LMFDB - Embedded newform 6050.2.a.ca.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6050,2,Mod(1,6050)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6050, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6050.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6050 = 2 \cdot 5^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6050.a (trivial)

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:	yes
Analytic conductor:	$48.3094932229$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 550)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	6050.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.61803 q^{3} +1.00000 q^{4} -1.61803 q^{6} +2.23607 q^{7} -1.00000 q^{8} -0.381966 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.61803 q^{3} +1.00000 q^{4} -1.61803 q^{6} +2.23607 q^{7} -1.00000 q^{8} -0.381966 q^{9} +1.61803 q^{12} -0.236068 q^{13} -2.23607 q^{14} +1.00000 q^{16} +0.381966 q^{17} +0.381966 q^{18} +2.61803 q^{19} +3.61803 q^{21} -3.85410 q^{23} -1.61803 q^{24} +0.236068 q^{26} -5.47214 q^{27} +2.23607 q^{28} -0.527864 q^{29} -10.2361 q^{31} -1.00000 q^{32} -0.381966 q^{34} -0.381966 q^{36} -8.85410 q^{37} -2.61803 q^{38} -0.381966 q^{39} -3.70820 q^{41} -3.61803 q^{42} -9.47214 q^{43} +3.85410 q^{46} +0.236068 q^{47} +1.61803 q^{48} -2.00000 q^{49} +0.618034 q^{51} -0.236068 q^{52} +1.38197 q^{53} +5.47214 q^{54} -2.23607 q^{56} +4.23607 q^{57} +0.527864 q^{58} -11.4721 q^{59} +2.23607 q^{61} +10.2361 q^{62} -0.854102 q^{63} +1.00000 q^{64} -9.32624 q^{67} +0.381966 q^{68} -6.23607 q^{69} +9.79837 q^{71} +0.381966 q^{72} +8.14590 q^{73} +8.85410 q^{74} +2.61803 q^{76} +0.381966 q^{78} -11.9443 q^{79} -7.70820 q^{81} +3.70820 q^{82} +9.00000 q^{83} +3.61803 q^{84} +9.47214 q^{86} -0.854102 q^{87} -13.1803 q^{89} -0.527864 q^{91} -3.85410 q^{92} -16.5623 q^{93} -0.236068 q^{94} -1.61803 q^{96} +12.0000 q^{97} +2.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + q^{3} + 2 q^{4} - q^{6} - 2 q^{8} - 3 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + q^3 + 2 * q^4 - q^6 - 2 * q^8 - 3 * q^9	$ 2 q - 2 q^{2} + q^{3} + 2 q^{4} - q^{6} - 2 q^{8} - 3 q^{9} + q^{12} + 4 q^{13} + 2 q^{16} + 3 q^{17} + 3 q^{18} + 3 q^{19} + 5 q^{21} - q^{23} - q^{24} - 4 q^{26} - 2 q^{27} - 10 q^{29} - 16 q^{31} - 2 q^{32} - 3 q^{34} - 3 q^{36} - 11 q^{37} - 3 q^{38} - 3 q^{39} + 6 q^{41} - 5 q^{42} - 10 q^{43} + q^{46} - 4 q^{47} + q^{48} - 4 q^{49} - q^{51} + 4 q^{52} + 5 q^{53} + 2 q^{54} + 4 q^{57} + 10 q^{58} - 14 q^{59} + 16 q^{62} + 5 q^{63} + 2 q^{64} - 3 q^{67} + 3 q^{68} - 8 q^{69} - 5 q^{71} + 3 q^{72} + 23 q^{73} + 11 q^{74} + 3 q^{76} + 3 q^{78} - 6 q^{79} - 2 q^{81} - 6 q^{82} + 18 q^{83} + 5 q^{84} + 10 q^{86} + 5 q^{87} - 4 q^{89} - 10 q^{91} - q^{92} - 13 q^{93} + 4 q^{94} - q^{96} + 24 q^{97} + 4 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + q^3 + 2 * q^4 - q^6 - 2 * q^8 - 3 * q^9 + q^12 + 4 * q^13 + 2 * q^16 + 3 * q^17 + 3 * q^18 + 3 * q^19 + 5 * q^21 - q^23 - q^24 - 4 * q^26 - 2 * q^27 - 10 * q^29 - 16 * q^31 - 2 * q^32 - 3 * q^34 - 3 * q^36 - 11 * q^37 - 3 * q^38 - 3 * q^39 + 6 * q^41 - 5 * q^42 - 10 * q^43 + q^46 - 4 * q^47 + q^48 - 4 * q^49 - q^51 + 4 * q^52 + 5 * q^53 + 2 * q^54 + 4 * q^57 + 10 * q^58 - 14 * q^59 + 16 * q^62 + 5 * q^63 + 2 * q^64 - 3 * q^67 + 3 * q^68 - 8 * q^69 - 5 * q^71 + 3 * q^72 + 23 * q^73 + 11 * q^74 + 3 * q^76 + 3 * q^78 - 6 * q^79 - 2 * q^81 - 6 * q^82 + 18 * q^83 + 5 * q^84 + 10 * q^86 + 5 * q^87 - 4 * q^89 - 10 * q^91 - q^92 - 13 * q^93 + 4 * q^94 - q^96 + 24 * q^97 + 4 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.61803	0.934172	0.467086	−	0.884212i	$-0.345304\pi$
$3$	1.61803	0.934172	0.467086	+	0.884212i	$0.345304\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.61803	−0.660560
$7$	2.23607	0.845154	0.422577	−	0.906327i	$-0.361126\pi$
$7$	2.23607	0.845154	0.422577	+	0.906327i	$0.361126\pi$
$8$	−1.00000	−0.353553
$9$	−0.381966	−0.127322
$10$	0	0
$11$	0	0
$11$	0	0
$12$	1.61803	0.467086
$13$	−0.236068	−0.0654735	−0.0327367	−	0.999464i	$-0.510422\pi$
$13$	−0.236068	−0.0654735	−0.0327367	+	0.999464i	$0.510422\pi$
$14$	−2.23607	−0.597614
$15$	0	0
$16$	1.00000	0.250000
$17$	0.381966	0.0926404	0.0463202	−	0.998927i	$-0.485251\pi$
$17$	0.381966	0.0926404	0.0463202	+	0.998927i	$0.485251\pi$
$18$	0.381966	0.0900303
$19$	2.61803	0.600618	0.300309	−	0.953842i	$-0.402910\pi$
$19$	2.61803	0.600618	0.300309	+	0.953842i	$0.402910\pi$
$20$	0	0
$21$	3.61803	0.789520
$22$	0	0
$23$	−3.85410	−0.803636	−0.401818	−	0.915720i	$-0.631622\pi$
$23$	−3.85410	−0.803636	−0.401818	+	0.915720i	$0.631622\pi$
$24$	−1.61803	−0.330280
$25$	0	0
$26$	0.236068	0.0462967
$27$	−5.47214	−1.05311
$28$	2.23607	0.422577
$29$	−0.527864	−0.0980219	−0.0490109	−	0.998798i	$-0.515607\pi$
$29$	−0.527864	−0.0980219	−0.0490109	+	0.998798i	$0.515607\pi$
$30$	0	0
$31$	−10.2361	−1.83845	−0.919226	−	0.393730i	$-0.871184\pi$
$31$	−10.2361	−1.83845	−0.919226	+	0.393730i	$0.871184\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−0.381966	−0.0655066
$35$	0	0
$36$	−0.381966	−0.0636610
$37$	−8.85410	−1.45561	−0.727803	−	0.685787i	$-0.759458\pi$
$37$	−8.85410	−1.45561	−0.727803	+	0.685787i	$0.759458\pi$
$38$	−2.61803	−0.424701
$39$	−0.381966	−0.0611635
$40$	0	0
$41$	−3.70820	−0.579124	−0.289562	−	0.957159i	$-0.593510\pi$
$41$	−3.70820	−0.579124	−0.289562	+	0.957159i	$0.593510\pi$
$42$	−3.61803	−0.558275
$43$	−9.47214	−1.44449	−0.722244	−	0.691639i	$-0.756889\pi$
$43$	−9.47214	−1.44449	−0.722244	+	0.691639i	$0.756889\pi$
$44$	0	0
$45$	0	0
$46$	3.85410	0.568256
$47$	0.236068	0.0344341	0.0172170	−	0.999852i	$-0.494519\pi$
$47$	0.236068	0.0344341	0.0172170	+	0.999852i	$0.494519\pi$
$48$	1.61803	0.233543
$49$	−2.00000	−0.285714
$50$	0	0
$51$	0.618034	0.0865421
$52$	−0.236068	−0.0327367
$53$	1.38197	0.189828	0.0949138	−	0.995485i	$-0.469742\pi$
$53$	1.38197	0.189828	0.0949138	+	0.995485i	$0.469742\pi$
$54$	5.47214	0.744663
$55$	0	0
$56$	−2.23607	−0.298807
$57$	4.23607	0.561081
$58$	0.527864	0.0693119
$59$	−11.4721	−1.49354	−0.746772	−	0.665080i	$-0.768398\pi$
$59$	−11.4721	−1.49354	−0.746772	+	0.665080i	$0.768398\pi$
$60$	0	0
$61$	2.23607	0.286299	0.143150	−	0.989701i	$-0.454277\pi$
$61$	2.23607	0.286299	0.143150	+	0.989701i	$0.454277\pi$
$62$	10.2361	1.29998
$63$	−0.854102	−0.107607
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−9.32624	−1.13938	−0.569691	−	0.821859i	$-0.692937\pi$
$67$	−9.32624	−1.13938	−0.569691	+	0.821859i	$0.692937\pi$
$68$	0.381966	0.0463202
$69$	−6.23607	−0.750734
$70$	0	0
$71$	9.79837	1.16285	0.581427	−	0.813599i	$-0.302495\pi$
$71$	9.79837	1.16285	0.581427	+	0.813599i	$0.302495\pi$
$72$	0.381966	0.0450151
$73$	8.14590	0.953405	0.476703	−	0.879065i	$-0.341832\pi$
$73$	8.14590	0.953405	0.476703	+	0.879065i	$0.341832\pi$
$74$	8.85410	1.02927
$75$	0	0
$76$	2.61803	0.300309
$77$	0	0
$78$	0.381966	0.0432491
$79$	−11.9443	−1.34384	−0.671918	−	0.740626i	$-0.734529\pi$
$79$	−11.9443	−1.34384	−0.671918	+	0.740626i	$0.734529\pi$
$80$	0	0
$81$	−7.70820	−0.856467
$82$	3.70820	0.409503
$83$	9.00000	0.987878	0.493939	−	0.869496i	$-0.335557\pi$
$83$	9.00000	0.987878	0.493939	+	0.869496i	$0.335557\pi$
$84$	3.61803	0.394760
$85$	0	0
$86$	9.47214	1.02141
$87$	−0.854102	−0.0915693
$88$	0	0
$89$	−13.1803	−1.39711	−0.698557	−	0.715555i	$-0.746174\pi$
$89$	−13.1803	−1.39711	−0.698557	+	0.715555i	$0.746174\pi$
$90$	0	0
$91$	−0.527864	−0.0553352
$92$	−3.85410	−0.401818
$93$	−16.5623	−1.71743
$94$	−0.236068	−0.0243486
$95$	0	0
$96$	−1.61803	−0.165140
$97$	12.0000	1.21842	0.609208	−	0.793011i	$-0.291488\pi$
$97$	12.0000	1.21842	0.609208	+	0.793011i	$0.291488\pi$
$98$	2.00000	0.202031
$99$	0	0
$100$	0	0
$101$	18.7082	1.86154	0.930768	−	0.365611i	$-0.119140\pi$
$101$	18.7082	1.86154	0.930768	+	0.365611i	$0.119140\pi$
$102$	−0.618034	−0.0611945
$103$	16.8885	1.66408	0.832039	−	0.554717i	$-0.187174\pi$
$103$	16.8885	1.66408	0.832039	+	0.554717i	$0.187174\pi$
$104$	0.236068	0.0231484
$105$	0	0
$106$	−1.38197	−0.134228
$107$	−8.18034	−0.790823	−0.395412	−	0.918504i	$-0.629398\pi$
$107$	−8.18034	−0.790823	−0.395412	+	0.918504i	$0.629398\pi$
$108$	−5.47214	−0.526557
$109$	−11.6180	−1.11281	−0.556403	−	0.830913i	$-0.687819\pi$
$109$	−11.6180	−1.11281	−0.556403	+	0.830913i	$0.687819\pi$
$110$	0	0
$111$	−14.3262	−1.35979
$112$	2.23607	0.211289
$113$	14.7082	1.38363	0.691816	−	0.722074i	$-0.256811\pi$
$113$	14.7082	1.38363	0.691816	+	0.722074i	$0.256811\pi$
$114$	−4.23607	−0.396744
$115$	0	0
$116$	−0.527864	−0.0490109
$117$	0.0901699	0.00833621
$118$	11.4721	1.05610
$119$	0.854102	0.0782954
$120$	0	0
$121$	0	0
$122$	−2.23607	−0.202444
$123$	−6.00000	−0.541002
$124$	−10.2361	−0.919226
$125$	0	0
$126$	0.854102	0.0760895
$127$	−2.90983	−0.258206	−0.129103	−	0.991631i	$-0.541210\pi$
$127$	−2.90983	−0.258206	−0.129103	+	0.991631i	$0.541210\pi$
$128$	−1.00000	−0.0883883
$129$	−15.3262	−1.34940
$130$	0	0
$131$	11.1459	0.973822	0.486911	−	0.873452i	$-0.338124\pi$
$131$	11.1459	0.973822	0.486911	+	0.873452i	$0.338124\pi$
$132$	0	0
$133$	5.85410	0.507615
$134$	9.32624	0.805664
$135$	0	0
$136$	−0.381966	−0.0327533
$137$	−15.7639	−1.34680	−0.673402	−	0.739277i	$-0.735168\pi$
$137$	−15.7639	−1.34680	−0.673402	+	0.739277i	$0.735168\pi$
$138$	6.23607	0.530849
$139$	−2.85410	−0.242082	−0.121041	−	0.992648i	$-0.538623\pi$
$139$	−2.85410	−0.242082	−0.121041	+	0.992648i	$0.538623\pi$
$140$	0	0
$141$	0.381966	0.0321673
$142$	−9.79837	−0.822261
$143$	0	0
$144$	−0.381966	−0.0318305
$145$	0	0
$146$	−8.14590	−0.674159
$147$	−3.23607	−0.266906
$148$	−8.85410	−0.727803
$149$	−10.7082	−0.877250	−0.438625	−	0.898670i	$-0.644534\pi$
$149$	−10.7082	−0.877250	−0.438625	+	0.898670i	$0.644534\pi$
$150$	0	0
$151$	−19.0344	−1.54900	−0.774500	−	0.632573i	$-0.781999\pi$
$151$	−19.0344	−1.54900	−0.774500	+	0.632573i	$0.781999\pi$
$152$	−2.61803	−0.212351
$153$	−0.145898	−0.0117952
$154$	0	0
$155$	0	0
$156$	−0.381966	−0.0305818
$157$	−6.76393	−0.539821	−0.269910	−	0.962885i	$-0.586994\pi$
$157$	−6.76393	−0.539821	−0.269910	+	0.962885i	$0.586994\pi$
$158$	11.9443	0.950235
$159$	2.23607	0.177332
$160$	0	0
$161$	−8.61803	−0.679196
$162$	7.70820	0.605614
$163$	−0.180340	−0.0141253	−0.00706266	−	0.999975i	$-0.502248\pi$
$163$	−0.180340	−0.0141253	−0.00706266	+	0.999975i	$0.502248\pi$
$164$	−3.70820	−0.289562
$165$	0	0
$166$	−9.00000	−0.698535
$167$	2.23607	0.173032	0.0865161	−	0.996250i	$-0.472427\pi$
$167$	2.23607	0.173032	0.0865161	+	0.996250i	$0.472427\pi$
$168$	−3.61803	−0.279137
$169$	−12.9443	−0.995713
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	−9.47214	−0.722244
$173$	13.3262	1.01318	0.506588	−	0.862189i	$-0.330907\pi$
$173$	13.3262	1.01318	0.506588	+	0.862189i	$0.330907\pi$
$174$	0.854102	0.0647493
$175$	0	0
$176$	0	0
$177$	−18.5623	−1.39523
$178$	13.1803	0.987908
$179$	14.6525	1.09518	0.547589	−	0.836748i	$-0.315546\pi$
$179$	14.6525	1.09518	0.547589	+	0.836748i	$0.315546\pi$
$180$	0	0
$181$	−7.05573	−0.524448	−0.262224	−	0.965007i	$-0.584456\pi$
$181$	−7.05573	−0.524448	−0.262224	+	0.965007i	$0.584456\pi$
$182$	0.527864	0.0391279
$183$	3.61803	0.267453
$184$	3.85410	0.284128
$185$	0	0
$186$	16.5623	1.21441
$187$	0	0
$188$	0.236068	0.0172170
$189$	−12.2361	−0.890043
$190$	0	0
$191$	−19.8541	−1.43659	−0.718296	−	0.695737i	$-0.755078\pi$
$191$	−19.8541	−1.43659	−0.718296	+	0.695737i	$0.755078\pi$
$192$	1.61803	0.116772
$193$	−2.43769	−0.175469	−0.0877345	−	0.996144i	$-0.527963\pi$
$193$	−2.43769	−0.175469	−0.0877345	+	0.996144i	$0.527963\pi$
$194$	−12.0000	−0.861550
$195$	0	0
$196$	−2.00000	−0.142857
$197$	27.5967	1.96619	0.983093	−	0.183105i	$-0.0586147\pi$
$197$	27.5967	1.96619	0.983093	+	0.183105i	$0.0586147\pi$
$198$	0	0
$199$	−1.29180	−0.0915730	−0.0457865	−	0.998951i	$-0.514579\pi$
$199$	−1.29180	−0.0915730	−0.0457865	+	0.998951i	$0.514579\pi$
$200$	0	0
$201$	−15.0902	−1.06438
$202$	−18.7082	−1.31630
$203$	−1.18034	−0.0828436
$204$	0.618034	0.0432710
$205$	0	0
$206$	−16.8885	−1.17668
$207$	1.47214	0.102321
$208$	−0.236068	−0.0163684
$209$	0	0
$210$	0	0
$211$	14.5066	0.998674	0.499337	−	0.866408i	$-0.333577\pi$
$211$	14.5066	0.998674	0.499337	+	0.866408i	$0.333577\pi$
$212$	1.38197	0.0949138
$213$	15.8541	1.08631
$214$	8.18034	0.559197
$215$	0	0
$216$	5.47214	0.372332
$217$	−22.8885	−1.55378
$218$	11.6180	0.786873
$219$	13.1803	0.890645
$220$	0	0
$221$	−0.0901699	−0.00606549
$222$	14.3262	0.961514
$223$	2.05573	0.137662	0.0688309	−	0.997628i	$-0.478073\pi$
$223$	2.05573	0.137662	0.0688309	+	0.997628i	$0.478073\pi$
$224$	−2.23607	−0.149404
$225$	0	0
$226$	−14.7082	−0.978375
$227$	19.7426	1.31037	0.655183	−	0.755470i	$-0.272592\pi$
$227$	19.7426	1.31037	0.655183	+	0.755470i	$0.272592\pi$
$228$	4.23607	0.280540
$229$	−15.4721	−1.02243	−0.511214	−	0.859454i	$-0.670804\pi$
$229$	−15.4721	−1.02243	−0.511214	+	0.859454i	$0.670804\pi$
$230$	0	0
$231$	0	0
$232$	0.527864	0.0346560
$233$	20.3262	1.33162	0.665808	−	0.746123i	$-0.268087\pi$
$233$	20.3262	1.33162	0.665808	+	0.746123i	$0.268087\pi$
$234$	−0.0901699	−0.00589459
$235$	0	0
$236$	−11.4721	−0.746772
$237$	−19.3262	−1.25537
$238$	−0.854102	−0.0553632
$239$	−17.4721	−1.13018	−0.565089	−	0.825030i	$-0.691158\pi$
$239$	−17.4721	−1.13018	−0.565089	+	0.825030i	$0.691158\pi$
$240$	0	0
$241$	18.0000	1.15948	0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000	1.15948	0.579741	+	0.814801i	$0.303154\pi$
$242$	0	0
$243$	3.94427	0.253025
$244$	2.23607	0.143150
$245$	0	0
$246$	6.00000	0.382546
$247$	−0.618034	−0.0393246
$248$	10.2361	0.649991
$249$	14.5623	0.922849
$250$	0	0
$251$	4.79837	0.302871	0.151435	−	0.988467i	$-0.451610\pi$
$251$	4.79837	0.302871	0.151435	+	0.988467i	$0.451610\pi$
$252$	−0.854102	−0.0538034
$253$	0	0
$254$	2.90983	0.182579
$255$	0	0
$256$	1.00000	0.0625000
$257$	13.8541	0.864195	0.432098	−	0.901827i	$-0.357774\pi$
$257$	13.8541	0.864195	0.432098	+	0.901827i	$0.357774\pi$
$258$	15.3262	0.954170
$259$	−19.7984	−1.23021
$260$	0	0
$261$	0.201626	0.0124803
$262$	−11.1459	−0.688596
$263$	−16.0902	−0.992162	−0.496081	−	0.868276i	$-0.665228\pi$
$263$	−16.0902	−0.992162	−0.496081	+	0.868276i	$0.665228\pi$
$264$	0	0
$265$	0	0
$266$	−5.85410	−0.358938
$267$	−21.3262	−1.30514
$268$	−9.32624	−0.569691
$269$	9.90983	0.604213	0.302107	−	0.953274i	$-0.402310\pi$
$269$	9.90983	0.604213	0.302107	+	0.953274i	$0.402310\pi$
$270$	0	0
$271$	27.9787	1.69959	0.849793	−	0.527117i	$-0.176727\pi$
$271$	27.9787	1.69959	0.849793	+	0.527117i	$0.176727\pi$
$272$	0.381966	0.0231601
$273$	−0.854102	−0.0516926
$274$	15.7639	0.952334
$275$	0	0
$276$	−6.23607	−0.375367
$277$	−19.2361	−1.15578	−0.577892	−	0.816113i	$-0.696124\pi$
$277$	−19.2361	−1.15578	−0.577892	+	0.816113i	$0.696124\pi$
$278$	2.85410	0.171178
$279$	3.90983	0.234075
$280$	0	0
$281$	−23.1246	−1.37950	−0.689749	−	0.724048i	$-0.742279\pi$
$281$	−23.1246	−1.37950	−0.689749	+	0.724048i	$0.742279\pi$
$282$	−0.381966	−0.0227457
$283$	−22.0000	−1.30776	−0.653882	−	0.756596i	$-0.726861\pi$
$283$	−22.0000	−1.30776	−0.653882	+	0.756596i	$0.726861\pi$
$284$	9.79837	0.581427
$285$	0	0
$286$	0	0
$287$	−8.29180	−0.489449
$288$	0.381966	0.0225076
$289$	−16.8541	−0.991418
$290$	0	0
$291$	19.4164	1.13821
$292$	8.14590	0.476703
$293$	−10.5066	−0.613801	−0.306900	−	0.951742i	$-0.599292\pi$
$293$	−10.5066	−0.613801	−0.306900	+	0.951742i	$0.599292\pi$
$294$	3.23607	0.188731
$295$	0	0
$296$	8.85410	0.514634
$297$	0	0
$298$	10.7082	0.620310
$299$	0.909830	0.0526168
$300$	0	0
$301$	−21.1803	−1.22081
$302$	19.0344	1.09531
$303$	30.2705	1.73900
$304$	2.61803	0.150155
$305$	0	0
$306$	0.145898	0.00834044
$307$	28.7984	1.64361	0.821805	−	0.569769i	$-0.192967\pi$
$307$	28.7984	1.64361	0.821805	+	0.569769i	$0.192967\pi$
$308$	0	0
$309$	27.3262	1.55454
$310$	0	0
$311$	4.23607	0.240205	0.120103	−	0.992761i	$-0.461678\pi$
$311$	4.23607	0.240205	0.120103	+	0.992761i	$0.461678\pi$
$312$	0.381966	0.0216246
$313$	1.00000	0.0565233	0.0282617	−	0.999601i	$-0.491003\pi$
$313$	1.00000	0.0565233	0.0282617	+	0.999601i	$0.491003\pi$
$314$	6.76393	0.381711
$315$	0	0
$316$	−11.9443	−0.671918
$317$	2.79837	0.157172	0.0785862	−	0.996907i	$-0.474959\pi$
$317$	2.79837	0.157172	0.0785862	+	0.996907i	$0.474959\pi$
$318$	−2.23607	−0.125392
$319$	0	0
$320$	0	0
$321$	−13.2361	−0.738765
$322$	8.61803	0.480264
$323$	1.00000	0.0556415
$324$	−7.70820	−0.428234
$325$	0	0
$326$	0.180340	0.00998810
$327$	−18.7984	−1.03955
$328$	3.70820	0.204751
$329$	0.527864	0.0291021
$330$	0	0
$331$	−16.9443	−0.931341	−0.465671	−	0.884958i	$-0.654187\pi$
$331$	−16.9443	−0.931341	−0.465671	+	0.884958i	$0.654187\pi$
$332$	9.00000	0.493939
$333$	3.38197	0.185331
$334$	−2.23607	−0.122352
$335$	0	0
$336$	3.61803	0.197380
$337$	−30.1246	−1.64099	−0.820496	−	0.571652i	$-0.806303\pi$
$337$	−30.1246	−1.64099	−0.820496	+	0.571652i	$0.806303\pi$
$338$	12.9443	0.704076
$339$	23.7984	1.29255
$340$	0	0
$341$	0	0
$342$	1.00000	0.0540738
$343$	−20.1246	−1.08663
$344$	9.47214	0.510703
$345$	0	0
$346$	−13.3262	−0.716423
$347$	−18.8541	−1.01214	−0.506071	−	0.862492i	$-0.668902\pi$
$347$	−18.8541	−1.01214	−0.506071	+	0.862492i	$0.668902\pi$
$348$	−0.854102	−0.0457847
$349$	−23.5967	−1.26310	−0.631552	−	0.775333i	$-0.717582\pi$
$349$	−23.5967	−1.26310	−0.631552	+	0.775333i	$0.717582\pi$
$350$	0	0
$351$	1.29180	0.0689510
$352$	0	0
$353$	4.43769	0.236195	0.118097	−	0.993002i	$-0.462321\pi$
$353$	4.43769	0.236195	0.118097	+	0.993002i	$0.462321\pi$
$354$	18.5623	0.986575
$355$	0	0
$356$	−13.1803	−0.698557
$357$	1.38197	0.0731414
$358$	−14.6525	−0.774407
$359$	0.944272	0.0498368	0.0249184	−	0.999689i	$-0.492067\pi$
$359$	0.944272	0.0498368	0.0249184	+	0.999689i	$0.492067\pi$
$360$	0	0
$361$	−12.1459	−0.639258
$362$	7.05573	0.370841
$363$	0	0
$364$	−0.527864	−0.0276676
$365$	0	0
$366$	−3.61803	−0.189118
$367$	−5.34752	−0.279138	−0.139569	−	0.990212i	$-0.544572\pi$
$367$	−5.34752	−0.279138	−0.139569	+	0.990212i	$0.544572\pi$
$368$	−3.85410	−0.200909
$369$	1.41641	0.0737352
$370$	0	0
$371$	3.09017	0.160434
$372$	−16.5623	−0.858716
$373$	−2.38197	−0.123334	−0.0616668	−	0.998097i	$-0.519642\pi$
$373$	−2.38197	−0.123334	−0.0616668	+	0.998097i	$0.519642\pi$
$374$	0	0
$375$	0	0
$376$	−0.236068	−0.0121743
$377$	0.124612	0.00641783
$378$	12.2361	0.629355
$379$	−5.90983	−0.303568	−0.151784	−	0.988414i	$-0.548502\pi$
$379$	−5.90983	−0.303568	−0.151784	+	0.988414i	$0.548502\pi$
$380$	0	0
$381$	−4.70820	−0.241209
$382$	19.8541	1.01582
$383$	2.00000	0.102195	0.0510976	−	0.998694i	$-0.483728\pi$
$383$	2.00000	0.102195	0.0510976	+	0.998694i	$0.483728\pi$
$384$	−1.61803	−0.0825700
$385$	0	0
$386$	2.43769	0.124075
$387$	3.61803	0.183915
$388$	12.0000	0.609208
$389$	1.76393	0.0894349	0.0447175	−	0.999000i	$-0.485761\pi$
$389$	1.76393	0.0894349	0.0447175	+	0.999000i	$0.485761\pi$
$390$	0	0
$391$	−1.47214	−0.0744491
$392$	2.00000	0.101015
$393$	18.0344	0.909717
$394$	−27.5967	−1.39030
$395$	0	0
$396$	0	0
$397$	17.4721	0.876901	0.438451	−	0.898755i	$-0.355527\pi$
$397$	17.4721	0.876901	0.438451	+	0.898755i	$0.355527\pi$
$398$	1.29180	0.0647519
$399$	9.47214	0.474200
$400$	0	0
$401$	−1.90983	−0.0953724	−0.0476862	−	0.998862i	$-0.515185\pi$
$401$	−1.90983	−0.0953724	−0.0476862	+	0.998862i	$0.515185\pi$
$402$	15.0902	0.752629
$403$	2.41641	0.120370
$404$	18.7082	0.930768
$405$	0	0
$406$	1.18034	0.0585793
$407$	0	0
$408$	−0.618034	−0.0305972
$409$	18.5066	0.915091	0.457546	−	0.889186i	$-0.348729\pi$
$409$	18.5066	0.915091	0.457546	+	0.889186i	$0.348729\pi$
$410$	0	0
$411$	−25.5066	−1.25815
$412$	16.8885	0.832039
$413$	−25.6525	−1.26228
$414$	−1.47214	−0.0723515
$415$	0	0
$416$	0.236068	0.0115742
$417$	−4.61803	−0.226146
$418$	0	0
$419$	−31.1803	−1.52326	−0.761630	−	0.648013i	$-0.775600\pi$
$419$	−31.1803	−1.52326	−0.761630	+	0.648013i	$0.775600\pi$
$420$	0	0
$421$	−17.6738	−0.861366	−0.430683	−	0.902503i	$-0.641727\pi$
$421$	−17.6738	−0.861366	−0.430683	+	0.902503i	$0.641727\pi$
$422$	−14.5066	−0.706169
$423$	−0.0901699	−0.00438421
$424$	−1.38197	−0.0671142
$425$	0	0
$426$	−15.8541	−0.768134
$427$	5.00000	0.241967
$428$	−8.18034	−0.395412
$429$	0	0
$430$	0	0
$431$	5.94427	0.286326	0.143163	−	0.989699i	$-0.454273\pi$
$431$	5.94427	0.286326	0.143163	+	0.989699i	$0.454273\pi$
$432$	−5.47214	−0.263278
$433$	8.47214	0.407145	0.203572	−	0.979060i	$-0.434745\pi$
$433$	8.47214	0.407145	0.203572	+	0.979060i	$0.434745\pi$
$434$	22.8885	1.09869
$435$	0	0
$436$	−11.6180	−0.556403
$437$	−10.0902	−0.482678
$438$	−13.1803	−0.629781
$439$	22.7082	1.08380	0.541902	−	0.840442i	$-0.317704\pi$
$439$	22.7082	1.08380	0.541902	+	0.840442i	$0.317704\pi$
$440$	0	0
$441$	0.763932	0.0363777
$442$	0.0901699	0.00428895
$443$	30.5066	1.44941	0.724706	−	0.689059i	$-0.241976\pi$
$443$	30.5066	1.44941	0.724706	+	0.689059i	$0.241976\pi$
$444$	−14.3262	−0.679893
$445$	0	0
$446$	−2.05573	−0.0973415
$447$	−17.3262	−0.819503
$448$	2.23607	0.105644
$449$	16.2918	0.768857	0.384429	−	0.923155i	$-0.374398\pi$
$449$	16.2918	0.768857	0.384429	+	0.923155i	$0.374398\pi$
$450$	0	0
$451$	0	0
$452$	14.7082	0.691816
$453$	−30.7984	−1.44703
$454$	−19.7426	−0.926568
$455$	0	0
$456$	−4.23607	−0.198372
$457$	12.1246	0.567165	0.283583	−	0.958948i	$-0.408477\pi$
$457$	12.1246	0.567165	0.283583	+	0.958948i	$0.408477\pi$
$458$	15.4721	0.722965
$459$	−2.09017	−0.0975608
$460$	0	0
$461$	−36.4164	−1.69608	−0.848041	−	0.529931i	$-0.822218\pi$
$461$	−36.4164	−1.69608	−0.848041	+	0.529931i	$0.822218\pi$
$462$	0	0
$463$	38.5967	1.79374	0.896871	−	0.442291i	$-0.145834\pi$
$463$	38.5967	1.79374	0.896871	+	0.442291i	$0.145834\pi$
$464$	−0.527864	−0.0245055
$465$	0	0
$466$	−20.3262	−0.941595
$467$	32.1803	1.48913	0.744564	−	0.667551i	$-0.232657\pi$
$467$	32.1803	1.48913	0.744564	+	0.667551i	$0.232657\pi$
$468$	0.0901699	0.00416811
$469$	−20.8541	−0.962953
$470$	0	0
$471$	−10.9443	−0.504285
$472$	11.4721	0.528048
$473$	0	0
$474$	19.3262	0.887684
$475$	0	0
$476$	0.854102	0.0391477
$477$	−0.527864	−0.0241692
$478$	17.4721	0.799157
$479$	9.94427	0.454365	0.227183	−	0.973852i	$-0.427049\pi$
$479$	9.94427	0.454365	0.227183	+	0.973852i	$0.427049\pi$
$480$	0	0
$481$	2.09017	0.0953035
$482$	−18.0000	−0.819878
$483$	−13.9443	−0.634486
$484$	0	0
$485$	0	0
$486$	−3.94427	−0.178916
$487$	10.8885	0.493407	0.246704	−	0.969091i	$-0.420653\pi$
$487$	10.8885	0.493407	0.246704	+	0.969091i	$0.420653\pi$
$488$	−2.23607	−0.101222
$489$	−0.291796	−0.0131955
$490$	0	0
$491$	12.1803	0.549691	0.274846	−	0.961488i	$-0.411373\pi$
$491$	12.1803	0.549691	0.274846	+	0.961488i	$0.411373\pi$
$492$	−6.00000	−0.270501
$493$	−0.201626	−0.00908078
$494$	0.618034	0.0278067
$495$	0	0
$496$	−10.2361	−0.459613
$497$	21.9098	0.982790
$498$	−14.5623	−0.652553
$499$	15.4508	0.691675	0.345838	−	0.938294i	$-0.387595\pi$
$499$	15.4508	0.691675	0.345838	+	0.938294i	$0.387595\pi$
$500$	0	0
$501$	3.61803	0.161642
$502$	−4.79837	−0.214162
$503$	−31.5066	−1.40481	−0.702404	−	0.711778i	$-0.747890\pi$
$503$	−31.5066	−1.40481	−0.702404	+	0.711778i	$0.747890\pi$
$504$	0.854102	0.0380447
$505$	0	0
$506$	0	0
$507$	−20.9443	−0.930168
$508$	−2.90983	−0.129103
$509$	10.2705	0.455232	0.227616	−	0.973751i	$-0.426907\pi$
$509$	10.2705	0.455232	0.227616	+	0.973751i	$0.426907\pi$
$510$	0	0
$511$	18.2148	0.805775
$512$	−1.00000	−0.0441942
$513$	−14.3262	−0.632519
$514$	−13.8541	−0.611078
$515$	0	0
$516$	−15.3262	−0.674700
$517$	0	0
$518$	19.7984	0.869891
$519$	21.5623	0.946480
$520$	0	0
$521$	−10.0344	−0.439617	−0.219808	−	0.975543i	$-0.570543\pi$
$521$	−10.0344	−0.439617	−0.219808	+	0.975543i	$0.570543\pi$
$522$	−0.201626	−0.00882494
$523$	−27.5066	−1.20278	−0.601389	−	0.798956i	$-0.705386\pi$
$523$	−27.5066	−1.20278	−0.601389	+	0.798956i	$0.705386\pi$
$524$	11.1459	0.486911
$525$	0	0
$526$	16.0902	0.701565
$527$	−3.90983	−0.170315
$528$	0	0
$529$	−8.14590	−0.354169
$530$	0	0
$531$	4.38197	0.190161
$532$	5.85410	0.253808
$533$	0.875388	0.0379173
$534$	21.3262	0.922877
$535$	0	0
$536$	9.32624	0.402832
$537$	23.7082	1.02308
$538$	−9.90983	−0.427243
$539$	0	0
$540$	0	0
$541$	32.1459	1.38206	0.691030	−	0.722826i	$-0.257157\pi$
$541$	32.1459	1.38206	0.691030	+	0.722826i	$0.257157\pi$
$542$	−27.9787	−1.20179
$543$	−11.4164	−0.489925
$544$	−0.381966	−0.0163767
$545$	0	0
$546$	0.854102	0.0365522
$547$	−17.9443	−0.767242	−0.383621	−	0.923491i	$-0.625323\pi$
$547$	−17.9443	−0.767242	−0.383621	+	0.923491i	$0.625323\pi$
$548$	−15.7639	−0.673402
$549$	−0.854102	−0.0364522
$550$	0	0
$551$	−1.38197	−0.0588737
$552$	6.23607	0.265425
$553$	−26.7082	−1.13575
$554$	19.2361	0.817262
$555$	0	0
$556$	−2.85410	−0.121041
$557$	5.88854	0.249506	0.124753	−	0.992188i	$-0.460186\pi$
$557$	5.88854	0.249506	0.124753	+	0.992188i	$0.460186\pi$
$558$	−3.90983	−0.165516
$559$	2.23607	0.0945756
$560$	0	0
$561$	0	0
$562$	23.1246	0.975453
$563$	−17.6180	−0.742512	−0.371256	−	0.928531i	$-0.621073\pi$
$563$	−17.6180	−0.742512	−0.371256	+	0.928531i	$0.621073\pi$
$564$	0.381966	0.0160837
$565$	0	0
$566$	22.0000	0.924729
$567$	−17.2361	−0.723847
$568$	−9.79837	−0.411131
$569$	−2.09017	−0.0876245	−0.0438122	−	0.999040i	$-0.513950\pi$
$569$	−2.09017	−0.0876245	−0.0438122	+	0.999040i	$0.513950\pi$
$570$	0	0
$571$	−8.52786	−0.356880	−0.178440	−	0.983951i	$-0.557105\pi$
$571$	−8.52786	−0.356880	−0.178440	+	0.983951i	$0.557105\pi$
$572$	0	0
$573$	−32.1246	−1.34202
$574$	8.29180	0.346093
$575$	0	0
$576$	−0.381966	−0.0159153
$577$	21.7639	0.906044	0.453022	−	0.891499i	$-0.350346\pi$
$577$	21.7639	0.906044	0.453022	+	0.891499i	$0.350346\pi$
$578$	16.8541	0.701038
$579$	−3.94427	−0.163918
$580$	0	0
$581$	20.1246	0.834910
$582$	−19.4164	−0.804836
$583$	0	0
$584$	−8.14590	−0.337080
$585$	0	0
$586$	10.5066	0.434023
$587$	−16.2148	−0.669256	−0.334628	−	0.942350i	$-0.608611\pi$
$587$	−16.2148	−0.669256	−0.334628	+	0.942350i	$0.608611\pi$
$588$	−3.23607	−0.133453
$589$	−26.7984	−1.10421
$590$	0	0
$591$	44.6525	1.83676
$592$	−8.85410	−0.363901
$593$	42.3951	1.74096	0.870479	−	0.492205i	$-0.163809\pi$
$593$	42.3951	1.74096	0.870479	+	0.492205i	$0.163809\pi$
$594$	0	0
$595$	0	0
$596$	−10.7082	−0.438625
$597$	−2.09017	−0.0855450
$598$	−0.909830	−0.0372057
$599$	3.05573	0.124854	0.0624268	−	0.998050i	$-0.480116\pi$
$599$	3.05573	0.124854	0.0624268	+	0.998050i	$0.480116\pi$
$600$	0	0
$601$	−6.56231	−0.267682	−0.133841	−	0.991003i	$-0.542731\pi$
$601$	−6.56231	−0.267682	−0.133841	+	0.991003i	$0.542731\pi$
$602$	21.1803	0.863246
$603$	3.56231	0.145068
$604$	−19.0344	−0.774500
$605$	0	0
$606$	−30.2705	−1.22966
$607$	−13.3820	−0.543157	−0.271579	−	0.962416i	$-0.587546\pi$
$607$	−13.3820	−0.543157	−0.271579	+	0.962416i	$0.587546\pi$
$608$	−2.61803	−0.106175
$609$	−1.90983	−0.0773902
$610$	0	0
$611$	−0.0557281	−0.00225452
$612$	−0.145898	−0.00589758
$613$	7.18034	0.290011	0.145006	−	0.989431i	$-0.453680\pi$
$613$	7.18034	0.290011	0.145006	+	0.989431i	$0.453680\pi$
$614$	−28.7984	−1.16221
$615$	0	0
$616$	0	0
$617$	−25.7984	−1.03860	−0.519302	−	0.854591i	$-0.673808\pi$
$617$	−25.7984	−1.03860	−0.519302	+	0.854591i	$0.673808\pi$
$618$	−27.3262	−1.09922
$619$	−26.2361	−1.05452	−0.527258	−	0.849705i	$-0.676780\pi$
$619$	−26.2361	−1.05452	−0.527258	+	0.849705i	$0.676780\pi$
$620$	0	0
$621$	21.0902	0.846319
$622$	−4.23607	−0.169851
$623$	−29.4721	−1.18078
$624$	−0.381966	−0.0152909
$625$	0	0
$626$	−1.00000	−0.0399680
$627$	0	0
$628$	−6.76393	−0.269910
$629$	−3.38197	−0.134848
$630$	0	0
$631$	−6.70820	−0.267049	−0.133525	−	0.991045i	$-0.542630\pi$
$631$	−6.70820	−0.267049	−0.133525	+	0.991045i	$0.542630\pi$
$632$	11.9443	0.475118
$633$	23.4721	0.932934
$634$	−2.79837	−0.111138
$635$	0	0
$636$	2.23607	0.0886659
$637$	0.472136	0.0187067
$638$	0	0
$639$	−3.74265	−0.148057
$640$	0	0
$641$	−42.0689	−1.66162	−0.830811	−	0.556555i	$-0.812123\pi$
$641$	−42.0689	−1.66162	−0.830811	+	0.556555i	$0.812123\pi$
$642$	13.2361	0.522386
$643$	14.1246	0.557020	0.278510	−	0.960433i	$-0.410159\pi$
$643$	14.1246	0.557020	0.278510	+	0.960433i	$0.410159\pi$
$644$	−8.61803	−0.339598
$645$	0	0
$646$	−1.00000	−0.0393445
$647$	1.05573	0.0415050	0.0207525	−	0.999785i	$-0.493394\pi$
$647$	1.05573	0.0415050	0.0207525	+	0.999785i	$0.493394\pi$
$648$	7.70820	0.302807
$649$	0	0
$650$	0	0
$651$	−37.0344	−1.45149
$652$	−0.180340	−0.00706266
$653$	10.4721	0.409806	0.204903	−	0.978782i	$-0.434312\pi$
$653$	10.4721	0.409806	0.204903	+	0.978782i	$0.434312\pi$
$654$	18.7984	0.735075
$655$	0	0
$656$	−3.70820	−0.144781
$657$	−3.11146	−0.121389
$658$	−0.527864	−0.0205783
$659$	43.4508	1.69260	0.846302	−	0.532703i	$-0.178824\pi$
$659$	43.4508	1.69260	0.846302	+	0.532703i	$0.178824\pi$
$660$	0	0
$661$	27.5967	1.07339	0.536695	−	0.843777i	$-0.319673\pi$
$661$	27.5967	1.07339	0.536695	+	0.843777i	$0.319673\pi$
$662$	16.9443	0.658558
$663$	−0.145898	−0.00566621
$664$	−9.00000	−0.349268
$665$	0	0
$666$	−3.38197	−0.131049
$667$	2.03444	0.0787739
$668$	2.23607	0.0865161
$669$	3.32624	0.128600
$670$	0	0
$671$	0	0
$672$	−3.61803	−0.139569
$673$	11.5836	0.446515	0.223257	−	0.974760i	$-0.428331\pi$
$673$	11.5836	0.446515	0.223257	+	0.974760i	$0.428331\pi$
$674$	30.1246	1.16036
$675$	0	0
$676$	−12.9443	−0.497857
$677$	−10.8328	−0.416339	−0.208169	−	0.978093i	$-0.566751\pi$
$677$	−10.8328	−0.416339	−0.208169	+	0.978093i	$0.566751\pi$
$678$	−23.7984	−0.913971
$679$	26.8328	1.02975
$680$	0	0
$681$	31.9443	1.22411
$682$	0	0
$683$	−6.11146	−0.233848	−0.116924	−	0.993141i	$-0.537303\pi$
$683$	−6.11146	−0.233848	−0.116924	+	0.993141i	$0.537303\pi$
$684$	−1.00000	−0.0382360
$685$	0	0
$686$	20.1246	0.768361
$687$	−25.0344	−0.955124
$688$	−9.47214	−0.361122
$689$	−0.326238	−0.0124287
$690$	0	0
$691$	−39.9230	−1.51874	−0.759371	−	0.650658i	$-0.774493\pi$
$691$	−39.9230	−1.51874	−0.759371	+	0.650658i	$0.774493\pi$
$692$	13.3262	0.506588
$693$	0	0
$694$	18.8541	0.715692
$695$	0	0
$696$	0.854102	0.0323747
$697$	−1.41641	−0.0536503
$698$	23.5967	0.893150
$699$	32.8885	1.24396
$700$	0	0
$701$	−12.6180	−0.476577	−0.238288	−	0.971194i	$-0.576586\pi$
$701$	−12.6180	−0.476577	−0.238288	+	0.971194i	$0.576586\pi$
$702$	−1.29180	−0.0487557
$703$	−23.1803	−0.874263
$704$	0	0
$705$	0	0
$706$	−4.43769	−0.167015
$707$	41.8328	1.57328
$708$	−18.5623	−0.697614
$709$	−46.1246	−1.73225	−0.866123	−	0.499831i	$-0.833396\pi$
$709$	−46.1246	−1.73225	−0.866123	+	0.499831i	$0.833396\pi$
$710$	0	0
$711$	4.56231	0.171100
$712$	13.1803	0.493954
$713$	39.4508	1.47745
$714$	−1.38197	−0.0517188
$715$	0	0
$716$	14.6525	0.547589
$717$	−28.2705	−1.05578
$718$	−0.944272	−0.0352399
$719$	19.9443	0.743796	0.371898	−	0.928274i	$-0.378707\pi$
$719$	19.9443	0.743796	0.371898	+	0.928274i	$0.378707\pi$
$720$	0	0
$721$	37.7639	1.40640
$722$	12.1459	0.452024
$723$	29.1246	1.08316
$724$	−7.05573	−0.262224
$725$	0	0
$726$	0	0
$727$	13.8197	0.512543	0.256271	−	0.966605i	$-0.417506\pi$
$727$	13.8197	0.512543	0.256271	+	0.966605i	$0.417506\pi$
$728$	0.527864	0.0195639
$729$	29.5066	1.09284
$730$	0	0
$731$	−3.61803	−0.133818
$732$	3.61803	0.133726
$733$	21.8328	0.806413	0.403207	−	0.915109i	$-0.367896\pi$
$733$	21.8328	0.806413	0.403207	+	0.915109i	$0.367896\pi$
$734$	5.34752	0.197381
$735$	0	0
$736$	3.85410	0.142064
$737$	0	0
$738$	−1.41641	−0.0521387
$739$	9.70820	0.357122	0.178561	−	0.983929i	$-0.442856\pi$
$739$	9.70820	0.357122	0.178561	+	0.983929i	$0.442856\pi$
$740$	0	0
$741$	−1.00000	−0.0367359
$742$	−3.09017	−0.113444
$743$	−42.5279	−1.56020	−0.780098	−	0.625657i	$-0.784831\pi$
$743$	−42.5279	−1.56020	−0.780098	+	0.625657i	$0.784831\pi$
$744$	16.5623	0.607204
$745$	0	0
$746$	2.38197	0.0872100
$747$	−3.43769	−0.125779
$748$	0	0
$749$	−18.2918	−0.668368
$750$	0	0
$751$	26.9098	0.981954	0.490977	−	0.871173i	$-0.336640\pi$
$751$	26.9098	0.981954	0.490977	+	0.871173i	$0.336640\pi$
$752$	0.236068	0.00860851
$753$	7.76393	0.282933
$754$	−0.124612	−0.00453809
$755$	0	0
$756$	−12.2361	−0.445021
$757$	15.2148	0.552991	0.276495	−	0.961015i	$-0.410827\pi$
$757$	15.2148	0.552991	0.276495	+	0.961015i	$0.410827\pi$
$758$	5.90983	0.214655
$759$	0	0
$760$	0	0
$761$	−21.2918	−0.771827	−0.385914	−	0.922535i	$-0.626114\pi$
$761$	−21.2918	−0.771827	−0.385914	+	0.922535i	$0.626114\pi$
$762$	4.70820	0.170560
$763$	−25.9787	−0.940493
$764$	−19.8541	−0.718296
$765$	0	0
$766$	−2.00000	−0.0722629
$767$	2.70820	0.0977876
$768$	1.61803	0.0583858
$769$	−42.2361	−1.52307	−0.761536	−	0.648123i	$-0.775554\pi$
$769$	−42.2361	−1.52307	−0.761536	+	0.648123i	$0.775554\pi$
$770$	0	0
$771$	22.4164	0.807307
$772$	−2.43769	−0.0877345
$773$	−41.4164	−1.48964	−0.744822	−	0.667263i	$-0.767466\pi$
$773$	−41.4164	−1.48964	−0.744822	+	0.667263i	$0.767466\pi$
$774$	−3.61803	−0.130048
$775$	0	0
$776$	−12.0000	−0.430775
$777$	−32.0344	−1.14923
$778$	−1.76393	−0.0632400
$779$	−9.70820	−0.347833
$780$	0	0
$781$	0	0
$782$	1.47214	0.0526435
$783$	2.88854	0.103228
$784$	−2.00000	−0.0714286
$785$	0	0
$786$	−18.0344	−0.643267
$787$	−18.0000	−0.641631	−0.320815	−	0.947142i	$-0.603957\pi$
$787$	−18.0000	−0.641631	−0.320815	+	0.947142i	$0.603957\pi$
$788$	27.5967	0.983093
$789$	−26.0344	−0.926851
$790$	0	0
$791$	32.8885	1.16938
$792$	0	0
$793$	−0.527864	−0.0187450
$794$	−17.4721	−0.620063
$795$	0	0
$796$	−1.29180	−0.0457865
$797$	−50.0132	−1.77156	−0.885778	−	0.464108i	$-0.846375\pi$
$797$	−50.0132	−1.77156	−0.885778	+	0.464108i	$0.846375\pi$
$798$	−9.47214	−0.335310
$799$	0.0901699	0.00318998
$800$	0	0
$801$	5.03444	0.177883
$802$	1.90983	0.0674384
$803$	0	0
$804$	−15.0902	−0.532189
$805$	0	0
$806$	−2.41641	−0.0851143
$807$	16.0344	0.564439
$808$	−18.7082	−0.658152
$809$	33.0689	1.16264	0.581320	−	0.813675i	$-0.302536\pi$
$809$	33.0689	1.16264	0.581320	+	0.813675i	$0.302536\pi$
$810$	0	0
$811$	13.6525	0.479403	0.239702	−	0.970847i	$-0.422950\pi$
$811$	13.6525	0.479403	0.239702	+	0.970847i	$0.422950\pi$
$812$	−1.18034	−0.0414218
$813$	45.2705	1.58771
$814$	0	0
$815$	0	0
$816$	0.618034	0.0216355
$817$	−24.7984	−0.867585
$818$	−18.5066	−0.647067
$819$	0.201626	0.00704539
$820$	0	0
$821$	−39.3607	−1.37370	−0.686849	−	0.726801i	$-0.741006\pi$
$821$	−39.3607	−1.37370	−0.686849	+	0.726801i	$0.741006\pi$
$822$	25.5066	0.889644
$823$	−2.21478	−0.0772024	−0.0386012	−	0.999255i	$-0.512290\pi$
$823$	−2.21478	−0.0772024	−0.0386012	+	0.999255i	$0.512290\pi$
$824$	−16.8885	−0.588340
$825$	0	0
$826$	25.6525	0.892564
$827$	−22.7082	−0.789642	−0.394821	−	0.918758i	$-0.629193\pi$
$827$	−22.7082	−0.789642	−0.394821	+	0.918758i	$0.629193\pi$
$828$	1.47214	0.0511603
$829$	37.0132	1.28552	0.642760	−	0.766068i	$-0.277789\pi$
$829$	37.0132	1.28552	0.642760	+	0.766068i	$0.277789\pi$
$830$	0	0
$831$	−31.1246	−1.07970
$832$	−0.236068	−0.00818418
$833$	−0.763932	−0.0264687
$834$	4.61803	0.159909
$835$	0	0
$836$	0	0
$837$	56.0132	1.93610
$838$	31.1803	1.07711
$839$	−29.2492	−1.00980	−0.504898	−	0.863179i	$-0.668470\pi$
$839$	−29.2492	−1.00980	−0.504898	+	0.863179i	$0.668470\pi$
$840$	0	0
$841$	−28.7214	−0.990392
$842$	17.6738	0.609078
$843$	−37.4164	−1.28869
$844$	14.5066	0.499337
$845$	0	0
$846$	0.0901699	0.00310011
$847$	0	0
$848$	1.38197	0.0474569
$849$	−35.5967	−1.22168
$850$	0	0
$851$	34.1246	1.16978
$852$	15.8541	0.543153
$853$	3.34752	0.114617	0.0573085	−	0.998357i	$-0.481748\pi$
$853$	3.34752	0.114617	0.0573085	+	0.998357i	$0.481748\pi$
$854$	−5.00000	−0.171096
$855$	0	0
$856$	8.18034	0.279598
$857$	−25.2705	−0.863224	−0.431612	−	0.902059i	$-0.642055\pi$
$857$	−25.2705	−0.863224	−0.431612	+	0.902059i	$0.642055\pi$
$858$	0	0
$859$	−1.18034	−0.0402727	−0.0201363	−	0.999797i	$-0.506410\pi$
$859$	−1.18034	−0.0402727	−0.0201363	+	0.999797i	$0.506410\pi$
$860$	0	0
$861$	−13.4164	−0.457230
$862$	−5.94427	−0.202463
$863$	−1.11146	−0.0378344	−0.0189172	−	0.999821i	$-0.506022\pi$
$863$	−1.11146	−0.0378344	−0.0189172	+	0.999821i	$0.506022\pi$
$864$	5.47214	0.186166
$865$	0	0
$866$	−8.47214	−0.287895
$867$	−27.2705	−0.926155
$868$	−22.8885	−0.776888
$869$	0	0
$870$	0	0
$871$	2.20163	0.0745993
$872$	11.6180	0.393436
$873$	−4.58359	−0.155131
$874$	10.0902	0.341305
$875$	0	0
$876$	13.1803	0.445322
$877$	−37.7771	−1.27564	−0.637821	−	0.770185i	$-0.720164\pi$
$877$	−37.7771	−1.27564	−0.637821	+	0.770185i	$0.720164\pi$
$878$	−22.7082	−0.766365
$879$	−17.0000	−0.573396
$880$	0	0
$881$	7.67376	0.258536	0.129268	−	0.991610i	$-0.458737\pi$
$881$	7.67376	0.258536	0.129268	+	0.991610i	$0.458737\pi$
$882$	−0.763932	−0.0257229
$883$	−38.9443	−1.31058	−0.655290	−	0.755378i	$-0.727453\pi$
$883$	−38.9443	−1.31058	−0.655290	+	0.755378i	$0.727453\pi$
$884$	−0.0901699	−0.00303274
$885$	0	0
$886$	−30.5066	−1.02489
$887$	40.2705	1.35215	0.676076	−	0.736832i	$-0.263679\pi$
$887$	40.2705	1.35215	0.676076	+	0.736832i	$0.263679\pi$
$888$	14.3262	0.480757
$889$	−6.50658	−0.218224
$890$	0	0
$891$	0	0
$892$	2.05573	0.0688309
$893$	0.618034	0.0206817
$894$	17.3262	0.579476
$895$	0	0
$896$	−2.23607	−0.0747018
$897$	1.47214	0.0491532
$898$	−16.2918	−0.543664
$899$	5.40325	0.180209
$900$	0	0
$901$	0.527864	0.0175857
$902$	0	0
$903$	−34.2705	−1.14045
$904$	−14.7082	−0.489188
$905$	0	0
$906$	30.7984	1.02321
$907$	−39.5967	−1.31479	−0.657394	−	0.753547i	$-0.728341\pi$
$907$	−39.5967	−1.31479	−0.657394	+	0.753547i	$0.728341\pi$
$908$	19.7426	0.655183
$909$	−7.14590	−0.237014
$910$	0	0
$911$	−12.6525	−0.419195	−0.209598	−	0.977788i	$-0.567215\pi$
$911$	−12.6525	−0.419195	−0.209598	+	0.977788i	$0.567215\pi$
$912$	4.23607	0.140270
$913$	0	0
$914$	−12.1246	−0.401047
$915$	0	0
$916$	−15.4721	−0.511214
$917$	24.9230	0.823029
$918$	2.09017	0.0689859
$919$	−21.2705	−0.701649	−0.350825	−	0.936441i	$-0.614099\pi$
$919$	−21.2705	−0.701649	−0.350825	+	0.936441i	$0.614099\pi$
$920$	0	0
$921$	46.5967	1.53542
$922$	36.4164	1.19931
$923$	−2.31308	−0.0761360
$924$	0	0
$925$	0	0
$926$	−38.5967	−1.26837
$927$	−6.45085	−0.211874
$928$	0.527864	0.0173280
$929$	−23.4164	−0.768267	−0.384134	−	0.923277i	$-0.625500\pi$
$929$	−23.4164	−0.768267	−0.384134	+	0.923277i	$0.625500\pi$
$930$	0	0
$931$	−5.23607	−0.171605
$932$	20.3262	0.665808
$933$	6.85410	0.224393
$934$	−32.1803	−1.05297
$935$	0	0
$936$	−0.0901699	−0.00294730
$937$	−23.6525	−0.772693	−0.386346	−	0.922354i	$-0.626263\pi$
$937$	−23.6525	−0.772693	−0.386346	+	0.922354i	$0.626263\pi$
$938$	20.8541	0.680911
$939$	1.61803	0.0528025
$940$	0	0
$941$	25.3607	0.826735	0.413367	−	0.910564i	$-0.364353\pi$
$941$	25.3607	0.826735	0.413367	+	0.910564i	$0.364353\pi$
$942$	10.9443	0.356584
$943$	14.2918	0.465405
$944$	−11.4721	−0.373386
$945$	0	0
$946$	0	0
$947$	−18.0000	−0.584921	−0.292461	−	0.956278i	$-0.594474\pi$
$947$	−18.0000	−0.584921	−0.292461	+	0.956278i	$0.594474\pi$
$948$	−19.3262	−0.627687
$949$	−1.92299	−0.0624228
$950$	0	0
$951$	4.52786	0.146826
$952$	−0.854102	−0.0276816
$953$	41.1246	1.33216	0.666078	−	0.745882i	$-0.267972\pi$
$953$	41.1246	1.33216	0.666078	+	0.745882i	$0.267972\pi$
$954$	0.527864	0.0170902
$955$	0	0
$956$	−17.4721	−0.565089
$957$	0	0
$958$	−9.94427	−0.321285
$959$	−35.2492	−1.13826
$960$	0	0
$961$	73.7771	2.37991
$962$	−2.09017	−0.0673898
$963$	3.12461	0.100689
$964$	18.0000	0.579741
$965$	0	0
$966$	13.9443	0.448650
$967$	−20.0557	−0.644949	−0.322474	−	0.946578i	$-0.604515\pi$
$967$	−20.0557	−0.644949	−0.322474	+	0.946578i	$0.604515\pi$
$968$	0	0
$969$	1.61803	0.0519787
$970$	0	0
$971$	50.5279	1.62152	0.810758	−	0.585381i	$-0.199055\pi$
$971$	50.5279	1.62152	0.810758	+	0.585381i	$0.199055\pi$
$972$	3.94427	0.126513
$973$	−6.38197	−0.204596
$974$	−10.8885	−0.348891
$975$	0	0
$976$	2.23607	0.0715748
$977$	8.49342	0.271729	0.135864	−	0.990727i	$-0.456619\pi$
$977$	8.49342	0.271729	0.135864	+	0.990727i	$0.456619\pi$
$978$	0.291796	0.00933061
$979$	0	0
$980$	0	0
$981$	4.43769	0.141685
$982$	−12.1803	−0.388690
$983$	−27.5967	−0.880200	−0.440100	−	0.897949i	$-0.645057\pi$
$983$	−27.5967	−0.880200	−0.440100	+	0.897949i	$0.645057\pi$
$984$	6.00000	0.191273
$985$	0	0
$986$	0.201626	0.00642108
$987$	0.854102	0.0271864
$988$	−0.618034	−0.0196623
$989$	36.5066	1.16084
$990$	0	0
$991$	−2.11146	−0.0670726	−0.0335363	−	0.999437i	$-0.510677\pi$
$991$	−2.11146	−0.0670726	−0.0335363	+	0.999437i	$0.510677\pi$
$992$	10.2361	0.324995
$993$	−27.4164	−0.870033
$994$	−21.9098	−0.694938
$995$	0	0
$996$	14.5623	0.461424
$997$	28.9098	0.915584	0.457792	−	0.889059i	$-0.348641\pi$
$997$	28.9098	0.915584	0.457792	+	0.889059i	$0.348641\pi$
$998$	−15.4508	−0.489088
$999$	48.4508	1.53292

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6050.2.a.ca.1.2		2
5.4	even	2		6050.2.a.co.1.1		2
11.5	even	5		550.2.h.g.201.1	yes	4
11.9	even	5		550.2.h.g.301.1	yes	4
11.10	odd	2		6050.2.a.cr.1.2		2
55.9	even	10		550.2.h.c.301.1	yes	4
55.27	odd	20		550.2.ba.e.399.2		8
55.38	odd	20		550.2.ba.e.399.1		8
55.42	odd	20		550.2.ba.e.499.1		8
55.49	even	10		550.2.h.c.201.1	✓	4
55.53	odd	20		550.2.ba.e.499.2		8
55.54	odd	2		6050.2.a.bx.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
550.2.h.c.201.1	✓	4	55.49	even	10
550.2.h.c.301.1	yes	4	55.9	even	10
550.2.h.g.201.1	yes	4	11.5	even	5
550.2.h.g.301.1	yes	4	11.9	even	5
550.2.ba.e.399.1		8	55.38	odd	20
550.2.ba.e.399.2		8	55.27	odd	20
550.2.ba.e.499.1		8	55.42	odd	20
550.2.ba.e.499.2		8	55.53	odd	20
6050.2.a.bx.1.1		2	55.54	odd	2
6050.2.a.ca.1.2		2	1.1	even	1	trivial
6050.2.a.co.1.1		2	5.4	even	2
6050.2.a.cr.1.2		2	11.10	odd	2

Overview	Random
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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:	\( N \)	\(=\)	\( 6050 = 2 \cdot 5^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6050.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.61803 q^{3} +1.00000 q^{4} -1.61803 q^{6} +2.23607 q^{7} -1.00000 q^{8} -0.381966 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.61803 q^{3} +1.00000 q^{4} -1.61803 q^{6} +2.23607 q^{7} -1.00000 q^{8} -0.381966 q^{9} +1.61803 q^{12} -0.236068 q^{13} -2.23607 q^{14} +1.00000 q^{16} +0.381966 q^{17} +0.381966 q^{18} +2.61803 q^{19} +3.61803 q^{21} -3.85410 q^{23} -1.61803 q^{24} +0.236068 q^{26} -5.47214 q^{27} +2.23607 q^{28} -0.527864 q^{29} -10.2361 q^{31} -1.00000 q^{32} -0.381966 q^{34} -0.381966 q^{36} -8.85410 q^{37} -2.61803 q^{38} -0.381966 q^{39} -3.70820 q^{41} -3.61803 q^{42} -9.47214 q^{43} +3.85410 q^{46} +0.236068 q^{47} +1.61803 q^{48} -2.00000 q^{49} +0.618034 q^{51} -0.236068 q^{52} +1.38197 q^{53} +5.47214 q^{54} -2.23607 q^{56} +4.23607 q^{57} +0.527864 q^{58} -11.4721 q^{59} +2.23607 q^{61} +10.2361 q^{62} -0.854102 q^{63} +1.00000 q^{64} -9.32624 q^{67} +0.381966 q^{68} -6.23607 q^{69} +9.79837 q^{71} +0.381966 q^{72} +8.14590 q^{73} +8.85410 q^{74} +2.61803 q^{76} +0.381966 q^{78} -11.9443 q^{79} -7.70820 q^{81} +3.70820 q^{82} +9.00000 q^{83} +3.61803 q^{84} +9.47214 q^{86} -0.854102 q^{87} -13.1803 q^{89} -0.527864 q^{91} -3.85410 q^{92} -16.5623 q^{93} -0.236068 q^{94} -1.61803 q^{96} +12.0000 q^{97} +2.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + q^{3} + 2 q^{4} - q^{6} - 2 q^{8} - 3 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + q^3 + 2 * q^4 - q^6 - 2 * q^8 - 3 * q^9	\( 2 q - 2 q^{2} + q^{3} + 2 q^{4} - q^{6} - 2 q^{8} - 3 q^{9} + q^{12} + 4 q^{13} + 2 q^{16} + 3 q^{17} + 3 q^{18} + 3 q^{19} + 5 q^{21} - q^{23} - q^{24} - 4 q^{26} - 2 q^{27} - 10 q^{29} - 16 q^{31} - 2 q^{32} - 3 q^{34} - 3 q^{36} - 11 q^{37} - 3 q^{38} - 3 q^{39} + 6 q^{41} - 5 q^{42} - 10 q^{43} + q^{46} - 4 q^{47} + q^{48} - 4 q^{49} - q^{51} + 4 q^{52} + 5 q^{53} + 2 q^{54} + 4 q^{57} + 10 q^{58} - 14 q^{59} + 16 q^{62} + 5 q^{63} + 2 q^{64} - 3 q^{67} + 3 q^{68} - 8 q^{69} - 5 q^{71} + 3 q^{72} + 23 q^{73} + 11 q^{74} + 3 q^{76} + 3 q^{78} - 6 q^{79} - 2 q^{81} - 6 q^{82} + 18 q^{83} + 5 q^{84} + 10 q^{86} + 5 q^{87} - 4 q^{89} - 10 q^{91} - q^{92} - 13 q^{93} + 4 q^{94} - q^{96} + 24 q^{97} + 4 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + q^3 + 2 * q^4 - q^6 - 2 * q^8 - 3 * q^9 + q^12 + 4 * q^13 + 2 * q^16 + 3 * q^17 + 3 * q^18 + 3 * q^19 + 5 * q^21 - q^23 - q^24 - 4 * q^26 - 2 * q^27 - 10 * q^29 - 16 * q^31 - 2 * q^32 - 3 * q^34 - 3 * q^36 - 11 * q^37 - 3 * q^38 - 3 * q^39 + 6 * q^41 - 5 * q^42 - 10 * q^43 + q^46 - 4 * q^47 + q^48 - 4 * q^49 - q^51 + 4 * q^52 + 5 * q^53 + 2 * q^54 + 4 * q^57 + 10 * q^58 - 14 * q^59 + 16 * q^62 + 5 * q^63 + 2 * q^64 - 3 * q^67 + 3 * q^68 - 8 * q^69 - 5 * q^71 + 3 * q^72 + 23 * q^73 + 11 * q^74 + 3 * q^76 + 3 * q^78 - 6 * q^79 - 2 * q^81 - 6 * q^82 + 18 * q^83 + 5 * q^84 + 10 * q^86 + 5 * q^87 - 4 * q^89 - 10 * q^91 - q^92 - 13 * q^93 + 4 * q^94 - q^96 + 24 * q^97 + 4 * q^98

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