LMFDB - Embedded newform 1950.2.a.be.1.1

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:	$\Q(\zeta_{10})^+$
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:	$\Z[a_1, \ldots, a_{7}]$
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			1.1
Root			$-0.618034$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.23607 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.23607 q^{7} -1.00000 q^{8} +1.00000 q^{9} -4.47214 q^{11} +1.00000 q^{12} +1.00000 q^{13} +1.23607 q^{14} +1.00000 q^{16} +2.76393 q^{17} -1.00000 q^{18} -7.23607 q^{19} -1.23607 q^{21} +4.47214 q^{22} +7.23607 q^{23} -1.00000 q^{24} -1.00000 q^{26} +1.00000 q^{27} -1.23607 q^{28} +9.70820 q^{29} -4.00000 q^{31} -1.00000 q^{32} -4.47214 q^{33} -2.76393 q^{34} +1.00000 q^{36} +6.94427 q^{37} +7.23607 q^{38} +1.00000 q^{39} +12.4721 q^{41} +1.23607 q^{42} +6.47214 q^{43} -4.47214 q^{44} -7.23607 q^{46} -4.94427 q^{47} +1.00000 q^{48} -5.47214 q^{49} +2.76393 q^{51} +1.00000 q^{52} +8.47214 q^{53} -1.00000 q^{54} +1.23607 q^{56} -7.23607 q^{57} -9.70820 q^{58} +0.472136 q^{59} +6.94427 q^{61} +4.00000 q^{62} -1.23607 q^{63} +1.00000 q^{64} +4.47214 q^{66} +2.76393 q^{68} +7.23607 q^{69} +6.47214 q^{71} -1.00000 q^{72} +8.76393 q^{73} -6.94427 q^{74} -7.23607 q^{76} +5.52786 q^{77} -1.00000 q^{78} -4.00000 q^{79} +1.00000 q^{81} -12.4721 q^{82} +12.9443 q^{83} -1.23607 q^{84} -6.47214 q^{86} +9.70820 q^{87} +4.47214 q^{88} -8.47214 q^{89} -1.23607 q^{91} +7.23607 q^{92} -4.00000 q^{93} +4.94427 q^{94} -1.00000 q^{96} +9.70820 q^{97} +5.47214 q^{98} -4.47214 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} + 2 q^{7} - 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^7 - 2 * q^8 + 2 * q^9	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9} + 2 q^{12} + 2 q^{13} - 2 q^{14} + 2 q^{16} + 10 q^{17} - 2 q^{18} - 10 q^{19} + 2 q^{21} + 10 q^{23} - 2 q^{24} - 2 q^{26} + 2 q^{27} + 2 q^{28} + 6 q^{29} - 8 q^{31} - 2 q^{32} - 10 q^{34} + 2 q^{36} - 4 q^{37} + 10 q^{38} + 2 q^{39} + 16 q^{41} - 2 q^{42} + 4 q^{43} - 10 q^{46} + 8 q^{47} + 2 q^{48} - 2 q^{49} + 10 q^{51} + 2 q^{52} + 8 q^{53} - 2 q^{54} - 2 q^{56} - 10 q^{57} - 6 q^{58} - 8 q^{59} - 4 q^{61} + 8 q^{62} + 2 q^{63} + 2 q^{64} + 10 q^{68} + 10 q^{69} + 4 q^{71} - 2 q^{72} + 22 q^{73} + 4 q^{74} - 10 q^{76} + 20 q^{77} - 2 q^{78} - 8 q^{79} + 2 q^{81} - 16 q^{82} + 8 q^{83} + 2 q^{84} - 4 q^{86} + 6 q^{87} - 8 q^{89} + 2 q^{91} + 10 q^{92} - 8 q^{93} - 8 q^{94} - 2 q^{96} + 6 q^{97} + 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 + 2 * q^7 - 2 * q^8 + 2 * q^9 + 2 * q^12 + 2 * q^13 - 2 * q^14 + 2 * q^16 + 10 * q^17 - 2 * q^18 - 10 * q^19 + 2 * q^21 + 10 * q^23 - 2 * q^24 - 2 * q^26 + 2 * q^27 + 2 * q^28 + 6 * q^29 - 8 * q^31 - 2 * q^32 - 10 * q^34 + 2 * q^36 - 4 * q^37 + 10 * q^38 + 2 * q^39 + 16 * q^41 - 2 * q^42 + 4 * q^43 - 10 * q^46 + 8 * q^47 + 2 * q^48 - 2 * q^49 + 10 * q^51 + 2 * q^52 + 8 * q^53 - 2 * q^54 - 2 * q^56 - 10 * q^57 - 6 * q^58 - 8 * q^59 - 4 * q^61 + 8 * q^62 + 2 * q^63 + 2 * q^64 + 10 * q^68 + 10 * q^69 + 4 * q^71 - 2 * q^72 + 22 * q^73 + 4 * q^74 - 10 * q^76 + 20 * q^77 - 2 * q^78 - 8 * q^79 + 2 * q^81 - 16 * q^82 + 8 * q^83 + 2 * q^84 - 4 * q^86 + 6 * q^87 - 8 * q^89 + 2 * q^91 + 10 * q^92 - 8 * q^93 - 8 * q^94 - 2 * q^96 + 6 * 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
$7$	−1.23607	−0.467190	−0.233595	−	0.972334i	$-0.575049\pi$
$7$	−1.23607	−0.467190	−0.233595	+	0.972334i	$0.575049\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.47214	−1.34840	−0.674200	−	0.738549i	$-0.735511\pi$
$11$	−4.47214	−1.34840	−0.674200	+	0.738549i	$0.735511\pi$
$12$	1.00000	0.288675
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	1.23607	0.330353
$15$	0	0
$16$	1.00000	0.250000
$17$	2.76393	0.670352	0.335176	−	0.942156i	$-0.391204\pi$
$17$	2.76393	0.670352	0.335176	+	0.942156i	$0.391204\pi$
$18$	−1.00000	−0.235702
$19$	−7.23607	−1.66007	−0.830034	−	0.557713i	$-0.811679\pi$
$19$	−7.23607	−1.66007	−0.830034	+	0.557713i	$0.811679\pi$
$20$	0	0
$21$	−1.23607	−0.269732
$22$	4.47214	0.953463
$23$	7.23607	1.50882	0.754412	−	0.656401i	$-0.227922\pi$
$23$	7.23607	1.50882	0.754412	+	0.656401i	$0.227922\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	−1.00000	−0.196116
$27$	1.00000	0.192450
$28$	−1.23607	−0.233595
$29$	9.70820	1.80277	0.901384	−	0.433020i	$-0.142552\pi$
$29$	9.70820	1.80277	0.901384	+	0.433020i	$0.142552\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	−1.00000	−0.176777
$33$	−4.47214	−0.778499
$34$	−2.76393	−0.474010
$35$	0	0
$36$	1.00000	0.166667
$37$	6.94427	1.14163	0.570816	−	0.821078i	$-0.306627\pi$
$37$	6.94427	1.14163	0.570816	+	0.821078i	$0.306627\pi$
$38$	7.23607	1.17385
$39$	1.00000	0.160128
$40$	0	0
$41$	12.4721	1.94782	0.973910	−	0.226934i	$-0.0728701\pi$
$41$	12.4721	1.94782	0.973910	+	0.226934i	$0.0728701\pi$
$42$	1.23607	0.190729
$43$	6.47214	0.986991	0.493496	−	0.869748i	$-0.335719\pi$
$43$	6.47214	0.986991	0.493496	+	0.869748i	$0.335719\pi$
$44$	−4.47214	−0.674200
$45$	0	0
$46$	−7.23607	−1.06690
$47$	−4.94427	−0.721196	−0.360598	−	0.932721i	$-0.617427\pi$
$47$	−4.94427	−0.721196	−0.360598	+	0.932721i	$0.617427\pi$
$48$	1.00000	0.144338
$49$	−5.47214	−0.781734
$50$	0	0
$51$	2.76393	0.387028
$52$	1.00000	0.138675
$53$	8.47214	1.16374	0.581869	−	0.813283i	$-0.302322\pi$
$53$	8.47214	1.16374	0.581869	+	0.813283i	$0.302322\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	1.23607	0.165177
$57$	−7.23607	−0.958441
$58$	−9.70820	−1.27475
$59$	0.472136	0.0614669	0.0307334	−	0.999528i	$-0.490216\pi$
$59$	0.472136	0.0614669	0.0307334	+	0.999528i	$0.490216\pi$
$60$	0	0
$61$	6.94427	0.889123	0.444561	−	0.895748i	$-0.353360\pi$
$61$	6.94427	0.889123	0.444561	+	0.895748i	$0.353360\pi$
$62$	4.00000	0.508001
$63$	−1.23607	−0.155730
$64$	1.00000	0.125000
$65$	0	0
$66$	4.47214	0.550482
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	2.76393	0.335176
$69$	7.23607	0.871120
$70$	0	0
$71$	6.47214	0.768101	0.384051	−	0.923312i	$-0.374529\pi$
$71$	6.47214	0.768101	0.384051	+	0.923312i	$0.374529\pi$
$72$	−1.00000	−0.117851
$73$	8.76393	1.02574	0.512870	−	0.858466i	$-0.328582\pi$
$73$	8.76393	1.02574	0.512870	+	0.858466i	$0.328582\pi$
$74$	−6.94427	−0.807255
$75$	0	0
$76$	−7.23607	−0.830034
$77$	5.52786	0.629959
$78$	−1.00000	−0.113228
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−12.4721	−1.37732
$83$	12.9443	1.42082	0.710409	−	0.703789i	$-0.248510\pi$
$83$	12.9443	1.42082	0.710409	+	0.703789i	$0.248510\pi$
$84$	−1.23607	−0.134866
$85$	0	0
$86$	−6.47214	−0.697908
$87$	9.70820	1.04083
$88$	4.47214	0.476731
$89$	−8.47214	−0.898045	−0.449022	−	0.893521i	$-0.648228\pi$
$89$	−8.47214	−0.898045	−0.449022	+	0.893521i	$0.648228\pi$
$90$	0	0
$91$	−1.23607	−0.129575
$92$	7.23607	0.754412
$93$	−4.00000	−0.414781
$94$	4.94427	0.509963
$95$	0	0
$96$	−1.00000	−0.102062
$97$	9.70820	0.985719	0.492859	−	0.870109i	$-0.335952\pi$
$97$	9.70820	0.985719	0.492859	+	0.870109i	$0.335952\pi$
$98$	5.47214	0.552769
$99$	−4.47214	−0.449467
$100$	0	0
$101$	−4.76393	−0.474029	−0.237014	−	0.971506i	$-0.576169\pi$
$101$	−4.76393	−0.474029	−0.237014	+	0.971506i	$0.576169\pi$
$102$	−2.76393	−0.273670
$103$	−0.472136	−0.0465209	−0.0232605	−	0.999729i	$-0.507405\pi$
$103$	−0.472136	−0.0465209	−0.0232605	+	0.999729i	$0.507405\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	−8.47214	−0.822887
$107$	−10.4721	−1.01238	−0.506190	−	0.862422i	$-0.668946\pi$
$107$	−10.4721	−1.01238	−0.506190	+	0.862422i	$0.668946\pi$
$108$	1.00000	0.0962250
$109$	−15.7082	−1.50457	−0.752287	−	0.658836i	$-0.771049\pi$
$109$	−15.7082	−1.50457	−0.752287	+	0.658836i	$0.771049\pi$
$110$	0	0
$111$	6.94427	0.659121
$112$	−1.23607	−0.116797
$113$	−12.6525	−1.19024	−0.595122	−	0.803635i	$-0.702896\pi$
$113$	−12.6525	−1.19024	−0.595122	+	0.803635i	$0.702896\pi$
$114$	7.23607	0.677720
$115$	0	0
$116$	9.70820	0.901384
$117$	1.00000	0.0924500
$118$	−0.472136	−0.0434636
$119$	−3.41641	−0.313182
$120$	0	0
$121$	9.00000	0.818182
$122$	−6.94427	−0.628705
$123$	12.4721	1.12457
$124$	−4.00000	−0.359211
$125$	0	0
$126$	1.23607	0.110118
$127$	−5.41641	−0.480628	−0.240314	−	0.970695i	$-0.577250\pi$
$127$	−5.41641	−0.480628	−0.240314	+	0.970695i	$0.577250\pi$
$128$	−1.00000	−0.0883883
$129$	6.47214	0.569840
$130$	0	0
$131$	−3.70820	−0.323987	−0.161994	−	0.986792i	$-0.551792\pi$
$131$	−3.70820	−0.323987	−0.161994	+	0.986792i	$0.551792\pi$
$132$	−4.47214	−0.389249
$133$	8.94427	0.775567
$134$	0	0
$135$	0	0
$136$	−2.76393	−0.237005
$137$	15.8885	1.35745	0.678725	−	0.734393i	$-0.262533\pi$
$137$	15.8885	1.35745	0.678725	+	0.734393i	$0.262533\pi$
$138$	−7.23607	−0.615975
$139$	2.47214	0.209684	0.104842	−	0.994489i	$-0.466566\pi$
$139$	2.47214	0.209684	0.104842	+	0.994489i	$0.466566\pi$
$140$	0	0
$141$	−4.94427	−0.416383
$142$	−6.47214	−0.543130
$143$	−4.47214	−0.373979
$144$	1.00000	0.0833333
$145$	0	0
$146$	−8.76393	−0.725308
$147$	−5.47214	−0.451334
$148$	6.94427	0.570816
$149$	22.4721	1.84099	0.920495	−	0.390755i	$-0.127786\pi$
$149$	22.4721	1.84099	0.920495	+	0.390755i	$0.127786\pi$
$150$	0	0
$151$	7.41641	0.603539	0.301769	−	0.953381i	$-0.402423\pi$
$151$	7.41641	0.603539	0.301769	+	0.953381i	$0.402423\pi$
$152$	7.23607	0.586923
$153$	2.76393	0.223451
$154$	−5.52786	−0.445448
$155$	0	0
$156$	1.00000	0.0800641
$157$	−1.05573	−0.0842563	−0.0421281	−	0.999112i	$-0.513414\pi$
$157$	−1.05573	−0.0842563	−0.0421281	+	0.999112i	$0.513414\pi$
$158$	4.00000	0.318223
$159$	8.47214	0.671884
$160$	0	0
$161$	−8.94427	−0.704907
$162$	−1.00000	−0.0785674
$163$	−4.94427	−0.387265	−0.193633	−	0.981074i	$-0.562027\pi$
$163$	−4.94427	−0.387265	−0.193633	+	0.981074i	$0.562027\pi$
$164$	12.4721	0.973910
$165$	0	0
$166$	−12.9443	−1.00467
$167$	7.41641	0.573899	0.286949	−	0.957946i	$-0.407359\pi$
$167$	7.41641	0.573899	0.286949	+	0.957946i	$0.407359\pi$
$168$	1.23607	0.0953647
$169$	1.00000	0.0769231
$170$	0	0
$171$	−7.23607	−0.553356
$172$	6.47214	0.493496
$173$	−5.05573	−0.384380	−0.192190	−	0.981358i	$-0.561559\pi$
$173$	−5.05573	−0.384380	−0.192190	+	0.981358i	$0.561559\pi$
$174$	−9.70820	−0.735977
$175$	0	0
$176$	−4.47214	−0.337100
$177$	0.472136	0.0354879
$178$	8.47214	0.635013
$179$	−14.1803	−1.05989	−0.529944	−	0.848033i	$-0.677787\pi$
$179$	−14.1803	−1.05989	−0.529944	+	0.848033i	$0.677787\pi$
$180$	0	0
$181$	−9.41641	−0.699916	−0.349958	−	0.936765i	$-0.613804\pi$
$181$	−9.41641	−0.699916	−0.349958	+	0.936765i	$0.613804\pi$
$182$	1.23607	0.0916235
$183$	6.94427	0.513335
$184$	−7.23607	−0.533450
$185$	0	0
$186$	4.00000	0.293294
$187$	−12.3607	−0.903902
$188$	−4.94427	−0.360598
$189$	−1.23607	−0.0899107
$190$	0	0
$191$	13.5279	0.978842	0.489421	−	0.872048i	$-0.337208\pi$
$191$	13.5279	0.978842	0.489421	+	0.872048i	$0.337208\pi$
$192$	1.00000	0.0721688
$193$	−21.7082	−1.56259	−0.781295	−	0.624161i	$-0.785441\pi$
$193$	−21.7082	−1.56259	−0.781295	+	0.624161i	$0.785441\pi$
$194$	−9.70820	−0.697008
$195$	0	0
$196$	−5.47214	−0.390867
$197$	2.94427	0.209771	0.104885	−	0.994484i	$-0.466552\pi$
$197$	2.94427	0.209771	0.104885	+	0.994484i	$0.466552\pi$
$198$	4.47214	0.317821
$199$	16.9443	1.20115	0.600574	−	0.799569i	$-0.294939\pi$
$199$	16.9443	1.20115	0.600574	+	0.799569i	$0.294939\pi$
$200$	0	0
$201$	0	0
$202$	4.76393	0.335189
$203$	−12.0000	−0.842235
$204$	2.76393	0.193514
$205$	0	0
$206$	0.472136	0.0328953
$207$	7.23607	0.502941
$208$	1.00000	0.0693375
$209$	32.3607	2.23844
$210$	0	0
$211$	−21.8885	−1.50687	−0.753435	−	0.657523i	$-0.771604\pi$
$211$	−21.8885	−1.50687	−0.753435	+	0.657523i	$0.771604\pi$
$212$	8.47214	0.581869
$213$	6.47214	0.443463
$214$	10.4721	0.715860
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	4.94427	0.335639
$218$	15.7082	1.06389
$219$	8.76393	0.592212
$220$	0	0
$221$	2.76393	0.185922
$222$	−6.94427	−0.466069
$223$	21.2361	1.42207	0.711036	−	0.703155i	$-0.248226\pi$
$223$	21.2361	1.42207	0.711036	+	0.703155i	$0.248226\pi$
$224$	1.23607	0.0825883
$225$	0	0
$226$	12.6525	0.841630
$227$	24.9443	1.65561	0.827805	−	0.561016i	$-0.189590\pi$
$227$	24.9443	1.65561	0.827805	+	0.561016i	$0.189590\pi$
$228$	−7.23607	−0.479220
$229$	−3.70820	−0.245045	−0.122523	−	0.992466i	$-0.539098\pi$
$229$	−3.70820	−0.245045	−0.122523	+	0.992466i	$0.539098\pi$
$230$	0	0
$231$	5.52786	0.363707
$232$	−9.70820	−0.637375
$233$	−2.18034	−0.142839	−0.0714194	−	0.997446i	$-0.522753\pi$
$233$	−2.18034	−0.142839	−0.0714194	+	0.997446i	$0.522753\pi$
$234$	−1.00000	−0.0653720
$235$	0	0
$236$	0.472136	0.0307334
$237$	−4.00000	−0.259828
$238$	3.41641	0.221453
$239$	13.8885	0.898375	0.449188	−	0.893437i	$-0.351713\pi$
$239$	13.8885	0.898375	0.449188	+	0.893437i	$0.351713\pi$
$240$	0	0
$241$	12.4721	0.803401	0.401700	−	0.915771i	$-0.368419\pi$
$241$	12.4721	0.803401	0.401700	+	0.915771i	$0.368419\pi$
$242$	−9.00000	−0.578542
$243$	1.00000	0.0641500
$244$	6.94427	0.444561
$245$	0	0
$246$	−12.4721	−0.795194
$247$	−7.23607	−0.460420
$248$	4.00000	0.254000
$249$	12.9443	0.820310
$250$	0	0
$251$	15.7082	0.991493	0.495747	−	0.868467i	$-0.334895\pi$
$251$	15.7082	0.991493	0.495747	+	0.868467i	$0.334895\pi$
$252$	−1.23607	−0.0778650
$253$	−32.3607	−2.03450
$254$	5.41641	0.339856
$255$	0	0
$256$	1.00000	0.0625000
$257$	5.23607	0.326617	0.163308	−	0.986575i	$-0.447783\pi$
$257$	5.23607	0.326617	0.163308	+	0.986575i	$0.447783\pi$
$258$	−6.47214	−0.402938
$259$	−8.58359	−0.533358
$260$	0	0
$261$	9.70820	0.600923
$262$	3.70820	0.229094
$263$	−12.1803	−0.751072	−0.375536	−	0.926808i	$-0.622541\pi$
$263$	−12.1803	−0.751072	−0.375536	+	0.926808i	$0.622541\pi$
$264$	4.47214	0.275241
$265$	0	0
$266$	−8.94427	−0.548408
$267$	−8.47214	−0.518486
$268$	0	0
$269$	11.2361	0.685075	0.342538	−	0.939504i	$-0.388714\pi$
$269$	11.2361	0.685075	0.342538	+	0.939504i	$0.388714\pi$
$270$	0	0
$271$	15.4164	0.936480	0.468240	−	0.883601i	$-0.344888\pi$
$271$	15.4164	0.936480	0.468240	+	0.883601i	$0.344888\pi$
$272$	2.76393	0.167588
$273$	−1.23607	−0.0748102
$274$	−15.8885	−0.959862
$275$	0	0
$276$	7.23607	0.435560
$277$	−32.8328	−1.97273	−0.986366	−	0.164564i	$-0.947378\pi$
$277$	−32.8328	−1.97273	−0.986366	+	0.164564i	$0.947378\pi$
$278$	−2.47214	−0.148269
$279$	−4.00000	−0.239474
$280$	0	0
$281$	15.5279	0.926315	0.463157	−	0.886276i	$-0.346716\pi$
$281$	15.5279	0.926315	0.463157	+	0.886276i	$0.346716\pi$
$282$	4.94427	0.294427
$283$	15.4164	0.916410	0.458205	−	0.888846i	$-0.348492\pi$
$283$	15.4164	0.916410	0.458205	+	0.888846i	$0.348492\pi$
$284$	6.47214	0.384051
$285$	0	0
$286$	4.47214	0.264443
$287$	−15.4164	−0.910002
$288$	−1.00000	−0.0589256
$289$	−9.36068	−0.550628
$290$	0	0
$291$	9.70820	0.569105
$292$	8.76393	0.512870
$293$	−14.9443	−0.873054	−0.436527	−	0.899691i	$-0.643792\pi$
$293$	−14.9443	−0.873054	−0.436527	+	0.899691i	$0.643792\pi$
$294$	5.47214	0.319141
$295$	0	0
$296$	−6.94427	−0.403628
$297$	−4.47214	−0.259500
$298$	−22.4721	−1.30178
$299$	7.23607	0.418473
$300$	0	0
$301$	−8.00000	−0.461112
$302$	−7.41641	−0.426766
$303$	−4.76393	−0.273681
$304$	−7.23607	−0.415017
$305$	0	0
$306$	−2.76393	−0.158003
$307$	20.3607	1.16205	0.581023	−	0.813887i	$-0.302653\pi$
$307$	20.3607	1.16205	0.581023	+	0.813887i	$0.302653\pi$
$308$	5.52786	0.314979
$309$	−0.472136	−0.0268589
$310$	0	0
$311$	−13.5279	−0.767095	−0.383547	−	0.923521i	$-0.625298\pi$
$311$	−13.5279	−0.767095	−0.383547	+	0.923521i	$0.625298\pi$
$312$	−1.00000	−0.0566139
$313$	7.41641	0.419200	0.209600	−	0.977787i	$-0.432784\pi$
$313$	7.41641	0.419200	0.209600	+	0.977787i	$0.432784\pi$
$314$	1.05573	0.0595782
$315$	0	0
$316$	−4.00000	−0.225018
$317$	−13.4164	−0.753541	−0.376770	−	0.926307i	$-0.622965\pi$
$317$	−13.4164	−0.753541	−0.376770	+	0.926307i	$0.622965\pi$
$318$	−8.47214	−0.475094
$319$	−43.4164	−2.43085
$320$	0	0
$321$	−10.4721	−0.584498
$322$	8.94427	0.498445
$323$	−20.0000	−1.11283
$324$	1.00000	0.0555556
$325$	0	0
$326$	4.94427	0.273838
$327$	−15.7082	−0.868666
$328$	−12.4721	−0.688659
$329$	6.11146	0.336935
$330$	0	0
$331$	−7.23607	−0.397730	−0.198865	−	0.980027i	$-0.563726\pi$
$331$	−7.23607	−0.397730	−0.198865	+	0.980027i	$0.563726\pi$
$332$	12.9443	0.710409
$333$	6.94427	0.380544
$334$	−7.41641	−0.405808
$335$	0	0
$336$	−1.23607	−0.0674330
$337$	2.47214	0.134666	0.0673329	−	0.997731i	$-0.478551\pi$
$337$	2.47214	0.134666	0.0673329	+	0.997731i	$0.478551\pi$
$338$	−1.00000	−0.0543928
$339$	−12.6525	−0.687188
$340$	0	0
$341$	17.8885	0.968719
$342$	7.23607	0.391282
$343$	15.4164	0.832408
$344$	−6.47214	−0.348954
$345$	0	0
$346$	5.05573	0.271798
$347$	10.4721	0.562174	0.281087	−	0.959682i	$-0.409305\pi$
$347$	10.4721	0.562174	0.281087	+	0.959682i	$0.409305\pi$
$348$	9.70820	0.520414
$349$	−10.7639	−0.576180	−0.288090	−	0.957603i	$-0.593020\pi$
$349$	−10.7639	−0.576180	−0.288090	+	0.957603i	$0.593020\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	4.47214	0.238366
$353$	−9.05573	−0.481988	−0.240994	−	0.970527i	$-0.577473\pi$
$353$	−9.05573	−0.481988	−0.240994	+	0.970527i	$0.577473\pi$
$354$	−0.472136	−0.0250937
$355$	0	0
$356$	−8.47214	−0.449022
$357$	−3.41641	−0.180815
$358$	14.1803	0.749454
$359$	−2.47214	−0.130474	−0.0652372	−	0.997870i	$-0.520780\pi$
$359$	−2.47214	−0.130474	−0.0652372	+	0.997870i	$0.520780\pi$
$360$	0	0
$361$	33.3607	1.75583
$362$	9.41641	0.494915
$363$	9.00000	0.472377
$364$	−1.23607	−0.0647876
$365$	0	0
$366$	−6.94427	−0.362983
$367$	−8.47214	−0.442242	−0.221121	−	0.975246i	$-0.570972\pi$
$367$	−8.47214	−0.442242	−0.221121	+	0.975246i	$0.570972\pi$
$368$	7.23607	0.377206
$369$	12.4721	0.649273
$370$	0	0
$371$	−10.4721	−0.543686
$372$	−4.00000	−0.207390
$373$	2.00000	0.103556	0.0517780	−	0.998659i	$-0.483511\pi$
$373$	2.00000	0.103556	0.0517780	+	0.998659i	$0.483511\pi$
$374$	12.3607	0.639156
$375$	0	0
$376$	4.94427	0.254981
$377$	9.70820	0.499998
$378$	1.23607	0.0635765
$379$	5.12461	0.263234	0.131617	−	0.991301i	$-0.457983\pi$
$379$	5.12461	0.263234	0.131617	+	0.991301i	$0.457983\pi$
$380$	0	0
$381$	−5.41641	−0.277491
$382$	−13.5279	−0.692146
$383$	14.4721	0.739492	0.369746	−	0.929133i	$-0.379445\pi$
$383$	14.4721	0.739492	0.369746	+	0.929133i	$0.379445\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	21.7082	1.10492
$387$	6.47214	0.328997
$388$	9.70820	0.492859
$389$	−18.6525	−0.945718	−0.472859	−	0.881138i	$-0.656778\pi$
$389$	−18.6525	−0.945718	−0.472859	+	0.881138i	$0.656778\pi$
$390$	0	0
$391$	20.0000	1.01144
$392$	5.47214	0.276385
$393$	−3.70820	−0.187054
$394$	−2.94427	−0.148330
$395$	0	0
$396$	−4.47214	−0.224733
$397$	−9.41641	−0.472596	−0.236298	−	0.971681i	$-0.575934\pi$
$397$	−9.41641	−0.472596	−0.236298	+	0.971681i	$0.575934\pi$
$398$	−16.9443	−0.849340
$399$	8.94427	0.447774
$400$	0	0
$401$	−15.5279	−0.775425	−0.387712	−	0.921780i	$-0.626735\pi$
$401$	−15.5279	−0.775425	−0.387712	+	0.921780i	$0.626735\pi$
$402$	0	0
$403$	−4.00000	−0.199254
$404$	−4.76393	−0.237014
$405$	0	0
$406$	12.0000	0.595550
$407$	−31.0557	−1.53938
$408$	−2.76393	−0.136835
$409$	6.00000	0.296681	0.148340	−	0.988936i	$-0.452607\pi$
$409$	6.00000	0.296681	0.148340	+	0.988936i	$0.452607\pi$
$410$	0	0
$411$	15.8885	0.783724
$412$	−0.472136	−0.0232605
$413$	−0.583592	−0.0287167
$414$	−7.23607	−0.355633
$415$	0	0
$416$	−1.00000	−0.0490290
$417$	2.47214	0.121061
$418$	−32.3607	−1.58281
$419$	−21.5967	−1.05507	−0.527535	−	0.849533i	$-0.676884\pi$
$419$	−21.5967	−1.05507	−0.527535	+	0.849533i	$0.676884\pi$
$420$	0	0
$421$	−0.291796	−0.0142213	−0.00711064	−	0.999975i	$-0.502263\pi$
$421$	−0.291796	−0.0142213	−0.00711064	+	0.999975i	$0.502263\pi$
$422$	21.8885	1.06552
$423$	−4.94427	−0.240399
$424$	−8.47214	−0.411443
$425$	0	0
$426$	−6.47214	−0.313576
$427$	−8.58359	−0.415389
$428$	−10.4721	−0.506190
$429$	−4.47214	−0.215917
$430$	0	0
$431$	38.8328	1.87051	0.935255	−	0.353973i	$-0.115170\pi$
$431$	38.8328	1.87051	0.935255	+	0.353973i	$0.115170\pi$
$432$	1.00000	0.0481125
$433$	−18.4721	−0.887714	−0.443857	−	0.896098i	$-0.646390\pi$
$433$	−18.4721	−0.887714	−0.443857	+	0.896098i	$0.646390\pi$
$434$	−4.94427	−0.237333
$435$	0	0
$436$	−15.7082	−0.752287
$437$	−52.3607	−2.50475
$438$	−8.76393	−0.418757
$439$	12.9443	0.617796	0.308898	−	0.951095i	$-0.400040\pi$
$439$	12.9443	0.617796	0.308898	+	0.951095i	$0.400040\pi$
$440$	0	0
$441$	−5.47214	−0.260578
$442$	−2.76393	−0.131467
$443$	−13.5279	−0.642728	−0.321364	−	0.946956i	$-0.604141\pi$
$443$	−13.5279	−0.642728	−0.321364	+	0.946956i	$0.604141\pi$
$444$	6.94427	0.329561
$445$	0	0
$446$	−21.2361	−1.00556
$447$	22.4721	1.06290
$448$	−1.23607	−0.0583987
$449$	−8.47214	−0.399825	−0.199912	−	0.979814i	$-0.564066\pi$
$449$	−8.47214	−0.399825	−0.199912	+	0.979814i	$0.564066\pi$
$450$	0	0
$451$	−55.7771	−2.62644
$452$	−12.6525	−0.595122
$453$	7.41641	0.348453
$454$	−24.9443	−1.17069
$455$	0	0
$456$	7.23607	0.338860
$457$	9.70820	0.454131	0.227065	−	0.973880i	$-0.427087\pi$
$457$	9.70820	0.454131	0.227065	+	0.973880i	$0.427087\pi$
$458$	3.70820	0.173273
$459$	2.76393	0.129009
$460$	0	0
$461$	6.11146	0.284639	0.142319	−	0.989821i	$-0.454544\pi$
$461$	6.11146	0.284639	0.142319	+	0.989821i	$0.454544\pi$
$462$	−5.52786	−0.257180
$463$	−35.7082	−1.65950	−0.829750	−	0.558135i	$-0.811517\pi$
$463$	−35.7082	−1.65950	−0.829750	+	0.558135i	$0.811517\pi$
$464$	9.70820	0.450692
$465$	0	0
$466$	2.18034	0.101002
$467$	−2.47214	−0.114397	−0.0571984	−	0.998363i	$-0.518217\pi$
$467$	−2.47214	−0.114397	−0.0571984	+	0.998363i	$0.518217\pi$
$468$	1.00000	0.0462250
$469$	0	0
$470$	0	0
$471$	−1.05573	−0.0486454
$472$	−0.472136	−0.0217318
$473$	−28.9443	−1.33086
$474$	4.00000	0.183726
$475$	0	0
$476$	−3.41641	−0.156591
$477$	8.47214	0.387912
$478$	−13.8885	−0.635247
$479$	−36.0000	−1.64488	−0.822441	−	0.568850i	$-0.807388\pi$
$479$	−36.0000	−1.64488	−0.822441	+	0.568850i	$0.807388\pi$
$480$	0	0
$481$	6.94427	0.316632
$482$	−12.4721	−0.568090
$483$	−8.94427	−0.406978
$484$	9.00000	0.409091
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	25.5967	1.15990	0.579950	−	0.814652i	$-0.303072\pi$
$487$	25.5967	1.15990	0.579950	+	0.814652i	$0.303072\pi$
$488$	−6.94427	−0.314352
$489$	−4.94427	−0.223588
$490$	0	0
$491$	6.18034	0.278915	0.139457	−	0.990228i	$-0.455464\pi$
$491$	6.18034	0.278915	0.139457	+	0.990228i	$0.455464\pi$
$492$	12.4721	0.562287
$493$	26.8328	1.20849
$494$	7.23607	0.325566
$495$	0	0
$496$	−4.00000	−0.179605
$497$	−8.00000	−0.358849
$498$	−12.9443	−0.580047
$499$	1.12461	0.0503445	0.0251723	−	0.999683i	$-0.491987\pi$
$499$	1.12461	0.0503445	0.0251723	+	0.999683i	$0.491987\pi$
$500$	0	0
$501$	7.41641	0.331341
$502$	−15.7082	−0.701091
$503$	−35.2361	−1.57110	−0.785549	−	0.618799i	$-0.787620\pi$
$503$	−35.2361	−1.57110	−0.785549	+	0.618799i	$0.787620\pi$
$504$	1.23607	0.0550588
$505$	0	0
$506$	32.3607	1.43861
$507$	1.00000	0.0444116
$508$	−5.41641	−0.240314
$509$	−32.9443	−1.46023	−0.730115	−	0.683325i	$-0.760533\pi$
$509$	−32.9443	−1.46023	−0.730115	+	0.683325i	$0.760533\pi$
$510$	0	0
$511$	−10.8328	−0.479216
$512$	−1.00000	−0.0441942
$513$	−7.23607	−0.319480
$514$	−5.23607	−0.230953
$515$	0	0
$516$	6.47214	0.284920
$517$	22.1115	0.972461
$518$	8.58359	0.377141
$519$	−5.05573	−0.221922
$520$	0	0
$521$	−32.8328	−1.43843	−0.719216	−	0.694787i	$-0.755499\pi$
$521$	−32.8328	−1.43843	−0.719216	+	0.694787i	$0.755499\pi$
$522$	−9.70820	−0.424917
$523$	8.00000	0.349816	0.174908	−	0.984585i	$-0.444037\pi$
$523$	8.00000	0.349816	0.174908	+	0.984585i	$0.444037\pi$
$524$	−3.70820	−0.161994
$525$	0	0
$526$	12.1803	0.531088
$527$	−11.0557	−0.481595
$528$	−4.47214	−0.194625
$529$	29.3607	1.27655
$530$	0	0
$531$	0.472136	0.0204890
$532$	8.94427	0.387783
$533$	12.4721	0.540228
$534$	8.47214	0.366625
$535$	0	0
$536$	0	0
$537$	−14.1803	−0.611927
$538$	−11.2361	−0.484421
$539$	24.4721	1.05409
$540$	0	0
$541$	−9.23607	−0.397090	−0.198545	−	0.980092i	$-0.563622\pi$
$541$	−9.23607	−0.397090	−0.198545	+	0.980092i	$0.563622\pi$
$542$	−15.4164	−0.662191
$543$	−9.41641	−0.404097
$544$	−2.76393	−0.118503
$545$	0	0
$546$	1.23607	0.0528988
$547$	12.0000	0.513083	0.256541	−	0.966533i	$-0.417417\pi$
$547$	12.0000	0.513083	0.256541	+	0.966533i	$0.417417\pi$
$548$	15.8885	0.678725
$549$	6.94427	0.296374
$550$	0	0
$551$	−70.2492	−2.99272
$552$	−7.23607	−0.307988
$553$	4.94427	0.210252
$554$	32.8328	1.39493
$555$	0	0
$556$	2.47214	0.104842
$557$	−43.3050	−1.83489	−0.917445	−	0.397863i	$-0.869752\pi$
$557$	−43.3050	−1.83489	−0.917445	+	0.397863i	$0.869752\pi$
$558$	4.00000	0.169334
$559$	6.47214	0.273742
$560$	0	0
$561$	−12.3607	−0.521868
$562$	−15.5279	−0.655003
$563$	7.41641	0.312564	0.156282	−	0.987712i	$-0.450049\pi$
$563$	7.41641	0.312564	0.156282	+	0.987712i	$0.450049\pi$
$564$	−4.94427	−0.208191
$565$	0	0
$566$	−15.4164	−0.648000
$567$	−1.23607	−0.0519100
$568$	−6.47214	−0.271565
$569$	17.0557	0.715013	0.357507	−	0.933911i	$-0.383627\pi$
$569$	17.0557	0.715013	0.357507	+	0.933911i	$0.383627\pi$
$570$	0	0
$571$	−35.4164	−1.48213	−0.741065	−	0.671433i	$-0.765679\pi$
$571$	−35.4164	−1.48213	−0.741065	+	0.671433i	$0.765679\pi$
$572$	−4.47214	−0.186989
$573$	13.5279	0.565135
$574$	15.4164	0.643468
$575$	0	0
$576$	1.00000	0.0416667
$577$	−2.87539	−0.119704	−0.0598520	−	0.998207i	$-0.519063\pi$
$577$	−2.87539	−0.119704	−0.0598520	+	0.998207i	$0.519063\pi$
$578$	9.36068	0.389353
$579$	−21.7082	−0.902162
$580$	0	0
$581$	−16.0000	−0.663792
$582$	−9.70820	−0.402418
$583$	−37.8885	−1.56918
$584$	−8.76393	−0.362654
$585$	0	0
$586$	14.9443	0.617342
$587$	−37.8885	−1.56383	−0.781914	−	0.623387i	$-0.785756\pi$
$587$	−37.8885	−1.56383	−0.781914	+	0.623387i	$0.785756\pi$
$588$	−5.47214	−0.225667
$589$	28.9443	1.19263
$590$	0	0
$591$	2.94427	0.121111
$592$	6.94427	0.285408
$593$	12.4721	0.512169	0.256085	−	0.966654i	$-0.417567\pi$
$593$	12.4721	0.512169	0.256085	+	0.966654i	$0.417567\pi$
$594$	4.47214	0.183494
$595$	0	0
$596$	22.4721	0.920495
$597$	16.9443	0.693483
$598$	−7.23607	−0.295905
$599$	−41.8885	−1.71152	−0.855760	−	0.517373i	$-0.826910\pi$
$599$	−41.8885	−1.71152	−0.855760	+	0.517373i	$0.826910\pi$
$600$	0	0
$601$	−13.4164	−0.547267	−0.273633	−	0.961834i	$-0.588225\pi$
$601$	−13.4164	−0.547267	−0.273633	+	0.961834i	$0.588225\pi$
$602$	8.00000	0.326056
$603$	0	0
$604$	7.41641	0.301769
$605$	0	0
$606$	4.76393	0.193522
$607$	−25.4164	−1.03162	−0.515810	−	0.856703i	$-0.672509\pi$
$607$	−25.4164	−1.03162	−0.515810	+	0.856703i	$0.672509\pi$
$608$	7.23607	0.293461
$609$	−12.0000	−0.486265
$610$	0	0
$611$	−4.94427	−0.200024
$612$	2.76393	0.111725
$613$	18.0000	0.727013	0.363507	−	0.931592i	$-0.381579\pi$
$613$	18.0000	0.727013	0.363507	+	0.931592i	$0.381579\pi$
$614$	−20.3607	−0.821690
$615$	0	0
$616$	−5.52786	−0.222724
$617$	0.111456	0.00448706	0.00224353	−	0.999997i	$-0.499286\pi$
$617$	0.111456	0.00448706	0.00224353	+	0.999997i	$0.499286\pi$
$618$	0.472136	0.0189921
$619$	−20.7639	−0.834573	−0.417286	−	0.908775i	$-0.637019\pi$
$619$	−20.7639	−0.834573	−0.417286	+	0.908775i	$0.637019\pi$
$620$	0	0
$621$	7.23607	0.290373
$622$	13.5279	0.542418
$623$	10.4721	0.419557
$624$	1.00000	0.0400320
$625$	0	0
$626$	−7.41641	−0.296419
$627$	32.3607	1.29236
$628$	−1.05573	−0.0421281
$629$	19.1935	0.765295
$630$	0	0
$631$	−39.4164	−1.56914	−0.784571	−	0.620039i	$-0.787117\pi$
$631$	−39.4164	−1.56914	−0.784571	+	0.620039i	$0.787117\pi$
$632$	4.00000	0.159111
$633$	−21.8885	−0.869992
$634$	13.4164	0.532834
$635$	0	0
$636$	8.47214	0.335942
$637$	−5.47214	−0.216814
$638$	43.4164	1.71887
$639$	6.47214	0.256034
$640$	0	0
$641$	39.3050	1.55245	0.776226	−	0.630455i	$-0.217131\pi$
$641$	39.3050	1.55245	0.776226	+	0.630455i	$0.217131\pi$
$642$	10.4721	0.413302
$643$	36.9443	1.45694	0.728470	−	0.685078i	$-0.240232\pi$
$643$	36.9443	1.45694	0.728470	+	0.685078i	$0.240232\pi$
$644$	−8.94427	−0.352454
$645$	0	0
$646$	20.0000	0.786889
$647$	9.70820	0.381669	0.190834	−	0.981622i	$-0.438881\pi$
$647$	9.70820	0.381669	0.190834	+	0.981622i	$0.438881\pi$
$648$	−1.00000	−0.0392837
$649$	−2.11146	−0.0828819
$650$	0	0
$651$	4.94427	0.193781
$652$	−4.94427	−0.193633
$653$	14.3607	0.561977	0.280988	−	0.959711i	$-0.409338\pi$
$653$	14.3607	0.561977	0.280988	+	0.959711i	$0.409338\pi$
$654$	15.7082	0.614239
$655$	0	0
$656$	12.4721	0.486955
$657$	8.76393	0.341914
$658$	−6.11146	−0.238249
$659$	48.0689	1.87250	0.936249	−	0.351337i	$-0.114273\pi$
$659$	48.0689	1.87250	0.936249	+	0.351337i	$0.114273\pi$
$660$	0	0
$661$	−28.2918	−1.10042	−0.550212	−	0.835025i	$-0.685453\pi$
$661$	−28.2918	−1.10042	−0.550212	+	0.835025i	$0.685453\pi$
$662$	7.23607	0.281238
$663$	2.76393	0.107342
$664$	−12.9443	−0.502335
$665$	0	0
$666$	−6.94427	−0.269085
$667$	70.2492	2.72006
$668$	7.41641	0.286949
$669$	21.2361	0.821034
$670$	0	0
$671$	−31.0557	−1.19889
$672$	1.23607	0.0476824
$673$	44.0000	1.69608	0.848038	−	0.529936i	$-0.177784\pi$
$673$	44.0000	1.69608	0.848038	+	0.529936i	$0.177784\pi$
$674$	−2.47214	−0.0952231
$675$	0	0
$676$	1.00000	0.0384615
$677$	36.4721	1.40174	0.700869	−	0.713290i	$-0.252796\pi$
$677$	36.4721	1.40174	0.700869	+	0.713290i	$0.252796\pi$
$678$	12.6525	0.485915
$679$	−12.0000	−0.460518
$680$	0	0
$681$	24.9443	0.955867
$682$	−17.8885	−0.684988
$683$	−6.11146	−0.233848	−0.116924	−	0.993141i	$-0.537303\pi$
$683$	−6.11146	−0.233848	−0.116924	+	0.993141i	$0.537303\pi$
$684$	−7.23607	−0.276678
$685$	0	0
$686$	−15.4164	−0.588601
$687$	−3.70820	−0.141477
$688$	6.47214	0.246748
$689$	8.47214	0.322763
$690$	0	0
$691$	13.7082	0.521485	0.260742	−	0.965408i	$-0.416033\pi$
$691$	13.7082	0.521485	0.260742	+	0.965408i	$0.416033\pi$
$692$	−5.05573	−0.192190
$693$	5.52786	0.209986
$694$	−10.4721	−0.397517
$695$	0	0
$696$	−9.70820	−0.367989
$697$	34.4721	1.30573
$698$	10.7639	0.407421
$699$	−2.18034	−0.0824680
$700$	0	0
$701$	34.0689	1.28676	0.643382	−	0.765545i	$-0.277531\pi$
$701$	34.0689	1.28676	0.643382	+	0.765545i	$0.277531\pi$
$702$	−1.00000	−0.0377426
$703$	−50.2492	−1.89519
$704$	−4.47214	−0.168550
$705$	0	0
$706$	9.05573	0.340817
$707$	5.88854	0.221461
$708$	0.472136	0.0177440
$709$	−34.5410	−1.29722	−0.648608	−	0.761123i	$-0.724648\pi$
$709$	−34.5410	−1.29722	−0.648608	+	0.761123i	$0.724648\pi$
$710$	0	0
$711$	−4.00000	−0.150012
$712$	8.47214	0.317507
$713$	−28.9443	−1.08397
$714$	3.41641	0.127856
$715$	0	0
$716$	−14.1803	−0.529944
$717$	13.8885	0.518677
$718$	2.47214	0.0922593
$719$	3.05573	0.113959	0.0569797	−	0.998375i	$-0.481853\pi$
$719$	3.05573	0.113959	0.0569797	+	0.998375i	$0.481853\pi$
$720$	0	0
$721$	0.583592	0.0217341
$722$	−33.3607	−1.24156
$723$	12.4721	0.463844
$724$	−9.41641	−0.349958
$725$	0	0
$726$	−9.00000	−0.334021
$727$	18.0000	0.667583	0.333792	−	0.942647i	$-0.391672\pi$
$727$	18.0000	0.667583	0.333792	+	0.942647i	$0.391672\pi$
$728$	1.23607	0.0458117
$729$	1.00000	0.0370370
$730$	0	0
$731$	17.8885	0.661632
$732$	6.94427	0.256668
$733$	6.58359	0.243171	0.121585	−	0.992581i	$-0.461202\pi$
$733$	6.58359	0.243171	0.121585	+	0.992581i	$0.461202\pi$
$734$	8.47214	0.312712
$735$	0	0
$736$	−7.23607	−0.266725
$737$	0	0
$738$	−12.4721	−0.459106
$739$	−13.7082	−0.504264	−0.252132	−	0.967693i	$-0.581132\pi$
$739$	−13.7082	−0.504264	−0.252132	+	0.967693i	$0.581132\pi$
$740$	0	0
$741$	−7.23607	−0.265824
$742$	10.4721	0.384444
$743$	−16.3607	−0.600215	−0.300108	−	0.953905i	$-0.597023\pi$
$743$	−16.3607	−0.600215	−0.300108	+	0.953905i	$0.597023\pi$
$744$	4.00000	0.146647
$745$	0	0
$746$	−2.00000	−0.0732252
$747$	12.9443	0.473606
$748$	−12.3607	−0.451951
$749$	12.9443	0.472973
$750$	0	0
$751$	6.83282	0.249333	0.124666	−	0.992199i	$-0.460214\pi$
$751$	6.83282	0.249333	0.124666	+	0.992199i	$0.460214\pi$
$752$	−4.94427	−0.180299
$753$	15.7082	0.572439
$754$	−9.70820	−0.353552
$755$	0	0
$756$	−1.23607	−0.0449554
$757$	−38.0000	−1.38113	−0.690567	−	0.723269i	$-0.742639\pi$
$757$	−38.0000	−1.38113	−0.690567	+	0.723269i	$0.742639\pi$
$758$	−5.12461	−0.186134
$759$	−32.3607	−1.17462
$760$	0	0
$761$	16.4721	0.597114	0.298557	−	0.954392i	$-0.403495\pi$
$761$	16.4721	0.597114	0.298557	+	0.954392i	$0.403495\pi$
$762$	5.41641	0.196216
$763$	19.4164	0.702921
$764$	13.5279	0.489421
$765$	0	0
$766$	−14.4721	−0.522900
$767$	0.472136	0.0170478
$768$	1.00000	0.0360844
$769$	48.8328	1.76096	0.880478	−	0.474087i	$-0.157222\pi$
$769$	48.8328	1.76096	0.880478	+	0.474087i	$0.157222\pi$
$770$	0	0
$771$	5.23607	0.188572
$772$	−21.7082	−0.781295
$773$	14.9443	0.537508	0.268754	−	0.963209i	$-0.413388\pi$
$773$	14.9443	0.537508	0.268754	+	0.963209i	$0.413388\pi$
$774$	−6.47214	−0.232636
$775$	0	0
$776$	−9.70820	−0.348504
$777$	−8.58359	−0.307935
$778$	18.6525	0.668724
$779$	−90.2492	−3.23351
$780$	0	0
$781$	−28.9443	−1.03571
$782$	−20.0000	−0.715199
$783$	9.70820	0.346943
$784$	−5.47214	−0.195433
$785$	0	0
$786$	3.70820	0.132267
$787$	11.0557	0.394094	0.197047	−	0.980394i	$-0.436865\pi$
$787$	11.0557	0.394094	0.197047	+	0.980394i	$0.436865\pi$
$788$	2.94427	0.104885
$789$	−12.1803	−0.433632
$790$	0	0
$791$	15.6393	0.556070
$792$	4.47214	0.158910
$793$	6.94427	0.246598
$794$	9.41641	0.334176
$795$	0	0
$796$	16.9443	0.600574
$797$	26.9443	0.954415	0.477208	−	0.878791i	$-0.341649\pi$
$797$	26.9443	0.954415	0.477208	+	0.878791i	$0.341649\pi$
$798$	−8.94427	−0.316624
$799$	−13.6656	−0.483455
$800$	0	0
$801$	−8.47214	−0.299348
$802$	15.5279	0.548308
$803$	−39.1935	−1.38311
$804$	0	0
$805$	0	0
$806$	4.00000	0.140894
$807$	11.2361	0.395528
$808$	4.76393	0.167595
$809$	40.4721	1.42292	0.711462	−	0.702724i	$-0.248033\pi$
$809$	40.4721	1.42292	0.711462	+	0.702724i	$0.248033\pi$
$810$	0	0
$811$	17.7082	0.621819	0.310910	−	0.950439i	$-0.399366\pi$
$811$	17.7082	0.621819	0.310910	+	0.950439i	$0.399366\pi$
$812$	−12.0000	−0.421117
$813$	15.4164	0.540677
$814$	31.0557	1.08850
$815$	0	0
$816$	2.76393	0.0967570
$817$	−46.8328	−1.63847
$818$	−6.00000	−0.209785
$819$	−1.23607	−0.0431917
$820$	0	0
$821$	−21.5279	−0.751328	−0.375664	−	0.926756i	$-0.622585\pi$
$821$	−21.5279	−0.751328	−0.375664	+	0.926756i	$0.622585\pi$
$822$	−15.8885	−0.554177
$823$	−32.2492	−1.12414	−0.562069	−	0.827091i	$-0.689994\pi$
$823$	−32.2492	−1.12414	−0.562069	+	0.827091i	$0.689994\pi$
$824$	0.472136	0.0164476
$825$	0	0
$826$	0.583592	0.0203058
$827$	−6.11146	−0.212516	−0.106258	−	0.994339i	$-0.533887\pi$
$827$	−6.11146	−0.212516	−0.106258	+	0.994339i	$0.533887\pi$
$828$	7.23607	0.251471
$829$	−46.7214	−1.62270	−0.811350	−	0.584561i	$-0.801267\pi$
$829$	−46.7214	−1.62270	−0.811350	+	0.584561i	$0.801267\pi$
$830$	0	0
$831$	−32.8328	−1.13896
$832$	1.00000	0.0346688
$833$	−15.1246	−0.524037
$834$	−2.47214	−0.0856031
$835$	0	0
$836$	32.3607	1.11922
$837$	−4.00000	−0.138260
$838$	21.5967	0.746047
$839$	19.7771	0.682781	0.341390	−	0.939922i	$-0.389102\pi$
$839$	19.7771	0.682781	0.341390	+	0.939922i	$0.389102\pi$
$840$	0	0
$841$	65.2492	2.24997
$842$	0.291796	0.0100560
$843$	15.5279	0.534808
$844$	−21.8885	−0.753435
$845$	0	0
$846$	4.94427	0.169988
$847$	−11.1246	−0.382246
$848$	8.47214	0.290934
$849$	15.4164	0.529090
$850$	0	0
$851$	50.2492	1.72252
$852$	6.47214	0.221732
$853$	14.5836	0.499333	0.249666	−	0.968332i	$-0.419679\pi$
$853$	14.5836	0.499333	0.249666	+	0.968332i	$0.419679\pi$
$854$	8.58359	0.293724
$855$	0	0
$856$	10.4721	0.357930
$857$	22.1803	0.757666	0.378833	−	0.925465i	$-0.376325\pi$
$857$	22.1803	0.757666	0.378833	+	0.925465i	$0.376325\pi$
$858$	4.47214	0.152676
$859$	36.0000	1.22830	0.614152	−	0.789188i	$-0.289498\pi$
$859$	36.0000	1.22830	0.614152	+	0.789188i	$0.289498\pi$
$860$	0	0
$861$	−15.4164	−0.525390
$862$	−38.8328	−1.32265
$863$	13.5279	0.460494	0.230247	−	0.973132i	$-0.426047\pi$
$863$	13.5279	0.460494	0.230247	+	0.973132i	$0.426047\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	18.4721	0.627709
$867$	−9.36068	−0.317905
$868$	4.94427	0.167820
$869$	17.8885	0.606827
$870$	0	0
$871$	0	0
$872$	15.7082	0.531947
$873$	9.70820	0.328573
$874$	52.3607	1.77113
$875$	0	0
$876$	8.76393	0.296106
$877$	15.5279	0.524339	0.262169	−	0.965022i	$-0.415562\pi$
$877$	15.5279	0.524339	0.262169	+	0.965022i	$0.415562\pi$
$878$	−12.9443	−0.436848
$879$	−14.9443	−0.504058
$880$	0	0
$881$	−10.3607	−0.349060	−0.174530	−	0.984652i	$-0.555841\pi$
$881$	−10.3607	−0.349060	−0.174530	+	0.984652i	$0.555841\pi$
$882$	5.47214	0.184256
$883$	−9.88854	−0.332776	−0.166388	−	0.986060i	$-0.553210\pi$
$883$	−9.88854	−0.332776	−0.166388	+	0.986060i	$0.553210\pi$
$884$	2.76393	0.0929611
$885$	0	0
$886$	13.5279	0.454477
$887$	12.1803	0.408976	0.204488	−	0.978869i	$-0.434447\pi$
$887$	12.1803	0.408976	0.204488	+	0.978869i	$0.434447\pi$
$888$	−6.94427	−0.233035
$889$	6.69505	0.224545
$890$	0	0
$891$	−4.47214	−0.149822
$892$	21.2361	0.711036
$893$	35.7771	1.19723
$894$	−22.4721	−0.751581
$895$	0	0
$896$	1.23607	0.0412941
$897$	7.23607	0.241605
$898$	8.47214	0.282719
$899$	−38.8328	−1.29515
$900$	0	0
$901$	23.4164	0.780114
$902$	55.7771	1.85717
$903$	−8.00000	−0.266223
$904$	12.6525	0.420815
$905$	0	0
$906$	−7.41641	−0.246394
$907$	54.4721	1.80872	0.904359	−	0.426773i	$-0.140350\pi$
$907$	54.4721	1.80872	0.904359	+	0.426773i	$0.140350\pi$
$908$	24.9443	0.827805
$909$	−4.76393	−0.158010
$910$	0	0
$911$	−28.9443	−0.958967	−0.479483	−	0.877551i	$-0.659176\pi$
$911$	−28.9443	−0.958967	−0.479483	+	0.877551i	$0.659176\pi$
$912$	−7.23607	−0.239610
$913$	−57.8885	−1.91583
$914$	−9.70820	−0.321119
$915$	0	0
$916$	−3.70820	−0.122523
$917$	4.58359	0.151364
$918$	−2.76393	−0.0912234
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	0	0
$921$	20.3607	0.670907
$922$	−6.11146	−0.201270
$923$	6.47214	0.213033
$924$	5.52786	0.181853
$925$	0	0
$926$	35.7082	1.17344
$927$	−0.472136	−0.0155070
$928$	−9.70820	−0.318687
$929$	23.5279	0.771924	0.385962	−	0.922515i	$-0.373870\pi$
$929$	23.5279	0.771924	0.385962	+	0.922515i	$0.373870\pi$
$930$	0	0
$931$	39.5967	1.29773
$932$	−2.18034	−0.0714194
$933$	−13.5279	−0.442882
$934$	2.47214	0.0808908
$935$	0	0
$936$	−1.00000	−0.0326860
$937$	−51.7771	−1.69148	−0.845742	−	0.533592i	$-0.820842\pi$
$937$	−51.7771	−1.69148	−0.845742	+	0.533592i	$0.820842\pi$
$938$	0	0
$939$	7.41641	0.242025
$940$	0	0
$941$	−46.4721	−1.51495	−0.757474	−	0.652865i	$-0.773567\pi$
$941$	−46.4721	−1.51495	−0.757474	+	0.652865i	$0.773567\pi$
$942$	1.05573	0.0343975
$943$	90.2492	2.93892
$944$	0.472136	0.0153667
$945$	0	0
$946$	28.9443	0.941059
$947$	−12.9443	−0.420632	−0.210316	−	0.977633i	$-0.567449\pi$
$947$	−12.9443	−0.420632	−0.210316	+	0.977633i	$0.567449\pi$
$948$	−4.00000	−0.129914
$949$	8.76393	0.284489
$950$	0	0
$951$	−13.4164	−0.435057
$952$	3.41641	0.110726
$953$	−11.1246	−0.360362	−0.180181	−	0.983634i	$-0.557668\pi$
$953$	−11.1246	−0.360362	−0.180181	+	0.983634i	$0.557668\pi$
$954$	−8.47214	−0.274296
$955$	0	0
$956$	13.8885	0.449188
$957$	−43.4164	−1.40345
$958$	36.0000	1.16311
$959$	−19.6393	−0.634187
$960$	0	0
$961$	−15.0000	−0.483871
$962$	−6.94427	−0.223892
$963$	−10.4721	−0.337460
$964$	12.4721	0.401700
$965$	0	0
$966$	8.94427	0.287777
$967$	10.5410	0.338976	0.169488	−	0.985532i	$-0.445789\pi$
$967$	10.5410	0.338976	0.169488	+	0.985532i	$0.445789\pi$
$968$	−9.00000	−0.289271
$969$	−20.0000	−0.642493
$970$	0	0
$971$	18.7639	0.602163	0.301082	−	0.953598i	$-0.402652\pi$
$971$	18.7639	0.602163	0.301082	+	0.953598i	$0.402652\pi$
$972$	1.00000	0.0320750
$973$	−3.05573	−0.0979621
$974$	−25.5967	−0.820173
$975$	0	0
$976$	6.94427	0.222281
$977$	34.3607	1.09930	0.549648	−	0.835397i	$-0.314762\pi$
$977$	34.3607	1.09930	0.549648	+	0.835397i	$0.314762\pi$
$978$	4.94427	0.158100
$979$	37.8885	1.21092
$980$	0	0
$981$	−15.7082	−0.501524
$982$	−6.18034	−0.197223
$983$	19.4164	0.619287	0.309644	−	0.950853i	$-0.399790\pi$
$983$	19.4164	0.619287	0.309644	+	0.950853i	$0.399790\pi$
$984$	−12.4721	−0.397597
$985$	0	0
$986$	−26.8328	−0.854531
$987$	6.11146	0.194530
$988$	−7.23607	−0.230210
$989$	46.8328	1.48920
$990$	0	0
$991$	7.05573	0.224133	0.112066	−	0.993701i	$-0.464253\pi$
$991$	7.05573	0.224133	0.112066	+	0.993701i	$0.464253\pi$
$992$	4.00000	0.127000
$993$	−7.23607	−0.229630
$994$	8.00000	0.253745
$995$	0	0
$996$	12.9443	0.410155
$997$	8.11146	0.256892	0.128446	−	0.991716i	$-0.459001\pi$
$997$	8.11146	0.256892	0.128446	+	0.991716i	$0.459001\pi$
$998$	−1.12461	−0.0355990
$999$	6.94427	0.219707

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1950.2.a.be.1.1		2
3.2	odd	2		5850.2.a.cm.1.1		2
5.2	odd	4		390.2.e.e.79.1	✓	4
5.3	odd	4		390.2.e.e.79.4	yes	4
5.4	even	2		1950.2.a.bf.1.2		2
15.2	even	4		1170.2.e.e.469.4		4
15.8	even	4		1170.2.e.e.469.1		4
15.14	odd	2		5850.2.a.cf.1.2		2
20.3	even	4		3120.2.l.k.1249.2		4
20.7	even	4		3120.2.l.k.1249.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.e.e.79.1	✓	4	5.2	odd	4
390.2.e.e.79.4	yes	4	5.3	odd	4
1170.2.e.e.469.1		4	15.8	even	4
1170.2.e.e.469.4		4	15.2	even	4
1950.2.a.be.1.1		2	1.1	even	1	trivial
1950.2.a.bf.1.2		2	5.4	even	2
3120.2.l.k.1249.2		4	20.3	even	4
3120.2.l.k.1249.3		4	20.7	even	4
5850.2.a.cf.1.2		2	15.14	odd	2
5850.2.a.cm.1.1		2	3.2	odd	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.23607 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.23607 q^{7} -1.00000 q^{8} +1.00000 q^{9} -4.47214 q^{11} +1.00000 q^{12} +1.00000 q^{13} +1.23607 q^{14} +1.00000 q^{16} +2.76393 q^{17} -1.00000 q^{18} -7.23607 q^{19} -1.23607 q^{21} +4.47214 q^{22} +7.23607 q^{23} -1.00000 q^{24} -1.00000 q^{26} +1.00000 q^{27} -1.23607 q^{28} +9.70820 q^{29} -4.00000 q^{31} -1.00000 q^{32} -4.47214 q^{33} -2.76393 q^{34} +1.00000 q^{36} +6.94427 q^{37} +7.23607 q^{38} +1.00000 q^{39} +12.4721 q^{41} +1.23607 q^{42} +6.47214 q^{43} -4.47214 q^{44} -7.23607 q^{46} -4.94427 q^{47} +1.00000 q^{48} -5.47214 q^{49} +2.76393 q^{51} +1.00000 q^{52} +8.47214 q^{53} -1.00000 q^{54} +1.23607 q^{56} -7.23607 q^{57} -9.70820 q^{58} +0.472136 q^{59} +6.94427 q^{61} +4.00000 q^{62} -1.23607 q^{63} +1.00000 q^{64} +4.47214 q^{66} +2.76393 q^{68} +7.23607 q^{69} +6.47214 q^{71} -1.00000 q^{72} +8.76393 q^{73} -6.94427 q^{74} -7.23607 q^{76} +5.52786 q^{77} -1.00000 q^{78} -4.00000 q^{79} +1.00000 q^{81} -12.4721 q^{82} +12.9443 q^{83} -1.23607 q^{84} -6.47214 q^{86} +9.70820 q^{87} +4.47214 q^{88} -8.47214 q^{89} -1.23607 q^{91} +7.23607 q^{92} -4.00000 q^{93} +4.94427 q^{94} -1.00000 q^{96} +9.70820 q^{97} +5.47214 q^{98} -4.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 + 2 * q^7 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9} + 2 q^{12} + 2 q^{13} - 2 q^{14} + 2 q^{16} + 10 q^{17} - 2 q^{18} - 10 q^{19} + 2 q^{21} + 10 q^{23} - 2 q^{24} - 2 q^{26} + 2 q^{27} + 2 q^{28} + 6 q^{29} - 8 q^{31} - 2 q^{32} - 10 q^{34} + 2 q^{36} - 4 q^{37} + 10 q^{38} + 2 q^{39} + 16 q^{41} - 2 q^{42} + 4 q^{43} - 10 q^{46} + 8 q^{47} + 2 q^{48} - 2 q^{49} + 10 q^{51} + 2 q^{52} + 8 q^{53} - 2 q^{54} - 2 q^{56} - 10 q^{57} - 6 q^{58} - 8 q^{59} - 4 q^{61} + 8 q^{62} + 2 q^{63} + 2 q^{64} + 10 q^{68} + 10 q^{69} + 4 q^{71} - 2 q^{72} + 22 q^{73} + 4 q^{74} - 10 q^{76} + 20 q^{77} - 2 q^{78} - 8 q^{79} + 2 q^{81} - 16 q^{82} + 8 q^{83} + 2 q^{84} - 4 q^{86} + 6 q^{87} - 8 q^{89} + 2 q^{91} + 10 q^{92} - 8 q^{93} - 8 q^{94} - 2 q^{96} + 6 q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 + 2 * q^7 - 2 * q^8 + 2 * q^9 + 2 * q^12 + 2 * q^13 - 2 * q^14 + 2 * q^16 + 10 * q^17 - 2 * q^18 - 10 * q^19 + 2 * q^21 + 10 * q^23 - 2 * q^24 - 2 * q^26 + 2 * q^27 + 2 * q^28 + 6 * q^29 - 8 * q^31 - 2 * q^32 - 10 * q^34 + 2 * q^36 - 4 * q^37 + 10 * q^38 + 2 * q^39 + 16 * q^41 - 2 * q^42 + 4 * q^43 - 10 * q^46 + 8 * q^47 + 2 * q^48 - 2 * q^49 + 10 * q^51 + 2 * q^52 + 8 * q^53 - 2 * q^54 - 2 * q^56 - 10 * q^57 - 6 * q^58 - 8 * q^59 - 4 * q^61 + 8 * q^62 + 2 * q^63 + 2 * q^64 + 10 * q^68 + 10 * q^69 + 4 * q^71 - 2 * q^72 + 22 * q^73 + 4 * q^74 - 10 * q^76 + 20 * q^77 - 2 * q^78 - 8 * q^79 + 2 * q^81 - 16 * q^82 + 8 * q^83 + 2 * q^84 - 4 * q^86 + 6 * q^87 - 8 * q^89 + 2 * q^91 + 10 * q^92 - 8 * q^93 - 8 * q^94 - 2 * q^96 + 6 * q^97 + 2 * q^98

Introduction

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