LMFDB - Embedded newform 325.2.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [325,2,Mod(1,325)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(325, 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("325.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 325 = 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	325.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:	$2.59513806569$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	325.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.41421 q^{2} -2.82843 q^{3} +3.82843 q^{4} -6.82843 q^{6} +2.41421 q^{7} +4.41421 q^{8} +5.00000 q^{9} +6.41421 q^{11} -10.8284 q^{12} +1.00000 q^{13} +5.82843 q^{14} +3.00000 q^{16} -3.82843 q^{17} +12.0711 q^{18} -3.65685 q^{19} -6.82843 q^{21} +15.4853 q^{22} -3.17157 q^{23} -12.4853 q^{24} +2.41421 q^{26} -5.65685 q^{27} +9.24264 q^{28} -0.171573 q^{29} +1.24264 q^{31} -1.58579 q^{32} -18.1421 q^{33} -9.24264 q^{34} +19.1421 q^{36} -6.00000 q^{37} -8.82843 q^{38} -2.82843 q^{39} -5.65685 q^{41} -16.4853 q^{42} +0.828427 q^{43} +24.5563 q^{44} -7.65685 q^{46} +3.58579 q^{47} -8.48528 q^{48} -1.17157 q^{49} +10.8284 q^{51} +3.82843 q^{52} +3.00000 q^{53} -13.6569 q^{54} +10.6569 q^{56} +10.3431 q^{57} -0.414214 q^{58} +13.2426 q^{59} +1.00000 q^{61} +3.00000 q^{62} +12.0711 q^{63} -9.82843 q^{64} -43.7990 q^{66} -2.75736 q^{67} -14.6569 q^{68} +8.97056 q^{69} -9.31371 q^{71} +22.0711 q^{72} -6.00000 q^{73} -14.4853 q^{74} -14.0000 q^{76} +15.4853 q^{77} -6.82843 q^{78} -6.00000 q^{79} +1.00000 q^{81} -13.6569 q^{82} +6.89949 q^{83} -26.1421 q^{84} +2.00000 q^{86} +0.485281 q^{87} +28.3137 q^{88} -8.48528 q^{89} +2.41421 q^{91} -12.1421 q^{92} -3.51472 q^{93} +8.65685 q^{94} +4.48528 q^{96} -0.828427 q^{97} -2.82843 q^{98} +32.0711 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} - 8 q^{6} + 2 q^{7} + 6 q^{8} + 10 q^{9} + 10 q^{11} - 16 q^{12} + 2 q^{13} + 6 q^{14} + 6 q^{16} - 2 q^{17} + 10 q^{18} + 4 q^{19} - 8 q^{21} + 14 q^{22} - 12 q^{23} - 8 q^{24}+ \cdots + 50 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 - 8 * q^6 + 2 * q^7 + 6 * q^8 + 10 * q^9 + 10 * q^11 - 16 * q^12 + 2 * q^13 + 6 * q^14 + 6 * q^16 - 2 * q^17 + 10 * q^18 + 4 * q^19 - 8 * q^21 + 14 * q^22 - 12 * q^23 - 8 * q^24 + 2 * q^26 + 10 * q^28 - 6 * q^29 - 6 * q^31 - 6 * q^32 - 8 * q^33 - 10 * q^34 + 10 * q^36 - 12 * q^37 - 12 * q^38 - 16 * q^42 - 4 * q^43 + 18 * q^44 - 4 * q^46 + 10 * q^47 - 8 * q^49 + 16 * q^51 + 2 * q^52 + 6 * q^53 - 16 * q^54 + 10 * q^56 + 32 * q^57 + 2 * q^58 + 18 * q^59 + 2 * q^61 + 6 * q^62 + 10 * q^63 - 14 * q^64 - 48 * q^66 - 14 * q^67 - 18 * q^68 - 16 * q^69 + 4 * q^71 + 30 * q^72 - 12 * q^73 - 12 * q^74 - 28 * q^76 + 14 * q^77 - 8 * q^78 - 12 * q^79 + 2 * q^81 - 16 * q^82 - 6 * q^83 - 24 * q^84 + 4 * q^86 - 16 * q^87 + 34 * q^88 + 2 * q^91 + 4 * q^92 - 24 * q^93 + 6 * q^94 - 8 * q^96 + 4 * q^97 + 50 * q^99

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 $

$2$	2.41421	1.70711	0.853553	−	0.521005i	$-0.174443\pi$
$2$	2.41421	1.70711	0.853553	+	0.521005i	$0.174443\pi$
$3$	−2.82843	−1.63299	−0.816497	−	0.577350i	$-0.804087\pi$
$3$	−2.82843	−1.63299	−0.816497	+	0.577350i	$0.804087\pi$
$4$	3.82843	1.91421
$5$	0	0
$5$	0	0
$6$	−6.82843	−2.78769
$7$	2.41421	0.912487	0.456243	−	0.889855i	$-0.349195\pi$
$7$	2.41421	0.912487	0.456243	+	0.889855i	$0.349195\pi$
$8$	4.41421	1.56066
$9$	5.00000	1.66667
$10$	0	0
$11$	6.41421	1.93396	0.966979	−	0.254856i	$-0.0820280\pi$
$11$	6.41421	1.93396	0.966979	+	0.254856i	$0.0820280\pi$
$12$	−10.8284	−3.12590
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	5.82843	1.55771
$15$	0	0
$16$	3.00000	0.750000
$17$	−3.82843	−0.928530	−0.464265	−	0.885696i	$-0.653681\pi$
$17$	−3.82843	−0.928530	−0.464265	+	0.885696i	$0.653681\pi$
$18$	12.0711	2.84518
$19$	−3.65685	−0.838940	−0.419470	−	0.907769i	$-0.637784\pi$
$19$	−3.65685	−0.838940	−0.419470	+	0.907769i	$0.637784\pi$
$20$	0	0
$21$	−6.82843	−1.49008
$22$	15.4853	3.30147
$23$	−3.17157	−0.661319	−0.330659	−	0.943750i	$-0.607271\pi$
$23$	−3.17157	−0.661319	−0.330659	+	0.943750i	$0.607271\pi$
$24$	−12.4853	−2.54855
$25$	0	0
$26$	2.41421	0.473466
$27$	−5.65685	−1.08866
$28$	9.24264	1.74669
$29$	−0.171573	−0.0318603	−0.0159301	−	0.999873i	$-0.505071\pi$
$29$	−0.171573	−0.0318603	−0.0159301	+	0.999873i	$0.505071\pi$
$30$	0	0
$31$	1.24264	0.223185	0.111592	−	0.993754i	$-0.464405\pi$
$31$	1.24264	0.223185	0.111592	+	0.993754i	$0.464405\pi$
$32$	−1.58579	−0.280330
$33$	−18.1421	−3.15814
$34$	−9.24264	−1.58510
$35$	0	0
$36$	19.1421	3.19036
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	−8.82843	−1.43216
$39$	−2.82843	−0.452911
$40$	0	0
$41$	−5.65685	−0.883452	−0.441726	−	0.897150i	$-0.645634\pi$
$41$	−5.65685	−0.883452	−0.441726	+	0.897150i	$0.645634\pi$
$42$	−16.4853	−2.54373
$43$	0.828427	0.126334	0.0631670	−	0.998003i	$-0.479880\pi$
$43$	0.828427	0.126334	0.0631670	+	0.998003i	$0.479880\pi$
$44$	24.5563	3.70201
$45$	0	0
$46$	−7.65685	−1.12894
$47$	3.58579	0.523041	0.261520	−	0.965198i	$-0.415776\pi$
$47$	3.58579	0.523041	0.261520	+	0.965198i	$0.415776\pi$
$48$	−8.48528	−1.22474
$49$	−1.17157	−0.167368
$50$	0	0
$51$	10.8284	1.51628
$52$	3.82843	0.530907
$53$	3.00000	0.412082	0.206041	−	0.978543i	$-0.433942\pi$
$53$	3.00000	0.412082	0.206041	+	0.978543i	$0.433942\pi$
$54$	−13.6569	−1.85846
$55$	0	0
$56$	10.6569	1.42408
$57$	10.3431	1.36998
$58$	−0.414214	−0.0543889
$59$	13.2426	1.72404	0.862022	−	0.506870i	$-0.169198\pi$
$59$	13.2426	1.72404	0.862022	+	0.506870i	$0.169198\pi$
$60$	0	0
$61$	1.00000	0.128037	0.0640184	−	0.997949i	$-0.479608\pi$
$61$	1.00000	0.128037	0.0640184	+	0.997949i	$0.479608\pi$
$62$	3.00000	0.381000
$63$	12.0711	1.52081
$64$	−9.82843	−1.22855
$65$	0	0
$66$	−43.7990	−5.39128
$67$	−2.75736	−0.336865	−0.168433	−	0.985713i	$-0.553871\pi$
$67$	−2.75736	−0.336865	−0.168433	+	0.985713i	$0.553871\pi$
$68$	−14.6569	−1.77740
$69$	8.97056	1.07993
$70$	0	0
$71$	−9.31371	−1.10533	−0.552667	−	0.833402i	$-0.686390\pi$
$71$	−9.31371	−1.10533	−0.552667	+	0.833402i	$0.686390\pi$
$72$	22.0711	2.60110
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	−14.4853	−1.68388
$75$	0	0
$76$	−14.0000	−1.60591
$77$	15.4853	1.76471
$78$	−6.82843	−0.773167
$79$	−6.00000	−0.675053	−0.337526	−	0.941316i	$-0.609590\pi$
$79$	−6.00000	−0.675053	−0.337526	+	0.941316i	$0.609590\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−13.6569	−1.50815
$83$	6.89949	0.757318	0.378659	−	0.925536i	$-0.376385\pi$
$83$	6.89949	0.757318	0.378659	+	0.925536i	$0.376385\pi$
$84$	−26.1421	−2.85234
$85$	0	0
$86$	2.00000	0.215666
$87$	0.485281	0.0520276
$88$	28.3137	3.01825
$89$	−8.48528	−0.899438	−0.449719	−	0.893170i	$-0.648476\pi$
$89$	−8.48528	−0.899438	−0.449719	+	0.893170i	$0.648476\pi$
$90$	0	0
$91$	2.41421	0.253078
$92$	−12.1421	−1.26591
$93$	−3.51472	−0.364459
$94$	8.65685	0.892886
$95$	0	0
$96$	4.48528	0.457777
$97$	−0.828427	−0.0841140	−0.0420570	−	0.999115i	$-0.513391\pi$
$97$	−0.828427	−0.0841140	−0.0420570	+	0.999115i	$0.513391\pi$
$98$	−2.82843	−0.285714
$99$	32.0711	3.22326
$100$	0	0
$101$	−6.65685	−0.662382	−0.331191	−	0.943564i	$-0.607450\pi$
$101$	−6.65685	−0.662382	−0.331191	+	0.943564i	$0.607450\pi$
$102$	26.1421	2.58846
$103$	13.6569	1.34565	0.672825	−	0.739802i	$-0.265081\pi$
$103$	13.6569	1.34565	0.672825	+	0.739802i	$0.265081\pi$
$104$	4.41421	0.432849
$105$	0	0
$106$	7.24264	0.703467
$107$	16.1421	1.56052	0.780260	−	0.625456i	$-0.215087\pi$
$107$	16.1421	1.56052	0.780260	+	0.625456i	$0.215087\pi$
$108$	−21.6569	−2.08393
$109$	−16.4853	−1.57900	−0.789502	−	0.613748i	$-0.789661\pi$
$109$	−16.4853	−1.57900	−0.789502	+	0.613748i	$0.789661\pi$
$110$	0	0
$111$	16.9706	1.61077
$112$	7.24264	0.684365
$113$	−11.6569	−1.09658	−0.548292	−	0.836287i	$-0.684722\pi$
$113$	−11.6569	−1.09658	−0.548292	+	0.836287i	$0.684722\pi$
$114$	24.9706	2.33871
$115$	0	0
$116$	−0.656854	−0.0609874
$117$	5.00000	0.462250
$118$	31.9706	2.94313
$119$	−9.24264	−0.847271
$120$	0	0
$121$	30.1421	2.74019
$122$	2.41421	0.218573
$123$	16.0000	1.44267
$124$	4.75736	0.427223
$125$	0	0
$126$	29.1421	2.59619
$127$	−3.65685	−0.324493	−0.162247	−	0.986750i	$-0.551874\pi$
$127$	−3.65685	−0.324493	−0.162247	+	0.986750i	$0.551874\pi$
$128$	−20.5563	−1.81694
$129$	−2.34315	−0.206302
$130$	0	0
$131$	8.48528	0.741362	0.370681	−	0.928760i	$-0.379124\pi$
$131$	8.48528	0.741362	0.370681	+	0.928760i	$0.379124\pi$
$132$	−69.4558	−6.04536
$133$	−8.82843	−0.765522
$134$	−6.65685	−0.575065
$135$	0	0
$136$	−16.8995	−1.44912
$137$	2.82843	0.241649	0.120824	−	0.992674i	$-0.461446\pi$
$137$	2.82843	0.241649	0.120824	+	0.992674i	$0.461446\pi$
$138$	21.6569	1.84355
$139$	−18.4853	−1.56790	−0.783951	−	0.620823i	$-0.786798\pi$
$139$	−18.4853	−1.56790	−0.783951	+	0.620823i	$0.786798\pi$
$140$	0	0
$141$	−10.1421	−0.854122
$142$	−22.4853	−1.88692
$143$	6.41421	0.536383
$144$	15.0000	1.25000
$145$	0	0
$146$	−14.4853	−1.19881
$147$	3.31371	0.273310
$148$	−22.9706	−1.88817
$149$	8.82843	0.723253	0.361626	−	0.932323i	$-0.382222\pi$
$149$	8.82843	0.723253	0.361626	+	0.932323i	$0.382222\pi$
$150$	0	0
$151$	−8.75736	−0.712664	−0.356332	−	0.934359i	$-0.615973\pi$
$151$	−8.75736	−0.712664	−0.356332	+	0.934359i	$0.615973\pi$
$152$	−16.1421	−1.30930
$153$	−19.1421	−1.54755
$154$	37.3848	3.01255
$155$	0	0
$156$	−10.8284	−0.866968
$157$	23.4853	1.87433	0.937165	−	0.348887i	$-0.113440\pi$
$157$	23.4853	1.87433	0.937165	+	0.348887i	$0.113440\pi$
$158$	−14.4853	−1.15239
$159$	−8.48528	−0.672927
$160$	0	0
$161$	−7.65685	−0.603445
$162$	2.41421	0.189679
$163$	10.0000	0.783260	0.391630	−	0.920123i	$-0.371911\pi$
$163$	10.0000	0.783260	0.391630	+	0.920123i	$0.371911\pi$
$164$	−21.6569	−1.69112
$165$	0	0
$166$	16.6569	1.29282
$167$	−5.31371	−0.411187	−0.205594	−	0.978637i	$-0.565912\pi$
$167$	−5.31371	−0.411187	−0.205594	+	0.978637i	$0.565912\pi$
$168$	−30.1421	−2.32552
$169$	1.00000	0.0769231
$170$	0	0
$171$	−18.2843	−1.39823
$172$	3.17157	0.241830
$173$	18.6569	1.41845	0.709227	−	0.704980i	$-0.249044\pi$
$173$	18.6569	1.41845	0.709227	+	0.704980i	$0.249044\pi$
$174$	1.17157	0.0888167
$175$	0	0
$176$	19.2426	1.45047
$177$	−37.4558	−2.81535
$178$	−20.4853	−1.53544
$179$	0.686292	0.0512958	0.0256479	−	0.999671i	$-0.491835\pi$
$179$	0.686292	0.0512958	0.0256479	+	0.999671i	$0.491835\pi$
$180$	0	0
$181$	−17.4853	−1.29967	−0.649835	−	0.760075i	$-0.725162\pi$
$181$	−17.4853	−1.29967	−0.649835	+	0.760075i	$0.725162\pi$
$182$	5.82843	0.432032
$183$	−2.82843	−0.209083
$184$	−14.0000	−1.03209
$185$	0	0
$186$	−8.48528	−0.622171
$187$	−24.5563	−1.79574
$188$	13.7279	1.00121
$189$	−13.6569	−0.993390
$190$	0	0
$191$	25.3137	1.83164	0.915818	−	0.401594i	$-0.131544\pi$
$191$	25.3137	1.83164	0.915818	+	0.401594i	$0.131544\pi$
$192$	27.7990	2.00622
$193$	1.65685	0.119263	0.0596315	−	0.998220i	$-0.481007\pi$
$193$	1.65685	0.119263	0.0596315	+	0.998220i	$0.481007\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	−4.48528	−0.320377
$197$	−12.0000	−0.854965	−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000	−0.854965	−0.427482	+	0.904024i	$0.640599\pi$
$198$	77.4264	5.50246
$199$	10.0000	0.708881	0.354441	−	0.935079i	$-0.384671\pi$
$199$	10.0000	0.708881	0.354441	+	0.935079i	$0.384671\pi$
$200$	0	0
$201$	7.79899	0.550098
$202$	−16.0711	−1.13076
$203$	−0.414214	−0.0290721
$204$	41.4558	2.90249
$205$	0	0
$206$	32.9706	2.29717
$207$	−15.8579	−1.10220
$208$	3.00000	0.208013
$209$	−23.4558	−1.62247
$210$	0	0
$211$	17.7990	1.22533	0.612666	−	0.790342i	$-0.290097\pi$
$211$	17.7990	1.22533	0.612666	+	0.790342i	$0.290097\pi$
$212$	11.4853	0.788812
$213$	26.3431	1.80500
$214$	38.9706	2.66397
$215$	0	0
$216$	−24.9706	−1.69903
$217$	3.00000	0.203653
$218$	−39.7990	−2.69553
$219$	16.9706	1.14676
$220$	0	0
$221$	−3.82843	−0.257528
$222$	40.9706	2.74976
$223$	−10.9706	−0.734643	−0.367322	−	0.930094i	$-0.619725\pi$
$223$	−10.9706	−0.734643	−0.367322	+	0.930094i	$0.619725\pi$
$224$	−3.82843	−0.255798
$225$	0	0
$226$	−28.1421	−1.87199
$227$	−8.89949	−0.590680	−0.295340	−	0.955392i	$-0.595433\pi$
$227$	−8.89949	−0.590680	−0.295340	+	0.955392i	$0.595433\pi$
$228$	39.5980	2.62244
$229$	−4.82843	−0.319071	−0.159536	−	0.987192i	$-0.551000\pi$
$229$	−4.82843	−0.319071	−0.159536	+	0.987192i	$0.551000\pi$
$230$	0	0
$231$	−43.7990	−2.88176
$232$	−0.757359	−0.0497231
$233$	20.6274	1.35135	0.675674	−	0.737201i	$-0.263853\pi$
$233$	20.6274	1.35135	0.675674	+	0.737201i	$0.263853\pi$
$234$	12.0711	0.789110
$235$	0	0
$236$	50.6985	3.30019
$237$	16.9706	1.10236
$238$	−22.3137	−1.44638
$239$	23.5858	1.52564	0.762819	−	0.646612i	$-0.223815\pi$
$239$	23.5858	1.52564	0.762819	+	0.646612i	$0.223815\pi$
$240$	0	0
$241$	4.97056	0.320182	0.160091	−	0.987102i	$-0.448821\pi$
$241$	4.97056	0.320182	0.160091	+	0.987102i	$0.448821\pi$
$242$	72.7696	4.67780
$243$	14.1421	0.907218
$244$	3.82843	0.245090
$245$	0	0
$246$	38.6274	2.46279
$247$	−3.65685	−0.232680
$248$	5.48528	0.348316
$249$	−19.5147	−1.23670
$250$	0	0
$251$	−5.65685	−0.357057	−0.178529	−	0.983935i	$-0.557134\pi$
$251$	−5.65685	−0.357057	−0.178529	+	0.983935i	$0.557134\pi$
$252$	46.2132	2.91116
$253$	−20.3431	−1.27896
$254$	−8.82843	−0.553945
$255$	0	0
$256$	−29.9706	−1.87316
$257$	−6.65685	−0.415243	−0.207622	−	0.978209i	$-0.566572\pi$
$257$	−6.65685	−0.415243	−0.207622	+	0.978209i	$0.566572\pi$
$258$	−5.65685	−0.352180
$259$	−14.4853	−0.900072
$260$	0	0
$261$	−0.857864	−0.0531005
$262$	20.4853	1.26558
$263$	−26.1421	−1.61199	−0.805997	−	0.591920i	$-0.798370\pi$
$263$	−26.1421	−1.61199	−0.805997	+	0.591920i	$0.798370\pi$
$264$	−80.0833	−4.92878
$265$	0	0
$266$	−21.3137	−1.30683
$267$	24.0000	1.46878
$268$	−10.5563	−0.644832
$269$	15.1421	0.923232	0.461616	−	0.887080i	$-0.347270\pi$
$269$	15.1421	0.923232	0.461616	+	0.887080i	$0.347270\pi$
$270$	0	0
$271$	−25.7279	−1.56286	−0.781430	−	0.623993i	$-0.785509\pi$
$271$	−25.7279	−1.56286	−0.781430	+	0.623993i	$0.785509\pi$
$272$	−11.4853	−0.696397
$273$	−6.82843	−0.413275
$274$	6.82843	0.412520
$275$	0	0
$276$	34.3431	2.06721
$277$	9.31371	0.559607	0.279803	−	0.960057i	$-0.409731\pi$
$277$	9.31371	0.559607	0.279803	+	0.960057i	$0.409731\pi$
$278$	−44.6274	−2.67657
$279$	6.21320	0.371975
$280$	0	0
$281$	4.82843	0.288040	0.144020	−	0.989575i	$-0.453997\pi$
$281$	4.82843	0.288040	0.144020	+	0.989575i	$0.453997\pi$
$282$	−24.4853	−1.45808
$283$	−22.4853	−1.33661	−0.668306	−	0.743887i	$-0.732980\pi$
$283$	−22.4853	−1.33661	−0.668306	+	0.743887i	$0.732980\pi$
$284$	−35.6569	−2.11585
$285$	0	0
$286$	15.4853	0.915664
$287$	−13.6569	−0.806139
$288$	−7.92893	−0.467217
$289$	−2.34315	−0.137832
$290$	0	0
$291$	2.34315	0.137358
$292$	−22.9706	−1.34425
$293$	8.82843	0.515762	0.257881	−	0.966177i	$-0.416976\pi$
$293$	8.82843	0.515762	0.257881	+	0.966177i	$0.416976\pi$
$294$	8.00000	0.466569
$295$	0	0
$296$	−26.4853	−1.53943
$297$	−36.2843	−2.10543
$298$	21.3137	1.23467
$299$	−3.17157	−0.183417
$300$	0	0
$301$	2.00000	0.115278
$302$	−21.1421	−1.21659
$303$	18.8284	1.08166
$304$	−10.9706	−0.629205
$305$	0	0
$306$	−46.2132	−2.64183
$307$	−9.31371	−0.531561	−0.265781	−	0.964034i	$-0.585630\pi$
$307$	−9.31371	−0.531561	−0.265781	+	0.964034i	$0.585630\pi$
$308$	59.2843	3.37803
$309$	−38.6274	−2.19744
$310$	0	0
$311$	9.51472	0.539530	0.269765	−	0.962926i	$-0.413054\pi$
$311$	9.51472	0.539530	0.269765	+	0.962926i	$0.413054\pi$
$312$	−12.4853	−0.706840
$313$	2.85786	0.161536	0.0807680	−	0.996733i	$-0.474263\pi$
$313$	2.85786	0.161536	0.0807680	+	0.996733i	$0.474263\pi$
$314$	56.6985	3.19968
$315$	0	0
$316$	−22.9706	−1.29220
$317$	20.1421	1.13130	0.565648	−	0.824647i	$-0.308626\pi$
$317$	20.1421	1.13130	0.565648	+	0.824647i	$0.308626\pi$
$318$	−20.4853	−1.14876
$319$	−1.10051	−0.0616165
$320$	0	0
$321$	−45.6569	−2.54832
$322$	−18.4853	−1.03014
$323$	14.0000	0.778981
$324$	3.82843	0.212690
$325$	0	0
$326$	24.1421	1.33711
$327$	46.6274	2.57850
$328$	−24.9706	−1.37877
$329$	8.65685	0.477268
$330$	0	0
$331$	27.6569	1.52016	0.760079	−	0.649831i	$-0.225160\pi$
$331$	27.6569	1.52016	0.760079	+	0.649831i	$0.225160\pi$
$332$	26.4142	1.44967
$333$	−30.0000	−1.64399
$334$	−12.8284	−0.701940
$335$	0	0
$336$	−20.4853	−1.11756
$337$	−20.7990	−1.13299	−0.566497	−	0.824064i	$-0.691702\pi$
$337$	−20.7990	−1.13299	−0.566497	+	0.824064i	$0.691702\pi$
$338$	2.41421	0.131316
$339$	32.9706	1.79072
$340$	0	0
$341$	7.97056	0.431630
$342$	−44.1421	−2.38693
$343$	−19.7279	−1.06521
$344$	3.65685	0.197164
$345$	0	0
$346$	45.0416	2.42145
$347$	−19.4558	−1.04444	−0.522222	−	0.852809i	$-0.674897\pi$
$347$	−19.4558	−1.04444	−0.522222	+	0.852809i	$0.674897\pi$
$348$	1.85786	0.0995920
$349$	35.4558	1.89791	0.948954	−	0.315415i	$-0.102144\pi$
$349$	35.4558	1.89791	0.948954	+	0.315415i	$0.102144\pi$
$350$	0	0
$351$	−5.65685	−0.301941
$352$	−10.1716	−0.542147
$353$	26.1421	1.39141	0.695703	−	0.718330i	$-0.255093\pi$
$353$	26.1421	1.39141	0.695703	+	0.718330i	$0.255093\pi$
$354$	−90.4264	−4.80611
$355$	0	0
$356$	−32.4853	−1.72172
$357$	26.1421	1.38359
$358$	1.65685	0.0875675
$359$	21.3848	1.12865	0.564323	−	0.825554i	$-0.309137\pi$
$359$	21.3848	1.12865	0.564323	+	0.825554i	$0.309137\pi$
$360$	0	0
$361$	−5.62742	−0.296180
$362$	−42.2132	−2.21868
$363$	−85.2548	−4.47472
$364$	9.24264	0.484446
$365$	0	0
$366$	−6.82843	−0.356928
$367$	−34.8284	−1.81803	−0.909015	−	0.416764i	$-0.863164\pi$
$367$	−34.8284	−1.81803	−0.909015	+	0.416764i	$0.863164\pi$
$368$	−9.51472	−0.495989
$369$	−28.2843	−1.47242
$370$	0	0
$371$	7.24264	0.376019
$372$	−13.4558	−0.697653
$373$	1.14214	0.0591375	0.0295688	−	0.999563i	$-0.490587\pi$
$373$	1.14214	0.0591375	0.0295688	+	0.999563i	$0.490587\pi$
$374$	−59.2843	−3.06552
$375$	0	0
$376$	15.8284	0.816289
$377$	−0.171573	−0.00883645
$378$	−32.9706	−1.69582
$379$	−31.8701	−1.63705	−0.818527	−	0.574467i	$-0.805209\pi$
$379$	−31.8701	−1.63705	−0.818527	+	0.574467i	$0.805209\pi$
$380$	0	0
$381$	10.3431	0.529895
$382$	61.1127	3.12680
$383$	27.6569	1.41320	0.706600	−	0.707614i	$-0.250228\pi$
$383$	27.6569	1.41320	0.706600	+	0.707614i	$0.250228\pi$
$384$	58.1421	2.96705
$385$	0	0
$386$	4.00000	0.203595
$387$	4.14214	0.210557
$388$	−3.17157	−0.161012
$389$	28.6274	1.45147	0.725734	−	0.687976i	$-0.241500\pi$
$389$	28.6274	1.45147	0.725734	+	0.687976i	$0.241500\pi$
$390$	0	0
$391$	12.1421	0.614054
$392$	−5.17157	−0.261204
$393$	−24.0000	−1.21064
$394$	−28.9706	−1.45952
$395$	0	0
$396$	122.782	6.17001
$397$	7.65685	0.384286	0.192143	−	0.981367i	$-0.438456\pi$
$397$	7.65685	0.384286	0.192143	+	0.981367i	$0.438456\pi$
$398$	24.1421	1.21014
$399$	24.9706	1.25009
$400$	0	0
$401$	−3.85786	−0.192653	−0.0963263	−	0.995350i	$-0.530709\pi$
$401$	−3.85786	−0.192653	−0.0963263	+	0.995350i	$0.530709\pi$
$402$	18.8284	0.939077
$403$	1.24264	0.0619003
$404$	−25.4853	−1.26794
$405$	0	0
$406$	−1.00000	−0.0496292
$407$	−38.4853	−1.90764
$408$	47.7990	2.36640
$409$	−10.8284	−0.535431	−0.267716	−	0.963498i	$-0.586269\pi$
$409$	−10.8284	−0.535431	−0.267716	+	0.963498i	$0.586269\pi$
$410$	0	0
$411$	−8.00000	−0.394611
$412$	52.2843	2.57586
$413$	31.9706	1.57317
$414$	−38.2843	−1.88157
$415$	0	0
$416$	−1.58579	−0.0777496
$417$	52.2843	2.56037
$418$	−56.6274	−2.76974
$419$	22.8284	1.11524	0.557621	−	0.830096i	$-0.311714\pi$
$419$	22.8284	1.11524	0.557621	+	0.830096i	$0.311714\pi$
$420$	0	0
$421$	−16.9706	−0.827095	−0.413547	−	0.910483i	$-0.635710\pi$
$421$	−16.9706	−0.827095	−0.413547	+	0.910483i	$0.635710\pi$
$422$	42.9706	2.09177
$423$	17.9289	0.871735
$424$	13.2426	0.643119
$425$	0	0
$426$	63.5980	3.08133
$427$	2.41421	0.116832
$428$	61.7990	2.98717
$429$	−18.1421	−0.875911
$430$	0	0
$431$	−8.34315	−0.401875	−0.200938	−	0.979604i	$-0.564399\pi$
$431$	−8.34315	−0.401875	−0.200938	+	0.979604i	$0.564399\pi$
$432$	−16.9706	−0.816497
$433$	16.3431	0.785401	0.392701	−	0.919666i	$-0.371541\pi$
$433$	16.3431	0.785401	0.392701	+	0.919666i	$0.371541\pi$
$434$	7.24264	0.347658
$435$	0	0
$436$	−63.1127	−3.02255
$437$	11.5980	0.554807
$438$	40.9706	1.95765
$439$	14.0000	0.668184	0.334092	−	0.942541i	$-0.391570\pi$
$439$	14.0000	0.668184	0.334092	+	0.942541i	$0.391570\pi$
$440$	0	0
$441$	−5.85786	−0.278946
$442$	−9.24264	−0.439628
$443$	−29.7990	−1.41579	−0.707896	−	0.706316i	$-0.750356\pi$
$443$	−29.7990	−1.41579	−0.707896	+	0.706316i	$0.750356\pi$
$444$	64.9706	3.08337
$445$	0	0
$446$	−26.4853	−1.25411
$447$	−24.9706	−1.18107
$448$	−23.7279	−1.12104
$449$	−5.17157	−0.244062	−0.122031	−	0.992526i	$-0.538941\pi$
$449$	−5.17157	−0.244062	−0.122031	+	0.992526i	$0.538941\pi$
$450$	0	0
$451$	−36.2843	−1.70856
$452$	−44.6274	−2.09910
$453$	24.7696	1.16378
$454$	−21.4853	−1.00835
$455$	0	0
$456$	45.6569	2.13808
$457$	8.48528	0.396925	0.198462	−	0.980109i	$-0.436405\pi$
$457$	8.48528	0.396925	0.198462	+	0.980109i	$0.436405\pi$
$458$	−11.6569	−0.544689
$459$	21.6569	1.01086
$460$	0	0
$461$	−19.4558	−0.906149	−0.453074	−	0.891473i	$-0.649673\pi$
$461$	−19.4558	−0.906149	−0.453074	+	0.891473i	$0.649673\pi$
$462$	−105.740	−4.91948
$463$	28.5563	1.32713	0.663563	−	0.748120i	$-0.269043\pi$
$463$	28.5563	1.32713	0.663563	+	0.748120i	$0.269043\pi$
$464$	−0.514719	−0.0238952
$465$	0	0
$466$	49.7990	2.30689
$467$	31.1127	1.43972	0.719862	−	0.694117i	$-0.244205\pi$
$467$	31.1127	1.43972	0.719862	+	0.694117i	$0.244205\pi$
$468$	19.1421	0.884846
$469$	−6.65685	−0.307385
$470$	0	0
$471$	−66.4264	−3.06077
$472$	58.4558	2.69065
$473$	5.31371	0.244325
$474$	40.9706	1.88184
$475$	0	0
$476$	−35.3848	−1.62186
$477$	15.0000	0.686803
$478$	56.9411	2.60443
$479$	14.2721	0.652108	0.326054	−	0.945351i	$-0.394281\pi$
$479$	14.2721	0.652108	0.326054	+	0.945351i	$0.394281\pi$
$480$	0	0
$481$	−6.00000	−0.273576
$482$	12.0000	0.546585
$483$	21.6569	0.985421
$484$	115.397	5.24532
$485$	0	0
$486$	34.1421	1.54872
$487$	−42.2132	−1.91286	−0.956431	−	0.291957i	$-0.905694\pi$
$487$	−42.2132	−1.91286	−0.956431	+	0.291957i	$0.905694\pi$
$488$	4.41421	0.199822
$489$	−28.2843	−1.27906
$490$	0	0
$491$	11.1716	0.504166	0.252083	−	0.967706i	$-0.418884\pi$
$491$	11.1716	0.504166	0.252083	+	0.967706i	$0.418884\pi$
$492$	61.2548	2.76158
$493$	0.656854	0.0295832
$494$	−8.82843	−0.397210
$495$	0	0
$496$	3.72792	0.167389
$497$	−22.4853	−1.00860
$498$	−47.1127	−2.11117
$499$	2.55635	0.114438	0.0572190	−	0.998362i	$-0.481777\pi$
$499$	2.55635	0.114438	0.0572190	+	0.998362i	$0.481777\pi$
$500$	0	0
$501$	15.0294	0.671466
$502$	−13.6569	−0.609535
$503$	−14.6274	−0.652204	−0.326102	−	0.945335i	$-0.605735\pi$
$503$	−14.6274	−0.652204	−0.326102	+	0.945335i	$0.605735\pi$
$504$	53.2843	2.37347
$505$	0	0
$506$	−49.1127	−2.18333
$507$	−2.82843	−0.125615
$508$	−14.0000	−0.621150
$509$	14.6274	0.648349	0.324174	−	0.945997i	$-0.394914\pi$
$509$	14.6274	0.648349	0.324174	+	0.945997i	$0.394914\pi$
$510$	0	0
$511$	−14.4853	−0.640791
$512$	−31.2426	−1.38074
$513$	20.6863	0.913322
$514$	−16.0711	−0.708864
$515$	0	0
$516$	−8.97056	−0.394907
$517$	23.0000	1.01154
$518$	−34.9706	−1.53652
$519$	−52.7696	−2.31633
$520$	0	0
$521$	0.343146	0.0150335	0.00751674	−	0.999972i	$-0.497607\pi$
$521$	0.343146	0.0150335	0.00751674	+	0.999972i	$0.497607\pi$
$522$	−2.07107	−0.0906482
$523$	6.00000	0.262362	0.131181	−	0.991358i	$-0.458123\pi$
$523$	6.00000	0.262362	0.131181	+	0.991358i	$0.458123\pi$
$524$	32.4853	1.41913
$525$	0	0
$526$	−63.1127	−2.75184
$527$	−4.75736	−0.207234
$528$	−54.4264	−2.36861
$529$	−12.9411	−0.562658
$530$	0	0
$531$	66.2132	2.87341
$532$	−33.7990	−1.46537
$533$	−5.65685	−0.245026
$534$	57.9411	2.50736
$535$	0	0
$536$	−12.1716	−0.525732
$537$	−1.94113	−0.0837657
$538$	36.5563	1.57606
$539$	−7.51472	−0.323682
$540$	0	0
$541$	33.7990	1.45313	0.726566	−	0.687097i	$-0.241115\pi$
$541$	33.7990	1.45313	0.726566	+	0.687097i	$0.241115\pi$
$542$	−62.1127	−2.66797
$543$	49.4558	2.12235
$544$	6.07107	0.260295
$545$	0	0
$546$	−16.4853	−0.705505
$547$	28.4853	1.21794	0.608971	−	0.793192i	$-0.291582\pi$
$547$	28.4853	1.21794	0.608971	+	0.793192i	$0.291582\pi$
$548$	10.8284	0.462567
$549$	5.00000	0.213395
$550$	0	0
$551$	0.627417	0.0267289
$552$	39.5980	1.68540
$553$	−14.4853	−0.615977
$554$	22.4853	0.955308
$555$	0	0
$556$	−70.7696	−3.00130
$557$	27.3137	1.15732	0.578659	−	0.815569i	$-0.303576\pi$
$557$	27.3137	1.15732	0.578659	+	0.815569i	$0.303576\pi$
$558$	15.0000	0.635001
$559$	0.828427	0.0350387
$560$	0	0
$561$	69.4558	2.93243
$562$	11.6569	0.491715
$563$	−18.0000	−0.758610	−0.379305	−	0.925272i	$-0.623837\pi$
$563$	−18.0000	−0.758610	−0.379305	+	0.925272i	$0.623837\pi$
$564$	−38.8284	−1.63497
$565$	0	0
$566$	−54.2843	−2.28174
$567$	2.41421	0.101387
$568$	−41.1127	−1.72505
$569$	−35.2843	−1.47919	−0.739597	−	0.673050i	$-0.764984\pi$
$569$	−35.2843	−1.47919	−0.739597	+	0.673050i	$0.764984\pi$
$570$	0	0
$571$	−2.00000	−0.0836974	−0.0418487	−	0.999124i	$-0.513325\pi$
$571$	−2.00000	−0.0836974	−0.0418487	+	0.999124i	$0.513325\pi$
$572$	24.5563	1.02675
$573$	−71.5980	−2.99105
$574$	−32.9706	−1.37616
$575$	0	0
$576$	−49.1421	−2.04759
$577$	−7.17157	−0.298556	−0.149278	−	0.988795i	$-0.547695\pi$
$577$	−7.17157	−0.298556	−0.149278	+	0.988795i	$0.547695\pi$
$578$	−5.65685	−0.235294
$579$	−4.68629	−0.194756
$580$	0	0
$581$	16.6569	0.691043
$582$	5.65685	0.234484
$583$	19.2426	0.796949
$584$	−26.4853	−1.09597
$585$	0	0
$586$	21.3137	0.880461
$587$	14.4142	0.594938	0.297469	−	0.954731i	$-0.403857\pi$
$587$	14.4142	0.594938	0.297469	+	0.954731i	$0.403857\pi$
$588$	12.6863	0.523174
$589$	−4.54416	−0.187239
$590$	0	0
$591$	33.9411	1.39615
$592$	−18.0000	−0.739795
$593$	19.6569	0.807210	0.403605	−	0.914933i	$-0.367757\pi$
$593$	19.6569	0.807210	0.403605	+	0.914933i	$0.367757\pi$
$594$	−87.5980	−3.59419
$595$	0	0
$596$	33.7990	1.38446
$597$	−28.2843	−1.15760
$598$	−7.65685	−0.313112
$599$	−3.51472	−0.143608	−0.0718038	−	0.997419i	$-0.522876\pi$
$599$	−3.51472	−0.143608	−0.0718038	+	0.997419i	$0.522876\pi$
$600$	0	0
$601$	13.8284	0.564073	0.282037	−	0.959404i	$-0.408990\pi$
$601$	13.8284	0.564073	0.282037	+	0.959404i	$0.408990\pi$
$602$	4.82843	0.196792
$603$	−13.7868	−0.561442
$604$	−33.5269	−1.36419
$605$	0	0
$606$	45.4558	1.84652
$607$	−21.5147	−0.873255	−0.436628	−	0.899642i	$-0.643827\pi$
$607$	−21.5147	−0.873255	−0.436628	+	0.899642i	$0.643827\pi$
$608$	5.79899	0.235180
$609$	1.17157	0.0474745
$610$	0	0
$611$	3.58579	0.145065
$612$	−73.2843	−2.96234
$613$	−12.8284	−0.518135	−0.259068	−	0.965859i	$-0.583415\pi$
$613$	−12.8284	−0.518135	−0.259068	+	0.965859i	$0.583415\pi$
$614$	−22.4853	−0.907432
$615$	0	0
$616$	68.3553	2.75412
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	−93.2548	−3.75126
$619$	22.9706	0.923265	0.461632	−	0.887071i	$-0.347264\pi$
$619$	22.9706	0.923265	0.461632	+	0.887071i	$0.347264\pi$
$620$	0	0
$621$	17.9411	0.719953
$622$	22.9706	0.921036
$623$	−20.4853	−0.820725
$624$	−8.48528	−0.339683
$625$	0	0
$626$	6.89949	0.275759
$627$	66.3431	2.64949
$628$	89.9117	3.58787
$629$	22.9706	0.915896
$630$	0	0
$631$	34.0000	1.35352	0.676759	−	0.736204i	$-0.263384\pi$
$631$	34.0000	1.35352	0.676759	+	0.736204i	$0.263384\pi$
$632$	−26.4853	−1.05353
$633$	−50.3431	−2.00096
$634$	48.6274	1.93124
$635$	0	0
$636$	−32.4853	−1.28813
$637$	−1.17157	−0.0464194
$638$	−2.65685	−0.105186
$639$	−46.5685	−1.84222
$640$	0	0
$641$	28.1127	1.11038	0.555192	−	0.831722i	$-0.312645\pi$
$641$	28.1127	1.11038	0.555192	+	0.831722i	$0.312645\pi$
$642$	−110.225	−4.35025
$643$	−1.02944	−0.0405970	−0.0202985	−	0.999794i	$-0.506462\pi$
$643$	−1.02944	−0.0405970	−0.0202985	+	0.999794i	$0.506462\pi$
$644$	−29.3137	−1.15512
$645$	0	0
$646$	33.7990	1.32980
$647$	18.8284	0.740222	0.370111	−	0.928988i	$-0.379320\pi$
$647$	18.8284	0.740222	0.370111	+	0.928988i	$0.379320\pi$
$648$	4.41421	0.173407
$649$	84.9411	3.33423
$650$	0	0
$651$	−8.48528	−0.332564
$652$	38.2843	1.49933
$653$	−22.4558	−0.878765	−0.439383	−	0.898300i	$-0.644803\pi$
$653$	−22.4558	−0.878765	−0.439383	+	0.898300i	$0.644803\pi$
$654$	112.569	4.40178
$655$	0	0
$656$	−16.9706	−0.662589
$657$	−30.0000	−1.17041
$658$	20.8995	0.814747
$659$	−8.62742	−0.336076	−0.168038	−	0.985780i	$-0.553743\pi$
$659$	−8.62742	−0.336076	−0.168038	+	0.985780i	$0.553743\pi$
$660$	0	0
$661$	−6.82843	−0.265595	−0.132798	−	0.991143i	$-0.542396\pi$
$661$	−6.82843	−0.265595	−0.132798	+	0.991143i	$0.542396\pi$
$662$	66.7696	2.59507
$663$	10.8284	0.420541
$664$	30.4558	1.18192
$665$	0	0
$666$	−72.4264	−2.80647
$667$	0.544156	0.0210698
$668$	−20.3431	−0.787100
$669$	31.0294	1.19967
$670$	0	0
$671$	6.41421	0.247618
$672$	10.8284	0.417716
$673$	24.4558	0.942704	0.471352	−	0.881945i	$-0.343766\pi$
$673$	24.4558	0.942704	0.471352	+	0.881945i	$0.343766\pi$
$674$	−50.2132	−1.93414
$675$	0	0
$676$	3.82843	0.147247
$677$	−41.3137	−1.58781	−0.793907	−	0.608039i	$-0.791957\pi$
$677$	−41.3137	−1.58781	−0.793907	+	0.608039i	$0.791957\pi$
$678$	79.5980	3.05694
$679$	−2.00000	−0.0767530
$680$	0	0
$681$	25.1716	0.964577
$682$	19.2426	0.736839
$683$	21.5858	0.825957	0.412979	−	0.910741i	$-0.364488\pi$
$683$	21.5858	0.825957	0.412979	+	0.910741i	$0.364488\pi$
$684$	−70.0000	−2.67652
$685$	0	0
$686$	−47.6274	−1.81842
$687$	13.6569	0.521041
$688$	2.48528	0.0947505
$689$	3.00000	0.114291
$690$	0	0
$691$	8.89949	0.338553	0.169276	−	0.985569i	$-0.445857\pi$
$691$	8.89949	0.338553	0.169276	+	0.985569i	$0.445857\pi$
$692$	71.4264	2.71522
$693$	77.4264	2.94119
$694$	−46.9706	−1.78298
$695$	0	0
$696$	2.14214	0.0811974
$697$	21.6569	0.820312
$698$	85.5980	3.23993
$699$	−58.3431	−2.20674
$700$	0	0
$701$	−15.8284	−0.597831	−0.298916	−	0.954280i	$-0.596625\pi$
$701$	−15.8284	−0.597831	−0.298916	+	0.954280i	$0.596625\pi$
$702$	−13.6569	−0.515445
$703$	21.9411	0.827525
$704$	−63.0416	−2.37597
$705$	0	0
$706$	63.1127	2.37528
$707$	−16.0711	−0.604415
$708$	−143.397	−5.38919
$709$	−49.2548	−1.84980	−0.924902	−	0.380205i	$-0.875853\pi$
$709$	−49.2548	−1.84980	−0.924902	+	0.380205i	$0.875853\pi$
$710$	0	0
$711$	−30.0000	−1.12509
$712$	−37.4558	−1.40372
$713$	−3.94113	−0.147596
$714$	63.1127	2.36193
$715$	0	0
$716$	2.62742	0.0981912
$717$	−66.7107	−2.49136
$718$	51.6274	1.92672
$719$	−49.4558	−1.84439	−0.922196	−	0.386723i	$-0.873607\pi$
$719$	−49.4558	−1.84439	−0.922196	+	0.386723i	$0.873607\pi$
$720$	0	0
$721$	32.9706	1.22789
$722$	−13.5858	−0.505611
$723$	−14.0589	−0.522855
$724$	−66.9411	−2.48785
$725$	0	0
$726$	−205.823	−7.63882
$727$	16.6863	0.618860	0.309430	−	0.950922i	$-0.399862\pi$
$727$	16.6863	0.618860	0.309430	+	0.950922i	$0.399862\pi$
$728$	10.6569	0.394969
$729$	−43.0000	−1.59259
$730$	0	0
$731$	−3.17157	−0.117305
$732$	−10.8284	−0.400230
$733$	−35.7990	−1.32227	−0.661133	−	0.750269i	$-0.729924\pi$
$733$	−35.7990	−1.32227	−0.661133	+	0.750269i	$0.729924\pi$
$734$	−84.0833	−3.10357
$735$	0	0
$736$	5.02944	0.185388
$737$	−17.6863	−0.651483
$738$	−68.2843	−2.51358
$739$	−27.7279	−1.01999	−0.509994	−	0.860178i	$-0.670352\pi$
$739$	−27.7279	−1.01999	−0.509994	+	0.860178i	$0.670352\pi$
$740$	0	0
$741$	10.3431	0.379965
$742$	17.4853	0.641905
$743$	−14.6985	−0.539235	−0.269618	−	0.962967i	$-0.586897\pi$
$743$	−14.6985	−0.539235	−0.269618	+	0.962967i	$0.586897\pi$
$744$	−15.5147	−0.568797
$745$	0	0
$746$	2.75736	0.100954
$747$	34.4975	1.26220
$748$	−94.0122	−3.43743
$749$	38.9706	1.42395
$750$	0	0
$751$	31.4558	1.14784	0.573920	−	0.818911i	$-0.305422\pi$
$751$	31.4558	1.14784	0.573920	+	0.818911i	$0.305422\pi$
$752$	10.7574	0.392281
$753$	16.0000	0.583072
$754$	−0.414214	−0.0150848
$755$	0	0
$756$	−52.2843	−1.90156
$757$	−16.3137	−0.592932	−0.296466	−	0.955043i	$-0.595808\pi$
$757$	−16.3137	−0.592932	−0.296466	+	0.955043i	$0.595808\pi$
$758$	−76.9411	−2.79463
$759$	57.5391	2.08854
$760$	0	0
$761$	−21.7990	−0.790213	−0.395106	−	0.918635i	$-0.629292\pi$
$761$	−21.7990	−0.790213	−0.395106	+	0.918635i	$0.629292\pi$
$762$	24.9706	0.904588
$763$	−39.7990	−1.44082
$764$	96.9117	3.50614
$765$	0	0
$766$	66.7696	2.41248
$767$	13.2426	0.478164
$768$	84.7696	3.05886
$769$	4.97056	0.179243	0.0896215	−	0.995976i	$-0.471434\pi$
$769$	4.97056	0.179243	0.0896215	+	0.995976i	$0.471434\pi$
$770$	0	0
$771$	18.8284	0.678089
$772$	6.34315	0.228295
$773$	−5.79899	−0.208575	−0.104288	−	0.994547i	$-0.533256\pi$
$773$	−5.79899	−0.208575	−0.104288	+	0.994547i	$0.533256\pi$
$774$	10.0000	0.359443
$775$	0	0
$776$	−3.65685	−0.131273
$777$	40.9706	1.46981
$778$	69.1127	2.47781
$779$	20.6863	0.741163
$780$	0	0
$781$	−59.7401	−2.13767
$782$	29.3137	1.04826
$783$	0.970563	0.0346851
$784$	−3.51472	−0.125526
$785$	0	0
$786$	−57.9411	−2.06669
$787$	38.7574	1.38155	0.690775	−	0.723069i	$-0.257269\pi$
$787$	38.7574	1.38155	0.690775	+	0.723069i	$0.257269\pi$
$788$	−45.9411	−1.63658
$789$	73.9411	2.63237
$790$	0	0
$791$	−28.1421	−1.00062
$792$	141.569	5.03042
$793$	1.00000	0.0355110
$794$	18.4853	0.656018
$795$	0	0
$796$	38.2843	1.35695
$797$	−11.4853	−0.406830	−0.203415	−	0.979093i	$-0.565204\pi$
$797$	−11.4853	−0.406830	−0.203415	+	0.979093i	$0.565204\pi$
$798$	60.2843	2.13404
$799$	−13.7279	−0.485659
$800$	0	0
$801$	−42.4264	−1.49906
$802$	−9.31371	−0.328878
$803$	−38.4853	−1.35812
$804$	29.8579	1.05301
$805$	0	0
$806$	3.00000	0.105670
$807$	−42.8284	−1.50763
$808$	−29.3848	−1.03375
$809$	17.3137	0.608718	0.304359	−	0.952557i	$-0.401558\pi$
$809$	17.3137	0.608718	0.304359	+	0.952557i	$0.401558\pi$
$810$	0	0
$811$	9.58579	0.336602	0.168301	−	0.985736i	$-0.446172\pi$
$811$	9.58579	0.336602	0.168301	+	0.985736i	$0.446172\pi$
$812$	−1.58579	−0.0556502
$813$	72.7696	2.55214
$814$	−92.9117	−3.25655
$815$	0	0
$816$	32.4853	1.13721
$817$	−3.02944	−0.105987
$818$	−26.1421	−0.914038
$819$	12.0711	0.421797
$820$	0	0
$821$	−20.8284	−0.726917	−0.363459	−	0.931610i	$-0.618404\pi$
$821$	−20.8284	−0.726917	−0.363459	+	0.931610i	$0.618404\pi$
$822$	−19.3137	−0.673643
$823$	−20.2843	−0.707065	−0.353533	−	0.935422i	$-0.615020\pi$
$823$	−20.2843	−0.707065	−0.353533	+	0.935422i	$0.615020\pi$
$824$	60.2843	2.10010
$825$	0	0
$826$	77.1838	2.68557
$827$	−32.3553	−1.12511	−0.562553	−	0.826761i	$-0.690181\pi$
$827$	−32.3553	−1.12511	−0.562553	+	0.826761i	$0.690181\pi$
$828$	−60.7107	−2.10984
$829$	7.97056	0.276829	0.138415	−	0.990374i	$-0.455799\pi$
$829$	7.97056	0.276829	0.138415	+	0.990374i	$0.455799\pi$
$830$	0	0
$831$	−26.3431	−0.913834
$832$	−9.82843	−0.340739
$833$	4.48528	0.155406
$834$	126.225	4.37083
$835$	0	0
$836$	−89.7990	−3.10576
$837$	−7.02944	−0.242973
$838$	55.1127	1.90384
$839$	−1.02944	−0.0355401	−0.0177701	−	0.999842i	$-0.505657\pi$
$839$	−1.02944	−0.0355401	−0.0177701	+	0.999842i	$0.505657\pi$
$840$	0	0
$841$	−28.9706	−0.998985
$842$	−40.9706	−1.41194
$843$	−13.6569	−0.470367
$844$	68.1421	2.34555
$845$	0	0
$846$	43.2843	1.48814
$847$	72.7696	2.50039
$848$	9.00000	0.309061
$849$	63.5980	2.18268
$850$	0	0
$851$	19.0294	0.652321
$852$	100.853	3.45516
$853$	−42.4264	−1.45265	−0.726326	−	0.687350i	$-0.758774\pi$
$853$	−42.4264	−1.45265	−0.726326	+	0.687350i	$0.758774\pi$
$854$	5.82843	0.199445
$855$	0	0
$856$	71.2548	2.43544
$857$	−4.62742	−0.158070	−0.0790348	−	0.996872i	$-0.525184\pi$
$857$	−4.62742	−0.158070	−0.0790348	+	0.996872i	$0.525184\pi$
$858$	−43.7990	−1.49527
$859$	24.1421	0.823719	0.411860	−	0.911247i	$-0.364879\pi$
$859$	24.1421	0.823719	0.411860	+	0.911247i	$0.364879\pi$
$860$	0	0
$861$	38.6274	1.31642
$862$	−20.1421	−0.686044
$863$	29.1838	0.993427	0.496713	−	0.867915i	$-0.334540\pi$
$863$	29.1838	0.993427	0.496713	+	0.867915i	$0.334540\pi$
$864$	8.97056	0.305185
$865$	0	0
$866$	39.4558	1.34076
$867$	6.62742	0.225079
$868$	11.4853	0.389836
$869$	−38.4853	−1.30552
$870$	0	0
$871$	−2.75736	−0.0934296
$872$	−72.7696	−2.46429
$873$	−4.14214	−0.140190
$874$	28.0000	0.947114
$875$	0	0
$876$	64.9706	2.19515
$877$	−15.7990	−0.533494	−0.266747	−	0.963767i	$-0.585949\pi$
$877$	−15.7990	−0.533494	−0.266747	+	0.963767i	$0.585949\pi$
$878$	33.7990	1.14066
$879$	−24.9706	−0.842236
$880$	0	0
$881$	−15.0000	−0.505363	−0.252681	−	0.967550i	$-0.581312\pi$
$881$	−15.0000	−0.505363	−0.252681	+	0.967550i	$0.581312\pi$
$882$	−14.1421	−0.476190
$883$	32.9706	1.10955	0.554774	−	0.832001i	$-0.312805\pi$
$883$	32.9706	1.10955	0.554774	+	0.832001i	$0.312805\pi$
$884$	−14.6569	−0.492963
$885$	0	0
$886$	−71.9411	−2.41691
$887$	8.62742	0.289680	0.144840	−	0.989455i	$-0.453733\pi$
$887$	8.62742	0.289680	0.144840	+	0.989455i	$0.453733\pi$
$888$	74.9117	2.51387
$889$	−8.82843	−0.296096
$890$	0	0
$891$	6.41421	0.214884
$892$	−42.0000	−1.40626
$893$	−13.1127	−0.438800
$894$	−60.2843	−2.01621
$895$	0	0
$896$	−49.6274	−1.65794
$897$	8.97056	0.299518
$898$	−12.4853	−0.416639
$899$	−0.213203	−0.00711073
$900$	0	0
$901$	−11.4853	−0.382630
$902$	−87.5980	−2.91669
$903$	−5.65685	−0.188248
$904$	−51.4558	−1.71140
$905$	0	0
$906$	59.7990	1.98669
$907$	−24.0000	−0.796907	−0.398453	−	0.917189i	$-0.630453\pi$
$907$	−24.0000	−0.796907	−0.398453	+	0.917189i	$0.630453\pi$
$908$	−34.0711	−1.13069
$909$	−33.2843	−1.10397
$910$	0	0
$911$	6.00000	0.198789	0.0993944	−	0.995048i	$-0.468309\pi$
$911$	6.00000	0.198789	0.0993944	+	0.995048i	$0.468309\pi$
$912$	31.0294	1.02749
$913$	44.2548	1.46462
$914$	20.4853	0.677593
$915$	0	0
$916$	−18.4853	−0.610771
$917$	20.4853	0.676484
$918$	52.2843	1.72564
$919$	−18.9706	−0.625781	−0.312891	−	0.949789i	$-0.601297\pi$
$919$	−18.9706	−0.625781	−0.312891	+	0.949789i	$0.601297\pi$
$920$	0	0
$921$	26.3431	0.868036
$922$	−46.9706	−1.54689
$923$	−9.31371	−0.306564
$924$	−167.681	−5.51631
$925$	0	0
$926$	68.9411	2.26555
$927$	68.2843	2.24275
$928$	0.272078	0.00893140
$929$	30.2843	0.993595	0.496797	−	0.867867i	$-0.334509\pi$
$929$	30.2843	0.993595	0.496797	+	0.867867i	$0.334509\pi$
$930$	0	0
$931$	4.28427	0.140411
$932$	78.9706	2.58677
$933$	−26.9117	−0.881049
$934$	75.1127	2.45776
$935$	0	0
$936$	22.0711	0.721415
$937$	1.97056	0.0643755	0.0321877	−	0.999482i	$-0.489753\pi$
$937$	1.97056	0.0643755	0.0321877	+	0.999482i	$0.489753\pi$
$938$	−16.0711	−0.524739
$939$	−8.08326	−0.263787
$940$	0	0
$941$	−35.6569	−1.16238	−0.581190	−	0.813768i	$-0.697413\pi$
$941$	−35.6569	−1.16238	−0.581190	+	0.813768i	$0.697413\pi$
$942$	−160.368	−5.22506
$943$	17.9411	0.584243
$944$	39.7279	1.29303
$945$	0	0
$946$	12.8284	0.417088
$947$	−26.4142	−0.858347	−0.429173	−	0.903222i	$-0.641195\pi$
$947$	−26.4142	−0.858347	−0.429173	+	0.903222i	$0.641195\pi$
$948$	64.9706	2.11015
$949$	−6.00000	−0.194768
$950$	0	0
$951$	−56.9706	−1.84740
$952$	−40.7990	−1.32230
$953$	−47.3431	−1.53359	−0.766797	−	0.641889i	$-0.778151\pi$
$953$	−47.3431	−1.53359	−0.766797	+	0.641889i	$0.778151\pi$
$954$	36.2132	1.17245
$955$	0	0
$956$	90.2965	2.92040
$957$	3.11270	0.100619
$958$	34.4558	1.11322
$959$	6.82843	0.220501
$960$	0	0
$961$	−29.4558	−0.950189
$962$	−14.4853	−0.467024
$963$	80.7107	2.60087
$964$	19.0294	0.612897
$965$	0	0
$966$	52.2843	1.68222
$967$	43.7279	1.40620	0.703098	−	0.711093i	$-0.251800\pi$
$967$	43.7279	1.40620	0.703098	+	0.711093i	$0.251800\pi$
$968$	133.054	4.27651
$969$	−39.5980	−1.27207
$970$	0	0
$971$	36.8284	1.18188	0.590940	−	0.806715i	$-0.298757\pi$
$971$	36.8284	1.18188	0.590940	+	0.806715i	$0.298757\pi$
$972$	54.1421	1.73661
$973$	−44.6274	−1.43069
$974$	−101.912	−3.26546
$975$	0	0
$976$	3.00000	0.0960277
$977$	−8.48528	−0.271468	−0.135734	−	0.990745i	$-0.543339\pi$
$977$	−8.48528	−0.271468	−0.135734	+	0.990745i	$0.543339\pi$
$978$	−68.2843	−2.18349
$979$	−54.4264	−1.73948
$980$	0	0
$981$	−82.4264	−2.63167
$982$	26.9706	0.860665
$983$	26.6985	0.851549	0.425775	−	0.904829i	$-0.360002\pi$
$983$	26.6985	0.851549	0.425775	+	0.904829i	$0.360002\pi$
$984$	70.6274	2.25152
$985$	0	0
$986$	1.58579	0.0505017
$987$	−24.4853	−0.779375
$988$	−14.0000	−0.445399
$989$	−2.62742	−0.0835470
$990$	0	0
$991$	−13.5147	−0.429309	−0.214655	−	0.976690i	$-0.568863\pi$
$991$	−13.5147	−0.429309	−0.214655	+	0.976690i	$0.568863\pi$
$992$	−1.97056	−0.0625654
$993$	−78.2254	−2.48241
$994$	−54.2843	−1.72179
$995$	0	0
$996$	−74.7107	−2.36730
$997$	12.6569	0.400847	0.200423	−	0.979709i	$-0.435768\pi$
$997$	12.6569	0.400847	0.200423	+	0.979709i	$0.435768\pi$
$998$	6.17157	0.195358
$999$	33.9411	1.07385

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.2.a.h.1.2	yes	2
3.2	odd	2		2925.2.a.w.1.1		2
4.3	odd	2		5200.2.a.br.1.2		2
5.2	odd	4		325.2.b.d.274.4		4
5.3	odd	4		325.2.b.d.274.1		4
5.4	even	2		325.2.a.f.1.1	✓	2
13.12	even	2		4225.2.a.s.1.1		2
15.2	even	4		2925.2.c.q.2224.1		4
15.8	even	4		2925.2.c.q.2224.4		4
15.14	odd	2		2925.2.a.bd.1.2		2
20.19	odd	2		5200.2.a.bt.1.1		2
65.64	even	2		4225.2.a.z.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
325.2.a.f.1.1	✓	2	5.4	even	2
325.2.a.h.1.2	yes	2	1.1	even	1	trivial
325.2.b.d.274.1		4	5.3	odd	4
325.2.b.d.274.4		4	5.2	odd	4
2925.2.a.w.1.1		2	3.2	odd	2
2925.2.a.bd.1.2		2	15.14	odd	2
2925.2.c.q.2224.1		4	15.2	even	4
2925.2.c.q.2224.4		4	15.8	even	4
4225.2.a.s.1.1		2	13.12	even	2
4225.2.a.z.1.2		2	65.64	even	2
5200.2.a.br.1.2		2	4.3	odd	2
5200.2.a.bt.1.1		2	20.19	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:	\( N \)	\(=\)	\( 325 = 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	325.a (trivial)

\(f(q)\)	\(=\)	\(q+2.41421 q^{2} -2.82843 q^{3} +3.82843 q^{4} -6.82843 q^{6} +2.41421 q^{7} +4.41421 q^{8} +5.00000 q^{9} +6.41421 q^{11} -10.8284 q^{12} +1.00000 q^{13} +5.82843 q^{14} +3.00000 q^{16} -3.82843 q^{17} +12.0711 q^{18} -3.65685 q^{19} -6.82843 q^{21} +15.4853 q^{22} -3.17157 q^{23} -12.4853 q^{24} +2.41421 q^{26} -5.65685 q^{27} +9.24264 q^{28} -0.171573 q^{29} +1.24264 q^{31} -1.58579 q^{32} -18.1421 q^{33} -9.24264 q^{34} +19.1421 q^{36} -6.00000 q^{37} -8.82843 q^{38} -2.82843 q^{39} -5.65685 q^{41} -16.4853 q^{42} +0.828427 q^{43} +24.5563 q^{44} -7.65685 q^{46} +3.58579 q^{47} -8.48528 q^{48} -1.17157 q^{49} +10.8284 q^{51} +3.82843 q^{52} +3.00000 q^{53} -13.6569 q^{54} +10.6569 q^{56} +10.3431 q^{57} -0.414214 q^{58} +13.2426 q^{59} +1.00000 q^{61} +3.00000 q^{62} +12.0711 q^{63} -9.82843 q^{64} -43.7990 q^{66} -2.75736 q^{67} -14.6569 q^{68} +8.97056 q^{69} -9.31371 q^{71} +22.0711 q^{72} -6.00000 q^{73} -14.4853 q^{74} -14.0000 q^{76} +15.4853 q^{77} -6.82843 q^{78} -6.00000 q^{79} +1.00000 q^{81} -13.6569 q^{82} +6.89949 q^{83} -26.1421 q^{84} +2.00000 q^{86} +0.485281 q^{87} +28.3137 q^{88} -8.48528 q^{89} +2.41421 q^{91} -12.1421 q^{92} -3.51472 q^{93} +8.65685 q^{94} +4.48528 q^{96} -0.828427 q^{97} -2.82843 q^{98} +32.0711 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 8 q^{6} + 2 q^{7} + 6 q^{8} + 10 q^{9} + 10 q^{11} - 16 q^{12} + 2 q^{13} + 6 q^{14} + 6 q^{16} - 2 q^{17} + 10 q^{18} + 4 q^{19} - 8 q^{21} + 14 q^{22} - 12 q^{23} - 8 q^{24}+ \cdots + 50 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - 8 * q^6 + 2 * q^7 + 6 * q^8 + 10 * q^9 + 10 * q^11 - 16 * q^12 + 2 * q^13 + 6 * q^14 + 6 * q^16 - 2 * q^17 + 10 * q^18 + 4 * q^19 - 8 * q^21 + 14 * q^22 - 12 * q^23 - 8 * q^24 + 2 * q^26 + 10 * q^28 - 6 * q^29 - 6 * q^31 - 6 * q^32 - 8 * q^33 - 10 * q^34 + 10 * q^36 - 12 * q^37 - 12 * q^38 - 16 * q^42 - 4 * q^43 + 18 * q^44 - 4 * q^46 + 10 * q^47 - 8 * q^49 + 16 * q^51 + 2 * q^52 + 6 * q^53 - 16 * q^54 + 10 * q^56 + 32 * q^57 + 2 * q^58 + 18 * q^59 + 2 * q^61 + 6 * q^62 + 10 * q^63 - 14 * q^64 - 48 * q^66 - 14 * q^67 - 18 * q^68 - 16 * q^69 + 4 * q^71 + 30 * q^72 - 12 * q^73 - 12 * q^74 - 28 * q^76 + 14 * q^77 - 8 * q^78 - 12 * q^79 + 2 * q^81 - 16 * q^82 - 6 * q^83 - 24 * q^84 + 4 * q^86 - 16 * q^87 + 34 * q^88 + 2 * q^91 + 4 * q^92 - 24 * q^93 + 6 * q^94 - 8 * q^96 + 4 * q^97 + 50 * q^99

Introduction

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