941\pi$
$941$	−54.1245	−1.76441	−0.882204	+	0.470867i	$0.843941\pi$
$942$	− 22.5311i	− 0.734104i
$943$	− 46.1245i	− 1.50202i
$944$	9.06226	0.294951
$945$	0	0
$946$	−55.7179	−1.81155
$947$	18.9377i	0.615394i	0.951484	+	0.307697i	$0.0995583\pi$
$947$	18.9377i	0.615394i	−0.951484	+	0.307697i	$0.900442\pi$
$948$	8.53113i	0.277078i
$949$	51.1868	1.66159
$950$	0	0
$951$	12.9377	0.419535
$952$	18.1245i	0.587419i
$953$	− 21.8755i	− 0.708616i	−0.935129	−	0.354308i	$-0.884716\pi$
$953$	− 21.8755i	− 0.708616i	0.935129	−	0.354308i	$-0.115284\pi$
$954$	2.53113	0.0819483
$955$	0	0
$956$	−8.00000	−0.258738
$957$	0	0
$958$	− 1.34436i	− 0.0434342i
$959$	77.3113	2.49651
$960$	0	0
$961$	1.00000	0.0322581
$962$	− 42.3735i	− 1.36618i
$963$	− 9.59339i	− 0.309143i
$964$	−2.00000	−0.0644157
$965$	0	0
$966$	−29.5934	−0.952152
$967$	− 0.937742i	− 0.0301558i	−0.999886	−	0.0150779i	$-0.995200\pi$
$967$	− 0.937742i	− 0.0301558i	0.999886	−	0.0150779i	$-0.00479962\pi$
$968$	− 31.6556i	− 1.01745i
$969$	18.1245	0.582243
$970$	0	0
$971$	15.1868	0.487367	0.243683	−	0.969855i	$-0.421644\pi$
$971$	15.1868	0.487367	0.243683	+	0.969855i	$0.421644\pi$
$972$	1.00000i	0.0320750i
$973$	27.1868i	0.871568i
$974$	−23.0623	−0.738962
$975$	0	0
$976$	5.06226	0.162039
$977$	− 38.0000i	− 1.21573i	−0.794041	−	0.607864i	$-0.792027\pi$
$977$	− 38.0000i	− 1.21573i	0.794041	−	0.607864i	$-0.207973\pi$
$978$	− 5.06226i	− 0.161873i
$979$	42.6556	1.36328
$980$	0	0
$981$	15.0623	0.480901
$982$	− 7.59339i	− 0.242315i
$983$	− 0.937742i	− 0.0299093i	−0.999888	−	0.0149547i	$-0.995240\pi$
$983$	− 0.937742i	− 0.0299093i	0.999888	−	0.0149547i	$-0.00476040\pi$
$984$	−7.06226	−0.225137
$985$	0	0
$986$	0	0
$987$	22.9377i	0.730116i
$988$	− 27.1868i	− 0.864926i
$989$	55.7179	1.77173
$990$	0	0
$991$	−32.5311	−1.03339	−0.516693	−	0.856171i	$-0.672837\pi$
$991$	−32.5311	−1.03339	−0.516693	+	0.856171i	$0.672837\pi$
$992$	1.00000i	0.0317500i
$993$	− 12.9377i	− 0.410567i
$994$	−56.7802	−1.80096
$995$	0	0
$996$	−8.00000	−0.253490
$997$	− 2.24903i	− 0.0712275i	−0.999366	−	0.0356138i	$-0.988661\pi$
$997$	− 2.24903i	− 0.0712275i	0.999366	−	0.0356138i	$-0.0113386\pi$
$998$	7.06226i	0.223552i
$999$	7.06226	0.223440

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4650.2.d.bg.3349.3		4
5.2	odd	4		4650.2.a.bz.1.2		2
5.3	odd	4		930.2.a.q.1.1	✓	2
5.4	even	2	inner	4650.2.d.bg.3349.2		4
15.8	even	4		2790.2.a.bf.1.1		2
20.3	even	4		7440.2.a.bd.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.a.q.1.1	✓	2	5.3	odd	4
2790.2.a.bf.1.1		2	15.8	even	4
4650.2.a.bz.1.2		2	5.2	odd	4
4650.2.d.bg.3349.2		4	5.4	even	2	inner
4650.2.d.bg.3349.3		4	1.1	even	1	trivial
7440.2.a.bd.1.2		2	20.3	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -4.53113i q^{7} -1.00000i q^{8} -1.00000 q^{9} +6.53113 q^{11} +1.00000i q^{12} -6.00000i q^{13} +4.53113 q^{14} +1.00000 q^{16} -4.00000i q^{17} -1.00000i q^{18} -4.53113 q^{19} -4.53113 q^{21} +6.53113i q^{22} -6.53113i q^{23} -1.00000 q^{24} +6.00000 q^{26} +1.00000i q^{27} +4.53113i q^{28} +1.00000 q^{31} +1.00000i q^{32} -6.53113i q^{33} +4.00000 q^{34} +1.00000 q^{36} -7.06226i q^{37} -4.53113i q^{38} -6.00000 q^{39} +7.06226 q^{41} -4.53113i q^{42} +8.53113i q^{43} -6.53113 q^{44} +6.53113 q^{46} -5.06226i q^{47} -1.00000i q^{48} -13.5311 q^{49} -4.00000 q^{51} +6.00000i q^{52} +2.53113i q^{53} -1.00000 q^{54} -4.53113 q^{56} +4.53113i q^{57} +9.06226 q^{59} +5.06226 q^{61} +1.00000i q^{62} +4.53113i q^{63} -1.00000 q^{64} +6.53113 q^{66} +5.06226i q^{67} +4.00000i q^{68} -6.53113 q^{69} -12.5311 q^{71} +1.00000i q^{72} +8.53113i q^{73} +7.06226 q^{74} +4.53113 q^{76} -29.5934i q^{77} -6.00000i q^{78} +8.53113 q^{79} +1.00000 q^{81} +7.06226i q^{82} +8.00000i q^{83} +4.53113 q^{84} -8.53113 q^{86} -6.53113i q^{88} +6.53113 q^{89} -27.1868 q^{91} +6.53113i q^{92} -1.00000i q^{93} +5.06226 q^{94} +1.00000 q^{96} +16.1245i q^{97} -13.5311i q^{98} -6.53113 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} + 4 q^{6} - 4 q^{9} + 10 q^{11} + 2 q^{14} + 4 q^{16} - 2 q^{19} - 2 q^{21} - 4 q^{24} + 24 q^{26} + 4 q^{31} + 16 q^{34} + 4 q^{36} - 24 q^{39} - 4 q^{41} - 10 q^{44} + 10 q^{46} - 38 q^{49}+ \cdots - 10 q^{99}+O(q^{100}) \) 4 * q - 4 * q^4 + 4 * q^6 - 4 * q^9 + 10 * q^11 + 2 * q^14 + 4 * q^16 - 2 * q^19 - 2 * q^21 - 4 * q^24 + 24 * q^26 + 4 * q^31 + 16 * q^34 + 4 * q^36 - 24 * q^39 - 4 * q^41 - 10 * q^44 + 10 * q^46 - 38 * q^49 - 16 * q^51 - 4 * q^54 - 2 * q^56 + 4 * q^59 - 12 * q^61 - 4 * q^64 + 10 * q^66 - 10 * q^69 - 34 * q^71 - 4 * q^74 + 2 * q^76 + 18 * q^79 + 4 * q^81 + 2 * q^84 - 18 * q^86 + 10 * q^89 - 12 * q^91 - 12 * q^94 + 4 * q^96 - 10 * q^99

Introduction

L-functions

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Database

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