LMFDB - Embedded newform 1950.2.a.bd.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1950,2,Mod(1,1950)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1950, 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("1950.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1950.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:	$15.5708283941$
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 390)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	1950.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +2.82843 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +2.82843 q^{7} -1.00000 q^{8} +1.00000 q^{9} +5.65685 q^{11} -1.00000 q^{12} +1.00000 q^{13} -2.82843 q^{14} +1.00000 q^{16} -0.828427 q^{17} -1.00000 q^{18} +2.82843 q^{19} -2.82843 q^{21} -5.65685 q^{22} +8.48528 q^{23} +1.00000 q^{24} -1.00000 q^{26} -1.00000 q^{27} +2.82843 q^{28} -8.82843 q^{29} +4.00000 q^{31} -1.00000 q^{32} -5.65685 q^{33} +0.828427 q^{34} +1.00000 q^{36} +11.6569 q^{37} -2.82843 q^{38} -1.00000 q^{39} -7.65685 q^{41} +2.82843 q^{42} -9.65685 q^{43} +5.65685 q^{44} -8.48528 q^{46} +8.00000 q^{47} -1.00000 q^{48} +1.00000 q^{49} +0.828427 q^{51} +1.00000 q^{52} -13.3137 q^{53} +1.00000 q^{54} -2.82843 q^{56} -2.82843 q^{57} +8.82843 q^{58} -2.34315 q^{59} +6.00000 q^{61} -4.00000 q^{62} +2.82843 q^{63} +1.00000 q^{64} +5.65685 q^{66} -5.65685 q^{67} -0.828427 q^{68} -8.48528 q^{69} -5.65685 q^{71} -1.00000 q^{72} +14.4853 q^{73} -11.6569 q^{74} +2.82843 q^{76} +16.0000 q^{77} +1.00000 q^{78} +2.34315 q^{79} +1.00000 q^{81} +7.65685 q^{82} -6.34315 q^{83} -2.82843 q^{84} +9.65685 q^{86} +8.82843 q^{87} -5.65685 q^{88} -15.6569 q^{89} +2.82843 q^{91} +8.48528 q^{92} -4.00000 q^{93} -8.00000 q^{94} +1.00000 q^{96} +3.17157 q^{97} -1.00000 q^{98} +5.65685 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - 2 * q^8 + 2 * q^9	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{8} + 2 q^{9} - 2 q^{12} + 2 q^{13} + 2 q^{16} + 4 q^{17} - 2 q^{18} + 2 q^{24} - 2 q^{26} - 2 q^{27} - 12 q^{29} + 8 q^{31} - 2 q^{32} - 4 q^{34} + 2 q^{36} + 12 q^{37} - 2 q^{39} - 4 q^{41} - 8 q^{43} + 16 q^{47} - 2 q^{48} + 2 q^{49} - 4 q^{51} + 2 q^{52} - 4 q^{53} + 2 q^{54} + 12 q^{58} - 16 q^{59} + 12 q^{61} - 8 q^{62} + 2 q^{64} + 4 q^{68} - 2 q^{72} + 12 q^{73} - 12 q^{74} + 32 q^{77} + 2 q^{78} + 16 q^{79} + 2 q^{81} + 4 q^{82} - 24 q^{83} + 8 q^{86} + 12 q^{87} - 20 q^{89} - 8 q^{93} - 16 q^{94} + 2 q^{96} + 12 q^{97} - 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - 2 * q^8 + 2 * q^9 - 2 * q^12 + 2 * q^13 + 2 * q^16 + 4 * q^17 - 2 * q^18 + 2 * q^24 - 2 * q^26 - 2 * q^27 - 12 * q^29 + 8 * q^31 - 2 * q^32 - 4 * q^34 + 2 * q^36 + 12 * q^37 - 2 * q^39 - 4 * q^41 - 8 * q^43 + 16 * q^47 - 2 * q^48 + 2 * q^49 - 4 * q^51 + 2 * q^52 - 4 * q^53 + 2 * q^54 + 12 * q^58 - 16 * q^59 + 12 * q^61 - 8 * q^62 + 2 * q^64 + 4 * q^68 - 2 * q^72 + 12 * q^73 - 12 * q^74 + 32 * q^77 + 2 * q^78 + 16 * q^79 + 2 * q^81 + 4 * q^82 - 24 * q^83 + 8 * q^86 + 12 * q^87 - 20 * q^89 - 8 * q^93 - 16 * q^94 + 2 * q^96 + 12 * q^97 - 2 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	2.82843	1.06904	0.534522	−	0.845154i	$-0.320491\pi$
$7$	2.82843	1.06904	0.534522	+	0.845154i	$0.320491\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	5.65685	1.70561	0.852803	−	0.522233i	$-0.174901\pi$
$11$	5.65685	1.70561	0.852803	+	0.522233i	$0.174901\pi$
$12$	−1.00000	−0.288675
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−2.82843	−0.755929
$15$	0	0
$16$	1.00000	0.250000
$17$	−0.828427	−0.200923	−0.100462	−	0.994941i	$-0.532032\pi$
$17$	−0.828427	−0.200923	−0.100462	+	0.994941i	$0.532032\pi$
$18$	−1.00000	−0.235702
$19$	2.82843	0.648886	0.324443	−	0.945905i	$-0.394823\pi$
$19$	2.82843	0.648886	0.324443	+	0.945905i	$0.394823\pi$
$20$	0	0
$21$	−2.82843	−0.617213
$22$	−5.65685	−1.20605
$23$	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
$26$	−1.00000	−0.196116
$27$	−1.00000	−0.192450
$28$	2.82843	0.534522
$29$	−8.82843	−1.63940	−0.819699	−	0.572795i	$-0.805859\pi$
$29$	−8.82843	−1.63940	−0.819699	+	0.572795i	$0.805859\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	−1.00000	−0.176777
$33$	−5.65685	−0.984732
$34$	0.828427	0.142074
$35$	0	0
$36$	1.00000	0.166667
$37$	11.6569	1.91638	0.958188	−	0.286141i	$-0.0923726\pi$
$37$	11.6569	1.91638	0.958188	+	0.286141i	$0.0923726\pi$
$38$	−2.82843	−0.458831
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−7.65685	−1.19580	−0.597900	−	0.801571i	$-0.703998\pi$
$41$	−7.65685	−1.19580	−0.597900	+	0.801571i	$0.703998\pi$
$42$	2.82843	0.436436
$43$	−9.65685	−1.47266	−0.736328	−	0.676625i	$-0.763442\pi$
$43$	−9.65685	−1.47266	−0.736328	+	0.676625i	$0.763442\pi$
$44$	5.65685	0.852803
$45$	0	0
$46$	−8.48528	−1.25109
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	−1.00000	−0.144338
$49$	1.00000	0.142857
$50$	0	0
$51$	0.828427	0.116003
$52$	1.00000	0.138675
$53$	−13.3137	−1.82878	−0.914389	−	0.404836i	$-0.867329\pi$
$53$	−13.3137	−1.82878	−0.914389	+	0.404836i	$0.867329\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	−2.82843	−0.377964
$57$	−2.82843	−0.374634
$58$	8.82843	1.15923
$59$	−2.34315	−0.305052	−0.152526	−	0.988299i	$-0.548741\pi$
$59$	−2.34315	−0.305052	−0.152526	+	0.988299i	$0.548741\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	−4.00000	−0.508001
$63$	2.82843	0.356348
$64$	1.00000	0.125000
$65$	0	0
$66$	5.65685	0.696311
$67$	−5.65685	−0.691095	−0.345547	−	0.938401i	$-0.612307\pi$
$67$	−5.65685	−0.691095	−0.345547	+	0.938401i	$0.612307\pi$
$68$	−0.828427	−0.100462
$69$	−8.48528	−1.02151
$70$	0	0
$71$	−5.65685	−0.671345	−0.335673	−	0.941979i	$-0.608964\pi$
$71$	−5.65685	−0.671345	−0.335673	+	0.941979i	$0.608964\pi$
$72$	−1.00000	−0.117851
$73$	14.4853	1.69537	0.847687	−	0.530497i	$-0.177995\pi$
$73$	14.4853	1.69537	0.847687	+	0.530497i	$0.177995\pi$
$74$	−11.6569	−1.35508
$75$	0	0
$76$	2.82843	0.324443
$77$	16.0000	1.82337
$78$	1.00000	0.113228
$79$	2.34315	0.263624	0.131812	−	0.991275i	$-0.457920\pi$
$79$	2.34315	0.263624	0.131812	+	0.991275i	$0.457920\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	7.65685	0.845558
$83$	−6.34315	−0.696251	−0.348125	−	0.937448i	$-0.613182\pi$
$83$	−6.34315	−0.696251	−0.348125	+	0.937448i	$0.613182\pi$
$84$	−2.82843	−0.308607
$85$	0	0
$86$	9.65685	1.04133
$87$	8.82843	0.946507
$88$	−5.65685	−0.603023
$89$	−15.6569	−1.65962	−0.829812	−	0.558044i	$-0.811552\pi$
$89$	−15.6569	−1.65962	−0.829812	+	0.558044i	$0.811552\pi$
$90$	0	0
$91$	2.82843	0.296500
$92$	8.48528	0.884652
$93$	−4.00000	−0.414781
$94$	−8.00000	−0.825137
$95$	0	0
$96$	1.00000	0.102062
$97$	3.17157	0.322024	0.161012	−	0.986952i	$-0.448524\pi$
$97$	3.17157	0.322024	0.161012	+	0.986952i	$0.448524\pi$
$98$	−1.00000	−0.101015
$99$	5.65685	0.568535
$100$	0	0
$101$	16.1421	1.60620	0.803101	−	0.595843i	$-0.203182\pi$
$101$	16.1421	1.60620	0.803101	+	0.595843i	$0.203182\pi$
$102$	−0.828427	−0.0820265
$103$	1.65685	0.163255	0.0816274	−	0.996663i	$-0.473988\pi$
$103$	1.65685	0.163255	0.0816274	+	0.996663i	$0.473988\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	13.3137	1.29314
$107$	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
$109$	8.82843	0.845610	0.422805	−	0.906221i	$-0.361046\pi$
$109$	8.82843	0.845610	0.422805	+	0.906221i	$0.361046\pi$
$110$	0	0
$111$	−11.6569	−1.10642
$112$	2.82843	0.267261
$113$	−6.48528	−0.610084	−0.305042	−	0.952339i	$-0.598670\pi$
$113$	−6.48528	−0.610084	−0.305042	+	0.952339i	$0.598670\pi$
$114$	2.82843	0.264906
$115$	0	0
$116$	−8.82843	−0.819699
$117$	1.00000	0.0924500
$118$	2.34315	0.215704
$119$	−2.34315	−0.214796
$120$	0	0
$121$	21.0000	1.90909
$122$	−6.00000	−0.543214
$123$	7.65685	0.690395
$124$	4.00000	0.359211
$125$	0	0
$126$	−2.82843	−0.251976
$127$	−9.65685	−0.856907	−0.428454	−	0.903564i	$-0.640941\pi$
$127$	−9.65685	−0.856907	−0.428454	+	0.903564i	$0.640941\pi$
$128$	−1.00000	−0.0883883
$129$	9.65685	0.850239
$130$	0	0
$131$	−6.14214	−0.536641	−0.268320	−	0.963330i	$-0.586469\pi$
$131$	−6.14214	−0.536641	−0.268320	+	0.963330i	$0.586469\pi$
$132$	−5.65685	−0.492366
$133$	8.00000	0.693688
$134$	5.65685	0.488678
$135$	0	0
$136$	0.828427	0.0710370
$137$	17.3137	1.47921	0.739605	−	0.673041i	$-0.235012\pi$
$137$	17.3137	1.47921	0.739605	+	0.673041i	$0.235012\pi$
$138$	8.48528	0.722315
$139$	6.34315	0.538019	0.269009	−	0.963138i	$-0.413304\pi$
$139$	6.34315	0.538019	0.269009	+	0.963138i	$0.413304\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	5.65685	0.474713
$143$	5.65685	0.473050
$144$	1.00000	0.0833333
$145$	0	0
$146$	−14.4853	−1.19881
$147$	−1.00000	−0.0824786
$148$	11.6569	0.958188
$149$	3.65685	0.299581	0.149791	−	0.988718i	$-0.452140\pi$
$149$	3.65685	0.299581	0.149791	+	0.988718i	$0.452140\pi$
$150$	0	0
$151$	12.0000	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	−2.82843	−0.229416
$153$	−0.828427	−0.0669744
$154$	−16.0000	−1.28932
$155$	0	0
$156$	−1.00000	−0.0800641
$157$	−5.31371	−0.424080	−0.212040	−	0.977261i	$-0.568011\pi$
$157$	−5.31371	−0.424080	−0.212040	+	0.977261i	$0.568011\pi$
$158$	−2.34315	−0.186411
$159$	13.3137	1.05585
$160$	0	0
$161$	24.0000	1.89146
$162$	−1.00000	−0.0785674
$163$	−11.3137	−0.886158	−0.443079	−	0.896483i	$-0.646114\pi$
$163$	−11.3137	−0.886158	−0.443079	+	0.896483i	$0.646114\pi$
$164$	−7.65685	−0.597900
$165$	0	0
$166$	6.34315	0.492324
$167$	−8.97056	−0.694163	−0.347081	−	0.937835i	$-0.612827\pi$
$167$	−8.97056	−0.694163	−0.347081	+	0.937835i	$0.612827\pi$
$168$	2.82843	0.218218
$169$	1.00000	0.0769231
$170$	0	0
$171$	2.82843	0.216295
$172$	−9.65685	−0.736328
$173$	9.31371	0.708108	0.354054	−	0.935225i	$-0.384803\pi$
$173$	9.31371	0.708108	0.354054	+	0.935225i	$0.384803\pi$
$174$	−8.82843	−0.669281
$175$	0	0
$176$	5.65685	0.426401
$177$	2.34315	0.176122
$178$	15.6569	1.17353
$179$	7.51472	0.561676	0.280838	−	0.959755i	$-0.409388\pi$
$179$	7.51472	0.561676	0.280838	+	0.959755i	$0.409388\pi$
$180$	0	0
$181$	−7.65685	−0.569129	−0.284565	−	0.958657i	$-0.591849\pi$
$181$	−7.65685	−0.569129	−0.284565	+	0.958657i	$0.591849\pi$
$182$	−2.82843	−0.209657
$183$	−6.00000	−0.443533
$184$	−8.48528	−0.625543
$185$	0	0
$186$	4.00000	0.293294
$187$	−4.68629	−0.342696
$188$	8.00000	0.583460
$189$	−2.82843	−0.205738
$190$	0	0
$191$	11.3137	0.818631	0.409316	−	0.912393i	$-0.365768\pi$
$191$	11.3137	0.818631	0.409316	+	0.912393i	$0.365768\pi$
$192$	−1.00000	−0.0721688
$193$	−2.48528	−0.178894	−0.0894472	−	0.995992i	$-0.528510\pi$
$193$	−2.48528	−0.178894	−0.0894472	+	0.995992i	$0.528510\pi$
$194$	−3.17157	−0.227706
$195$	0	0
$196$	1.00000	0.0714286
$197$	13.3137	0.948562	0.474281	−	0.880373i	$-0.342708\pi$
$197$	13.3137	0.948562	0.474281	+	0.880373i	$0.342708\pi$
$198$	−5.65685	−0.402015
$199$	10.3431	0.733206	0.366603	−	0.930377i	$-0.380521\pi$
$199$	10.3431	0.733206	0.366603	+	0.930377i	$0.380521\pi$
$200$	0	0
$201$	5.65685	0.399004
$202$	−16.1421	−1.13576
$203$	−24.9706	−1.75259
$204$	0.828427	0.0580015
$205$	0	0
$206$	−1.65685	−0.115439
$207$	8.48528	0.589768
$208$	1.00000	0.0693375
$209$	16.0000	1.10674
$210$	0	0
$211$	0.686292	0.0472463	0.0236231	−	0.999721i	$-0.492480\pi$
$211$	0.686292	0.0472463	0.0236231	+	0.999721i	$0.492480\pi$
$212$	−13.3137	−0.914389
$213$	5.65685	0.387601
$214$	−4.00000	−0.273434
$215$	0	0
$216$	1.00000	0.0680414
$217$	11.3137	0.768025
$218$	−8.82843	−0.597937
$219$	−14.4853	−0.978825
$220$	0	0
$221$	−0.828427	−0.0557260
$222$	11.6569	0.782357
$223$	−10.8284	−0.725125	−0.362563	−	0.931959i	$-0.618098\pi$
$223$	−10.8284	−0.725125	−0.362563	+	0.931959i	$0.618098\pi$
$224$	−2.82843	−0.188982
$225$	0	0
$226$	6.48528	0.431394
$227$	−4.00000	−0.265489	−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000	−0.265489	−0.132745	+	0.991150i	$0.542379\pi$
$228$	−2.82843	−0.187317
$229$	4.14214	0.273720	0.136860	−	0.990590i	$-0.456299\pi$
$229$	4.14214	0.273720	0.136860	+	0.990590i	$0.456299\pi$
$230$	0	0
$231$	−16.0000	−1.05272
$232$	8.82843	0.579615
$233$	−5.51472	−0.361281	−0.180641	−	0.983549i	$-0.557817\pi$
$233$	−5.51472	−0.361281	−0.180641	+	0.983549i	$0.557817\pi$
$234$	−1.00000	−0.0653720
$235$	0	0
$236$	−2.34315	−0.152526
$237$	−2.34315	−0.152204
$238$	2.34315	0.151884
$239$	16.0000	1.03495	0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000	1.03495	0.517477	+	0.855697i	$0.326871\pi$
$240$	0	0
$241$	5.31371	0.342286	0.171143	−	0.985246i	$-0.445254\pi$
$241$	5.31371	0.342286	0.171143	+	0.985246i	$0.445254\pi$
$242$	−21.0000	−1.34993
$243$	−1.00000	−0.0641500
$244$	6.00000	0.384111
$245$	0	0
$246$	−7.65685	−0.488183
$247$	2.82843	0.179969
$248$	−4.00000	−0.254000
$249$	6.34315	0.401981
$250$	0	0
$251$	10.8284	0.683484	0.341742	−	0.939794i	$-0.388983\pi$
$251$	10.8284	0.683484	0.341742	+	0.939794i	$0.388983\pi$
$252$	2.82843	0.178174
$253$	48.0000	3.01773
$254$	9.65685	0.605925
$255$	0	0
$256$	1.00000	0.0625000
$257$	4.82843	0.301189	0.150595	−	0.988596i	$-0.451881\pi$
$257$	4.82843	0.301189	0.150595	+	0.988596i	$0.451881\pi$
$258$	−9.65685	−0.601209
$259$	32.9706	2.04869
$260$	0	0
$261$	−8.82843	−0.546466
$262$	6.14214	0.379462
$263$	−16.4853	−1.01653	−0.508263	−	0.861202i	$-0.669712\pi$
$263$	−16.4853	−1.01653	−0.508263	+	0.861202i	$0.669712\pi$
$264$	5.65685	0.348155
$265$	0	0
$266$	−8.00000	−0.490511
$267$	15.6569	0.958184
$268$	−5.65685	−0.345547
$269$	−14.4853	−0.883183	−0.441592	−	0.897216i	$-0.645586\pi$
$269$	−14.4853	−0.883183	−0.441592	+	0.897216i	$0.645586\pi$
$270$	0	0
$271$	7.31371	0.444276	0.222138	−	0.975015i	$-0.428696\pi$
$271$	7.31371	0.444276	0.222138	+	0.975015i	$0.428696\pi$
$272$	−0.828427	−0.0502308
$273$	−2.82843	−0.171184
$274$	−17.3137	−1.04596
$275$	0	0
$276$	−8.48528	−0.510754
$277$	−26.0000	−1.56219	−0.781094	−	0.624413i	$-0.785338\pi$
$277$	−26.0000	−1.56219	−0.781094	+	0.624413i	$0.785338\pi$
$278$	−6.34315	−0.380437
$279$	4.00000	0.239474
$280$	0	0
$281$	8.34315	0.497710	0.248855	−	0.968541i	$-0.419946\pi$
$281$	8.34315	0.497710	0.248855	+	0.968541i	$0.419946\pi$
$282$	8.00000	0.476393
$283$	17.6569	1.04959	0.524796	−	0.851228i	$-0.324142\pi$
$283$	17.6569	1.04959	0.524796	+	0.851228i	$0.324142\pi$
$284$	−5.65685	−0.335673
$285$	0	0
$286$	−5.65685	−0.334497
$287$	−21.6569	−1.27836
$288$	−1.00000	−0.0589256
$289$	−16.3137	−0.959630
$290$	0	0
$291$	−3.17157	−0.185921
$292$	14.4853	0.847687
$293$	16.6274	0.971384	0.485692	−	0.874130i	$-0.338568\pi$
$293$	16.6274	0.971384	0.485692	+	0.874130i	$0.338568\pi$
$294$	1.00000	0.0583212
$295$	0	0
$296$	−11.6569	−0.677541
$297$	−5.65685	−0.328244
$298$	−3.65685	−0.211836
$299$	8.48528	0.490716
$300$	0	0
$301$	−27.3137	−1.57434
$302$	−12.0000	−0.690522
$303$	−16.1421	−0.927341
$304$	2.82843	0.162221
$305$	0	0
$306$	0.828427	0.0473580
$307$	21.6569	1.23602	0.618011	−	0.786169i	$-0.287939\pi$
$307$	21.6569	1.23602	0.618011	+	0.786169i	$0.287939\pi$
$308$	16.0000	0.911685
$309$	−1.65685	−0.0942551
$310$	0	0
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	1.00000	0.0566139
$313$	30.9706	1.75056	0.875280	−	0.483617i	$-0.160677\pi$
$313$	30.9706	1.75056	0.875280	+	0.483617i	$0.160677\pi$
$314$	5.31371	0.299870
$315$	0	0
$316$	2.34315	0.131812
$317$	−25.3137	−1.42176	−0.710880	−	0.703314i	$-0.751703\pi$
$317$	−25.3137	−1.42176	−0.710880	+	0.703314i	$0.751703\pi$
$318$	−13.3137	−0.746596
$319$	−49.9411	−2.79617
$320$	0	0
$321$	−4.00000	−0.223258
$322$	−24.0000	−1.33747
$323$	−2.34315	−0.130376
$324$	1.00000	0.0555556
$325$	0	0
$326$	11.3137	0.626608
$327$	−8.82843	−0.488213
$328$	7.65685	0.422779
$329$	22.6274	1.24749
$330$	0	0
$331$	−8.48528	−0.466393	−0.233197	−	0.972430i	$-0.574919\pi$
$331$	−8.48528	−0.466393	−0.233197	+	0.972430i	$0.574919\pi$
$332$	−6.34315	−0.348125
$333$	11.6569	0.638792
$334$	8.97056	0.490847
$335$	0	0
$336$	−2.82843	−0.154303
$337$	−10.9706	−0.597605	−0.298802	−	0.954315i	$-0.596587\pi$
$337$	−10.9706	−0.597605	−0.298802	+	0.954315i	$0.596587\pi$
$338$	−1.00000	−0.0543928
$339$	6.48528	0.352232
$340$	0	0
$341$	22.6274	1.22534
$342$	−2.82843	−0.152944
$343$	−16.9706	−0.916324
$344$	9.65685	0.520663
$345$	0	0
$346$	−9.31371	−0.500708
$347$	9.65685	0.518407	0.259204	−	0.965823i	$-0.416540\pi$
$347$	9.65685	0.518407	0.259204	+	0.965823i	$0.416540\pi$
$348$	8.82843	0.473253
$349$	12.1421	0.649954	0.324977	−	0.945722i	$-0.394644\pi$
$349$	12.1421	0.649954	0.324977	+	0.945722i	$0.394644\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	−5.65685	−0.301511
$353$	−5.31371	−0.282820	−0.141410	−	0.989951i	$-0.545164\pi$
$353$	−5.31371	−0.282820	−0.141410	+	0.989951i	$0.545164\pi$
$354$	−2.34315	−0.124537
$355$	0	0
$356$	−15.6569	−0.829812
$357$	2.34315	0.124012
$358$	−7.51472	−0.397165
$359$	−28.2843	−1.49279	−0.746393	−	0.665505i	$-0.768216\pi$
$359$	−28.2843	−1.49279	−0.746393	+	0.665505i	$0.768216\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	7.65685	0.402435
$363$	−21.0000	−1.10221
$364$	2.82843	0.148250
$365$	0	0
$366$	6.00000	0.313625
$367$	−25.6569	−1.33928	−0.669638	−	0.742687i	$-0.733551\pi$
$367$	−25.6569	−1.33928	−0.669638	+	0.742687i	$0.733551\pi$
$368$	8.48528	0.442326
$369$	−7.65685	−0.398600
$370$	0	0
$371$	−37.6569	−1.95505
$372$	−4.00000	−0.207390
$373$	2.68629	0.139091	0.0695455	−	0.997579i	$-0.477845\pi$
$373$	2.68629	0.139091	0.0695455	+	0.997579i	$0.477845\pi$
$374$	4.68629	0.242322
$375$	0	0
$376$	−8.00000	−0.412568
$377$	−8.82843	−0.454687
$378$	2.82843	0.145479
$379$	−7.51472	−0.386005	−0.193003	−	0.981198i	$-0.561823\pi$
$379$	−7.51472	−0.386005	−0.193003	+	0.981198i	$0.561823\pi$
$380$	0	0
$381$	9.65685	0.494736
$382$	−11.3137	−0.578860
$383$	29.6569	1.51539	0.757697	−	0.652606i	$-0.226324\pi$
$383$	29.6569	1.51539	0.757697	+	0.652606i	$0.226324\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	2.48528	0.126497
$387$	−9.65685	−0.490885
$388$	3.17157	0.161012
$389$	−6.48528	−0.328817	−0.164408	−	0.986392i	$-0.552571\pi$
$389$	−6.48528	−0.328817	−0.164408	+	0.986392i	$0.552571\pi$
$390$	0	0
$391$	−7.02944	−0.355494
$392$	−1.00000	−0.0505076
$393$	6.14214	0.309830
$394$	−13.3137	−0.670735
$395$	0	0
$396$	5.65685	0.284268
$397$	−30.2843	−1.51992	−0.759962	−	0.649968i	$-0.774782\pi$
$397$	−30.2843	−1.51992	−0.759962	+	0.649968i	$0.774782\pi$
$398$	−10.3431	−0.518455
$399$	−8.00000	−0.400501
$400$	0	0
$401$	−26.9706	−1.34685	−0.673423	−	0.739258i	$-0.735177\pi$
$401$	−26.9706	−1.34685	−0.673423	+	0.739258i	$0.735177\pi$
$402$	−5.65685	−0.282138
$403$	4.00000	0.199254
$404$	16.1421	0.803101
$405$	0	0
$406$	24.9706	1.23927
$407$	65.9411	3.26858
$408$	−0.828427	−0.0410133
$409$	−3.65685	−0.180820	−0.0904099	−	0.995905i	$-0.528818\pi$
$409$	−3.65685	−0.180820	−0.0904099	+	0.995905i	$0.528818\pi$
$410$	0	0
$411$	−17.3137	−0.854022
$412$	1.65685	0.0816274
$413$	−6.62742	−0.326114
$414$	−8.48528	−0.417029
$415$	0	0
$416$	−1.00000	−0.0490290
$417$	−6.34315	−0.310625
$418$	−16.0000	−0.782586
$419$	−10.8284	−0.529003	−0.264502	−	0.964385i	$-0.585207\pi$
$419$	−10.8284	−0.529003	−0.264502	+	0.964385i	$0.585207\pi$
$420$	0	0
$421$	−24.1421	−1.17662	−0.588308	−	0.808637i	$-0.700206\pi$
$421$	−24.1421	−1.17662	−0.588308	+	0.808637i	$0.700206\pi$
$422$	−0.686292	−0.0334081
$423$	8.00000	0.388973
$424$	13.3137	0.646571
$425$	0	0
$426$	−5.65685	−0.274075
$427$	16.9706	0.821263
$428$	4.00000	0.193347
$429$	−5.65685	−0.273115
$430$	0	0
$431$	−16.0000	−0.770693	−0.385346	−	0.922772i	$-0.625918\pi$
$431$	−16.0000	−0.770693	−0.385346	+	0.922772i	$0.625918\pi$
$432$	−1.00000	−0.0481125
$433$	22.9706	1.10389	0.551947	−	0.833879i	$-0.313885\pi$
$433$	22.9706	1.10389	0.551947	+	0.833879i	$0.313885\pi$
$434$	−11.3137	−0.543075
$435$	0	0
$436$	8.82843	0.422805
$437$	24.0000	1.14808
$438$	14.4853	0.692134
$439$	22.6274	1.07995	0.539974	−	0.841682i	$-0.318434\pi$
$439$	22.6274	1.07995	0.539974	+	0.841682i	$0.318434\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0.828427	0.0394043
$443$	−30.3431	−1.44165	−0.720823	−	0.693119i	$-0.756236\pi$
$443$	−30.3431	−1.44165	−0.720823	+	0.693119i	$0.756236\pi$
$444$	−11.6569	−0.553210
$445$	0	0
$446$	10.8284	0.512741
$447$	−3.65685	−0.172963
$448$	2.82843	0.133631
$449$	26.2843	1.24043	0.620216	−	0.784431i	$-0.287045\pi$
$449$	26.2843	1.24043	0.620216	+	0.784431i	$0.287045\pi$
$450$	0	0
$451$	−43.3137	−2.03956
$452$	−6.48528	−0.305042
$453$	−12.0000	−0.563809
$454$	4.00000	0.187729
$455$	0	0
$456$	2.82843	0.132453
$457$	−20.8284	−0.974313	−0.487156	−	0.873315i	$-0.661966\pi$
$457$	−20.8284	−0.974313	−0.487156	+	0.873315i	$0.661966\pi$
$458$	−4.14214	−0.193549
$459$	0.828427	0.0386677
$460$	0	0
$461$	14.0000	0.652045	0.326023	−	0.945362i	$-0.394291\pi$
$461$	14.0000	0.652045	0.326023	+	0.945362i	$0.394291\pi$
$462$	16.0000	0.744387
$463$	3.79899	0.176554	0.0882770	−	0.996096i	$-0.471864\pi$
$463$	3.79899	0.176554	0.0882770	+	0.996096i	$0.471864\pi$
$464$	−8.82843	−0.409849
$465$	0	0
$466$	5.51472	0.255464
$467$	−7.31371	−0.338438	−0.169219	−	0.985578i	$-0.554125\pi$
$467$	−7.31371	−0.338438	−0.169219	+	0.985578i	$0.554125\pi$
$468$	1.00000	0.0462250
$469$	−16.0000	−0.738811
$470$	0	0
$471$	5.31371	0.244843
$472$	2.34315	0.107852
$473$	−54.6274	−2.51177
$474$	2.34315	0.107624
$475$	0	0
$476$	−2.34315	−0.107398
$477$	−13.3137	−0.609593
$478$	−16.0000	−0.731823
$479$	−11.3137	−0.516937	−0.258468	−	0.966020i	$-0.583218\pi$
$479$	−11.3137	−0.516937	−0.258468	+	0.966020i	$0.583218\pi$
$480$	0	0
$481$	11.6569	0.531507
$482$	−5.31371	−0.242033
$483$	−24.0000	−1.09204
$484$	21.0000	0.954545
$485$	0	0
$486$	1.00000	0.0453609
$487$	−16.4853	−0.747019	−0.373510	−	0.927626i	$-0.621846\pi$
$487$	−16.4853	−0.747019	−0.373510	+	0.927626i	$0.621846\pi$
$488$	−6.00000	−0.271607
$489$	11.3137	0.511624
$490$	0	0
$491$	−38.1421	−1.72133	−0.860665	−	0.509171i	$-0.829952\pi$
$491$	−38.1421	−1.72133	−0.860665	+	0.509171i	$0.829952\pi$
$492$	7.65685	0.345198
$493$	7.31371	0.329393
$494$	−2.82843	−0.127257
$495$	0	0
$496$	4.00000	0.179605
$497$	−16.0000	−0.717698
$498$	−6.34315	−0.284243
$499$	0.485281	0.0217242	0.0108621	−	0.999941i	$-0.496542\pi$
$499$	0.485281	0.0217242	0.0108621	+	0.999941i	$0.496542\pi$
$500$	0	0
$501$	8.97056	0.400775
$502$	−10.8284	−0.483296
$503$	23.5147	1.04847	0.524235	−	0.851574i	$-0.324351\pi$
$503$	23.5147	1.04847	0.524235	+	0.851574i	$0.324351\pi$
$504$	−2.82843	−0.125988
$505$	0	0
$506$	−48.0000	−2.13386
$507$	−1.00000	−0.0444116
$508$	−9.65685	−0.428454
$509$	−37.3137	−1.65390	−0.826951	−	0.562275i	$-0.809926\pi$
$509$	−37.3137	−1.65390	−0.826951	+	0.562275i	$0.809926\pi$
$510$	0	0
$511$	40.9706	1.81243
$512$	−1.00000	−0.0441942
$513$	−2.82843	−0.124878
$514$	−4.82843	−0.212973
$515$	0	0
$516$	9.65685	0.425119
$517$	45.2548	1.99031
$518$	−32.9706	−1.44864
$519$	−9.31371	−0.408826
$520$	0	0
$521$	26.9706	1.18160	0.590801	−	0.806817i	$-0.298812\pi$
$521$	26.9706	1.18160	0.590801	+	0.806817i	$0.298812\pi$
$522$	8.82843	0.386410
$523$	10.6274	0.464704	0.232352	−	0.972632i	$-0.425358\pi$
$523$	10.6274	0.464704	0.232352	+	0.972632i	$0.425358\pi$
$524$	−6.14214	−0.268320
$525$	0	0
$526$	16.4853	0.718792
$527$	−3.31371	−0.144347
$528$	−5.65685	−0.246183
$529$	49.0000	2.13043
$530$	0	0
$531$	−2.34315	−0.101684
$532$	8.00000	0.346844
$533$	−7.65685	−0.331655
$534$	−15.6569	−0.677538
$535$	0	0
$536$	5.65685	0.244339
$537$	−7.51472	−0.324284
$538$	14.4853	0.624505
$539$	5.65685	0.243658
$540$	0	0
$541$	14.4853	0.622771	0.311385	−	0.950284i	$-0.399207\pi$
$541$	14.4853	0.622771	0.311385	+	0.950284i	$0.399207\pi$
$542$	−7.31371	−0.314151
$543$	7.65685	0.328587
$544$	0.828427	0.0355185
$545$	0	0
$546$	2.82843	0.121046
$547$	−0.686292	−0.0293437	−0.0146719	−	0.999892i	$-0.504670\pi$
$547$	−0.686292	−0.0293437	−0.0146719	+	0.999892i	$0.504670\pi$
$548$	17.3137	0.739605
$549$	6.00000	0.256074
$550$	0	0
$551$	−24.9706	−1.06378
$552$	8.48528	0.361158
$553$	6.62742	0.281826
$554$	26.0000	1.10463
$555$	0	0
$556$	6.34315	0.269009
$557$	−10.6863	−0.452793	−0.226396	−	0.974035i	$-0.572694\pi$
$557$	−10.6863	−0.452793	−0.226396	+	0.974035i	$0.572694\pi$
$558$	−4.00000	−0.169334
$559$	−9.65685	−0.408441
$560$	0	0
$561$	4.68629	0.197855
$562$	−8.34315	−0.351934
$563$	30.3431	1.27881	0.639406	−	0.768870i	$-0.279181\pi$
$563$	30.3431	1.27881	0.639406	+	0.768870i	$0.279181\pi$
$564$	−8.00000	−0.336861
$565$	0	0
$566$	−17.6569	−0.742173
$567$	2.82843	0.118783
$568$	5.65685	0.237356
$569$	31.6569	1.32712	0.663562	−	0.748121i	$-0.269044\pi$
$569$	31.6569	1.32712	0.663562	+	0.748121i	$0.269044\pi$
$570$	0	0
$571$	20.9706	0.877591	0.438795	−	0.898587i	$-0.355405\pi$
$571$	20.9706	0.877591	0.438795	+	0.898587i	$0.355405\pi$
$572$	5.65685	0.236525
$573$	−11.3137	−0.472637
$574$	21.6569	0.903940
$575$	0	0
$576$	1.00000	0.0416667
$577$	23.4558	0.976480	0.488240	−	0.872710i	$-0.337639\pi$
$577$	23.4558	0.976480	0.488240	+	0.872710i	$0.337639\pi$
$578$	16.3137	0.678561
$579$	2.48528	0.103285
$580$	0	0
$581$	−17.9411	−0.744323
$582$	3.17157	0.131466
$583$	−75.3137	−3.11918
$584$	−14.4853	−0.599405
$585$	0	0
$586$	−16.6274	−0.686872
$587$	−2.62742	−0.108445	−0.0542226	−	0.998529i	$-0.517268\pi$
$587$	−2.62742	−0.108445	−0.0542226	+	0.998529i	$0.517268\pi$
$588$	−1.00000	−0.0412393
$589$	11.3137	0.466173
$590$	0	0
$591$	−13.3137	−0.547653
$592$	11.6569	0.479094
$593$	0.343146	0.0140913	0.00704565	−	0.999975i	$-0.497757\pi$
$593$	0.343146	0.0140913	0.00704565	+	0.999975i	$0.497757\pi$
$594$	5.65685	0.232104
$595$	0	0
$596$	3.65685	0.149791
$597$	−10.3431	−0.423317
$598$	−8.48528	−0.346989
$599$	40.0000	1.63436	0.817178	−	0.576386i	$-0.195537\pi$
$599$	40.0000	1.63436	0.817178	+	0.576386i	$0.195537\pi$
$600$	0	0
$601$	29.3137	1.19573	0.597866	−	0.801596i	$-0.296016\pi$
$601$	29.3137	1.19573	0.597866	+	0.801596i	$0.296016\pi$
$602$	27.3137	1.11322
$603$	−5.65685	−0.230365
$604$	12.0000	0.488273
$605$	0	0
$606$	16.1421	0.655729
$607$	−28.9706	−1.17588	−0.587939	−	0.808905i	$-0.700061\pi$
$607$	−28.9706	−1.17588	−0.587939	+	0.808905i	$0.700061\pi$
$608$	−2.82843	−0.114708
$609$	24.9706	1.01186
$610$	0	0
$611$	8.00000	0.323645
$612$	−0.828427	−0.0334872
$613$	−22.2843	−0.900053	−0.450027	−	0.893015i	$-0.648586\pi$
$613$	−22.2843	−0.900053	−0.450027	+	0.893015i	$0.648586\pi$
$614$	−21.6569	−0.874000
$615$	0	0
$616$	−16.0000	−0.644658
$617$	−2.00000	−0.0805170	−0.0402585	−	0.999189i	$-0.512818\pi$
$617$	−2.00000	−0.0805170	−0.0402585	+	0.999189i	$0.512818\pi$
$618$	1.65685	0.0666485
$619$	−34.8284	−1.39987	−0.699936	−	0.714205i	$-0.746788\pi$
$619$	−34.8284	−1.39987	−0.699936	+	0.714205i	$0.746788\pi$
$620$	0	0
$621$	−8.48528	−0.340503
$622$	24.0000	0.962312
$623$	−44.2843	−1.77421
$624$	−1.00000	−0.0400320
$625$	0	0
$626$	−30.9706	−1.23783
$627$	−16.0000	−0.638978
$628$	−5.31371	−0.212040
$629$	−9.65685	−0.385044
$630$	0	0
$631$	−33.6569	−1.33986	−0.669929	−	0.742425i	$-0.733676\pi$
$631$	−33.6569	−1.33986	−0.669929	+	0.742425i	$0.733676\pi$
$632$	−2.34315	−0.0932053
$633$	−0.686292	−0.0272776
$634$	25.3137	1.00534
$635$	0	0
$636$	13.3137	0.527923
$637$	1.00000	0.0396214
$638$	49.9411	1.97719
$639$	−5.65685	−0.223782
$640$	0	0
$641$	−4.62742	−0.182772	−0.0913860	−	0.995816i	$-0.529130\pi$
$641$	−4.62742	−0.182772	−0.0913860	+	0.995816i	$0.529130\pi$
$642$	4.00000	0.157867
$643$	−39.5980	−1.56159	−0.780796	−	0.624786i	$-0.785186\pi$
$643$	−39.5980	−1.56159	−0.780796	+	0.624786i	$0.785186\pi$
$644$	24.0000	0.945732
$645$	0	0
$646$	2.34315	0.0921898
$647$	−8.48528	−0.333591	−0.166795	−	0.985992i	$-0.553342\pi$
$647$	−8.48528	−0.333591	−0.166795	+	0.985992i	$0.553342\pi$
$648$	−1.00000	−0.0392837
$649$	−13.2548	−0.520298
$650$	0	0
$651$	−11.3137	−0.443419
$652$	−11.3137	−0.443079
$653$	42.2843	1.65471	0.827356	−	0.561678i	$-0.189844\pi$
$653$	42.2843	1.65471	0.827356	+	0.561678i	$0.189844\pi$
$654$	8.82843	0.345219
$655$	0	0
$656$	−7.65685	−0.298950
$657$	14.4853	0.565125
$658$	−22.6274	−0.882109
$659$	−7.51472	−0.292732	−0.146366	−	0.989231i	$-0.546758\pi$
$659$	−7.51472	−0.292732	−0.146366	+	0.989231i	$0.546758\pi$
$660$	0	0
$661$	−8.14214	−0.316692	−0.158346	−	0.987384i	$-0.550616\pi$
$661$	−8.14214	−0.316692	−0.158346	+	0.987384i	$0.550616\pi$
$662$	8.48528	0.329790
$663$	0.828427	0.0321734
$664$	6.34315	0.246162
$665$	0	0
$666$	−11.6569	−0.451694
$667$	−74.9117	−2.90059
$668$	−8.97056	−0.347081
$669$	10.8284	0.418651
$670$	0	0
$671$	33.9411	1.31028
$672$	2.82843	0.109109
$673$	−32.6274	−1.25769	−0.628847	−	0.777529i	$-0.716473\pi$
$673$	−32.6274	−1.25769	−0.628847	+	0.777529i	$0.716473\pi$
$674$	10.9706	0.422570
$675$	0	0
$676$	1.00000	0.0384615
$677$	−12.3431	−0.474386	−0.237193	−	0.971463i	$-0.576227\pi$
$677$	−12.3431	−0.474386	−0.237193	+	0.971463i	$0.576227\pi$
$678$	−6.48528	−0.249066
$679$	8.97056	0.344259
$680$	0	0
$681$	4.00000	0.153280
$682$	−22.6274	−0.866449
$683$	−33.6569	−1.28784	−0.643922	−	0.765091i	$-0.722694\pi$
$683$	−33.6569	−1.28784	−0.643922	+	0.765091i	$0.722694\pi$
$684$	2.82843	0.108148
$685$	0	0
$686$	16.9706	0.647939
$687$	−4.14214	−0.158032
$688$	−9.65685	−0.368164
$689$	−13.3137	−0.507212
$690$	0	0
$691$	−27.7990	−1.05752	−0.528762	−	0.848770i	$-0.677343\pi$
$691$	−27.7990	−1.05752	−0.528762	+	0.848770i	$0.677343\pi$
$692$	9.31371	0.354054
$693$	16.0000	0.607790
$694$	−9.65685	−0.366569
$695$	0	0
$696$	−8.82843	−0.334641
$697$	6.34315	0.240264
$698$	−12.1421	−0.459587
$699$	5.51472	0.208586
$700$	0	0
$701$	0.142136	0.00536839	0.00268419	−	0.999996i	$-0.499146\pi$
$701$	0.142136	0.00536839	0.00268419	+	0.999996i	$0.499146\pi$
$702$	1.00000	0.0377426
$703$	32.9706	1.24351
$704$	5.65685	0.213201
$705$	0	0
$706$	5.31371	0.199984
$707$	45.6569	1.71710
$708$	2.34315	0.0880608
$709$	−7.17157	−0.269334	−0.134667	−	0.990891i	$-0.542996\pi$
$709$	−7.17157	−0.269334	−0.134667	+	0.990891i	$0.542996\pi$
$710$	0	0
$711$	2.34315	0.0878748
$712$	15.6569	0.586765
$713$	33.9411	1.27111
$714$	−2.34315	−0.0876900
$715$	0	0
$716$	7.51472	0.280838
$717$	−16.0000	−0.597531
$718$	28.2843	1.05556
$719$	−29.6569	−1.10601	−0.553007	−	0.833177i	$-0.686520\pi$
$719$	−29.6569	−1.10601	−0.553007	+	0.833177i	$0.686520\pi$
$720$	0	0
$721$	4.68629	0.174527
$722$	11.0000	0.409378
$723$	−5.31371	−0.197619
$724$	−7.65685	−0.284565
$725$	0	0
$726$	21.0000	0.779383
$727$	45.9411	1.70386	0.851931	−	0.523654i	$-0.175432\pi$
$727$	45.9411	1.70386	0.851931	+	0.523654i	$0.175432\pi$
$728$	−2.82843	−0.104828
$729$	1.00000	0.0370370
$730$	0	0
$731$	8.00000	0.295891
$732$	−6.00000	−0.221766
$733$	0.343146	0.0126744	0.00633719	−	0.999980i	$-0.497983\pi$
$733$	0.343146	0.0126744	0.00633719	+	0.999980i	$0.497983\pi$
$734$	25.6569	0.947012
$735$	0	0
$736$	−8.48528	−0.312772
$737$	−32.0000	−1.17874
$738$	7.65685	0.281853
$739$	−14.1421	−0.520227	−0.260113	−	0.965578i	$-0.583760\pi$
$739$	−14.1421	−0.520227	−0.260113	+	0.965578i	$0.583760\pi$
$740$	0	0
$741$	−2.82843	−0.103905
$742$	37.6569	1.38243
$743$	36.2843	1.33114	0.665570	−	0.746335i	$-0.268188\pi$
$743$	36.2843	1.33114	0.665570	+	0.746335i	$0.268188\pi$
$744$	4.00000	0.146647
$745$	0	0
$746$	−2.68629	−0.0983521
$747$	−6.34315	−0.232084
$748$	−4.68629	−0.171348
$749$	11.3137	0.413394
$750$	0	0
$751$	11.3137	0.412843	0.206422	−	0.978463i	$-0.433818\pi$
$751$	11.3137	0.412843	0.206422	+	0.978463i	$0.433818\pi$
$752$	8.00000	0.291730
$753$	−10.8284	−0.394610
$754$	8.82843	0.321512
$755$	0	0
$756$	−2.82843	−0.102869
$757$	−19.9411	−0.724773	−0.362386	−	0.932028i	$-0.618038\pi$
$757$	−19.9411	−0.724773	−0.362386	+	0.932028i	$0.618038\pi$
$758$	7.51472	0.272947
$759$	−48.0000	−1.74229
$760$	0	0
$761$	27.6569	1.00256	0.501280	−	0.865285i	$-0.332863\pi$
$761$	27.6569	1.00256	0.501280	+	0.865285i	$0.332863\pi$
$762$	−9.65685	−0.349831
$763$	24.9706	0.903995
$764$	11.3137	0.409316
$765$	0	0
$766$	−29.6569	−1.07155
$767$	−2.34315	−0.0846061
$768$	−1.00000	−0.0360844
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	0	0
$771$	−4.82843	−0.173892
$772$	−2.48528	−0.0894472
$773$	53.3137	1.91756	0.958780	−	0.284148i	$-0.0917107\pi$
$773$	53.3137	1.91756	0.958780	+	0.284148i	$0.0917107\pi$
$774$	9.65685	0.347108
$775$	0	0
$776$	−3.17157	−0.113853
$777$	−32.9706	−1.18281
$778$	6.48528	0.232509
$779$	−21.6569	−0.775937
$780$	0	0
$781$	−32.0000	−1.14505
$782$	7.02944	0.251372
$783$	8.82843	0.315502
$784$	1.00000	0.0357143
$785$	0	0
$786$	−6.14214	−0.219083
$787$	24.0000	0.855508	0.427754	−	0.903895i	$-0.359305\pi$
$787$	24.0000	0.855508	0.427754	+	0.903895i	$0.359305\pi$
$788$	13.3137	0.474281
$789$	16.4853	0.586892
$790$	0	0
$791$	−18.3431	−0.652207
$792$	−5.65685	−0.201008
$793$	6.00000	0.213066
$794$	30.2843	1.07475
$795$	0	0
$796$	10.3431	0.366603
$797$	−16.6274	−0.588973	−0.294487	−	0.955656i	$-0.595149\pi$
$797$	−16.6274	−0.588973	−0.294487	+	0.955656i	$0.595149\pi$
$798$	8.00000	0.283197
$799$	−6.62742	−0.234461
$800$	0	0
$801$	−15.6569	−0.553208
$802$	26.9706	0.952364
$803$	81.9411	2.89164
$804$	5.65685	0.199502
$805$	0	0
$806$	−4.00000	−0.140894
$807$	14.4853	0.509906
$808$	−16.1421	−0.567878
$809$	13.3137	0.468085	0.234043	−	0.972226i	$-0.424804\pi$
$809$	13.3137	0.468085	0.234043	+	0.972226i	$0.424804\pi$
$810$	0	0
$811$	−1.85786	−0.0652384	−0.0326192	−	0.999468i	$-0.510385\pi$
$811$	−1.85786	−0.0652384	−0.0326192	+	0.999468i	$0.510385\pi$
$812$	−24.9706	−0.876295
$813$	−7.31371	−0.256503
$814$	−65.9411	−2.31124
$815$	0	0
$816$	0.828427	0.0290008
$817$	−27.3137	−0.955586
$818$	3.65685	0.127859
$819$	2.82843	0.0988332
$820$	0	0
$821$	34.2843	1.19653	0.598265	−	0.801299i	$-0.295857\pi$
$821$	34.2843	1.19653	0.598265	+	0.801299i	$0.295857\pi$
$822$	17.3137	0.603885
$823$	52.9706	1.84644	0.923219	−	0.384275i	$-0.125548\pi$
$823$	52.9706	1.84644	0.923219	+	0.384275i	$0.125548\pi$
$824$	−1.65685	−0.0577193
$825$	0	0
$826$	6.62742	0.230597
$827$	−9.65685	−0.335802	−0.167901	−	0.985804i	$-0.553699\pi$
$827$	−9.65685	−0.335802	−0.167901	+	0.985804i	$0.553699\pi$
$828$	8.48528	0.294884
$829$	−53.3137	−1.85166	−0.925831	−	0.377938i	$-0.876633\pi$
$829$	−53.3137	−1.85166	−0.925831	+	0.377938i	$0.876633\pi$
$830$	0	0
$831$	26.0000	0.901930
$832$	1.00000	0.0346688
$833$	−0.828427	−0.0287033
$834$	6.34315	0.219645
$835$	0	0
$836$	16.0000	0.553372
$837$	−4.00000	−0.138260
$838$	10.8284	0.374062
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	48.9411	1.68763
$842$	24.1421	0.831993
$843$	−8.34315	−0.287353
$844$	0.686292	0.0236231
$845$	0	0
$846$	−8.00000	−0.275046
$847$	59.3970	2.04090
$848$	−13.3137	−0.457195
$849$	−17.6569	−0.605982
$850$	0	0
$851$	98.9117	3.39065
$852$	5.65685	0.193801
$853$	−38.2843	−1.31083	−0.655414	−	0.755270i	$-0.727506\pi$
$853$	−38.2843	−1.31083	−0.655414	+	0.755270i	$0.727506\pi$
$854$	−16.9706	−0.580721
$855$	0	0
$856$	−4.00000	−0.136717
$857$	20.8284	0.711486	0.355743	−	0.934584i	$-0.384228\pi$
$857$	20.8284	0.711486	0.355743	+	0.934584i	$0.384228\pi$
$858$	5.65685	0.193122
$859$	37.9411	1.29453	0.647267	−	0.762263i	$-0.275912\pi$
$859$	37.9411	1.29453	0.647267	+	0.762263i	$0.275912\pi$
$860$	0	0
$861$	21.6569	0.738064
$862$	16.0000	0.544962
$863$	28.2843	0.962808	0.481404	−	0.876499i	$-0.340127\pi$
$863$	28.2843	0.962808	0.481404	+	0.876499i	$0.340127\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	−22.9706	−0.780571
$867$	16.3137	0.554043
$868$	11.3137	0.384012
$869$	13.2548	0.449639
$870$	0	0
$871$	−5.65685	−0.191675
$872$	−8.82843	−0.298968
$873$	3.17157	0.107341
$874$	−24.0000	−0.811812
$875$	0	0
$876$	−14.4853	−0.489412
$877$	51.2548	1.73075	0.865376	−	0.501122i	$-0.167079\pi$
$877$	51.2548	1.73075	0.865376	+	0.501122i	$0.167079\pi$
$878$	−22.6274	−0.763638
$879$	−16.6274	−0.560829
$880$	0	0
$881$	−10.2843	−0.346486	−0.173243	−	0.984879i	$-0.555425\pi$
$881$	−10.2843	−0.346486	−0.173243	+	0.984879i	$0.555425\pi$
$882$	−1.00000	−0.0336718
$883$	−31.3137	−1.05379	−0.526895	−	0.849930i	$-0.676644\pi$
$883$	−31.3137	−1.05379	−0.526895	+	0.849930i	$0.676644\pi$
$884$	−0.828427	−0.0278630
$885$	0	0
$886$	30.3431	1.01940
$887$	40.4853	1.35936	0.679681	−	0.733508i	$-0.262118\pi$
$887$	40.4853	1.35936	0.679681	+	0.733508i	$0.262118\pi$
$888$	11.6569	0.391178
$889$	−27.3137	−0.916072
$890$	0	0
$891$	5.65685	0.189512
$892$	−10.8284	−0.362563
$893$	22.6274	0.757198
$894$	3.65685	0.122304
$895$	0	0
$896$	−2.82843	−0.0944911
$897$	−8.48528	−0.283315
$898$	−26.2843	−0.877117
$899$	−35.3137	−1.17778
$900$	0	0
$901$	11.0294	0.367444
$902$	43.3137	1.44219
$903$	27.3137	0.908943
$904$	6.48528	0.215697
$905$	0	0
$906$	12.0000	0.398673
$907$	−8.28427	−0.275075	−0.137537	−	0.990497i	$-0.543919\pi$
$907$	−8.28427	−0.275075	−0.137537	+	0.990497i	$0.543919\pi$
$908$	−4.00000	−0.132745
$909$	16.1421	0.535401
$910$	0	0
$911$	24.9706	0.827312	0.413656	−	0.910433i	$-0.364252\pi$
$911$	24.9706	0.827312	0.413656	+	0.910433i	$0.364252\pi$
$912$	−2.82843	−0.0936586
$913$	−35.8823	−1.18753
$914$	20.8284	0.688943
$915$	0	0
$916$	4.14214	0.136860
$917$	−17.3726	−0.573693
$918$	−0.828427	−0.0273422
$919$	41.9411	1.38351	0.691755	−	0.722132i	$-0.256838\pi$
$919$	41.9411	1.38351	0.691755	+	0.722132i	$0.256838\pi$
$920$	0	0
$921$	−21.6569	−0.713618
$922$	−14.0000	−0.461065
$923$	−5.65685	−0.186198
$924$	−16.0000	−0.526361
$925$	0	0
$926$	−3.79899	−0.124843
$927$	1.65685	0.0544182
$928$	8.82843	0.289807
$929$	−33.5980	−1.10231	−0.551157	−	0.834402i	$-0.685813\pi$
$929$	−33.5980	−1.10231	−0.551157	+	0.834402i	$0.685813\pi$
$930$	0	0
$931$	2.82843	0.0926980
$932$	−5.51472	−0.180641
$933$	24.0000	0.785725
$934$	7.31371	0.239312
$935$	0	0
$936$	−1.00000	−0.0326860
$937$	−16.6274	−0.543194	−0.271597	−	0.962411i	$-0.587552\pi$
$937$	−16.6274	−0.543194	−0.271597	+	0.962411i	$0.587552\pi$
$938$	16.0000	0.522419
$939$	−30.9706	−1.01069
$940$	0	0
$941$	54.9706	1.79199	0.895995	−	0.444065i	$-0.146464\pi$
$941$	54.9706	1.79199	0.895995	+	0.444065i	$0.146464\pi$
$942$	−5.31371	−0.173130
$943$	−64.9706	−2.11573
$944$	−2.34315	−0.0762629
$945$	0	0
$946$	54.6274	1.77609
$947$	−30.3431	−0.986020	−0.493010	−	0.870024i	$-0.664103\pi$
$947$	−30.3431	−0.986020	−0.493010	+	0.870024i	$0.664103\pi$
$948$	−2.34315	−0.0761018
$949$	14.4853	0.470212
$950$	0	0
$951$	25.3137	0.820853
$952$	2.34315	0.0759418
$953$	27.8579	0.902405	0.451202	−	0.892422i	$-0.350995\pi$
$953$	27.8579	0.902405	0.451202	+	0.892422i	$0.350995\pi$
$954$	13.3137	0.431047
$955$	0	0
$956$	16.0000	0.517477
$957$	49.9411	1.61437
$958$	11.3137	0.365529
$959$	48.9706	1.58134
$960$	0	0
$961$	−15.0000	−0.483871
$962$	−11.6569	−0.375832
$963$	4.00000	0.128898
$964$	5.31371	0.171143
$965$	0	0
$966$	24.0000	0.772187
$967$	7.51472	0.241657	0.120829	−	0.992673i	$-0.461445\pi$
$967$	7.51472	0.241657	0.120829	+	0.992673i	$0.461445\pi$
$968$	−21.0000	−0.674966
$969$	2.34315	0.0752727
$970$	0	0
$971$	15.5147	0.497891	0.248946	−	0.968517i	$-0.419916\pi$
$971$	15.5147	0.497891	0.248946	+	0.968517i	$0.419916\pi$
$972$	−1.00000	−0.0320750
$973$	17.9411	0.575166
$974$	16.4853	0.528222
$975$	0	0
$976$	6.00000	0.192055
$977$	8.34315	0.266921	0.133460	−	0.991054i	$-0.457391\pi$
$977$	8.34315	0.266921	0.133460	+	0.991054i	$0.457391\pi$
$978$	−11.3137	−0.361773
$979$	−88.5685	−2.83066
$980$	0	0
$981$	8.82843	0.281870
$982$	38.1421	1.21716
$983$	2.34315	0.0747347	0.0373674	−	0.999302i	$-0.488103\pi$
$983$	2.34315	0.0747347	0.0373674	+	0.999302i	$0.488103\pi$
$984$	−7.65685	−0.244092
$985$	0	0
$986$	−7.31371	−0.232916
$987$	−22.6274	−0.720239
$988$	2.82843	0.0899843
$989$	−81.9411	−2.60558
$990$	0	0
$991$	−42.9117	−1.36313	−0.681567	−	0.731755i	$-0.738701\pi$
$991$	−42.9117	−1.36313	−0.681567	+	0.731755i	$0.738701\pi$
$992$	−4.00000	−0.127000
$993$	8.48528	0.269272
$994$	16.0000	0.507489
$995$	0	0
$996$	6.34315	0.200990
$997$	−61.3137	−1.94182	−0.970912	−	0.239435i	$-0.923038\pi$
$997$	−61.3137	−1.94182	−0.970912	+	0.239435i	$0.923038\pi$
$998$	−0.485281	−0.0153613
$999$	−11.6569	−0.368807

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1950.2.a.bd.1.2		2
3.2	odd	2		5850.2.a.cl.1.2		2
5.2	odd	4		1950.2.e.o.1249.2		4
5.3	odd	4		1950.2.e.o.1249.3		4
5.4	even	2		390.2.a.h.1.1	✓	2
15.2	even	4		5850.2.e.bk.5149.4		4
15.8	even	4		5850.2.e.bk.5149.1		4
15.14	odd	2		1170.2.a.o.1.1		2
20.19	odd	2		3120.2.a.bc.1.2		2
60.59	even	2		9360.2.a.ch.1.2		2
65.34	odd	4		5070.2.b.q.1351.4		4
65.44	odd	4		5070.2.b.q.1351.1		4
65.64	even	2		5070.2.a.bc.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.a.h.1.1	✓	2	5.4	even	2
1170.2.a.o.1.1		2	15.14	odd	2
1950.2.a.bd.1.2		2	1.1	even	1	trivial
1950.2.e.o.1249.2		4	5.2	odd	4
1950.2.e.o.1249.3		4	5.3	odd	4
3120.2.a.bc.1.2		2	20.19	odd	2
5070.2.a.bc.1.2		2	65.64	even	2
5070.2.b.q.1351.1		4	65.44	odd	4
5070.2.b.q.1351.4		4	65.34	odd	4
5850.2.a.cl.1.2		2	3.2	odd	2
5850.2.e.bk.5149.1		4	15.8	even	4
5850.2.e.bk.5149.4		4	15.2	even	4
9360.2.a.ch.1.2		2	60.59	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1950.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +2.82843 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +2.82843 q^{7} -1.00000 q^{8} +1.00000 q^{9} +5.65685 q^{11} -1.00000 q^{12} +1.00000 q^{13} -2.82843 q^{14} +1.00000 q^{16} -0.828427 q^{17} -1.00000 q^{18} +2.82843 q^{19} -2.82843 q^{21} -5.65685 q^{22} +8.48528 q^{23} +1.00000 q^{24} -1.00000 q^{26} -1.00000 q^{27} +2.82843 q^{28} -8.82843 q^{29} +4.00000 q^{31} -1.00000 q^{32} -5.65685 q^{33} +0.828427 q^{34} +1.00000 q^{36} +11.6569 q^{37} -2.82843 q^{38} -1.00000 q^{39} -7.65685 q^{41} +2.82843 q^{42} -9.65685 q^{43} +5.65685 q^{44} -8.48528 q^{46} +8.00000 q^{47} -1.00000 q^{48} +1.00000 q^{49} +0.828427 q^{51} +1.00000 q^{52} -13.3137 q^{53} +1.00000 q^{54} -2.82843 q^{56} -2.82843 q^{57} +8.82843 q^{58} -2.34315 q^{59} +6.00000 q^{61} -4.00000 q^{62} +2.82843 q^{63} +1.00000 q^{64} +5.65685 q^{66} -5.65685 q^{67} -0.828427 q^{68} -8.48528 q^{69} -5.65685 q^{71} -1.00000 q^{72} +14.4853 q^{73} -11.6569 q^{74} +2.82843 q^{76} +16.0000 q^{77} +1.00000 q^{78} +2.34315 q^{79} +1.00000 q^{81} +7.65685 q^{82} -6.34315 q^{83} -2.82843 q^{84} +9.65685 q^{86} +8.82843 q^{87} -5.65685 q^{88} -15.6569 q^{89} +2.82843 q^{91} +8.48528 q^{92} -4.00000 q^{93} -8.00000 q^{94} +1.00000 q^{96} +3.17157 q^{97} -1.00000 q^{98} +5.65685 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{8} + 2 q^{9} - 2 q^{12} + 2 q^{13} + 2 q^{16} + 4 q^{17} - 2 q^{18} + 2 q^{24} - 2 q^{26} - 2 q^{27} - 12 q^{29} + 8 q^{31} - 2 q^{32} - 4 q^{34} + 2 q^{36} + 12 q^{37} - 2 q^{39} - 4 q^{41} - 8 q^{43} + 16 q^{47} - 2 q^{48} + 2 q^{49} - 4 q^{51} + 2 q^{52} - 4 q^{53} + 2 q^{54} + 12 q^{58} - 16 q^{59} + 12 q^{61} - 8 q^{62} + 2 q^{64} + 4 q^{68} - 2 q^{72} + 12 q^{73} - 12 q^{74} + 32 q^{77} + 2 q^{78} + 16 q^{79} + 2 q^{81} + 4 q^{82} - 24 q^{83} + 8 q^{86} + 12 q^{87} - 20 q^{89} - 8 q^{93} - 16 q^{94} + 2 q^{96} + 12 q^{97} - 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - 2 * q^8 + 2 * q^9 - 2 * q^12 + 2 * q^13 + 2 * q^16 + 4 * q^17 - 2 * q^18 + 2 * q^24 - 2 * q^26 - 2 * q^27 - 12 * q^29 + 8 * q^31 - 2 * q^32 - 4 * q^34 + 2 * q^36 + 12 * q^37 - 2 * q^39 - 4 * q^41 - 8 * q^43 + 16 * q^47 - 2 * q^48 + 2 * q^49 - 4 * q^51 + 2 * q^52 - 4 * q^53 + 2 * q^54 + 12 * q^58 - 16 * q^59 + 12 * q^61 - 8 * q^62 + 2 * q^64 + 4 * q^68 - 2 * q^72 + 12 * q^73 - 12 * q^74 + 32 * q^77 + 2 * q^78 + 16 * q^79 + 2 * q^81 + 4 * q^82 - 24 * q^83 + 8 * q^86 + 12 * q^87 - 20 * q^89 - 8 * q^93 - 16 * q^94 + 2 * q^96 + 12 * q^97 - 2 * q^98

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