LMFDB - Embedded newform 4650.2.a.cc.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4650,2,Mod(1,4650)]

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, 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("4650.1");

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.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:	$37.1304369399$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 930)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	4650.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +0.828427 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +0.828427 q^{7} -1.00000 q^{8} +1.00000 q^{9} -0.828427 q^{11} +1.00000 q^{12} +4.82843 q^{13} -0.828427 q^{14} +1.00000 q^{16} +0.828427 q^{17} -1.00000 q^{18} +0.828427 q^{21} +0.828427 q^{22} +8.48528 q^{23} -1.00000 q^{24} -4.82843 q^{26} +1.00000 q^{27} +0.828427 q^{28} +9.65685 q^{29} +1.00000 q^{31} -1.00000 q^{32} -0.828427 q^{33} -0.828427 q^{34} +1.00000 q^{36} -10.4853 q^{37} +4.82843 q^{39} +7.65685 q^{41} -0.828427 q^{42} -9.65685 q^{43} -0.828427 q^{44} -8.48528 q^{46} -5.65685 q^{47} +1.00000 q^{48} -6.31371 q^{49} +0.828427 q^{51} +4.82843 q^{52} +0.343146 q^{53} -1.00000 q^{54} -0.828427 q^{56} -9.65685 q^{58} +3.17157 q^{59} +0.828427 q^{61} -1.00000 q^{62} +0.828427 q^{63} +1.00000 q^{64} +0.828427 q^{66} +9.17157 q^{67} +0.828427 q^{68} +8.48528 q^{69} -2.82843 q^{71} -1.00000 q^{72} -13.6569 q^{73} +10.4853 q^{74} -0.686292 q^{77} -4.82843 q^{78} +11.3137 q^{79} +1.00000 q^{81} -7.65685 q^{82} -1.65685 q^{83} +0.828427 q^{84} +9.65685 q^{86} +9.65685 q^{87} +0.828427 q^{88} +4.82843 q^{89} +4.00000 q^{91} +8.48528 q^{92} +1.00000 q^{93} +5.65685 q^{94} -1.00000 q^{96} -11.3137 q^{97} +6.31371 q^{98} -0.828427 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 4 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - 4 * q^7 - 2 * q^8 + 2 * q^9	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 4 q^{7} - 2 q^{8} + 2 q^{9} + 4 q^{11} + 2 q^{12} + 4 q^{13} + 4 q^{14} + 2 q^{16} - 4 q^{17} - 2 q^{18} - 4 q^{21} - 4 q^{22} - 2 q^{24} - 4 q^{26} + 2 q^{27} - 4 q^{28} + 8 q^{29} + 2 q^{31} - 2 q^{32} + 4 q^{33} + 4 q^{34} + 2 q^{36} - 4 q^{37} + 4 q^{39} + 4 q^{41} + 4 q^{42} - 8 q^{43} + 4 q^{44} + 2 q^{48} + 10 q^{49} - 4 q^{51} + 4 q^{52} + 12 q^{53} - 2 q^{54} + 4 q^{56} - 8 q^{58} + 12 q^{59} - 4 q^{61} - 2 q^{62} - 4 q^{63} + 2 q^{64} - 4 q^{66} + 24 q^{67} - 4 q^{68} - 2 q^{72} - 16 q^{73} + 4 q^{74} - 24 q^{77} - 4 q^{78} + 2 q^{81} - 4 q^{82} + 8 q^{83} - 4 q^{84} + 8 q^{86} + 8 q^{87} - 4 q^{88} + 4 q^{89} + 8 q^{91} + 2 q^{93} - 2 q^{96} - 10 q^{98} + 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - 4 * q^7 - 2 * q^8 + 2 * q^9 + 4 * q^11 + 2 * q^12 + 4 * q^13 + 4 * q^14 + 2 * q^16 - 4 * q^17 - 2 * q^18 - 4 * q^21 - 4 * q^22 - 2 * q^24 - 4 * q^26 + 2 * q^27 - 4 * q^28 + 8 * q^29 + 2 * q^31 - 2 * q^32 + 4 * q^33 + 4 * q^34 + 2 * q^36 - 4 * q^37 + 4 * q^39 + 4 * q^41 + 4 * q^42 - 8 * q^43 + 4 * q^44 + 2 * q^48 + 10 * q^49 - 4 * q^51 + 4 * q^52 + 12 * q^53 - 2 * q^54 + 4 * q^56 - 8 * q^58 + 12 * q^59 - 4 * q^61 - 2 * q^62 - 4 * q^63 + 2 * q^64 - 4 * q^66 + 24 * q^67 - 4 * q^68 - 2 * q^72 - 16 * q^73 + 4 * q^74 - 24 * q^77 - 4 * q^78 + 2 * q^81 - 4 * q^82 + 8 * q^83 - 4 * q^84 + 8 * q^86 + 8 * q^87 - 4 * q^88 + 4 * q^89 + 8 * q^91 + 2 * q^93 - 2 * q^96 - 10 * q^98 + 4 * q^99

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	0.828427	0.313116	0.156558	−	0.987669i	$-0.449960\pi$
$7$	0.828427	0.313116	0.156558	+	0.987669i	$0.449960\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.828427	−0.249780	−0.124890	−	0.992171i	$-0.539858\pi$
$11$	−0.828427	−0.249780	−0.124890	+	0.992171i	$0.539858\pi$
$12$	1.00000	0.288675
$13$	4.82843	1.33916	0.669582	−	0.742738i	$-0.266473\pi$
$13$	4.82843	1.33916	0.669582	+	0.742738i	$0.266473\pi$
$14$	−0.828427	−0.221406
$15$	0	0
$16$	1.00000	0.250000
$17$	0.828427	0.200923	0.100462	−	0.994941i	$-0.467968\pi$
$17$	0.828427	0.200923	0.100462	+	0.994941i	$0.467968\pi$
$18$	−1.00000	−0.235702
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0.828427	0.180778
$22$	0.828427	0.176621
$23$	8.48528	1.76930	0.884652	−	0.466252i	$-0.154396\pi$
$23$	8.48528	1.76930	0.884652	+	0.466252i	$0.154396\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	−4.82843	−0.946932
$27$	1.00000	0.192450
$28$	0.828427	0.156558
$29$	9.65685	1.79323	0.896616	−	0.442808i	$-0.146018\pi$
$29$	9.65685	1.79323	0.896616	+	0.442808i	$0.146018\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	−1.00000	−0.176777
$33$	−0.828427	−0.144211
$34$	−0.828427	−0.142074
$35$	0	0
$36$	1.00000	0.166667
$37$	−10.4853	−1.72377	−0.861885	−	0.507104i	$-0.830716\pi$
$37$	−10.4853	−1.72377	−0.861885	+	0.507104i	$0.830716\pi$
$38$	0	0
$39$	4.82843	0.773167
$40$	0	0
$41$	7.65685	1.19580	0.597900	−	0.801571i	$-0.296002\pi$
$41$	7.65685	1.19580	0.597900	+	0.801571i	$0.296002\pi$
$42$	−0.828427	−0.127829
$43$	−9.65685	−1.47266	−0.736328	−	0.676625i	$-0.763442\pi$
$43$	−9.65685	−1.47266	−0.736328	+	0.676625i	$0.763442\pi$
$44$	−0.828427	−0.124890
$45$	0	0
$46$	−8.48528	−1.25109
$47$	−5.65685	−0.825137	−0.412568	−	0.910927i	$-0.635368\pi$
$47$	−5.65685	−0.825137	−0.412568	+	0.910927i	$0.635368\pi$
$48$	1.00000	0.144338
$49$	−6.31371	−0.901958
$50$	0	0
$51$	0.828427	0.116003
$52$	4.82843	0.669582
$53$	0.343146	0.0471347	0.0235673	−	0.999722i	$-0.492498\pi$
$53$	0.343146	0.0471347	0.0235673	+	0.999722i	$0.492498\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	−0.828427	−0.110703
$57$	0	0
$58$	−9.65685	−1.26801
$59$	3.17157	0.412904	0.206452	−	0.978457i	$-0.433808\pi$
$59$	3.17157	0.412904	0.206452	+	0.978457i	$0.433808\pi$
$60$	0	0
$61$	0.828427	0.106069	0.0530346	−	0.998593i	$-0.483111\pi$
$61$	0.828427	0.106069	0.0530346	+	0.998593i	$0.483111\pi$
$62$	−1.00000	−0.127000
$63$	0.828427	0.104372
$64$	1.00000	0.125000
$65$	0	0
$66$	0.828427	0.101972
$67$	9.17157	1.12049	0.560243	−	0.828328i	$-0.310708\pi$
$67$	9.17157	1.12049	0.560243	+	0.828328i	$0.310708\pi$
$68$	0.828427	0.100462
$69$	8.48528	1.02151
$70$	0	0
$71$	−2.82843	−0.335673	−0.167836	−	0.985815i	$-0.553678\pi$
$71$	−2.82843	−0.335673	−0.167836	+	0.985815i	$0.553678\pi$
$72$	−1.00000	−0.117851
$73$	−13.6569	−1.59841	−0.799207	−	0.601056i	$-0.794747\pi$
$73$	−13.6569	−1.59841	−0.799207	+	0.601056i	$0.794747\pi$
$74$	10.4853	1.21889
$75$	0	0
$76$	0	0
$77$	−0.686292	−0.0782102
$78$	−4.82843	−0.546712
$79$	11.3137	1.27289	0.636446	−	0.771321i	$-0.280404\pi$
$79$	11.3137	1.27289	0.636446	+	0.771321i	$0.280404\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−7.65685	−0.845558
$83$	−1.65685	−0.181863	−0.0909317	−	0.995857i	$-0.528984\pi$
$83$	−1.65685	−0.181863	−0.0909317	+	0.995857i	$0.528984\pi$
$84$	0.828427	0.0903888
$85$	0	0
$86$	9.65685	1.04133
$87$	9.65685	1.03532
$88$	0.828427	0.0883106
$89$	4.82843	0.511812	0.255906	−	0.966702i	$-0.417626\pi$
$89$	4.82843	0.511812	0.255906	+	0.966702i	$0.417626\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	8.48528	0.884652
$93$	1.00000	0.103695
$94$	5.65685	0.583460
$95$	0	0
$96$	−1.00000	−0.102062
$97$	−11.3137	−1.14873	−0.574367	−	0.818598i	$-0.694752\pi$
$97$	−11.3137	−1.14873	−0.574367	+	0.818598i	$0.694752\pi$
$98$	6.31371	0.637781
$99$	−0.828427	−0.0832601
$100$	0	0
$101$	11.3137	1.12576	0.562878	−	0.826540i	$-0.309694\pi$
$101$	11.3137	1.12576	0.562878	+	0.826540i	$0.309694\pi$
$102$	−0.828427	−0.0820265
$103$	−1.51472	−0.149250	−0.0746248	−	0.997212i	$-0.523776\pi$
$103$	−1.51472	−0.149250	−0.0746248	+	0.997212i	$0.523776\pi$
$104$	−4.82843	−0.473466
$105$	0	0
$106$	−0.343146	−0.0333293
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	1.00000	0.0962250
$109$	−11.6569	−1.11652	−0.558262	−	0.829665i	$-0.688532\pi$
$109$	−11.6569	−1.11652	−0.558262	+	0.829665i	$0.688532\pi$
$110$	0	0
$111$	−10.4853	−0.995219
$112$	0.828427	0.0782790
$113$	14.0000	1.31701	0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000	1.31701	0.658505	+	0.752577i	$0.271189\pi$
$114$	0	0
$115$	0	0
$116$	9.65685	0.896616
$117$	4.82843	0.446388
$118$	−3.17157	−0.291967
$119$	0.686292	0.0629122
$120$	0	0
$121$	−10.3137	−0.937610
$122$	−0.828427	−0.0750023
$123$	7.65685	0.690395
$124$	1.00000	0.0898027
$125$	0	0
$126$	−0.828427	−0.0738022
$127$	16.1421	1.43238	0.716191	−	0.697904i	$-0.245884\pi$
$127$	16.1421	1.43238	0.716191	+	0.697904i	$0.245884\pi$
$128$	−1.00000	−0.0883883
$129$	−9.65685	−0.850239
$130$	0	0
$131$	18.4853	1.61507	0.807533	−	0.589822i	$-0.200802\pi$
$131$	18.4853	1.61507	0.807533	+	0.589822i	$0.200802\pi$
$132$	−0.828427	−0.0721053
$133$	0	0
$134$	−9.17157	−0.792303
$135$	0	0
$136$	−0.828427	−0.0710370
$137$	−6.48528	−0.554075	−0.277037	−	0.960859i	$-0.589353\pi$
$137$	−6.48528	−0.554075	−0.277037	+	0.960859i	$0.589353\pi$
$138$	−8.48528	−0.722315
$139$	−14.1421	−1.19952	−0.599760	−	0.800180i	$-0.704737\pi$
$139$	−14.1421	−1.19952	−0.599760	+	0.800180i	$0.704737\pi$
$140$	0	0
$141$	−5.65685	−0.476393
$142$	2.82843	0.237356
$143$	−4.00000	−0.334497
$144$	1.00000	0.0833333
$145$	0	0
$146$	13.6569	1.13025
$147$	−6.31371	−0.520746
$148$	−10.4853	−0.861885
$149$	−0.686292	−0.0562232	−0.0281116	−	0.999605i	$-0.508949\pi$
$149$	−0.686292	−0.0562232	−0.0281116	+	0.999605i	$0.508949\pi$
$150$	0	0
$151$	21.6569	1.76241	0.881205	−	0.472735i	$-0.156733\pi$
$151$	21.6569	1.76241	0.881205	+	0.472735i	$0.156733\pi$
$152$	0	0
$153$	0.828427	0.0669744
$154$	0.686292	0.0553029
$155$	0	0
$156$	4.82843	0.386584
$157$	17.3137	1.38178	0.690892	−	0.722958i	$-0.257218\pi$
$157$	17.3137	1.38178	0.690892	+	0.722958i	$0.257218\pi$
$158$	−11.3137	−0.900070
$159$	0.343146	0.0272132
$160$	0	0
$161$	7.02944	0.553997
$162$	−1.00000	−0.0785674
$163$	−14.8284	−1.16145	−0.580726	−	0.814099i	$-0.697231\pi$
$163$	−14.8284	−1.16145	−0.580726	+	0.814099i	$0.697231\pi$
$164$	7.65685	0.597900
$165$	0	0
$166$	1.65685	0.128597
$167$	−0.485281	−0.0375522	−0.0187761	−	0.999824i	$-0.505977\pi$
$167$	−0.485281	−0.0375522	−0.0187761	+	0.999824i	$0.505977\pi$
$168$	−0.828427	−0.0639145
$169$	10.3137	0.793362
$170$	0	0
$171$	0	0
$172$	−9.65685	−0.736328
$173$	−1.31371	−0.0998794	−0.0499397	−	0.998752i	$-0.515903\pi$
$173$	−1.31371	−0.0998794	−0.0499397	+	0.998752i	$0.515903\pi$
$174$	−9.65685	−0.732084
$175$	0	0
$176$	−0.828427	−0.0624450
$177$	3.17157	0.238390
$178$	−4.82843	−0.361906
$179$	−11.1716	−0.835003	−0.417501	−	0.908676i	$-0.637094\pi$
$179$	−11.1716	−0.835003	−0.417501	+	0.908676i	$0.637094\pi$
$180$	0	0
$181$	24.8284	1.84548	0.922741	−	0.385420i	$-0.125943\pi$
$181$	24.8284	1.84548	0.922741	+	0.385420i	$0.125943\pi$
$182$	−4.00000	−0.296500
$183$	0.828427	0.0612391
$184$	−8.48528	−0.625543
$185$	0	0
$186$	−1.00000	−0.0733236
$187$	−0.686292	−0.0501866
$188$	−5.65685	−0.412568
$189$	0.828427	0.0602592
$190$	0	0
$191$	−0.485281	−0.0351137	−0.0175569	−	0.999846i	$-0.505589\pi$
$191$	−0.485281	−0.0351137	−0.0175569	+	0.999846i	$0.505589\pi$
$192$	1.00000	0.0721688
$193$	15.3137	1.10230	0.551152	−	0.834405i	$-0.314188\pi$
$193$	15.3137	1.10230	0.551152	+	0.834405i	$0.314188\pi$
$194$	11.3137	0.812277
$195$	0	0
$196$	−6.31371	−0.450979
$197$	−26.9706	−1.92157	−0.960787	−	0.277289i	$-0.910564\pi$
$197$	−26.9706	−1.92157	−0.960787	+	0.277289i	$0.910564\pi$
$198$	0.828427	0.0588738
$199$	13.6569	0.968109	0.484054	−	0.875038i	$-0.339164\pi$
$199$	13.6569	0.968109	0.484054	+	0.875038i	$0.339164\pi$
$200$	0	0
$201$	9.17157	0.646913
$202$	−11.3137	−0.796030
$203$	8.00000	0.561490
$204$	0.828427	0.0580015
$205$	0	0
$206$	1.51472	0.105535
$207$	8.48528	0.589768
$208$	4.82843	0.334791
$209$	0	0
$210$	0	0
$211$	23.3137	1.60498	0.802491	−	0.596664i	$-0.203508\pi$
$211$	23.3137	1.60498	0.802491	+	0.596664i	$0.203508\pi$
$212$	0.343146	0.0235673
$213$	−2.82843	−0.193801
$214$	−4.00000	−0.273434
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	0.828427	0.0562373
$218$	11.6569	0.789502
$219$	−13.6569	−0.922845
$220$	0	0
$221$	4.00000	0.269069
$222$	10.4853	0.703726
$223$	−17.7990	−1.19191	−0.595954	−	0.803018i	$-0.703226\pi$
$223$	−17.7990	−1.19191	−0.595954	+	0.803018i	$0.703226\pi$
$224$	−0.828427	−0.0553516
$225$	0	0
$226$	−14.0000	−0.931266
$227$	17.6569	1.17193	0.585963	−	0.810338i	$-0.300716\pi$
$227$	17.6569	1.17193	0.585963	+	0.810338i	$0.300716\pi$
$228$	0	0
$229$	18.4853	1.22154	0.610771	−	0.791807i	$-0.290860\pi$
$229$	18.4853	1.22154	0.610771	+	0.791807i	$0.290860\pi$
$230$	0	0
$231$	−0.686292	−0.0451547
$232$	−9.65685	−0.634004
$233$	−6.97056	−0.456657	−0.228328	−	0.973584i	$-0.573326\pi$
$233$	−6.97056	−0.456657	−0.228328	+	0.973584i	$0.573326\pi$
$234$	−4.82843	−0.315644
$235$	0	0
$236$	3.17157	0.206452
$237$	11.3137	0.734904
$238$	−0.686292	−0.0444857
$239$	0.686292	0.0443925	0.0221963	−	0.999754i	$-0.492934\pi$
$239$	0.686292	0.0443925	0.0221963	+	0.999754i	$0.492934\pi$
$240$	0	0
$241$	22.0000	1.41714	0.708572	−	0.705638i	$-0.249340\pi$
$241$	22.0000	1.41714	0.708572	+	0.705638i	$0.249340\pi$
$242$	10.3137	0.662990
$243$	1.00000	0.0641500
$244$	0.828427	0.0530346
$245$	0	0
$246$	−7.65685	−0.488183
$247$	0	0
$248$	−1.00000	−0.0635001
$249$	−1.65685	−0.104999
$250$	0	0
$251$	3.17157	0.200188	0.100094	−	0.994978i	$-0.468086\pi$
$251$	3.17157	0.200188	0.100094	+	0.994978i	$0.468086\pi$
$252$	0.828427	0.0521860
$253$	−7.02944	−0.441937
$254$	−16.1421	−1.01285
$255$	0	0
$256$	1.00000	0.0625000
$257$	−15.6569	−0.976648	−0.488324	−	0.872662i	$-0.662392\pi$
$257$	−15.6569	−0.976648	−0.488324	+	0.872662i	$0.662392\pi$
$258$	9.65685	0.601209
$259$	−8.68629	−0.539740
$260$	0	0
$261$	9.65685	0.597744
$262$	−18.4853	−1.14202
$263$	24.4853	1.50983	0.754914	−	0.655824i	$-0.227679\pi$
$263$	24.4853	1.50983	0.754914	+	0.655824i	$0.227679\pi$
$264$	0.828427	0.0509862
$265$	0	0
$266$	0	0
$267$	4.82843	0.295495
$268$	9.17157	0.560243
$269$	−4.97056	−0.303061	−0.151530	−	0.988453i	$-0.548420\pi$
$269$	−4.97056	−0.303061	−0.151530	+	0.988453i	$0.548420\pi$
$270$	0	0
$271$	−13.6569	−0.829595	−0.414797	−	0.909914i	$-0.636148\pi$
$271$	−13.6569	−0.829595	−0.414797	+	0.909914i	$0.636148\pi$
$272$	0.828427	0.0502308
$273$	4.00000	0.242091
$274$	6.48528	0.391790
$275$	0	0
$276$	8.48528	0.510754
$277$	4.14214	0.248877	0.124438	−	0.992227i	$-0.460287\pi$
$277$	4.14214	0.248877	0.124438	+	0.992227i	$0.460287\pi$
$278$	14.1421	0.848189
$279$	1.00000	0.0598684
$280$	0	0
$281$	−21.3137	−1.27147	−0.635735	−	0.771908i	$-0.719303\pi$
$281$	−21.3137	−1.27147	−0.635735	+	0.771908i	$0.719303\pi$
$282$	5.65685	0.336861
$283$	−14.8284	−0.881458	−0.440729	−	0.897640i	$-0.645280\pi$
$283$	−14.8284	−0.881458	−0.440729	+	0.897640i	$0.645280\pi$
$284$	−2.82843	−0.167836
$285$	0	0
$286$	4.00000	0.236525
$287$	6.34315	0.374424
$288$	−1.00000	−0.0589256
$289$	−16.3137	−0.959630
$290$	0	0
$291$	−11.3137	−0.663221
$292$	−13.6569	−0.799207
$293$	−2.00000	−0.116841	−0.0584206	−	0.998292i	$-0.518606\pi$
$293$	−2.00000	−0.116841	−0.0584206	+	0.998292i	$0.518606\pi$
$294$	6.31371	0.368223
$295$	0	0
$296$	10.4853	0.609445
$297$	−0.828427	−0.0480702
$298$	0.686292	0.0397558
$299$	40.9706	2.36939
$300$	0	0
$301$	−8.00000	−0.461112
$302$	−21.6569	−1.24621
$303$	11.3137	0.649956
$304$	0	0
$305$	0	0
$306$	−0.828427	−0.0473580
$307$	−0.485281	−0.0276965	−0.0138482	−	0.999904i	$-0.504408\pi$
$307$	−0.485281	−0.0276965	−0.0138482	+	0.999904i	$0.504408\pi$
$308$	−0.686292	−0.0391051
$309$	−1.51472	−0.0861693
$310$	0	0
$311$	−16.4853	−0.934795	−0.467397	−	0.884047i	$-0.654808\pi$
$311$	−16.4853	−0.934795	−0.467397	+	0.884047i	$0.654808\pi$
$312$	−4.82843	−0.273356
$313$	0.970563	0.0548595	0.0274297	−	0.999624i	$-0.491268\pi$
$313$	0.970563	0.0548595	0.0274297	+	0.999624i	$0.491268\pi$
$314$	−17.3137	−0.977069
$315$	0	0
$316$	11.3137	0.636446
$317$	13.3137	0.747772	0.373886	−	0.927475i	$-0.378025\pi$
$317$	13.3137	0.747772	0.373886	+	0.927475i	$0.378025\pi$
$318$	−0.343146	−0.0192427
$319$	−8.00000	−0.447914
$320$	0	0
$321$	4.00000	0.223258
$322$	−7.02944	−0.391735
$323$	0	0
$324$	1.00000	0.0555556
$325$	0	0
$326$	14.8284	0.821271
$327$	−11.6569	−0.644626
$328$	−7.65685	−0.422779
$329$	−4.68629	−0.258364
$330$	0	0
$331$	12.4853	0.686253	0.343127	−	0.939289i	$-0.388514\pi$
$331$	12.4853	0.686253	0.343127	+	0.939289i	$0.388514\pi$
$332$	−1.65685	−0.0909317
$333$	−10.4853	−0.574590
$334$	0.485281	0.0265534
$335$	0	0
$336$	0.828427	0.0451944
$337$	−5.65685	−0.308148	−0.154074	−	0.988059i	$-0.549239\pi$
$337$	−5.65685	−0.308148	−0.154074	+	0.988059i	$0.549239\pi$
$338$	−10.3137	−0.560992
$339$	14.0000	0.760376
$340$	0	0
$341$	−0.828427	−0.0448618
$342$	0	0
$343$	−11.0294	−0.595534
$344$	9.65685	0.520663
$345$	0	0
$346$	1.31371	0.0706254
$347$	17.6569	0.947870	0.473935	−	0.880560i	$-0.342833\pi$
$347$	17.6569	0.947870	0.473935	+	0.880560i	$0.342833\pi$
$348$	9.65685	0.517662
$349$	−25.3137	−1.35501	−0.677506	−	0.735517i	$-0.736939\pi$
$349$	−25.3137	−1.35501	−0.677506	+	0.735517i	$0.736939\pi$
$350$	0	0
$351$	4.82843	0.257722
$352$	0.828427	0.0441553
$353$	−23.1716	−1.23330	−0.616649	−	0.787238i	$-0.711510\pi$
$353$	−23.1716	−1.23330	−0.616649	+	0.787238i	$0.711510\pi$
$354$	−3.17157	−0.168567
$355$	0	0
$356$	4.82843	0.255906
$357$	0.686292	0.0363224
$358$	11.1716	0.590436
$359$	10.1421	0.535281	0.267641	−	0.963519i	$-0.413756\pi$
$359$	10.1421	0.535281	0.267641	+	0.963519i	$0.413756\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	−24.8284	−1.30495
$363$	−10.3137	−0.541329
$364$	4.00000	0.209657
$365$	0	0
$366$	−0.828427	−0.0433026
$367$	−28.1421	−1.46901	−0.734504	−	0.678605i	$-0.762585\pi$
$367$	−28.1421	−1.46901	−0.734504	+	0.678605i	$0.762585\pi$
$368$	8.48528	0.442326
$369$	7.65685	0.398600
$370$	0	0
$371$	0.284271	0.0147586
$372$	1.00000	0.0518476
$373$	0.343146	0.0177674	0.00888371	−	0.999961i	$-0.497172\pi$
$373$	0.343146	0.0177674	0.00888371	+	0.999961i	$0.497172\pi$
$374$	0.686292	0.0354873
$375$	0	0
$376$	5.65685	0.291730
$377$	46.6274	2.40143
$378$	−0.828427	−0.0426097
$379$	−9.65685	−0.496039	−0.248020	−	0.968755i	$-0.579780\pi$
$379$	−9.65685	−0.496039	−0.248020	+	0.968755i	$0.579780\pi$
$380$	0	0
$381$	16.1421	0.826987
$382$	0.485281	0.0248292
$383$	11.5147	0.588375	0.294187	−	0.955748i	$-0.404951\pi$
$383$	11.5147	0.588375	0.294187	+	0.955748i	$0.404951\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−15.3137	−0.779447
$387$	−9.65685	−0.490885
$388$	−11.3137	−0.574367
$389$	12.6863	0.643221	0.321610	−	0.946872i	$-0.395776\pi$
$389$	12.6863	0.643221	0.321610	+	0.946872i	$0.395776\pi$
$390$	0	0
$391$	7.02944	0.355494
$392$	6.31371	0.318890
$393$	18.4853	0.932459
$394$	26.9706	1.35876
$395$	0	0
$396$	−0.828427	−0.0416300
$397$	15.6569	0.785795	0.392897	−	0.919582i	$-0.371473\pi$
$397$	15.6569	0.785795	0.392897	+	0.919582i	$0.371473\pi$
$398$	−13.6569	−0.684556
$399$	0	0
$400$	0	0
$401$	2.48528	0.124109	0.0620545	−	0.998073i	$-0.480235\pi$
$401$	2.48528	0.124109	0.0620545	+	0.998073i	$0.480235\pi$
$402$	−9.17157	−0.457436
$403$	4.82843	0.240521
$404$	11.3137	0.562878
$405$	0	0
$406$	−8.00000	−0.397033
$407$	8.68629	0.430563
$408$	−0.828427	−0.0410133
$409$	33.3137	1.64726	0.823628	−	0.567130i	$-0.191946\pi$
$409$	33.3137	1.64726	0.823628	+	0.567130i	$0.191946\pi$
$410$	0	0
$411$	−6.48528	−0.319895
$412$	−1.51472	−0.0746248
$413$	2.62742	0.129287
$414$	−8.48528	−0.417029
$415$	0	0
$416$	−4.82843	−0.236733
$417$	−14.1421	−0.692543
$418$	0	0
$419$	−9.79899	−0.478712	−0.239356	−	0.970932i	$-0.576936\pi$
$419$	−9.79899	−0.478712	−0.239356	+	0.970932i	$0.576936\pi$
$420$	0	0
$421$	14.0000	0.682318	0.341159	−	0.940006i	$-0.389181\pi$
$421$	14.0000	0.682318	0.341159	+	0.940006i	$0.389181\pi$
$422$	−23.3137	−1.13489
$423$	−5.65685	−0.275046
$424$	−0.343146	−0.0166646
$425$	0	0
$426$	2.82843	0.137038
$427$	0.686292	0.0332120
$428$	4.00000	0.193347
$429$	−4.00000	−0.193122
$430$	0	0
$431$	−1.17157	−0.0564327	−0.0282163	−	0.999602i	$-0.508983\pi$
$431$	−1.17157	−0.0564327	−0.0282163	+	0.999602i	$0.508983\pi$
$432$	1.00000	0.0481125
$433$	−13.6569	−0.656307	−0.328153	−	0.944624i	$-0.606426\pi$
$433$	−13.6569	−0.656307	−0.328153	+	0.944624i	$0.606426\pi$
$434$	−0.828427	−0.0397658
$435$	0	0
$436$	−11.6569	−0.558262
$437$	0	0
$438$	13.6569	0.652550
$439$	−0.970563	−0.0463224	−0.0231612	−	0.999732i	$-0.507373\pi$
$439$	−0.970563	−0.0463224	−0.0231612	+	0.999732i	$0.507373\pi$
$440$	0	0
$441$	−6.31371	−0.300653
$442$	−4.00000	−0.190261
$443$	6.34315	0.301372	0.150686	−	0.988582i	$-0.451852\pi$
$443$	6.34315	0.301372	0.150686	+	0.988582i	$0.451852\pi$
$444$	−10.4853	−0.497609
$445$	0	0
$446$	17.7990	0.842807
$447$	−0.686292	−0.0324605
$448$	0.828427	0.0391395
$449$	−12.1421	−0.573023	−0.286511	−	0.958077i	$-0.592496\pi$
$449$	−12.1421	−0.573023	−0.286511	+	0.958077i	$0.592496\pi$
$450$	0	0
$451$	−6.34315	−0.298687
$452$	14.0000	0.658505
$453$	21.6569	1.01753
$454$	−17.6569	−0.828677
$455$	0	0
$456$	0	0
$457$	−31.3137	−1.46479	−0.732397	−	0.680878i	$-0.761598\pi$
$457$	−31.3137	−1.46479	−0.732397	+	0.680878i	$0.761598\pi$
$458$	−18.4853	−0.863760
$459$	0.828427	0.0386677
$460$	0	0
$461$	−19.3137	−0.899529	−0.449765	−	0.893147i	$-0.648492\pi$
$461$	−19.3137	−0.899529	−0.449765	+	0.893147i	$0.648492\pi$
$462$	0.686292	0.0319292
$463$	3.17157	0.147395	0.0736977	−	0.997281i	$-0.476520\pi$
$463$	3.17157	0.147395	0.0736977	+	0.997281i	$0.476520\pi$
$464$	9.65685	0.448308
$465$	0	0
$466$	6.97056	0.322905
$467$	36.0000	1.66588	0.832941	−	0.553362i	$-0.186655\pi$
$467$	36.0000	1.66588	0.832941	+	0.553362i	$0.186655\pi$
$468$	4.82843	0.223194
$469$	7.59798	0.350842
$470$	0	0
$471$	17.3137	0.797774
$472$	−3.17157	−0.145983
$473$	8.00000	0.367840
$474$	−11.3137	−0.519656
$475$	0	0
$476$	0.686292	0.0314561
$477$	0.343146	0.0157116
$478$	−0.686292	−0.0313902
$479$	21.1716	0.967354	0.483677	−	0.875247i	$-0.339301\pi$
$479$	21.1716	0.967354	0.483677	+	0.875247i	$0.339301\pi$
$480$	0	0
$481$	−50.6274	−2.30841
$482$	−22.0000	−1.00207
$483$	7.02944	0.319850
$484$	−10.3137	−0.468805
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	20.1421	0.912727	0.456364	−	0.889793i	$-0.349152\pi$
$487$	20.1421	0.912727	0.456364	+	0.889793i	$0.349152\pi$
$488$	−0.828427	−0.0375011
$489$	−14.8284	−0.670565
$490$	0	0
$491$	−43.4558	−1.96113	−0.980567	−	0.196183i	$-0.937145\pi$
$491$	−43.4558	−1.96113	−0.980567	+	0.196183i	$0.937145\pi$
$492$	7.65685	0.345198
$493$	8.00000	0.360302
$494$	0	0
$495$	0	0
$496$	1.00000	0.0449013
$497$	−2.34315	−0.105104
$498$	1.65685	0.0742454
$499$	18.1421	0.812154	0.406077	−	0.913839i	$-0.366897\pi$
$499$	18.1421	0.812154	0.406077	+	0.913839i	$0.366897\pi$
$500$	0	0
$501$	−0.485281	−0.0216808
$502$	−3.17157	−0.141554
$503$	12.6863	0.565654	0.282827	−	0.959171i	$-0.408728\pi$
$503$	12.6863	0.565654	0.282827	+	0.959171i	$0.408728\pi$
$504$	−0.828427	−0.0369011
$505$	0	0
$506$	7.02944	0.312497
$507$	10.3137	0.458048
$508$	16.1421	0.716191
$509$	20.9706	0.929504	0.464752	−	0.885441i	$-0.346144\pi$
$509$	20.9706	0.929504	0.464752	+	0.885441i	$0.346144\pi$
$510$	0	0
$511$	−11.3137	−0.500489
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	15.6569	0.690594
$515$	0	0
$516$	−9.65685	−0.425119
$517$	4.68629	0.206103
$518$	8.68629	0.381654
$519$	−1.31371	−0.0576654
$520$	0	0
$521$	34.2843	1.50202	0.751011	−	0.660290i	$-0.229567\pi$
$521$	34.2843	1.50202	0.751011	+	0.660290i	$0.229567\pi$
$522$	−9.65685	−0.422669
$523$	22.6274	0.989428	0.494714	−	0.869056i	$-0.335273\pi$
$523$	22.6274	0.989428	0.494714	+	0.869056i	$0.335273\pi$
$524$	18.4853	0.807533
$525$	0	0
$526$	−24.4853	−1.06761
$527$	0.828427	0.0360869
$528$	−0.828427	−0.0360527
$529$	49.0000	2.13043
$530$	0	0
$531$	3.17157	0.137635
$532$	0	0
$533$	36.9706	1.60137
$534$	−4.82843	−0.208946
$535$	0	0
$536$	−9.17157	−0.396152
$537$	−11.1716	−0.482089
$538$	4.97056	0.214296
$539$	5.23045	0.225291
$540$	0	0
$541$	22.2843	0.958076	0.479038	−	0.877794i	$-0.340986\pi$
$541$	22.2843	0.958076	0.479038	+	0.877794i	$0.340986\pi$
$542$	13.6569	0.586612
$543$	24.8284	1.06549
$544$	−0.828427	−0.0355185
$545$	0	0
$546$	−4.00000	−0.171184
$547$	9.85786	0.421492	0.210746	−	0.977541i	$-0.432411\pi$
$547$	9.85786	0.421492	0.210746	+	0.977541i	$0.432411\pi$
$548$	−6.48528	−0.277037
$549$	0.828427	0.0353564
$550$	0	0
$551$	0	0
$552$	−8.48528	−0.361158
$553$	9.37258	0.398563
$554$	−4.14214	−0.175982
$555$	0	0
$556$	−14.1421	−0.599760
$557$	17.3137	0.733605	0.366803	−	0.930299i	$-0.380452\pi$
$557$	17.3137	0.733605	0.366803	+	0.930299i	$0.380452\pi$
$558$	−1.00000	−0.0423334
$559$	−46.6274	−1.97213
$560$	0	0
$561$	−0.686292	−0.0289752
$562$	21.3137	0.899065
$563$	4.97056	0.209484	0.104742	−	0.994499i	$-0.466598\pi$
$563$	4.97056	0.209484	0.104742	+	0.994499i	$0.466598\pi$
$564$	−5.65685	−0.238197
$565$	0	0
$566$	14.8284	0.623285
$567$	0.828427	0.0347907
$568$	2.82843	0.118678
$569$	−19.1716	−0.803714	−0.401857	−	0.915702i	$-0.631635\pi$
$569$	−19.1716	−0.803714	−0.401857	+	0.915702i	$0.631635\pi$
$570$	0	0
$571$	−33.1716	−1.38819	−0.694094	−	0.719885i	$-0.744195\pi$
$571$	−33.1716	−1.38819	−0.694094	+	0.719885i	$0.744195\pi$
$572$	−4.00000	−0.167248
$573$	−0.485281	−0.0202729
$574$	−6.34315	−0.264758
$575$	0	0
$576$	1.00000	0.0416667
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	16.3137	0.678561
$579$	15.3137	0.636416
$580$	0	0
$581$	−1.37258	−0.0569443
$582$	11.3137	0.468968
$583$	−0.284271	−0.0117733
$584$	13.6569	0.565125
$585$	0	0
$586$	2.00000	0.0826192
$587$	−7.31371	−0.301869	−0.150935	−	0.988544i	$-0.548228\pi$
$587$	−7.31371	−0.301869	−0.150935	+	0.988544i	$0.548228\pi$
$588$	−6.31371	−0.260373
$589$	0	0
$590$	0	0
$591$	−26.9706	−1.10942
$592$	−10.4853	−0.430942
$593$	−26.9706	−1.10755	−0.553774	−	0.832667i	$-0.686813\pi$
$593$	−26.9706	−1.10755	−0.553774	+	0.832667i	$0.686813\pi$
$594$	0.828427	0.0339908
$595$	0	0
$596$	−0.686292	−0.0281116
$597$	13.6569	0.558938
$598$	−40.9706	−1.67541
$599$	21.4558	0.876662	0.438331	−	0.898814i	$-0.355570\pi$
$599$	21.4558	0.876662	0.438331	+	0.898814i	$0.355570\pi$
$600$	0	0
$601$	18.0000	0.734235	0.367118	−	0.930175i	$-0.380345\pi$
$601$	18.0000	0.734235	0.367118	+	0.930175i	$0.380345\pi$
$602$	8.00000	0.326056
$603$	9.17157	0.373495
$604$	21.6569	0.881205
$605$	0	0
$606$	−11.3137	−0.459588
$607$	−9.79899	−0.397729	−0.198864	−	0.980027i	$-0.563725\pi$
$607$	−9.79899	−0.397729	−0.198864	+	0.980027i	$0.563725\pi$
$608$	0	0
$609$	8.00000	0.324176
$610$	0	0
$611$	−27.3137	−1.10499
$612$	0.828427	0.0334872
$613$	−49.1127	−1.98364	−0.991822	−	0.127632i	$-0.959262\pi$
$613$	−49.1127	−1.98364	−0.991822	+	0.127632i	$0.959262\pi$
$614$	0.485281	0.0195844
$615$	0	0
$616$	0.686292	0.0276515
$617$	41.5980	1.67467	0.837336	−	0.546689i	$-0.184112\pi$
$617$	41.5980	1.67467	0.837336	+	0.546689i	$0.184112\pi$
$618$	1.51472	0.0609309
$619$	−15.5147	−0.623589	−0.311795	−	0.950150i	$-0.600930\pi$
$619$	−15.5147	−0.623589	−0.311795	+	0.950150i	$0.600930\pi$
$620$	0	0
$621$	8.48528	0.340503
$622$	16.4853	0.661000
$623$	4.00000	0.160257
$624$	4.82843	0.193292
$625$	0	0
$626$	−0.970563	−0.0387915
$627$	0	0
$628$	17.3137	0.690892
$629$	−8.68629	−0.346345
$630$	0	0
$631$	14.6274	0.582308	0.291154	−	0.956676i	$-0.405961\pi$
$631$	14.6274	0.582308	0.291154	+	0.956676i	$0.405961\pi$
$632$	−11.3137	−0.450035
$633$	23.3137	0.926637
$634$	−13.3137	−0.528755
$635$	0	0
$636$	0.343146	0.0136066
$637$	−30.4853	−1.20787
$638$	8.00000	0.316723
$639$	−2.82843	−0.111891
$640$	0	0
$641$	−19.4558	−0.768460	−0.384230	−	0.923237i	$-0.625533\pi$
$641$	−19.4558	−0.768460	−0.384230	+	0.923237i	$0.625533\pi$
$642$	−4.00000	−0.157867
$643$	−8.00000	−0.315489	−0.157745	−	0.987480i	$-0.550422\pi$
$643$	−8.00000	−0.315489	−0.157745	+	0.987480i	$0.550422\pi$
$644$	7.02944	0.276999
$645$	0	0
$646$	0	0
$647$	18.1421	0.713241	0.356620	−	0.934249i	$-0.383929\pi$
$647$	18.1421	0.713241	0.356620	+	0.934249i	$0.383929\pi$
$648$	−1.00000	−0.0392837
$649$	−2.62742	−0.103135
$650$	0	0
$651$	0.828427	0.0324686
$652$	−14.8284	−0.580726
$653$	−14.0000	−0.547862	−0.273931	−	0.961749i	$-0.588324\pi$
$653$	−14.0000	−0.547862	−0.273931	+	0.961749i	$0.588324\pi$
$654$	11.6569	0.455819
$655$	0	0
$656$	7.65685	0.298950
$657$	−13.6569	−0.532805
$658$	4.68629	0.182691
$659$	−11.1716	−0.435183	−0.217591	−	0.976040i	$-0.569820\pi$
$659$	−11.1716	−0.435183	−0.217591	+	0.976040i	$0.569820\pi$
$660$	0	0
$661$	24.6274	0.957896	0.478948	−	0.877843i	$-0.341018\pi$
$661$	24.6274	0.957896	0.478948	+	0.877843i	$0.341018\pi$
$662$	−12.4853	−0.485254
$663$	4.00000	0.155347
$664$	1.65685	0.0642984
$665$	0	0
$666$	10.4853	0.406296
$667$	81.9411	3.17277
$668$	−0.485281	−0.0187761
$669$	−17.7990	−0.688149
$670$	0	0
$671$	−0.686292	−0.0264940
$672$	−0.828427	−0.0319573
$673$	7.02944	0.270965	0.135482	−	0.990780i	$-0.456742\pi$
$673$	7.02944	0.270965	0.135482	+	0.990780i	$0.456742\pi$
$674$	5.65685	0.217894
$675$	0	0
$676$	10.3137	0.396681
$677$	−38.0000	−1.46046	−0.730229	−	0.683202i	$-0.760587\pi$
$677$	−38.0000	−1.46046	−0.730229	+	0.683202i	$0.760587\pi$
$678$	−14.0000	−0.537667
$679$	−9.37258	−0.359687
$680$	0	0
$681$	17.6569	0.676612
$682$	0.828427	0.0317221
$683$	24.2843	0.929212	0.464606	−	0.885518i	$-0.346196\pi$
$683$	24.2843	0.929212	0.464606	+	0.885518i	$0.346196\pi$
$684$	0	0
$685$	0	0
$686$	11.0294	0.421106
$687$	18.4853	0.705257
$688$	−9.65685	−0.368164
$689$	1.65685	0.0631211
$690$	0	0
$691$	−44.2843	−1.68465	−0.842327	−	0.538968i	$-0.818815\pi$
$691$	−44.2843	−1.68465	−0.842327	+	0.538968i	$0.818815\pi$
$692$	−1.31371	−0.0499397
$693$	−0.686292	−0.0260701
$694$	−17.6569	−0.670245
$695$	0	0
$696$	−9.65685	−0.366042
$697$	6.34315	0.240264
$698$	25.3137	0.958138
$699$	−6.97056	−0.263651
$700$	0	0
$701$	16.0000	0.604312	0.302156	−	0.953259i	$-0.402294\pi$
$701$	16.0000	0.604312	0.302156	+	0.953259i	$0.402294\pi$
$702$	−4.82843	−0.182237
$703$	0	0
$704$	−0.828427	−0.0312225
$705$	0	0
$706$	23.1716	0.872074
$707$	9.37258	0.352492
$708$	3.17157	0.119195
$709$	−40.8284	−1.53334	−0.766672	−	0.642039i	$-0.778089\pi$
$709$	−40.8284	−1.53334	−0.766672	+	0.642039i	$0.778089\pi$
$710$	0	0
$711$	11.3137	0.424297
$712$	−4.82843	−0.180953
$713$	8.48528	0.317776
$714$	−0.686292	−0.0256838
$715$	0	0
$716$	−11.1716	−0.417501
$717$	0.686292	0.0256300
$718$	−10.1421	−0.378501
$719$	−35.3137	−1.31698	−0.658490	−	0.752590i	$-0.728804\pi$
$719$	−35.3137	−1.31698	−0.658490	+	0.752590i	$0.728804\pi$
$720$	0	0
$721$	−1.25483	−0.0467325
$722$	19.0000	0.707107
$723$	22.0000	0.818189
$724$	24.8284	0.922741
$725$	0	0
$726$	10.3137	0.382778
$727$	−29.5147	−1.09464	−0.547320	−	0.836923i	$-0.684352\pi$
$727$	−29.5147	−1.09464	−0.547320	+	0.836923i	$0.684352\pi$
$728$	−4.00000	−0.148250
$729$	1.00000	0.0370370
$730$	0	0
$731$	−8.00000	−0.295891
$732$	0.828427	0.0306195
$733$	−16.3431	−0.603648	−0.301824	−	0.953364i	$-0.597595\pi$
$733$	−16.3431	−0.603648	−0.301824	+	0.953364i	$0.597595\pi$
$734$	28.1421	1.03875
$735$	0	0
$736$	−8.48528	−0.312772
$737$	−7.59798	−0.279875
$738$	−7.65685	−0.281853
$739$	−3.51472	−0.129291	−0.0646455	−	0.997908i	$-0.520592\pi$
$739$	−3.51472	−0.129291	−0.0646455	+	0.997908i	$0.520592\pi$
$740$	0	0
$741$	0	0
$742$	−0.284271	−0.0104359
$743$	−21.4558	−0.787139	−0.393569	−	0.919295i	$-0.628760\pi$
$743$	−21.4558	−0.787139	−0.393569	+	0.919295i	$0.628760\pi$
$744$	−1.00000	−0.0366618
$745$	0	0
$746$	−0.343146	−0.0125635
$747$	−1.65685	−0.0606211
$748$	−0.686292	−0.0250933
$749$	3.31371	0.121080
$750$	0	0
$751$	−4.28427	−0.156335	−0.0781676	−	0.996940i	$-0.524907\pi$
$751$	−4.28427	−0.156335	−0.0781676	+	0.996940i	$0.524907\pi$
$752$	−5.65685	−0.206284
$753$	3.17157	0.115579
$754$	−46.6274	−1.69807
$755$	0	0
$756$	0.828427	0.0301296
$757$	−9.51472	−0.345818	−0.172909	−	0.984938i	$-0.555317\pi$
$757$	−9.51472	−0.345818	−0.172909	+	0.984938i	$0.555317\pi$
$758$	9.65685	0.350753
$759$	−7.02944	−0.255152
$760$	0	0
$761$	33.1127	1.20033	0.600167	−	0.799875i	$-0.295101\pi$
$761$	33.1127	1.20033	0.600167	+	0.799875i	$0.295101\pi$
$762$	−16.1421	−0.584768
$763$	−9.65685	−0.349602
$764$	−0.485281	−0.0175569
$765$	0	0
$766$	−11.5147	−0.416044
$767$	15.3137	0.552946
$768$	1.00000	0.0360844
$769$	2.00000	0.0721218	0.0360609	−	0.999350i	$-0.488519\pi$
$769$	2.00000	0.0721218	0.0360609	+	0.999350i	$0.488519\pi$
$770$	0	0
$771$	−15.6569	−0.563868
$772$	15.3137	0.551152
$773$	−0.343146	−0.0123421	−0.00617105	−	0.999981i	$-0.501964\pi$
$773$	−0.343146	−0.0123421	−0.00617105	+	0.999981i	$0.501964\pi$
$774$	9.65685	0.347108
$775$	0	0
$776$	11.3137	0.406138
$777$	−8.68629	−0.311619
$778$	−12.6863	−0.454826
$779$	0	0
$780$	0	0
$781$	2.34315	0.0838443
$782$	−7.02944	−0.251372
$783$	9.65685	0.345108
$784$	−6.31371	−0.225490
$785$	0	0
$786$	−18.4853	−0.659348
$787$	−24.9706	−0.890104	−0.445052	−	0.895505i	$-0.646815\pi$
$787$	−24.9706	−0.890104	−0.445052	+	0.895505i	$0.646815\pi$
$788$	−26.9706	−0.960787
$789$	24.4853	0.871699
$790$	0	0
$791$	11.5980	0.412377
$792$	0.828427	0.0294369
$793$	4.00000	0.142044
$794$	−15.6569	−0.555641
$795$	0	0
$796$	13.6569	0.484054
$797$	−1.31371	−0.0465339	−0.0232670	−	0.999729i	$-0.507407\pi$
$797$	−1.31371	−0.0465339	−0.0232670	+	0.999729i	$0.507407\pi$
$798$	0	0
$799$	−4.68629	−0.165789
$800$	0	0
$801$	4.82843	0.170604
$802$	−2.48528	−0.0877583
$803$	11.3137	0.399252
$804$	9.17157	0.323456
$805$	0	0
$806$	−4.82843	−0.170074
$807$	−4.97056	−0.174972
$808$	−11.3137	−0.398015
$809$	−18.4853	−0.649908	−0.324954	−	0.945730i	$-0.605349\pi$
$809$	−18.4853	−0.649908	−0.324954	+	0.945730i	$0.605349\pi$
$810$	0	0
$811$	−20.9706	−0.736376	−0.368188	−	0.929751i	$-0.620022\pi$
$811$	−20.9706	−0.736376	−0.368188	+	0.929751i	$0.620022\pi$
$812$	8.00000	0.280745
$813$	−13.6569	−0.478967
$814$	−8.68629	−0.304454
$815$	0	0
$816$	0.828427	0.0290008
$817$	0	0
$818$	−33.3137	−1.16479
$819$	4.00000	0.139771
$820$	0	0
$821$	25.6569	0.895430	0.447715	−	0.894176i	$-0.352238\pi$
$821$	25.6569	0.895430	0.447715	+	0.894176i	$0.352238\pi$
$822$	6.48528	0.226200
$823$	−31.1716	−1.08657	−0.543286	−	0.839547i	$-0.682820\pi$
$823$	−31.1716	−1.08657	−0.543286	+	0.839547i	$0.682820\pi$
$824$	1.51472	0.0527677
$825$	0	0
$826$	−2.62742	−0.0914195
$827$	−39.3137	−1.36707	−0.683536	−	0.729917i	$-0.739559\pi$
$827$	−39.3137	−1.36707	−0.683536	+	0.729917i	$0.739559\pi$
$828$	8.48528	0.294884
$829$	19.1716	0.665856	0.332928	−	0.942952i	$-0.391963\pi$
$829$	19.1716	0.665856	0.332928	+	0.942952i	$0.391963\pi$
$830$	0	0
$831$	4.14214	0.143689
$832$	4.82843	0.167396
$833$	−5.23045	−0.181224
$834$	14.1421	0.489702
$835$	0	0
$836$	0	0
$837$	1.00000	0.0345651
$838$	9.79899	0.338500
$839$	40.7696	1.40752	0.703761	−	0.710437i	$-0.251503\pi$
$839$	40.7696	1.40752	0.703761	+	0.710437i	$0.251503\pi$
$840$	0	0
$841$	64.2548	2.21568
$842$	−14.0000	−0.482472
$843$	−21.3137	−0.734083
$844$	23.3137	0.802491
$845$	0	0
$846$	5.65685	0.194487
$847$	−8.54416	−0.293581
$848$	0.343146	0.0117837
$849$	−14.8284	−0.508910
$850$	0	0
$851$	−88.9706	−3.04987
$852$	−2.82843	−0.0969003
$853$	21.3137	0.729767	0.364884	−	0.931053i	$-0.381109\pi$
$853$	21.3137	0.729767	0.364884	+	0.931053i	$0.381109\pi$
$854$	−0.686292	−0.0234844
$855$	0	0
$856$	−4.00000	−0.136717
$857$	6.68629	0.228399	0.114200	−	0.993458i	$-0.463570\pi$
$857$	6.68629	0.228399	0.114200	+	0.993458i	$0.463570\pi$
$858$	4.00000	0.136558
$859$	−42.4264	−1.44757	−0.723785	−	0.690025i	$-0.757599\pi$
$859$	−42.4264	−1.44757	−0.723785	+	0.690025i	$0.757599\pi$
$860$	0	0
$861$	6.34315	0.216174
$862$	1.17157	0.0399039
$863$	−3.79899	−0.129319	−0.0646596	−	0.997907i	$-0.520596\pi$
$863$	−3.79899	−0.129319	−0.0646596	+	0.997907i	$0.520596\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	13.6569	0.464079
$867$	−16.3137	−0.554043
$868$	0.828427	0.0281186
$869$	−9.37258	−0.317943
$870$	0	0
$871$	44.2843	1.50052
$872$	11.6569	0.394751
$873$	−11.3137	−0.382911
$874$	0	0
$875$	0	0
$876$	−13.6569	−0.461422
$877$	−23.6569	−0.798835	−0.399418	−	0.916769i	$-0.630788\pi$
$877$	−23.6569	−0.798835	−0.399418	+	0.916769i	$0.630788\pi$
$878$	0.970563	0.0327549
$879$	−2.00000	−0.0674583
$880$	0	0
$881$	−36.1421	−1.21766	−0.608830	−	0.793301i	$-0.708361\pi$
$881$	−36.1421	−1.21766	−0.608830	+	0.793301i	$0.708361\pi$
$882$	6.31371	0.212594
$883$	0.970563	0.0326620	0.0163310	−	0.999867i	$-0.494801\pi$
$883$	0.970563	0.0326620	0.0163310	+	0.999867i	$0.494801\pi$
$884$	4.00000	0.134535
$885$	0	0
$886$	−6.34315	−0.213102
$887$	−20.0000	−0.671534	−0.335767	−	0.941945i	$-0.608996\pi$
$887$	−20.0000	−0.671534	−0.335767	+	0.941945i	$0.608996\pi$
$888$	10.4853	0.351863
$889$	13.3726	0.448502
$890$	0	0
$891$	−0.828427	−0.0277534
$892$	−17.7990	−0.595954
$893$	0	0
$894$	0.686292	0.0229530
$895$	0	0
$896$	−0.828427	−0.0276758
$897$	40.9706	1.36797
$898$	12.1421	0.405188
$899$	9.65685	0.322074
$900$	0	0
$901$	0.284271	0.00947045
$902$	6.34315	0.211204
$903$	−8.00000	−0.266223
$904$	−14.0000	−0.465633
$905$	0	0
$906$	−21.6569	−0.719501
$907$	38.1421	1.26649	0.633244	−	0.773952i	$-0.281723\pi$
$907$	38.1421	1.26649	0.633244	+	0.773952i	$0.281723\pi$
$908$	17.6569	0.585963
$909$	11.3137	0.375252
$910$	0	0
$911$	−49.9411	−1.65462	−0.827312	−	0.561743i	$-0.810131\pi$
$911$	−49.9411	−1.65462	−0.827312	+	0.561743i	$0.810131\pi$
$912$	0	0
$913$	1.37258	0.0454259
$914$	31.3137	1.03577
$915$	0	0
$916$	18.4853	0.610771
$917$	15.3137	0.505703
$918$	−0.828427	−0.0273422
$919$	−16.9706	−0.559807	−0.279904	−	0.960028i	$-0.590303\pi$
$919$	−16.9706	−0.559807	−0.279904	+	0.960028i	$0.590303\pi$
$920$	0	0
$921$	−0.485281	−0.0159906
$922$	19.3137	0.636063
$923$	−13.6569	−0.449521
$924$	−0.686292	−0.0225773
$925$	0	0
$926$	−3.17157	−0.104224
$927$	−1.51472	−0.0497499
$928$	−9.65685	−0.317002
$929$	−36.1421	−1.18579	−0.592893	−	0.805282i	$-0.702014\pi$
$929$	−36.1421	−1.18579	−0.592893	+	0.805282i	$0.702014\pi$
$930$	0	0
$931$	0	0
$932$	−6.97056	−0.228328
$933$	−16.4853	−0.539704
$934$	−36.0000	−1.17796
$935$	0	0
$936$	−4.82843	−0.157822
$937$	−22.6274	−0.739205	−0.369603	−	0.929190i	$-0.620506\pi$
$937$	−22.6274	−0.739205	−0.369603	+	0.929190i	$0.620506\pi$
$938$	−7.59798	−0.248083
$939$	0.970563	0.0316731
$940$	0	0
$941$	−40.0000	−1.30396	−0.651981	−	0.758235i	$-0.726062\pi$
$941$	−40.0000	−1.30396	−0.651981	+	0.758235i	$0.726062\pi$
$942$	−17.3137	−0.564111
$943$	64.9706	2.11573
$944$	3.17157	0.103226
$945$	0	0
$946$	−8.00000	−0.260102
$947$	54.9117	1.78439	0.892195	−	0.451650i	$-0.149165\pi$
$947$	54.9117	1.78439	0.892195	+	0.451650i	$0.149165\pi$
$948$	11.3137	0.367452
$949$	−65.9411	−2.14054
$950$	0	0
$951$	13.3137	0.431727
$952$	−0.686292	−0.0222428
$953$	5.79899	0.187848	0.0939239	−	0.995579i	$-0.470059\pi$
$953$	5.79899	0.187848	0.0939239	+	0.995579i	$0.470059\pi$
$954$	−0.343146	−0.0111098
$955$	0	0
$956$	0.686292	0.0221963
$957$	−8.00000	−0.258603
$958$	−21.1716	−0.684022
$959$	−5.37258	−0.173490
$960$	0	0
$961$	1.00000	0.0322581
$962$	50.6274	1.63229
$963$	4.00000	0.128898
$964$	22.0000	0.708572
$965$	0	0
$966$	−7.02944	−0.226168
$967$	7.17157	0.230622	0.115311	−	0.993329i	$-0.463213\pi$
$967$	7.17157	0.230622	0.115311	+	0.993329i	$0.463213\pi$
$968$	10.3137	0.331495
$969$	0	0
$970$	0	0
$971$	−7.45584	−0.239269	−0.119635	−	0.992818i	$-0.538172\pi$
$971$	−7.45584	−0.239269	−0.119635	+	0.992818i	$0.538172\pi$
$972$	1.00000	0.0320750
$973$	−11.7157	−0.375589
$974$	−20.1421	−0.645396
$975$	0	0
$976$	0.828427	0.0265173
$977$	−27.6569	−0.884821	−0.442411	−	0.896813i	$-0.645877\pi$
$977$	−27.6569	−0.884821	−0.442411	+	0.896813i	$0.645877\pi$
$978$	14.8284	0.474161
$979$	−4.00000	−0.127841
$980$	0	0
$981$	−11.6569	−0.372175
$982$	43.4558	1.38673
$983$	36.4853	1.16370	0.581850	−	0.813296i	$-0.302329\pi$
$983$	36.4853	1.16370	0.581850	+	0.813296i	$0.302329\pi$
$984$	−7.65685	−0.244092
$985$	0	0
$986$	−8.00000	−0.254772
$987$	−4.68629	−0.149166
$988$	0	0
$989$	−81.9411	−2.60558
$990$	0	0
$991$	−33.9411	−1.07818	−0.539088	−	0.842250i	$-0.681231\pi$
$991$	−33.9411	−1.07818	−0.539088	+	0.842250i	$0.681231\pi$
$992$	−1.00000	−0.0317500
$993$	12.4853	0.396208
$994$	2.34315	0.0743201
$995$	0	0
$996$	−1.65685	−0.0524994
$997$	10.2843	0.325706	0.162853	−	0.986650i	$-0.447930\pi$
$997$	10.2843	0.325706	0.162853	+	0.986650i	$0.447930\pi$
$998$	−18.1421	−0.574279
$999$	−10.4853	−0.331740

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4650.2.a.cc.1.2		2
5.2	odd	4		930.2.d.g.559.2	✓	4
5.3	odd	4		930.2.d.g.559.3	yes	4
5.4	even	2		4650.2.a.cf.1.1		2
15.2	even	4		2790.2.d.i.559.4		4
15.8	even	4		2790.2.d.i.559.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.d.g.559.2	✓	4	5.2	odd	4
930.2.d.g.559.3	yes	4	5.3	odd	4
2790.2.d.i.559.1		4	15.8	even	4
2790.2.d.i.559.4		4	15.2	even	4
4650.2.a.cc.1.2		2	1.1	even	1	trivial
4650.2.a.cf.1.1		2	5.4	even	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:	\( N \)	\(=\)	\( 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4650.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +0.828427 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +0.828427 q^{7} -1.00000 q^{8} +1.00000 q^{9} -0.828427 q^{11} +1.00000 q^{12} +4.82843 q^{13} -0.828427 q^{14} +1.00000 q^{16} +0.828427 q^{17} -1.00000 q^{18} +0.828427 q^{21} +0.828427 q^{22} +8.48528 q^{23} -1.00000 q^{24} -4.82843 q^{26} +1.00000 q^{27} +0.828427 q^{28} +9.65685 q^{29} +1.00000 q^{31} -1.00000 q^{32} -0.828427 q^{33} -0.828427 q^{34} +1.00000 q^{36} -10.4853 q^{37} +4.82843 q^{39} +7.65685 q^{41} -0.828427 q^{42} -9.65685 q^{43} -0.828427 q^{44} -8.48528 q^{46} -5.65685 q^{47} +1.00000 q^{48} -6.31371 q^{49} +0.828427 q^{51} +4.82843 q^{52} +0.343146 q^{53} -1.00000 q^{54} -0.828427 q^{56} -9.65685 q^{58} +3.17157 q^{59} +0.828427 q^{61} -1.00000 q^{62} +0.828427 q^{63} +1.00000 q^{64} +0.828427 q^{66} +9.17157 q^{67} +0.828427 q^{68} +8.48528 q^{69} -2.82843 q^{71} -1.00000 q^{72} -13.6569 q^{73} +10.4853 q^{74} -0.686292 q^{77} -4.82843 q^{78} +11.3137 q^{79} +1.00000 q^{81} -7.65685 q^{82} -1.65685 q^{83} +0.828427 q^{84} +9.65685 q^{86} +9.65685 q^{87} +0.828427 q^{88} +4.82843 q^{89} +4.00000 q^{91} +8.48528 q^{92} +1.00000 q^{93} +5.65685 q^{94} -1.00000 q^{96} -11.3137 q^{97} +6.31371 q^{98} -0.828427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 4 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - 4 * q^7 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 4 q^{7} - 2 q^{8} + 2 q^{9} + 4 q^{11} + 2 q^{12} + 4 q^{13} + 4 q^{14} + 2 q^{16} - 4 q^{17} - 2 q^{18} - 4 q^{21} - 4 q^{22} - 2 q^{24} - 4 q^{26} + 2 q^{27} - 4 q^{28} + 8 q^{29} + 2 q^{31} - 2 q^{32} + 4 q^{33} + 4 q^{34} + 2 q^{36} - 4 q^{37} + 4 q^{39} + 4 q^{41} + 4 q^{42} - 8 q^{43} + 4 q^{44} + 2 q^{48} + 10 q^{49} - 4 q^{51} + 4 q^{52} + 12 q^{53} - 2 q^{54} + 4 q^{56} - 8 q^{58} + 12 q^{59} - 4 q^{61} - 2 q^{62} - 4 q^{63} + 2 q^{64} - 4 q^{66} + 24 q^{67} - 4 q^{68} - 2 q^{72} - 16 q^{73} + 4 q^{74} - 24 q^{77} - 4 q^{78} + 2 q^{81} - 4 q^{82} + 8 q^{83} - 4 q^{84} + 8 q^{86} + 8 q^{87} - 4 q^{88} + 4 q^{89} + 8 q^{91} + 2 q^{93} - 2 q^{96} - 10 q^{98} + 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - 4 * q^7 - 2 * q^8 + 2 * q^9 + 4 * q^11 + 2 * q^12 + 4 * q^13 + 4 * q^14 + 2 * q^16 - 4 * q^17 - 2 * q^18 - 4 * q^21 - 4 * q^22 - 2 * q^24 - 4 * q^26 + 2 * q^27 - 4 * q^28 + 8 * q^29 + 2 * q^31 - 2 * q^32 + 4 * q^33 + 4 * q^34 + 2 * q^36 - 4 * q^37 + 4 * q^39 + 4 * q^41 + 4 * q^42 - 8 * q^43 + 4 * q^44 + 2 * q^48 + 10 * q^49 - 4 * q^51 + 4 * q^52 + 12 * q^53 - 2 * q^54 + 4 * q^56 - 8 * q^58 + 12 * q^59 - 4 * q^61 - 2 * q^62 - 4 * q^63 + 2 * q^64 - 4 * q^66 + 24 * q^67 - 4 * q^68 - 2 * q^72 - 16 * q^73 + 4 * q^74 - 24 * q^77 - 4 * q^78 + 2 * q^81 - 4 * q^82 + 8 * q^83 - 4 * q^84 + 8 * q^86 + 8 * q^87 - 4 * q^88 + 4 * q^89 + 8 * q^91 + 2 * q^93 - 2 * q^96 - 10 * q^98 + 4 * q^99

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