LMFDB - Embedded newform 6025.2.a.b.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6025,2,Mod(1,6025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6025, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6025 = 5^{2} \cdot 241 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6025.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$48.1098672178$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1205)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	6025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.23607 q^{2} -2.61803 q^{3} +3.00000 q^{4} -5.85410 q^{6} +3.23607 q^{7} +2.23607 q^{8} +3.85410 q^{9} +O(q^{10})$	\(q+2.23607 q^{2} -2.61803 q^{3} +3.00000 q^{4} -5.85410 q^{6} +3.23607 q^{7} +2.23607 q^{8} +3.85410 q^{9} +0.618034 q^{11} -7.85410 q^{12} +3.85410 q^{13} +7.23607 q^{14} -1.00000 q^{16} -6.09017 q^{17} +8.61803 q^{18} -7.61803 q^{19} -8.47214 q^{21} +1.38197 q^{22} -4.47214 q^{23} -5.85410 q^{24} +8.61803 q^{26} -2.23607 q^{27} +9.70820 q^{28} +4.09017 q^{29} -6.47214 q^{31} -6.70820 q^{32} -1.61803 q^{33} -13.6180 q^{34} +11.5623 q^{36} -4.47214 q^{37} -17.0344 q^{38} -10.0902 q^{39} +4.61803 q^{41} -18.9443 q^{42} -10.4721 q^{43} +1.85410 q^{44} -10.0000 q^{46} +5.61803 q^{47} +2.61803 q^{48} +3.47214 q^{49} +15.9443 q^{51} +11.5623 q^{52} +1.52786 q^{53} -5.00000 q^{54} +7.23607 q^{56} +19.9443 q^{57} +9.14590 q^{58} -5.70820 q^{59} +6.14590 q^{61} -14.4721 q^{62} +12.4721 q^{63} -13.0000 q^{64} -3.61803 q^{66} +4.09017 q^{67} -18.2705 q^{68} +11.7082 q^{69} -12.5623 q^{71} +8.61803 q^{72} +7.61803 q^{73} -10.0000 q^{74} -22.8541 q^{76} +2.00000 q^{77} -22.5623 q^{78} -6.18034 q^{79} -5.70820 q^{81} +10.3262 q^{82} -12.0902 q^{83} -25.4164 q^{84} -23.4164 q^{86} -10.7082 q^{87} +1.38197 q^{88} +2.94427 q^{89} +12.4721 q^{91} -13.4164 q^{92} +16.9443 q^{93} +12.5623 q^{94} +17.5623 q^{96} -2.76393 q^{97} +7.76393 q^{98} +2.38197 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{3} + 6 q^{4} - 5 q^{6} + 2 q^{7} + q^{9}+O(q^{10}) $ 2 * q - 3 * q^3 + 6 * q^4 - 5 * q^6 + 2 * q^7 + q^9	$ 2 q - 3 q^{3} + 6 q^{4} - 5 q^{6} + 2 q^{7} + q^{9} - q^{11} - 9 q^{12} + q^{13} + 10 q^{14} - 2 q^{16} - q^{17} + 15 q^{18} - 13 q^{19} - 8 q^{21} + 5 q^{22} - 5 q^{24} + 15 q^{26} + 6 q^{28} - 3 q^{29} - 4 q^{31} - q^{33} - 25 q^{34} + 3 q^{36} - 5 q^{38} - 9 q^{39} + 7 q^{41} - 20 q^{42} - 12 q^{43} - 3 q^{44} - 20 q^{46} + 9 q^{47} + 3 q^{48} - 2 q^{49} + 14 q^{51} + 3 q^{52} + 12 q^{53} - 10 q^{54} + 10 q^{56} + 22 q^{57} + 25 q^{58} + 2 q^{59} + 19 q^{61} - 20 q^{62} + 16 q^{63} - 26 q^{64} - 5 q^{66} - 3 q^{67} - 3 q^{68} + 10 q^{69} - 5 q^{71} + 15 q^{72} + 13 q^{73} - 20 q^{74} - 39 q^{76} + 4 q^{77} - 25 q^{78} + 10 q^{79} + 2 q^{81} + 5 q^{82} - 13 q^{83} - 24 q^{84} - 20 q^{86} - 8 q^{87} + 5 q^{88} - 12 q^{89} + 16 q^{91} + 16 q^{93} + 5 q^{94} + 15 q^{96} - 10 q^{97} + 20 q^{98} + 7 q^{99}+O(q^{100}) $ 2 * q - 3 * q^3 + 6 * q^4 - 5 * q^6 + 2 * q^7 + q^9 - q^11 - 9 * q^12 + q^13 + 10 * q^14 - 2 * q^16 - q^17 + 15 * q^18 - 13 * q^19 - 8 * q^21 + 5 * q^22 - 5 * q^24 + 15 * q^26 + 6 * q^28 - 3 * q^29 - 4 * q^31 - q^33 - 25 * q^34 + 3 * q^36 - 5 * q^38 - 9 * q^39 + 7 * q^41 - 20 * q^42 - 12 * q^43 - 3 * q^44 - 20 * q^46 + 9 * q^47 + 3 * q^48 - 2 * q^49 + 14 * q^51 + 3 * q^52 + 12 * q^53 - 10 * q^54 + 10 * q^56 + 22 * q^57 + 25 * q^58 + 2 * q^59 + 19 * q^61 - 20 * q^62 + 16 * q^63 - 26 * q^64 - 5 * q^66 - 3 * q^67 - 3 * q^68 + 10 * q^69 - 5 * q^71 + 15 * q^72 + 13 * q^73 - 20 * q^74 - 39 * q^76 + 4 * q^77 - 25 * q^78 + 10 * q^79 + 2 * q^81 + 5 * q^82 - 13 * q^83 - 24 * q^84 - 20 * q^86 - 8 * q^87 + 5 * q^88 - 12 * q^89 + 16 * q^91 + 16 * q^93 + 5 * q^94 + 15 * q^96 - 10 * q^97 + 20 * q^98 + 7 * 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 $

$2$	2.23607	1.58114	0.790569	−	0.612372i	$-0.209785\pi$
$2$	2.23607	1.58114	0.790569	+	0.612372i	$0.209785\pi$
$3$	−2.61803	−1.51152	−0.755761	−	0.654847i	$-0.772733\pi$
$3$	−2.61803	−1.51152	−0.755761	+	0.654847i	$0.772733\pi$
$4$	3.00000	1.50000
$5$	0	0
$5$	0	0
$6$	−5.85410	−2.38993
$7$	3.23607	1.22312	0.611559	−	0.791199i	$-0.290543\pi$
$7$	3.23607	1.22312	0.611559	+	0.791199i	$0.290543\pi$
$8$	2.23607	0.790569
$9$	3.85410	1.28470
$10$	0	0
$11$	0.618034	0.186344	0.0931721	−	0.995650i	$-0.470299\pi$
$11$	0.618034	0.186344	0.0931721	+	0.995650i	$0.470299\pi$
$12$	−7.85410	−2.26728
$13$	3.85410	1.06894	0.534468	−	0.845189i	$-0.320512\pi$
$13$	3.85410	1.06894	0.534468	+	0.845189i	$0.320512\pi$
$14$	7.23607	1.93392
$15$	0	0
$16$	−1.00000	−0.250000
$17$	−6.09017	−1.47708	−0.738542	−	0.674208i	$-0.764485\pi$
$17$	−6.09017	−1.47708	−0.738542	+	0.674208i	$0.764485\pi$
$18$	8.61803	2.03129
$19$	−7.61803	−1.74770	−0.873848	−	0.486198i	$-0.838383\pi$
$19$	−7.61803	−1.74770	−0.873848	+	0.486198i	$0.838383\pi$
$20$	0	0
$21$	−8.47214	−1.84877
$22$	1.38197	0.294636
$23$	−4.47214	−0.932505	−0.466252	−	0.884652i	$-0.654396\pi$
$23$	−4.47214	−0.932505	−0.466252	+	0.884652i	$0.654396\pi$
$24$	−5.85410	−1.19496
$25$	0	0
$26$	8.61803	1.69014
$27$	−2.23607	−0.430331
$28$	9.70820	1.83468
$29$	4.09017	0.759525	0.379763	−	0.925084i	$-0.376006\pi$
$29$	4.09017	0.759525	0.379763	+	0.925084i	$0.376006\pi$
$30$	0	0
$31$	−6.47214	−1.16243	−0.581215	−	0.813750i	$-0.697422\pi$
$31$	−6.47214	−1.16243	−0.581215	+	0.813750i	$0.697422\pi$
$32$	−6.70820	−1.18585
$33$	−1.61803	−0.281664
$34$	−13.6180	−2.33547
$35$	0	0
$36$	11.5623	1.92705
$37$	−4.47214	−0.735215	−0.367607	−	0.929981i	$-0.619823\pi$
$37$	−4.47214	−0.735215	−0.367607	+	0.929981i	$0.619823\pi$
$38$	−17.0344	−2.76335
$39$	−10.0902	−1.61572
$40$	0	0
$41$	4.61803	0.721216	0.360608	−	0.932718i	$-0.382569\pi$
$41$	4.61803	0.721216	0.360608	+	0.932718i	$0.382569\pi$
$42$	−18.9443	−2.92316
$43$	−10.4721	−1.59699	−0.798493	−	0.602004i	$-0.794369\pi$
$43$	−10.4721	−1.59699	−0.798493	+	0.602004i	$0.794369\pi$
$44$	1.85410	0.279516
$45$	0	0
$46$	−10.0000	−1.47442
$47$	5.61803	0.819474	0.409737	−	0.912204i	$-0.365620\pi$
$47$	5.61803	0.819474	0.409737	+	0.912204i	$0.365620\pi$
$48$	2.61803	0.377881
$49$	3.47214	0.496019
$50$	0	0
$51$	15.9443	2.23264
$52$	11.5623	1.60340
$53$	1.52786	0.209868	0.104934	−	0.994479i	$-0.466537\pi$
$53$	1.52786	0.209868	0.104934	+	0.994479i	$0.466537\pi$
$54$	−5.00000	−0.680414
$55$	0	0
$56$	7.23607	0.966960
$57$	19.9443	2.64168
$58$	9.14590	1.20092
$59$	−5.70820	−0.743145	−0.371572	−	0.928404i	$-0.621181\pi$
$59$	−5.70820	−0.743145	−0.371572	+	0.928404i	$0.621181\pi$
$60$	0	0
$61$	6.14590	0.786902	0.393451	−	0.919346i	$-0.371281\pi$
$61$	6.14590	0.786902	0.393451	+	0.919346i	$0.371281\pi$
$62$	−14.4721	−1.83796
$63$	12.4721	1.57134
$64$	−13.0000	−1.62500
$65$	0	0
$66$	−3.61803	−0.445349
$67$	4.09017	0.499694	0.249847	−	0.968285i	$-0.419620\pi$
$67$	4.09017	0.499694	0.249847	+	0.968285i	$0.419620\pi$
$68$	−18.2705	−2.21562
$69$	11.7082	1.40950
$70$	0	0
$71$	−12.5623	−1.49087	−0.745436	−	0.666578i	$-0.767759\pi$
$71$	−12.5623	−1.49087	−0.745436	+	0.666578i	$0.767759\pi$
$72$	8.61803	1.01565
$73$	7.61803	0.891623	0.445812	−	0.895127i	$-0.352915\pi$
$73$	7.61803	0.891623	0.445812	+	0.895127i	$0.352915\pi$
$74$	−10.0000	−1.16248
$75$	0	0
$76$	−22.8541	−2.62155
$77$	2.00000	0.227921
$78$	−22.5623	−2.55468
$79$	−6.18034	−0.695343	−0.347671	−	0.937616i	$-0.613027\pi$
$79$	−6.18034	−0.695343	−0.347671	+	0.937616i	$0.613027\pi$
$80$	0	0
$81$	−5.70820	−0.634245
$82$	10.3262	1.14034
$83$	−12.0902	−1.32707	−0.663534	−	0.748146i	$-0.730944\pi$
$83$	−12.0902	−1.32707	−0.663534	+	0.748146i	$0.730944\pi$
$84$	−25.4164	−2.77316
$85$	0	0
$86$	−23.4164	−2.52506
$87$	−10.7082	−1.14804
$88$	1.38197	0.147318
$89$	2.94427	0.312092	0.156046	−	0.987750i	$-0.450125\pi$
$89$	2.94427	0.312092	0.156046	+	0.987750i	$0.450125\pi$
$90$	0	0
$91$	12.4721	1.30744
$92$	−13.4164	−1.39876
$93$	16.9443	1.75704
$94$	12.5623	1.29570
$95$	0	0
$96$	17.5623	1.79245
$97$	−2.76393	−0.280635	−0.140317	−	0.990107i	$-0.544812\pi$
$97$	−2.76393	−0.280635	−0.140317	+	0.990107i	$0.544812\pi$
$98$	7.76393	0.784276
$99$	2.38197	0.239397
$100$	0	0
$101$	16.1803	1.61000	0.805002	−	0.593272i	$-0.202164\pi$
$101$	16.1803	1.61000	0.805002	+	0.593272i	$0.202164\pi$
$102$	35.6525	3.53012
$103$	15.7082	1.54778	0.773888	−	0.633323i	$-0.218309\pi$
$103$	15.7082	1.54778	0.773888	+	0.633323i	$0.218309\pi$
$104$	8.61803	0.845068
$105$	0	0
$106$	3.41641	0.331831
$107$	7.56231	0.731076	0.365538	−	0.930796i	$-0.380885\pi$
$107$	7.56231	0.731076	0.365538	+	0.930796i	$0.380885\pi$
$108$	−6.70820	−0.645497
$109$	−4.94427	−0.473575	−0.236788	−	0.971561i	$-0.576095\pi$
$109$	−4.94427	−0.473575	−0.236788	+	0.971561i	$0.576095\pi$
$110$	0	0
$111$	11.7082	1.11129
$112$	−3.23607	−0.305780
$113$	−14.7639	−1.38887	−0.694437	−	0.719554i	$-0.744346\pi$
$113$	−14.7639	−1.38887	−0.694437	+	0.719554i	$0.744346\pi$
$114$	44.5967	4.17687
$115$	0	0
$116$	12.2705	1.13929
$117$	14.8541	1.37326
$118$	−12.7639	−1.17502
$119$	−19.7082	−1.80665
$120$	0	0
$121$	−10.6180	−0.965276
$122$	13.7426	1.24420
$123$	−12.0902	−1.09013
$124$	−19.4164	−1.74364
$125$	0	0
$126$	27.8885	2.48451
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	−15.6525	−1.38350
$129$	27.4164	2.41388
$130$	0	0
$131$	3.79837	0.331865	0.165933	−	0.986137i	$-0.446937\pi$
$131$	3.79837	0.331865	0.165933	+	0.986137i	$0.446937\pi$
$132$	−4.85410	−0.422495
$133$	−24.6525	−2.13764
$134$	9.14590	0.790085
$135$	0	0
$136$	−13.6180	−1.16774
$137$	19.8541	1.69625	0.848125	−	0.529796i	$-0.177731\pi$
$137$	19.8541	1.69625	0.848125	+	0.529796i	$0.177731\pi$
$138$	26.1803	2.22862
$139$	−12.6180	−1.07025	−0.535124	−	0.844774i	$-0.679735\pi$
$139$	−12.6180	−1.07025	−0.535124	+	0.844774i	$0.679735\pi$
$140$	0	0
$141$	−14.7082	−1.23865
$142$	−28.0902	−2.35727
$143$	2.38197	0.199190
$144$	−3.85410	−0.321175
$145$	0	0
$146$	17.0344	1.40978
$147$	−9.09017	−0.749745
$148$	−13.4164	−1.10282
$149$	−23.4164	−1.91835	−0.959173	−	0.282819i	$-0.908731\pi$
$149$	−23.4164	−1.91835	−0.959173	+	0.282819i	$0.908731\pi$
$150$	0	0
$151$	−21.4164	−1.74284	−0.871421	−	0.490535i	$-0.836801\pi$
$151$	−21.4164	−1.74284	−0.871421	+	0.490535i	$0.836801\pi$
$152$	−17.0344	−1.38168
$153$	−23.4721	−1.89761
$154$	4.47214	0.360375
$155$	0	0
$156$	−30.2705	−2.42358
$157$	−7.03444	−0.561410	−0.280705	−	0.959794i	$-0.590568\pi$
$157$	−7.03444	−0.561410	−0.280705	+	0.959794i	$0.590568\pi$
$158$	−13.8197	−1.09943
$159$	−4.00000	−0.317221
$160$	0	0
$161$	−14.4721	−1.14056
$162$	−12.7639	−1.00283
$163$	10.4721	0.820241	0.410120	−	0.912031i	$-0.365487\pi$
$163$	10.4721	0.820241	0.410120	+	0.912031i	$0.365487\pi$
$164$	13.8541	1.08182
$165$	0	0
$166$	−27.0344	−2.09828
$167$	−7.70820	−0.596479	−0.298239	−	0.954491i	$-0.596399\pi$
$167$	−7.70820	−0.596479	−0.298239	+	0.954491i	$0.596399\pi$
$168$	−18.9443	−1.46158
$169$	1.85410	0.142623
$170$	0	0
$171$	−29.3607	−2.24527
$172$	−31.4164	−2.39548
$173$	10.6180	0.807274	0.403637	−	0.914919i	$-0.367746\pi$
$173$	10.6180	0.807274	0.403637	+	0.914919i	$0.367746\pi$
$174$	−23.9443	−1.81521
$175$	0	0
$176$	−0.618034	−0.0465861
$177$	14.9443	1.12328
$178$	6.58359	0.493461
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	−5.85410	−0.435132	−0.217566	−	0.976046i	$-0.569812\pi$
$181$	−5.85410	−0.435132	−0.217566	+	0.976046i	$0.569812\pi$
$182$	27.8885	2.06724
$183$	−16.0902	−1.18942
$184$	−10.0000	−0.737210
$185$	0	0
$186$	37.8885	2.77812
$187$	−3.76393	−0.275246
$188$	16.8541	1.22921
$189$	−7.23607	−0.526346
$190$	0	0
$191$	6.18034	0.447194	0.223597	−	0.974682i	$-0.428220\pi$
$191$	6.18034	0.447194	0.223597	+	0.974682i	$0.428220\pi$
$192$	34.0344	2.45622
$193$	−11.5279	−0.829794	−0.414897	−	0.909868i	$-0.636182\pi$
$193$	−11.5279	−0.829794	−0.414897	+	0.909868i	$0.636182\pi$
$194$	−6.18034	−0.443723
$195$	0	0
$196$	10.4164	0.744029
$197$	−10.0000	−0.712470	−0.356235	−	0.934396i	$-0.615940\pi$
$197$	−10.0000	−0.712470	−0.356235	+	0.934396i	$0.615940\pi$
$198$	5.32624	0.378519
$199$	0.326238	0.0231264	0.0115632	−	0.999933i	$-0.496319\pi$
$199$	0.326238	0.0231264	0.0115632	+	0.999933i	$0.496319\pi$
$200$	0	0
$201$	−10.7082	−0.755298
$202$	36.1803	2.54564
$203$	13.2361	0.928990
$204$	47.8328	3.34897
$205$	0	0
$206$	35.1246	2.44725
$207$	−17.2361	−1.19799
$208$	−3.85410	−0.267234
$209$	−4.70820	−0.325673
$210$	0	0
$211$	−21.1246	−1.45428	−0.727139	−	0.686490i	$-0.759150\pi$
$211$	−21.1246	−1.45428	−0.727139	+	0.686490i	$0.759150\pi$
$212$	4.58359	0.314802
$213$	32.8885	2.25349
$214$	16.9098	1.15593
$215$	0	0
$216$	−5.00000	−0.340207
$217$	−20.9443	−1.42179
$218$	−11.0557	−0.748788
$219$	−19.9443	−1.34771
$220$	0	0
$221$	−23.4721	−1.57891
$222$	26.1803	1.75711
$223$	25.7984	1.72759	0.863793	−	0.503846i	$-0.168082\pi$
$223$	25.7984	1.72759	0.863793	+	0.503846i	$0.168082\pi$
$224$	−21.7082	−1.45044
$225$	0	0
$226$	−33.0132	−2.19600
$227$	23.1246	1.53483	0.767417	−	0.641148i	$-0.221542\pi$
$227$	23.1246	1.53483	0.767417	+	0.641148i	$0.221542\pi$
$228$	59.8328	3.96253
$229$	−7.09017	−0.468532	−0.234266	−	0.972173i	$-0.575269\pi$
$229$	−7.09017	−0.468532	−0.234266	+	0.972173i	$0.575269\pi$
$230$	0	0
$231$	−5.23607	−0.344508
$232$	9.14590	0.600458
$233$	−6.94427	−0.454934	−0.227467	−	0.973786i	$-0.573044\pi$
$233$	−6.94427	−0.454934	−0.227467	+	0.973786i	$0.573044\pi$
$234$	33.2148	2.17132
$235$	0	0
$236$	−17.1246	−1.11472
$237$	16.1803	1.05103
$238$	−44.0689	−2.85656
$239$	1.52786	0.0988293	0.0494147	−	0.998778i	$-0.484264\pi$
$239$	1.52786	0.0988293	0.0494147	+	0.998778i	$0.484264\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	−23.7426	−1.52624
$243$	21.6525	1.38901
$244$	18.4377	1.18035
$245$	0	0
$246$	−27.0344	−1.72365
$247$	−29.3607	−1.86818
$248$	−14.4721	−0.918982
$249$	31.6525	2.00589
$250$	0	0
$251$	−24.7639	−1.56309	−0.781543	−	0.623852i	$-0.785567\pi$
$251$	−24.7639	−1.56309	−0.781543	+	0.623852i	$0.785567\pi$
$252$	37.4164	2.35701
$253$	−2.76393	−0.173767
$254$	4.47214	0.280607
$255$	0	0
$256$	−9.00000	−0.562500
$257$	−18.7639	−1.17046	−0.585231	−	0.810867i	$-0.698996\pi$
$257$	−18.7639	−1.17046	−0.585231	+	0.810867i	$0.698996\pi$
$258$	61.3050	3.81668
$259$	−14.4721	−0.899255
$260$	0	0
$261$	15.7639	0.975763
$262$	8.49342	0.524725
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	−3.61803	−0.222675
$265$	0	0
$266$	−55.1246	−3.37991
$267$	−7.70820	−0.471734
$268$	12.2705	0.749541
$269$	−12.6525	−0.771435	−0.385718	−	0.922617i	$-0.626046\pi$
$269$	−12.6525	−0.771435	−0.385718	+	0.922617i	$0.626046\pi$
$270$	0	0
$271$	19.2361	1.16851	0.584254	−	0.811571i	$-0.301387\pi$
$271$	19.2361	1.16851	0.584254	+	0.811571i	$0.301387\pi$
$272$	6.09017	0.369271
$273$	−32.6525	−1.97622
$274$	44.3951	2.68201
$275$	0	0
$276$	35.1246	2.11425
$277$	12.9443	0.777746	0.388873	−	0.921291i	$-0.372865\pi$
$277$	12.9443	0.777746	0.388873	+	0.921291i	$0.372865\pi$
$278$	−28.2148	−1.69221
$279$	−24.9443	−1.49337
$280$	0	0
$281$	−16.9098	−1.00876	−0.504378	−	0.863483i	$-0.668278\pi$
$281$	−16.9098	−1.00876	−0.504378	+	0.863483i	$0.668278\pi$
$282$	−32.8885	−1.95848
$283$	3.23607	0.192364	0.0961821	−	0.995364i	$-0.469337\pi$
$283$	3.23607	0.192364	0.0961821	+	0.995364i	$0.469337\pi$
$284$	−37.6869	−2.23631
$285$	0	0
$286$	5.32624	0.314947
$287$	14.9443	0.882132
$288$	−25.8541	−1.52347
$289$	20.0902	1.18177
$290$	0	0
$291$	7.23607	0.424186
$292$	22.8541	1.33744
$293$	12.3820	0.723362	0.361681	−	0.932302i	$-0.382203\pi$
$293$	12.3820	0.723362	0.361681	+	0.932302i	$0.382203\pi$
$294$	−20.3262	−1.18545
$295$	0	0
$296$	−10.0000	−0.581238
$297$	−1.38197	−0.0801898
$298$	−52.3607	−3.03317
$299$	−17.2361	−0.996788
$300$	0	0
$301$	−33.8885	−1.95330
$302$	−47.8885	−2.75568
$303$	−42.3607	−2.43356
$304$	7.61803	0.436924
$305$	0	0
$306$	−52.4853	−3.00038
$307$	−15.1246	−0.863207	−0.431604	−	0.902063i	$-0.642052\pi$
$307$	−15.1246	−0.863207	−0.431604	+	0.902063i	$0.642052\pi$
$308$	6.00000	0.341882
$309$	−41.1246	−2.33950
$310$	0	0
$311$	−12.5623	−0.712343	−0.356172	−	0.934421i	$-0.615918\pi$
$311$	−12.5623	−0.712343	−0.356172	+	0.934421i	$0.615918\pi$
$312$	−22.5623	−1.27734
$313$	11.2361	0.635100	0.317550	−	0.948242i	$-0.397140\pi$
$313$	11.2361	0.635100	0.317550	+	0.948242i	$0.397140\pi$
$314$	−15.7295	−0.887666
$315$	0	0
$316$	−18.5410	−1.04301
$317$	−15.8541	−0.890455	−0.445228	−	0.895417i	$-0.646877\pi$
$317$	−15.8541	−0.890455	−0.445228	+	0.895417i	$0.646877\pi$
$318$	−8.94427	−0.501570
$319$	2.52786	0.141533
$320$	0	0
$321$	−19.7984	−1.10504
$322$	−32.3607	−1.80339
$323$	46.3951	2.58149
$324$	−17.1246	−0.951367
$325$	0	0
$326$	23.4164	1.29691
$327$	12.9443	0.715820
$328$	10.3262	0.570171
$329$	18.1803	1.00231
$330$	0	0
$331$	10.6525	0.585513	0.292757	−	0.956187i	$-0.405427\pi$
$331$	10.6525	0.585513	0.292757	+	0.956187i	$0.405427\pi$
$332$	−36.2705	−1.99060
$333$	−17.2361	−0.944531
$334$	−17.2361	−0.943116
$335$	0	0
$336$	8.47214	0.462193
$337$	0.291796	0.0158951	0.00794757	−	0.999968i	$-0.497470\pi$
$337$	0.291796	0.0158951	0.00794757	+	0.999968i	$0.497470\pi$
$338$	4.14590	0.225507
$339$	38.6525	2.09931
$340$	0	0
$341$	−4.00000	−0.216612
$342$	−65.6525	−3.55008
$343$	−11.4164	−0.616428
$344$	−23.4164	−1.26253
$345$	0	0
$346$	23.7426	1.27641
$347$	−1.03444	−0.0555317	−0.0277659	−	0.999614i	$-0.508839\pi$
$347$	−1.03444	−0.0555317	−0.0277659	+	0.999614i	$0.508839\pi$
$348$	−32.1246	−1.72206
$349$	13.6180	0.728957	0.364478	−	0.931212i	$-0.381247\pi$
$349$	13.6180	0.728957	0.364478	+	0.931212i	$0.381247\pi$
$350$	0	0
$351$	−8.61803	−0.459997
$352$	−4.14590	−0.220977
$353$	−23.2705	−1.23856	−0.619282	−	0.785169i	$-0.712576\pi$
$353$	−23.2705	−1.23856	−0.619282	+	0.785169i	$0.712576\pi$
$354$	33.4164	1.77606
$355$	0	0
$356$	8.83282	0.468138
$357$	51.5967	2.73079
$358$	−8.94427	−0.472719
$359$	−29.2361	−1.54302	−0.771510	−	0.636217i	$-0.780498\pi$
$359$	−29.2361	−1.54302	−0.771510	+	0.636217i	$0.780498\pi$
$360$	0	0
$361$	39.0344	2.05444
$362$	−13.0902	−0.688004
$363$	27.7984	1.45904
$364$	37.4164	1.96115
$365$	0	0
$366$	−35.9787	−1.88064
$367$	10.6525	0.556055	0.278027	−	0.960573i	$-0.410319\pi$
$367$	10.6525	0.556055	0.278027	+	0.960573i	$0.410319\pi$
$368$	4.47214	0.233126
$369$	17.7984	0.926546
$370$	0	0
$371$	4.94427	0.256694
$372$	50.8328	2.63556
$373$	5.90983	0.305999	0.153000	−	0.988226i	$-0.451107\pi$
$373$	5.90983	0.305999	0.153000	+	0.988226i	$0.451107\pi$
$374$	−8.41641	−0.435202
$375$	0	0
$376$	12.5623	0.647851
$377$	15.7639	0.811884
$378$	−16.1803	−0.832227
$379$	34.5066	1.77248	0.886242	−	0.463223i	$-0.153307\pi$
$379$	34.5066	1.77248	0.886242	+	0.463223i	$0.153307\pi$
$380$	0	0
$381$	−5.23607	−0.268252
$382$	13.8197	0.707075
$383$	−11.5279	−0.589046	−0.294523	−	0.955644i	$-0.595161\pi$
$383$	−11.5279	−0.589046	−0.294523	+	0.955644i	$0.595161\pi$
$384$	40.9787	2.09119
$385$	0	0
$386$	−25.7771	−1.31202
$387$	−40.3607	−2.05165
$388$	−8.29180	−0.420952
$389$	−25.1246	−1.27387	−0.636934	−	0.770918i	$-0.719798\pi$
$389$	−25.1246	−1.27387	−0.636934	+	0.770918i	$0.719798\pi$
$390$	0	0
$391$	27.2361	1.37739
$392$	7.76393	0.392138
$393$	−9.94427	−0.501622
$394$	−22.3607	−1.12651
$395$	0	0
$396$	7.14590	0.359095
$397$	19.5279	0.980075	0.490038	−	0.871701i	$-0.336983\pi$
$397$	19.5279	0.980075	0.490038	+	0.871701i	$0.336983\pi$
$398$	0.729490	0.0365660
$399$	64.5410	3.23109
$400$	0	0
$401$	27.5066	1.37361	0.686806	−	0.726840i	$-0.259012\pi$
$401$	27.5066	1.37361	0.686806	+	0.726840i	$0.259012\pi$
$402$	−23.9443	−1.19423
$403$	−24.9443	−1.24256
$404$	48.5410	2.41501
$405$	0	0
$406$	29.5967	1.46886
$407$	−2.76393	−0.137003
$408$	35.6525	1.76506
$409$	23.4164	1.15787	0.578933	−	0.815375i	$-0.303469\pi$
$409$	23.4164	1.15787	0.578933	+	0.815375i	$0.303469\pi$
$410$	0	0
$411$	−51.9787	−2.56392
$412$	47.1246	2.32166
$413$	−18.4721	−0.908954
$414$	−38.5410	−1.89419
$415$	0	0
$416$	−25.8541	−1.26760
$417$	33.0344	1.61770
$418$	−10.5279	−0.514935
$419$	−37.9230	−1.85266	−0.926330	−	0.376714i	$-0.877054\pi$
$419$	−37.9230	−1.85266	−0.926330	+	0.376714i	$0.877054\pi$
$420$	0	0
$421$	−19.5279	−0.951730	−0.475865	−	0.879518i	$-0.657865\pi$
$421$	−19.5279	−0.951730	−0.475865	+	0.879518i	$0.657865\pi$
$422$	−47.2361	−2.29942
$423$	21.6525	1.05278
$424$	3.41641	0.165915
$425$	0	0
$426$	73.5410	3.56307
$427$	19.8885	0.962474
$428$	22.6869	1.09661
$429$	−6.23607	−0.301080
$430$	0	0
$431$	16.9787	0.817836	0.408918	−	0.912571i	$-0.365906\pi$
$431$	16.9787	0.817836	0.408918	+	0.912571i	$0.365906\pi$
$432$	2.23607	0.107583
$433$	−12.2918	−0.590706	−0.295353	−	0.955388i	$-0.595437\pi$
$433$	−12.2918	−0.590706	−0.295353	+	0.955388i	$0.595437\pi$
$434$	−46.8328	−2.24805
$435$	0	0
$436$	−14.8328	−0.710363
$437$	34.0689	1.62974
$438$	−44.5967	−2.13092
$439$	−27.0902	−1.29294	−0.646472	−	0.762938i	$-0.723756\pi$
$439$	−27.0902	−1.29294	−0.646472	+	0.762938i	$0.723756\pi$
$440$	0	0
$441$	13.3820	0.637236
$442$	−52.4853	−2.49647
$443$	−14.0000	−0.665160	−0.332580	−	0.943075i	$-0.607919\pi$
$443$	−14.0000	−0.665160	−0.332580	+	0.943075i	$0.607919\pi$
$444$	35.1246	1.66694
$445$	0	0
$446$	57.6869	2.73155
$447$	61.3050	2.89962
$448$	−42.0689	−1.98757
$449$	−29.1246	−1.37448	−0.687238	−	0.726433i	$-0.741177\pi$
$449$	−29.1246	−1.37448	−0.687238	+	0.726433i	$0.741177\pi$
$450$	0	0
$451$	2.85410	0.134394
$452$	−44.2918	−2.08331
$453$	56.0689	2.63435
$454$	51.7082	2.42679
$455$	0	0
$456$	44.5967	2.08843
$457$	−14.2918	−0.668542	−0.334271	−	0.942477i	$-0.608490\pi$
$457$	−14.2918	−0.668542	−0.334271	+	0.942477i	$0.608490\pi$
$458$	−15.8541	−0.740814
$459$	13.6180	0.635635
$460$	0	0
$461$	9.52786	0.443757	0.221878	−	0.975074i	$-0.428781\pi$
$461$	9.52786	0.443757	0.221878	+	0.975074i	$0.428781\pi$
$462$	−11.7082	−0.544715
$463$	−11.8197	−0.549306	−0.274653	−	0.961543i	$-0.588563\pi$
$463$	−11.8197	−0.549306	−0.274653	+	0.961543i	$0.588563\pi$
$464$	−4.09017	−0.189881
$465$	0	0
$466$	−15.5279	−0.719314
$467$	4.14590	0.191849	0.0959246	−	0.995389i	$-0.469419\pi$
$467$	4.14590	0.191849	0.0959246	+	0.995389i	$0.469419\pi$
$468$	44.5623	2.05989
$469$	13.2361	0.611185
$470$	0	0
$471$	18.4164	0.848583
$472$	−12.7639	−0.587508
$473$	−6.47214	−0.297589
$474$	36.1803	1.66182
$475$	0	0
$476$	−59.1246	−2.70997
$477$	5.88854	0.269618
$478$	3.41641	0.156263
$479$	0.472136	0.0215724	0.0107862	−	0.999942i	$-0.496567\pi$
$479$	0.472136	0.0215724	0.0107862	+	0.999942i	$0.496567\pi$
$480$	0	0
$481$	−17.2361	−0.785897
$482$	2.23607	0.101850
$483$	37.8885	1.72399
$484$	−31.8541	−1.44791
$485$	0	0
$486$	48.4164	2.19621
$487$	−9.96556	−0.451583	−0.225791	−	0.974176i	$-0.572497\pi$
$487$	−9.96556	−0.451583	−0.225791	+	0.974176i	$0.572497\pi$
$488$	13.7426	0.622100
$489$	−27.4164	−1.23981
$490$	0	0
$491$	33.3050	1.50303	0.751516	−	0.659715i	$-0.229323\pi$
$491$	33.3050	1.50303	0.751516	+	0.659715i	$0.229323\pi$
$492$	−36.2705	−1.63520
$493$	−24.9098	−1.12188
$494$	−65.6525	−2.95384
$495$	0	0
$496$	6.47214	0.290607
$497$	−40.6525	−1.82351
$498$	70.7771	3.17160
$499$	34.4721	1.54318	0.771592	−	0.636117i	$-0.219461\pi$
$499$	34.4721	1.54318	0.771592	+	0.636117i	$0.219461\pi$
$500$	0	0
$501$	20.1803	0.901591
$502$	−55.3738	−2.47146
$503$	14.1803	0.632270	0.316135	−	0.948714i	$-0.397615\pi$
$503$	14.1803	0.632270	0.316135	+	0.948714i	$0.397615\pi$
$504$	27.8885	1.24225
$505$	0	0
$506$	−6.18034	−0.274750
$507$	−4.85410	−0.215578
$508$	6.00000	0.266207
$509$	37.5623	1.66492	0.832460	−	0.554085i	$-0.186932\pi$
$509$	37.5623	1.66492	0.832460	+	0.554085i	$0.186932\pi$
$510$	0	0
$511$	24.6525	1.09056
$512$	11.1803	0.494106
$513$	17.0344	0.752089
$514$	−41.9574	−1.85066
$515$	0	0
$516$	82.2492	3.62082
$517$	3.47214	0.152704
$518$	−32.3607	−1.42185
$519$	−27.7984	−1.22021
$520$	0	0
$521$	16.6525	0.729558	0.364779	−	0.931094i	$-0.381145\pi$
$521$	16.6525	0.729558	0.364779	+	0.931094i	$0.381145\pi$
$522$	35.2492	1.54282
$523$	−4.50658	−0.197059	−0.0985294	−	0.995134i	$-0.531414\pi$
$523$	−4.50658	−0.197059	−0.0985294	+	0.995134i	$0.531414\pi$
$524$	11.3951	0.497798
$525$	0	0
$526$	35.7771	1.55996
$527$	39.4164	1.71701
$528$	1.61803	0.0704159
$529$	−3.00000	−0.130435
$530$	0	0
$531$	−22.0000	−0.954719
$532$	−73.9574	−3.20646
$533$	17.7984	0.770933
$534$	−17.2361	−0.745878
$535$	0	0
$536$	9.14590	0.395043
$537$	10.4721	0.451906
$538$	−28.2918	−1.21975
$539$	2.14590	0.0924304
$540$	0	0
$541$	39.5066	1.69852	0.849260	−	0.527975i	$-0.177048\pi$
$541$	39.5066	1.69852	0.849260	+	0.527975i	$0.177048\pi$
$542$	43.0132	1.84757
$543$	15.3262	0.657712
$544$	40.8541	1.75161
$545$	0	0
$546$	−73.0132	−3.12467
$547$	−3.12461	−0.133599	−0.0667994	−	0.997766i	$-0.521279\pi$
$547$	−3.12461	−0.133599	−0.0667994	+	0.997766i	$0.521279\pi$
$548$	59.5623	2.54438
$549$	23.6869	1.01093
$550$	0	0
$551$	−31.1591	−1.32742
$552$	26.1803	1.11431
$553$	−20.0000	−0.850487
$554$	28.9443	1.22972
$555$	0	0
$556$	−37.8541	−1.60537
$557$	1.88854	0.0800202	0.0400101	−	0.999199i	$-0.487261\pi$
$557$	1.88854	0.0800202	0.0400101	+	0.999199i	$0.487261\pi$
$558$	−55.7771	−2.36123
$559$	−40.3607	−1.70707
$560$	0	0
$561$	9.85410	0.416041
$562$	−37.8115	−1.59498
$563$	−38.9787	−1.64276	−0.821378	−	0.570384i	$-0.806795\pi$
$563$	−38.9787	−1.64276	−0.821378	+	0.570384i	$0.806795\pi$
$564$	−44.1246	−1.85798
$565$	0	0
$566$	7.23607	0.304155
$567$	−18.4721	−0.775757
$568$	−28.0902	−1.17864
$569$	47.1591	1.97701	0.988505	−	0.151187i	$-0.0483095\pi$
$569$	47.1591	1.97701	0.988505	+	0.151187i	$0.0483095\pi$
$570$	0	0
$571$	37.1459	1.55451	0.777254	−	0.629187i	$-0.216612\pi$
$571$	37.1459	1.55451	0.777254	+	0.629187i	$0.216612\pi$
$572$	7.14590	0.298785
$573$	−16.1803	−0.675943
$574$	33.4164	1.39477
$575$	0	0
$576$	−50.1033	−2.08764
$577$	24.2148	1.00807	0.504037	−	0.863682i	$-0.331847\pi$
$577$	24.2148	1.00807	0.504037	+	0.863682i	$0.331847\pi$
$578$	44.9230	1.86855
$579$	30.1803	1.25425
$580$	0	0
$581$	−39.1246	−1.62316
$582$	16.1803	0.670697
$583$	0.944272	0.0391077
$584$	17.0344	0.704890
$585$	0	0
$586$	27.6869	1.14374
$587$	−4.11146	−0.169698	−0.0848490	−	0.996394i	$-0.527041\pi$
$587$	−4.11146	−0.169698	−0.0848490	+	0.996394i	$0.527041\pi$
$588$	−27.2705	−1.12462
$589$	49.3050	2.03158
$590$	0	0
$591$	26.1803	1.07692
$592$	4.47214	0.183804
$593$	−31.8885	−1.30951	−0.654753	−	0.755843i	$-0.727227\pi$
$593$	−31.8885	−1.30951	−0.654753	+	0.755843i	$0.727227\pi$
$594$	−3.09017	−0.126791
$595$	0	0
$596$	−70.2492	−2.87752
$597$	−0.854102	−0.0349561
$598$	−38.5410	−1.57606
$599$	9.32624	0.381060	0.190530	−	0.981681i	$-0.438979\pi$
$599$	9.32624	0.381060	0.190530	+	0.981681i	$0.438979\pi$
$600$	0	0
$601$	0.0344419	0.00140491	0.000702456	−	1.00000i	$-0.499776\pi$
$601$	0.0344419	0.00140491	0.000702456		1.00000i	$0.499776\pi$
$602$	−75.7771	−3.08844
$603$	15.7639	0.641957
$604$	−64.2492	−2.61426
$605$	0	0
$606$	−94.7214	−3.84779
$607$	20.3607	0.826414	0.413207	−	0.910637i	$-0.364409\pi$
$607$	20.3607	0.826414	0.413207	+	0.910637i	$0.364409\pi$
$608$	51.1033	2.07251
$609$	−34.6525	−1.40419
$610$	0	0
$611$	21.6525	0.875965
$612$	−70.4164	−2.84641
$613$	34.2705	1.38417	0.692086	−	0.721815i	$-0.256692\pi$
$613$	34.2705	1.38417	0.692086	+	0.721815i	$0.256692\pi$
$614$	−33.8197	−1.36485
$615$	0	0
$616$	4.47214	0.180187
$617$	26.4721	1.06573	0.532864	−	0.846201i	$-0.321116\pi$
$617$	26.4721	1.06573	0.532864	+	0.846201i	$0.321116\pi$
$618$	−91.9574	−3.69907
$619$	−10.4934	−0.421766	−0.210883	−	0.977511i	$-0.567634\pi$
$619$	−10.4934	−0.421766	−0.210883	+	0.977511i	$0.567634\pi$
$620$	0	0
$621$	10.0000	0.401286
$622$	−28.0902	−1.12631
$623$	9.52786	0.381726
$624$	10.0902	0.403930
$625$	0	0
$626$	25.1246	1.00418
$627$	12.3262	0.492263
$628$	−21.1033	−0.842114
$629$	27.2361	1.08597
$630$	0	0
$631$	−25.3262	−1.00822	−0.504111	−	0.863639i	$-0.668180\pi$
$631$	−25.3262	−1.00822	−0.504111	+	0.863639i	$0.668180\pi$
$632$	−13.8197	−0.549717
$633$	55.3050	2.19817
$634$	−35.4508	−1.40793
$635$	0	0
$636$	−12.0000	−0.475831
$637$	13.3820	0.530213
$638$	5.65248	0.223784
$639$	−48.4164	−1.91532
$640$	0	0
$641$	−6.38197	−0.252073	−0.126036	−	0.992026i	$-0.540226\pi$
$641$	−6.38197	−0.252073	−0.126036	+	0.992026i	$0.540226\pi$
$642$	−44.2705	−1.74722
$643$	41.8885	1.65192	0.825961	−	0.563727i	$-0.190633\pi$
$643$	41.8885	1.65192	0.825961	+	0.563727i	$0.190633\pi$
$644$	−43.4164	−1.71085
$645$	0	0
$646$	103.743	4.08170
$647$	−39.7082	−1.56109	−0.780545	−	0.625099i	$-0.785058\pi$
$647$	−39.7082	−1.56109	−0.780545	+	0.625099i	$0.785058\pi$
$648$	−12.7639	−0.501415
$649$	−3.52786	−0.138481
$650$	0	0
$651$	54.8328	2.14907
$652$	31.4164	1.23036
$653$	37.9098	1.48353	0.741763	−	0.670662i	$-0.233990\pi$
$653$	37.9098	1.48353	0.741763	+	0.670662i	$0.233990\pi$
$654$	28.9443	1.13181
$655$	0	0
$656$	−4.61803	−0.180304
$657$	29.3607	1.14547
$658$	40.6525	1.58480
$659$	−41.1246	−1.60199	−0.800994	−	0.598673i	$-0.795695\pi$
$659$	−41.1246	−1.60199	−0.800994	+	0.598673i	$0.795695\pi$
$660$	0	0
$661$	33.7082	1.31110	0.655549	−	0.755153i	$-0.272437\pi$
$661$	33.7082	1.31110	0.655549	+	0.755153i	$0.272437\pi$
$662$	23.8197	0.925777
$663$	61.4508	2.38655
$664$	−27.0344	−1.04914
$665$	0	0
$666$	−38.5410	−1.49343
$667$	−18.2918	−0.708261
$668$	−23.1246	−0.894718
$669$	−67.5410	−2.61129
$670$	0	0
$671$	3.79837	0.146635
$672$	56.8328	2.19237
$673$	42.1803	1.62593	0.812966	−	0.582311i	$-0.197851\pi$
$673$	42.1803	1.62593	0.812966	+	0.582311i	$0.197851\pi$
$674$	0.652476	0.0251324
$675$	0	0
$676$	5.56231	0.213935
$677$	30.9787	1.19061	0.595304	−	0.803500i	$-0.297031\pi$
$677$	30.9787	1.19061	0.595304	+	0.803500i	$0.297031\pi$
$678$	86.4296	3.31931
$679$	−8.94427	−0.343250
$680$	0	0
$681$	−60.5410	−2.31994
$682$	−8.94427	−0.342494
$683$	14.4721	0.553761	0.276880	−	0.960904i	$-0.410699\pi$
$683$	14.4721	0.553761	0.276880	+	0.960904i	$0.410699\pi$
$684$	−88.0820	−3.36790
$685$	0	0
$686$	−25.5279	−0.974658
$687$	18.5623	0.708196
$688$	10.4721	0.399246
$689$	5.88854	0.224336
$690$	0	0
$691$	−19.7082	−0.749735	−0.374868	−	0.927078i	$-0.622312\pi$
$691$	−19.7082	−0.749735	−0.374868	+	0.927078i	$0.622312\pi$
$692$	31.8541	1.21091
$693$	7.70820	0.292810
$694$	−2.31308	−0.0878034
$695$	0	0
$696$	−23.9443	−0.907605
$697$	−28.1246	−1.06530
$698$	30.4508	1.15258
$699$	18.1803	0.687644
$700$	0	0
$701$	−33.7082	−1.27314	−0.636571	−	0.771218i	$-0.719648\pi$
$701$	−33.7082	−1.27314	−0.636571	+	0.771218i	$0.719648\pi$
$702$	−19.2705	−0.727319
$703$	34.0689	1.28493
$704$	−8.03444	−0.302809
$705$	0	0
$706$	−52.0344	−1.95834
$707$	52.3607	1.96923
$708$	44.8328	1.68492
$709$	−42.8328	−1.60862	−0.804310	−	0.594210i	$-0.797465\pi$
$709$	−42.8328	−1.60862	−0.804310	+	0.594210i	$0.797465\pi$
$710$	0	0
$711$	−23.8197	−0.893307
$712$	6.58359	0.246731
$713$	28.9443	1.08397
$714$	115.374	4.31776
$715$	0	0
$716$	−12.0000	−0.448461
$717$	−4.00000	−0.149383
$718$	−65.3738	−2.43973
$719$	−53.2361	−1.98537	−0.992685	−	0.120732i	$-0.961476\pi$
$719$	−53.2361	−1.98537	−0.992685	+	0.120732i	$0.961476\pi$
$720$	0	0
$721$	50.8328	1.89311
$722$	87.2837	3.24836
$723$	−2.61803	−0.0973657
$724$	−17.5623	−0.652698
$725$	0	0
$726$	62.1591	2.30694
$727$	13.9787	0.518442	0.259221	−	0.965818i	$-0.416534\pi$
$727$	13.9787	0.518442	0.259221	+	0.965818i	$0.416534\pi$
$728$	27.8885	1.03362
$729$	−39.5623	−1.46527
$730$	0	0
$731$	63.7771	2.35888
$732$	−48.2705	−1.78413
$733$	17.7082	0.654067	0.327034	−	0.945013i	$-0.393951\pi$
$733$	17.7082	0.654067	0.327034	+	0.945013i	$0.393951\pi$
$734$	23.8197	0.879200
$735$	0	0
$736$	30.0000	1.10581
$737$	2.52786	0.0931151
$738$	39.7984	1.46500
$739$	−10.1803	−0.374490	−0.187245	−	0.982313i	$-0.559956\pi$
$739$	−10.1803	−0.374490	−0.187245	+	0.982313i	$0.559956\pi$
$740$	0	0
$741$	76.8673	2.82379
$742$	11.0557	0.405869
$743$	−19.5623	−0.717671	−0.358836	−	0.933401i	$-0.616826\pi$
$743$	−19.5623	−0.717671	−0.358836	+	0.933401i	$0.616826\pi$
$744$	37.8885	1.38906
$745$	0	0
$746$	13.2148	0.483828
$747$	−46.5967	−1.70489
$748$	−11.2918	−0.412869
$749$	24.4721	0.894192
$750$	0	0
$751$	28.6180	1.04429	0.522143	−	0.852858i	$-0.325133\pi$
$751$	28.6180	1.04429	0.522143	+	0.852858i	$0.325133\pi$
$752$	−5.61803	−0.204869
$753$	64.8328	2.36264
$754$	35.2492	1.28370
$755$	0	0
$756$	−21.7082	−0.789520
$757$	14.6738	0.533327	0.266663	−	0.963790i	$-0.414079\pi$
$757$	14.6738	0.533327	0.266663	+	0.963790i	$0.414079\pi$
$758$	77.1591	2.80254
$759$	7.23607	0.262653
$760$	0	0
$761$	−0.111456	−0.00404028	−0.00202014	−	0.999998i	$-0.500643\pi$
$761$	−0.111456	−0.00404028	−0.00202014	+	0.999998i	$0.500643\pi$
$762$	−11.7082	−0.424143
$763$	−16.0000	−0.579239
$764$	18.5410	0.670791
$765$	0	0
$766$	−25.7771	−0.931364
$767$	−22.0000	−0.794374
$768$	23.5623	0.850231
$769$	34.8328	1.25610	0.628052	−	0.778172i	$-0.283853\pi$
$769$	34.8328	1.25610	0.628052	+	0.778172i	$0.283853\pi$
$770$	0	0
$771$	49.1246	1.76918
$772$	−34.5836	−1.24469
$773$	−29.1246	−1.04754	−0.523770	−	0.851860i	$-0.675475\pi$
$773$	−29.1246	−1.04754	−0.523770	+	0.851860i	$0.675475\pi$
$774$	−90.2492	−3.24394
$775$	0	0
$776$	−6.18034	−0.221861
$777$	37.8885	1.35924
$778$	−56.1803	−2.01416
$779$	−35.1803	−1.26047
$780$	0	0
$781$	−7.76393	−0.277815
$782$	60.9017	2.17784
$783$	−9.14590	−0.326848
$784$	−3.47214	−0.124005
$785$	0	0
$786$	−22.2361	−0.793134
$787$	−25.3820	−0.904769	−0.452385	−	0.891823i	$-0.649427\pi$
$787$	−25.3820	−0.904769	−0.452385	+	0.891823i	$0.649427\pi$
$788$	−30.0000	−1.06871
$789$	−41.8885	−1.49127
$790$	0	0
$791$	−47.7771	−1.69876
$792$	5.32624	0.189260
$793$	23.6869	0.841147
$794$	43.6656	1.54964
$795$	0	0
$796$	0.978714	0.0346896
$797$	22.2705	0.788862	0.394431	−	0.918926i	$-0.370942\pi$
$797$	22.2705	0.788862	0.394431	+	0.918926i	$0.370942\pi$
$798$	144.318	5.10881
$799$	−34.2148	−1.21043
$800$	0	0
$801$	11.3475	0.400945
$802$	61.5066	2.17187
$803$	4.70820	0.166149
$804$	−32.1246	−1.13295
$805$	0	0
$806$	−55.7771	−1.96466
$807$	33.1246	1.16604
$808$	36.1803	1.27282
$809$	0.944272	0.0331988	0.0165994	−	0.999862i	$-0.494716\pi$
$809$	0.944272	0.0331988	0.0165994	+	0.999862i	$0.494716\pi$
$810$	0	0
$811$	−30.5066	−1.07123	−0.535615	−	0.844462i	$-0.679920\pi$
$811$	−30.5066	−1.07123	−0.535615	+	0.844462i	$0.679920\pi$
$812$	39.7082	1.39348
$813$	−50.3607	−1.76623
$814$	−6.18034	−0.216621
$815$	0	0
$816$	−15.9443	−0.558161
$817$	79.7771	2.79105
$818$	52.3607	1.83075
$819$	48.0689	1.67966
$820$	0	0
$821$	−8.61803	−0.300771	−0.150386	−	0.988627i	$-0.548052\pi$
$821$	−8.61803	−0.300771	−0.150386	+	0.988627i	$0.548052\pi$
$822$	−116.228	−4.05391
$823$	15.4164	0.537382	0.268691	−	0.963226i	$-0.413409\pi$
$823$	15.4164	0.537382	0.268691	+	0.963226i	$0.413409\pi$
$824$	35.1246	1.22362
$825$	0	0
$826$	−41.3050	−1.43718
$827$	11.8885	0.413405	0.206703	−	0.978404i	$-0.433727\pi$
$827$	11.8885	0.413405	0.206703	+	0.978404i	$0.433727\pi$
$828$	−51.7082	−1.79698
$829$	31.6869	1.10053	0.550266	−	0.834989i	$-0.314526\pi$
$829$	31.6869	1.10053	0.550266	+	0.834989i	$0.314526\pi$
$830$	0	0
$831$	−33.8885	−1.17558
$832$	−50.1033	−1.73702
$833$	−21.1459	−0.732662
$834$	73.8673	2.55781
$835$	0	0
$836$	−14.1246	−0.488510
$837$	14.4721	0.500230
$838$	−84.7984	−2.92931
$839$	0.472136	0.0162999	0.00814997	−	0.999967i	$-0.497406\pi$
$839$	0.472136	0.0162999	0.00814997	+	0.999967i	$0.497406\pi$
$840$	0	0
$841$	−12.2705	−0.423121
$842$	−43.6656	−1.50482
$843$	44.2705	1.52476
$844$	−63.3738	−2.18142
$845$	0	0
$846$	48.4164	1.66459
$847$	−34.3607	−1.18065
$848$	−1.52786	−0.0524671
$849$	−8.47214	−0.290763
$850$	0	0
$851$	20.0000	0.685591
$852$	98.6656	3.38023
$853$	4.21478	0.144311	0.0721557	−	0.997393i	$-0.477012\pi$
$853$	4.21478	0.144311	0.0721557	+	0.997393i	$0.477012\pi$
$854$	44.4721	1.52181
$855$	0	0
$856$	16.9098	0.577966
$857$	−52.1803	−1.78245	−0.891223	−	0.453565i	$-0.850152\pi$
$857$	−52.1803	−1.78245	−0.891223	+	0.453565i	$0.850152\pi$
$858$	−13.9443	−0.476050
$859$	27.6180	0.942315	0.471158	−	0.882049i	$-0.343836\pi$
$859$	27.6180	0.942315	0.471158	+	0.882049i	$0.343836\pi$
$860$	0	0
$861$	−39.1246	−1.33336
$862$	37.9656	1.29311
$863$	−0.291796	−0.00993285	−0.00496643	−	0.999988i	$-0.501581\pi$
$863$	−0.291796	−0.00993285	−0.00496643	+	0.999988i	$0.501581\pi$
$864$	15.0000	0.510310
$865$	0	0
$866$	−27.4853	−0.933988
$867$	−52.5967	−1.78628
$868$	−62.8328	−2.13268
$869$	−3.81966	−0.129573
$870$	0	0
$871$	15.7639	0.534140
$872$	−11.0557	−0.374394
$873$	−10.6525	−0.360532
$874$	76.1803	2.57684
$875$	0	0
$876$	−59.8328	−2.02156
$877$	10.7639	0.363472	0.181736	−	0.983347i	$-0.441828\pi$
$877$	10.7639	0.363472	0.181736	+	0.983347i	$0.441828\pi$
$878$	−60.5755	−2.04432
$879$	−32.4164	−1.09338
$880$	0	0
$881$	11.9098	0.401252	0.200626	−	0.979668i	$-0.435702\pi$
$881$	11.9098	0.401252	0.200626	+	0.979668i	$0.435702\pi$
$882$	29.9230	1.00756
$883$	15.8541	0.533533	0.266767	−	0.963761i	$-0.414045\pi$
$883$	15.8541	0.533533	0.266767	+	0.963761i	$0.414045\pi$
$884$	−70.4164	−2.36836
$885$	0	0
$886$	−31.3050	−1.05171
$887$	−47.2705	−1.58719	−0.793594	−	0.608447i	$-0.791793\pi$
$887$	−47.2705	−1.58719	−0.793594	+	0.608447i	$0.791793\pi$
$888$	26.1803	0.878555
$889$	6.47214	0.217068
$890$	0	0
$891$	−3.52786	−0.118188
$892$	77.3951	2.59138
$893$	−42.7984	−1.43219
$894$	137.082	4.58471
$895$	0	0
$896$	−50.6525	−1.69218
$897$	45.1246	1.50667
$898$	−65.1246	−2.17324
$899$	−26.4721	−0.882895
$900$	0	0
$901$	−9.30495	−0.309993
$902$	6.38197	0.212496
$903$	88.7214	2.95246
$904$	−33.0132	−1.09800
$905$	0	0
$906$	125.374	4.16527
$907$	37.2361	1.23640	0.618202	−	0.786020i	$-0.287862\pi$
$907$	37.2361	1.23640	0.618202	+	0.786020i	$0.287862\pi$
$908$	69.3738	2.30225
$909$	62.3607	2.06837
$910$	0	0
$911$	−38.0689	−1.26128	−0.630639	−	0.776076i	$-0.717207\pi$
$911$	−38.0689	−1.26128	−0.630639	+	0.776076i	$0.717207\pi$
$912$	−19.9443	−0.660421
$913$	−7.47214	−0.247292
$914$	−31.9574	−1.05706
$915$	0	0
$916$	−21.2705	−0.702797
$917$	12.2918	0.405911
$918$	30.4508	1.00503
$919$	27.3050	0.900707	0.450354	−	0.892850i	$-0.351298\pi$
$919$	27.3050	0.900707	0.450354	+	0.892850i	$0.351298\pi$
$920$	0	0
$921$	39.5967	1.30476
$922$	21.3050	0.701641
$923$	−48.4164	−1.59365
$924$	−15.7082	−0.516762
$925$	0	0
$926$	−26.4296	−0.868529
$927$	60.5410	1.98843
$928$	−27.4377	−0.900686
$929$	−27.1246	−0.889930	−0.444965	−	0.895548i	$-0.646784\pi$
$929$	−27.1246	−0.889930	−0.444965	+	0.895548i	$0.646784\pi$
$930$	0	0
$931$	−26.4508	−0.866892
$932$	−20.8328	−0.682402
$933$	32.8885	1.07672
$934$	9.27051	0.303340
$935$	0	0
$936$	33.2148	1.08566
$937$	20.0000	0.653372	0.326686	−	0.945133i	$-0.394068\pi$
$937$	20.0000	0.653372	0.326686	+	0.945133i	$0.394068\pi$
$938$	29.5967	0.966368
$939$	−29.4164	−0.959968
$940$	0	0
$941$	9.34752	0.304721	0.152360	−	0.988325i	$-0.451313\pi$
$941$	9.34752	0.304721	0.152360	+	0.988325i	$0.451313\pi$
$942$	41.1803	1.34173
$943$	−20.6525	−0.672537
$944$	5.70820	0.185786
$945$	0	0
$946$	−14.4721	−0.470530
$947$	−20.1803	−0.655773	−0.327886	−	0.944717i	$-0.606336\pi$
$947$	−20.1803	−0.655773	−0.327886	+	0.944717i	$0.606336\pi$
$948$	48.5410	1.57654
$949$	29.3607	0.953088
$950$	0	0
$951$	41.5066	1.34594
$952$	−44.0689	−1.42828
$953$	−14.9787	−0.485208	−0.242604	−	0.970125i	$-0.578002\pi$
$953$	−14.9787	−0.485208	−0.242604	+	0.970125i	$0.578002\pi$
$954$	13.1672	0.426303
$955$	0	0
$956$	4.58359	0.148244
$957$	−6.61803	−0.213931
$958$	1.05573	0.0341090
$959$	64.2492	2.07472
$960$	0	0
$961$	10.8885	0.351243
$962$	−38.5410	−1.24261
$963$	29.1459	0.939213
$964$	3.00000	0.0966235
$965$	0	0
$966$	84.7214	2.72587
$967$	−20.9443	−0.673522	−0.336761	−	0.941590i	$-0.609332\pi$
$967$	−20.9443	−0.673522	−0.336761	+	0.941590i	$0.609332\pi$
$968$	−23.7426	−0.763118
$969$	−121.464	−3.90199
$970$	0	0
$971$	−60.9443	−1.95579	−0.977897	−	0.209085i	$-0.932951\pi$
$971$	−60.9443	−1.95579	−0.977897	+	0.209085i	$0.932951\pi$
$972$	64.9574	2.08351
$973$	−40.8328	−1.30904
$974$	−22.2837	−0.714015
$975$	0	0
$976$	−6.14590	−0.196725
$977$	−4.11146	−0.131537	−0.0657686	−	0.997835i	$-0.520950\pi$
$977$	−4.11146	−0.131537	−0.0657686	+	0.997835i	$0.520950\pi$
$978$	−61.3050	−1.96032
$979$	1.81966	0.0581566
$980$	0	0
$981$	−19.0557	−0.608403
$982$	74.4721	2.37650
$983$	−29.1246	−0.928931	−0.464465	−	0.885591i	$-0.653754\pi$
$983$	−29.1246	−0.928931	−0.464465	+	0.885591i	$0.653754\pi$
$984$	−27.0344	−0.861827
$985$	0	0
$986$	−55.7001	−1.77385
$987$	−47.5967	−1.51502
$988$	−88.0820	−2.80226
$989$	46.8328	1.48920
$990$	0	0
$991$	4.36068	0.138522	0.0692608	−	0.997599i	$-0.477936\pi$
$991$	4.36068	0.138522	0.0692608	+	0.997599i	$0.477936\pi$
$992$	43.4164	1.37847
$993$	−27.8885	−0.885016
$994$	−90.9017	−2.88323
$995$	0	0
$996$	94.9574	3.00884
$997$	−11.2016	−0.354759	−0.177380	−	0.984143i	$-0.556762\pi$
$997$	−11.2016	−0.354759	−0.177380	+	0.984143i	$0.556762\pi$
$998$	77.0820	2.43999
$999$	10.0000	0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.b.1.2		2
5.2	odd	4		1205.2.b.a.724.4	yes	4
5.3	odd	4		1205.2.b.a.724.1	✓	4
5.4	even	2		6025.2.a.c.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1205.2.b.a.724.1	✓	4	5.3	odd	4
1205.2.b.a.724.4	yes	4	5.2	odd	4
6025.2.a.b.1.2		2	1.1	even	1	trivial
6025.2.a.c.1.1		2	5.4	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:	\( N \)	\(=\)	\( 6025 = 5^{2} \cdot 241 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6025.a (trivial)

\(f(q)\)	\(=\)	\(q+2.23607 q^{2} -2.61803 q^{3} +3.00000 q^{4} -5.85410 q^{6} +3.23607 q^{7} +2.23607 q^{8} +3.85410 q^{9} +O(q^{10})\)	\(q+2.23607 q^{2} -2.61803 q^{3} +3.00000 q^{4} -5.85410 q^{6} +3.23607 q^{7} +2.23607 q^{8} +3.85410 q^{9} +0.618034 q^{11} -7.85410 q^{12} +3.85410 q^{13} +7.23607 q^{14} -1.00000 q^{16} -6.09017 q^{17} +8.61803 q^{18} -7.61803 q^{19} -8.47214 q^{21} +1.38197 q^{22} -4.47214 q^{23} -5.85410 q^{24} +8.61803 q^{26} -2.23607 q^{27} +9.70820 q^{28} +4.09017 q^{29} -6.47214 q^{31} -6.70820 q^{32} -1.61803 q^{33} -13.6180 q^{34} +11.5623 q^{36} -4.47214 q^{37} -17.0344 q^{38} -10.0902 q^{39} +4.61803 q^{41} -18.9443 q^{42} -10.4721 q^{43} +1.85410 q^{44} -10.0000 q^{46} +5.61803 q^{47} +2.61803 q^{48} +3.47214 q^{49} +15.9443 q^{51} +11.5623 q^{52} +1.52786 q^{53} -5.00000 q^{54} +7.23607 q^{56} +19.9443 q^{57} +9.14590 q^{58} -5.70820 q^{59} +6.14590 q^{61} -14.4721 q^{62} +12.4721 q^{63} -13.0000 q^{64} -3.61803 q^{66} +4.09017 q^{67} -18.2705 q^{68} +11.7082 q^{69} -12.5623 q^{71} +8.61803 q^{72} +7.61803 q^{73} -10.0000 q^{74} -22.8541 q^{76} +2.00000 q^{77} -22.5623 q^{78} -6.18034 q^{79} -5.70820 q^{81} +10.3262 q^{82} -12.0902 q^{83} -25.4164 q^{84} -23.4164 q^{86} -10.7082 q^{87} +1.38197 q^{88} +2.94427 q^{89} +12.4721 q^{91} -13.4164 q^{92} +16.9443 q^{93} +12.5623 q^{94} +17.5623 q^{96} -2.76393 q^{97} +7.76393 q^{98} +2.38197 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{3} + 6 q^{4} - 5 q^{6} + 2 q^{7} + q^{9}+O(q^{10}) \) 2 * q - 3 * q^3 + 6 * q^4 - 5 * q^6 + 2 * q^7 + q^9	\( 2 q - 3 q^{3} + 6 q^{4} - 5 q^{6} + 2 q^{7} + q^{9} - q^{11} - 9 q^{12} + q^{13} + 10 q^{14} - 2 q^{16} - q^{17} + 15 q^{18} - 13 q^{19} - 8 q^{21} + 5 q^{22} - 5 q^{24} + 15 q^{26} + 6 q^{28} - 3 q^{29} - 4 q^{31} - q^{33} - 25 q^{34} + 3 q^{36} - 5 q^{38} - 9 q^{39} + 7 q^{41} - 20 q^{42} - 12 q^{43} - 3 q^{44} - 20 q^{46} + 9 q^{47} + 3 q^{48} - 2 q^{49} + 14 q^{51} + 3 q^{52} + 12 q^{53} - 10 q^{54} + 10 q^{56} + 22 q^{57} + 25 q^{58} + 2 q^{59} + 19 q^{61} - 20 q^{62} + 16 q^{63} - 26 q^{64} - 5 q^{66} - 3 q^{67} - 3 q^{68} + 10 q^{69} - 5 q^{71} + 15 q^{72} + 13 q^{73} - 20 q^{74} - 39 q^{76} + 4 q^{77} - 25 q^{78} + 10 q^{79} + 2 q^{81} + 5 q^{82} - 13 q^{83} - 24 q^{84} - 20 q^{86} - 8 q^{87} + 5 q^{88} - 12 q^{89} + 16 q^{91} + 16 q^{93} + 5 q^{94} + 15 q^{96} - 10 q^{97} + 20 q^{98} + 7 q^{99}+O(q^{100}) \) 2 * q - 3 * q^3 + 6 * q^4 - 5 * q^6 + 2 * q^7 + q^9 - q^11 - 9 * q^12 + q^13 + 10 * q^14 - 2 * q^16 - q^17 + 15 * q^18 - 13 * q^19 - 8 * q^21 + 5 * q^22 - 5 * q^24 + 15 * q^26 + 6 * q^28 - 3 * q^29 - 4 * q^31 - q^33 - 25 * q^34 + 3 * q^36 - 5 * q^38 - 9 * q^39 + 7 * q^41 - 20 * q^42 - 12 * q^43 - 3 * q^44 - 20 * q^46 + 9 * q^47 + 3 * q^48 - 2 * q^49 + 14 * q^51 + 3 * q^52 + 12 * q^53 - 10 * q^54 + 10 * q^56 + 22 * q^57 + 25 * q^58 + 2 * q^59 + 19 * q^61 - 20 * q^62 + 16 * q^63 - 26 * q^64 - 5 * q^66 - 3 * q^67 - 3 * q^68 + 10 * q^69 - 5 * q^71 + 15 * q^72 + 13 * q^73 - 20 * q^74 - 39 * q^76 + 4 * q^77 - 25 * q^78 + 10 * q^79 + 2 * q^81 + 5 * q^82 - 13 * q^83 - 24 * q^84 - 20 * q^86 - 8 * q^87 + 5 * q^88 - 12 * q^89 + 16 * q^91 + 16 * q^93 + 5 * q^94 + 15 * q^96 - 10 * q^97 + 20 * q^98 + 7 * q^99

Introduction

L-functions

Modular forms

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Database

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