LMFDB - Embedded newform 4225.2.a.bx.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4225,2,Mod(1,4225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4225, 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("4225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4225 = 5^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4225.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:	$33.7367948540$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 5 x^{11} - 5 x^{10} + 48 x^{9} + 2 x^{8} - 171 x^{7} + 6 x^{6} + 260 x^{5} + 27 x^{4} + \cdots + 1 $ x^12 - 5x^11 - 5x^10 + 48x^9 + 2x^8 - 171x^7 + 6x^6 + 260x^5 + 27x^4 - 138x^3 - 36x^2 + 3*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$0.162095$ of defining polynomial
Character	$\chi$	$=$	4225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.837905 q^{2} -0.531386 q^{3} -1.29791 q^{4} +0.445251 q^{6} -1.15977 q^{7} +2.76334 q^{8} -2.71763 q^{9} +O(q^{10})$	\(q-0.837905 q^{2} -0.531386 q^{3} -1.29791 q^{4} +0.445251 q^{6} -1.15977 q^{7} +2.76334 q^{8} -2.71763 q^{9} -0.824761 q^{11} +0.689694 q^{12} +0.971780 q^{14} +0.280412 q^{16} +3.49695 q^{17} +2.27712 q^{18} -2.73674 q^{19} +0.616287 q^{21} +0.691072 q^{22} -6.44528 q^{23} -1.46840 q^{24} +3.03827 q^{27} +1.50529 q^{28} +5.08958 q^{29} +9.28317 q^{31} -5.76164 q^{32} +0.438266 q^{33} -2.93011 q^{34} +3.52725 q^{36} +6.05550 q^{37} +2.29313 q^{38} +1.20843 q^{41} -0.516390 q^{42} +0.920038 q^{43} +1.07047 q^{44} +5.40054 q^{46} +2.97808 q^{47} -0.149007 q^{48} -5.65493 q^{49} -1.85823 q^{51} -4.76748 q^{53} -2.54578 q^{54} -3.20485 q^{56} +1.45427 q^{57} -4.26459 q^{58} -7.48896 q^{59} +10.9080 q^{61} -7.77841 q^{62} +3.15183 q^{63} +4.26688 q^{64} -0.367226 q^{66} -14.8138 q^{67} -4.53874 q^{68} +3.42493 q^{69} +0.539241 q^{71} -7.50973 q^{72} -0.0375099 q^{73} -5.07394 q^{74} +3.55206 q^{76} +0.956536 q^{77} +8.77305 q^{79} +6.53839 q^{81} -1.01255 q^{82} -9.33001 q^{83} -0.799888 q^{84} -0.770905 q^{86} -2.70453 q^{87} -2.27910 q^{88} +16.8351 q^{89} +8.36543 q^{92} -4.93294 q^{93} -2.49535 q^{94} +3.06165 q^{96} -16.3108 q^{97} +4.73829 q^{98} +2.24139 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 7 q^{2} + q^{3} + 13 q^{4} - 3 q^{6} - 12 q^{7} - 18 q^{8} + 11 q^{9}+O(q^{10}) $ 12 * q - 7 * q^2 + q^3 + 13 * q^4 - 3 * q^6 - 12 * q^7 - 18 * q^8 + 11 * q^9	$ 12 q - 7 q^{2} + q^{3} + 13 q^{4} - 3 q^{6} - 12 q^{7} - 18 q^{8} + 11 q^{9} + 3 q^{11} + 8 q^{12} + 2 q^{14} + 11 q^{16} - 2 q^{17} - 14 q^{18} - 5 q^{21} - 12 q^{22} + 3 q^{23} - 4 q^{24} - 5 q^{27} - 29 q^{28} + 9 q^{29} - 4 q^{31} - 48 q^{32} - 30 q^{33} - 3 q^{34} + 44 q^{36} - 17 q^{37} - 15 q^{38} + 9 q^{41} + 80 q^{42} - q^{43} + 10 q^{44} + 3 q^{46} - 61 q^{47} - 35 q^{48} + 8 q^{49} - 26 q^{51} - 23 q^{53} - 48 q^{54} + 51 q^{56} - 10 q^{57} + 17 q^{58} - q^{59} + 13 q^{61} - 29 q^{62} - 46 q^{63} + 50 q^{64} + 29 q^{66} - 43 q^{67} + 26 q^{68} - 31 q^{69} + 19 q^{71} - 48 q^{72} - 21 q^{73} + 7 q^{74} + 46 q^{76} + 42 q^{77} - 7 q^{79} - 16 q^{81} - 15 q^{82} - 75 q^{83} - 83 q^{84} + 17 q^{86} - 5 q^{87} - 38 q^{88} - 19 q^{89} + 35 q^{92} - 11 q^{93} + 26 q^{94} + 90 q^{96} - 17 q^{97} - 11 q^{98} - 6 q^{99}+O(q^{100}) $ 12 * q - 7 * q^2 + q^3 + 13 * q^4 - 3 * q^6 - 12 * q^7 - 18 * q^8 + 11 * q^9 + 3 * q^11 + 8 * q^12 + 2 * q^14 + 11 * q^16 - 2 * q^17 - 14 * q^18 - 5 * q^21 - 12 * q^22 + 3 * q^23 - 4 * q^24 - 5 * q^27 - 29 * q^28 + 9 * q^29 - 4 * q^31 - 48 * q^32 - 30 * q^33 - 3 * q^34 + 44 * q^36 - 17 * q^37 - 15 * q^38 + 9 * q^41 + 80 * q^42 - q^43 + 10 * q^44 + 3 * q^46 - 61 * q^47 - 35 * q^48 + 8 * q^49 - 26 * q^51 - 23 * q^53 - 48 * q^54 + 51 * q^56 - 10 * q^57 + 17 * q^58 - q^59 + 13 * q^61 - 29 * q^62 - 46 * q^63 + 50 * q^64 + 29 * q^66 - 43 * q^67 + 26 * q^68 - 31 * q^69 + 19 * q^71 - 48 * q^72 - 21 * q^73 + 7 * q^74 + 46 * q^76 + 42 * q^77 - 7 * q^79 - 16 * q^81 - 15 * q^82 - 75 * q^83 - 83 * q^84 + 17 * q^86 - 5 * q^87 - 38 * q^88 - 19 * q^89 + 35 * q^92 - 11 * q^93 + 26 * q^94 + 90 * q^96 - 17 * q^97 - 11 * q^98 - 6 * 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 $

$2$	−0.837905	−0.592489	−0.296244	−	0.955112i	$-0.595734\pi$
$2$	−0.837905	−0.592489	−0.296244	+	0.955112i	$0.595734\pi$
$3$	−0.531386	−0.306796	−0.153398	−	0.988165i	$-0.549022\pi$
$3$	−0.531386	−0.306796	−0.153398	+	0.988165i	$0.549022\pi$
$4$	−1.29791	−0.648957
$5$	0	0
$5$	0	0
$6$	0.445251	0.181773
$7$	−1.15977	−0.438353	−0.219177	−	0.975685i	$-0.570337\pi$
$7$	−1.15977	−0.438353	−0.219177	+	0.975685i	$0.570337\pi$
$8$	2.76334	0.976988
$9$	−2.71763	−0.905876
$10$	0	0
$11$	−0.824761	−0.248675	−0.124337	−	0.992240i	$-0.539681\pi$
$11$	−0.824761	−0.248675	−0.124337	+	0.992240i	$0.539681\pi$
$12$	0.689694	0.199097
$13$	0	0
$13$	0	0
$14$	0.971780	0.259719
$15$	0	0
$16$	0.280412	0.0701030
$17$	3.49695	0.848135	0.424067	−	0.905631i	$-0.360602\pi$
$17$	3.49695	0.848135	0.424067	+	0.905631i	$0.360602\pi$
$18$	2.27712	0.536721
$19$	−2.73674	−0.627852	−0.313926	−	0.949447i	$-0.601644\pi$
$19$	−2.73674	−0.627852	−0.313926	+	0.949447i	$0.601644\pi$
$20$	0	0
$21$	0.616287	0.134485
$22$	0.691072	0.147337
$23$	−6.44528	−1.34393	−0.671967	−	0.740581i	$-0.734550\pi$
$23$	−6.44528	−1.34393	−0.671967	+	0.740581i	$0.734550\pi$
$24$	−1.46840	−0.299736
$25$	0	0
$26$	0	0
$27$	3.03827	0.584715
$28$	1.50529	0.284473
$29$	5.08958	0.945111	0.472556	−	0.881301i	$-0.343332\pi$
$29$	5.08958	0.945111	0.472556	+	0.881301i	$0.343332\pi$
$30$	0	0
$31$	9.28317	1.66731	0.833653	−	0.552289i	$-0.186245\pi$
$31$	9.28317	1.66731	0.833653	+	0.552289i	$0.186245\pi$
$32$	−5.76164	−1.01852
$33$	0.438266	0.0762924
$34$	−2.93011	−0.502510
$35$	0	0
$36$	3.52725	0.587875
$37$	6.05550	0.995519	0.497759	−	0.867315i	$-0.334156\pi$
$37$	6.05550	0.995519	0.497759	+	0.867315i	$0.334156\pi$
$38$	2.29313	0.371995
$39$	0	0
$40$	0	0
$41$	1.20843	0.188725	0.0943626	−	0.995538i	$-0.469919\pi$
$41$	1.20843	0.188725	0.0943626	+	0.995538i	$0.469919\pi$
$42$	−0.516390	−0.0796808
$43$	0.920038	0.140304	0.0701522	−	0.997536i	$-0.477651\pi$
$43$	0.920038	0.140304	0.0701522	+	0.997536i	$0.477651\pi$
$44$	1.07047	0.161379
$45$	0	0
$46$	5.40054	0.796266
$47$	2.97808	0.434398	0.217199	−	0.976127i	$-0.430308\pi$
$47$	2.97808	0.434398	0.217199	+	0.976127i	$0.430308\pi$
$48$	−0.149007	−0.0215073
$49$	−5.65493	−0.807846
$50$	0	0
$51$	−1.85823	−0.260204
$52$	0	0
$53$	−4.76748	−0.654864	−0.327432	−	0.944875i	$-0.606183\pi$
$53$	−4.76748	−0.654864	−0.327432	+	0.944875i	$0.606183\pi$
$54$	−2.54578	−0.346437
$55$	0	0
$56$	−3.20485	−0.428266
$57$	1.45427	0.192622
$58$	−4.26459	−0.559968
$59$	−7.48896	−0.974980	−0.487490	−	0.873129i	$-0.662087\pi$
$59$	−7.48896	−0.974980	−0.487490	+	0.873129i	$0.662087\pi$
$60$	0	0
$61$	10.9080	1.39663	0.698314	−	0.715791i	$-0.253934\pi$
$61$	10.9080	1.39663	0.698314	+	0.715791i	$0.253934\pi$
$62$	−7.77841	−0.987859
$63$	3.15183	0.397094
$64$	4.26688	0.533360
$65$	0	0
$66$	−0.367226	−0.0452024
$67$	−14.8138	−1.80979	−0.904894	−	0.425637i	$-0.860050\pi$
$67$	−14.8138	−1.80979	−0.904894	+	0.425637i	$0.860050\pi$
$68$	−4.53874	−0.550403
$69$	3.42493	0.412313
$70$	0	0
$71$	0.539241	0.0639961	0.0319981	−	0.999488i	$-0.489813\pi$
$71$	0.539241	0.0639961	0.0319981	+	0.999488i	$0.489813\pi$
$72$	−7.50973	−0.885031
$73$	−0.0375099	−0.00439020	−0.00219510	−	0.999998i	$-0.500699\pi$
$73$	−0.0375099	−0.00439020	−0.00219510	+	0.999998i	$0.500699\pi$
$74$	−5.07394	−0.589833
$75$	0	0
$76$	3.55206	0.407449
$77$	0.956536	0.109007
$78$	0	0
$79$	8.77305	0.987045	0.493522	−	0.869733i	$-0.335709\pi$
$79$	8.77305	0.987045	0.493522	+	0.869733i	$0.335709\pi$
$80$	0	0
$81$	6.53839	0.726488
$82$	−1.01255	−0.111818
$83$	−9.33001	−1.02410	−0.512051	−	0.858955i	$-0.671114\pi$
$83$	−9.33001	−1.02410	−0.512051	+	0.858955i	$0.671114\pi$
$84$	−0.799888	−0.0872750
$85$	0	0
$86$	−0.770905	−0.0831288
$87$	−2.70453	−0.289956
$88$	−2.27910	−0.242952
$89$	16.8351	1.78451	0.892257	−	0.451528i	$-0.149121\pi$
$89$	16.8351	1.78451	0.892257	+	0.451528i	$0.149121\pi$
$90$	0	0
$91$	0	0
$92$	8.36543	0.872156
$93$	−4.93294	−0.511522
$94$	−2.49535	−0.257376
$95$	0	0
$96$	3.06165	0.312479
$97$	−16.3108	−1.65611	−0.828053	−	0.560649i	$-0.810552\pi$
$97$	−16.3108	−1.65611	−0.828053	+	0.560649i	$0.810552\pi$
$98$	4.73829	0.478640
$99$	2.24139	0.225269
$100$	0	0
$101$	19.1127	1.90178	0.950892	−	0.309522i	$-0.100169\pi$
$101$	19.1127	1.90178	0.950892	+	0.309522i	$0.100169\pi$
$102$	1.55702	0.154168
$103$	13.3539	1.31580	0.657899	−	0.753106i	$-0.271445\pi$
$103$	13.3539	1.31580	0.657899	+	0.753106i	$0.271445\pi$
$104$	0	0
$105$	0	0
$106$	3.99470	0.387999
$107$	14.3671	1.38892	0.694459	−	0.719533i	$-0.255644\pi$
$107$	14.3671	1.38892	0.694459	+	0.719533i	$0.255644\pi$
$108$	−3.94341	−0.379455
$109$	−4.13158	−0.395734	−0.197867	−	0.980229i	$-0.563401\pi$
$109$	−4.13158	−0.395734	−0.197867	+	0.980229i	$0.563401\pi$
$110$	0	0
$111$	−3.21781	−0.305421
$112$	−0.325214	−0.0307299
$113$	1.06728	0.100401	0.0502006	−	0.998739i	$-0.484014\pi$
$113$	1.06728	0.100401	0.0502006	+	0.998739i	$0.484014\pi$
$114$	−1.21854	−0.114127
$115$	0	0
$116$	−6.60584	−0.613337
$117$	0	0
$118$	6.27504	0.577664
$119$	−4.05567	−0.371783
$120$	0	0
$121$	−10.3198	−0.938161
$122$	−9.13989	−0.827486
$123$	−0.642143	−0.0579001
$124$	−12.0488	−1.08201
$125$	0	0
$126$	−2.64094	−0.235274
$127$	−19.7907	−1.75614	−0.878068	−	0.478535i	$-0.841168\pi$
$127$	−19.7907	−1.75614	−0.878068	+	0.478535i	$0.841168\pi$
$128$	7.94803	0.702514
$129$	−0.488895	−0.0430448
$130$	0	0
$131$	−7.18162	−0.627461	−0.313731	−	0.949512i	$-0.601579\pi$
$131$	−7.18162	−0.627461	−0.313731	+	0.949512i	$0.601579\pi$
$132$	−0.568832	−0.0495105
$133$	3.17400	0.275221
$134$	12.4125	1.07228
$135$	0	0
$136$	9.66326	0.828618
$137$	−13.4595	−1.14992	−0.574962	−	0.818180i	$-0.694983\pi$
$137$	−13.4595	−1.14992	−0.574962	+	0.818180i	$0.694983\pi$
$138$	−2.86977	−0.244291
$139$	5.58027	0.473312	0.236656	−	0.971594i	$-0.423949\pi$
$139$	5.58027	0.473312	0.236656	+	0.971594i	$0.423949\pi$
$140$	0	0
$141$	−1.58251	−0.133271
$142$	−0.451833	−0.0379170
$143$	0	0
$144$	−0.762056	−0.0635047
$145$	0	0
$146$	0.0314298	0.00260115
$147$	3.00495	0.247844
$148$	−7.85953	−0.646049
$149$	−7.62151	−0.624379	−0.312189	−	0.950020i	$-0.601062\pi$
$149$	−7.62151	−0.624379	−0.312189	+	0.950020i	$0.601062\pi$
$150$	0	0
$151$	−13.5402	−1.10189	−0.550943	−	0.834543i	$-0.685732\pi$
$151$	−13.5402	−1.10189	−0.550943	+	0.834543i	$0.685732\pi$
$152$	−7.56255	−0.613404
$153$	−9.50341	−0.768305
$154$	−0.801487	−0.0645856
$155$	0	0
$156$	0	0
$157$	13.1574	1.05007	0.525036	−	0.851080i	$-0.324052\pi$
$157$	13.1574	1.05007	0.525036	+	0.851080i	$0.324052\pi$
$158$	−7.35098	−0.584813
$159$	2.53337	0.200909
$160$	0	0
$161$	7.47507	0.589118
$162$	−5.47856	−0.430436
$163$	−14.7371	−1.15430	−0.577151	−	0.816637i	$-0.695836\pi$
$163$	−14.7371	−1.15430	−0.577151	+	0.816637i	$0.695836\pi$
$164$	−1.56844	−0.122475
$165$	0	0
$166$	7.81766	0.606768
$167$	8.72394	0.675079	0.337539	−	0.941311i	$-0.390405\pi$
$167$	8.72394	0.675079	0.337539	+	0.941311i	$0.390405\pi$
$168$	1.70301	0.131390
$169$	0	0
$170$	0	0
$171$	7.43745	0.568756
$172$	−1.19413	−0.0910516
$173$	16.5655	1.25945	0.629726	−	0.776817i	$-0.283167\pi$
$173$	16.5655	1.25945	0.629726	+	0.776817i	$0.283167\pi$
$174$	2.26614	0.171796
$175$	0	0
$176$	−0.231273	−0.0174329
$177$	3.97953	0.299120
$178$	−14.1062	−1.05730
$179$	−13.7396	−1.02694	−0.513472	−	0.858106i	$-0.671641\pi$
$179$	−13.7396	−1.02694	−0.513472	+	0.858106i	$0.671641\pi$
$180$	0	0
$181$	−12.7160	−0.945169	−0.472585	−	0.881285i	$-0.656679\pi$
$181$	−12.7160	−0.945169	−0.472585	+	0.881285i	$0.656679\pi$
$182$	0	0
$183$	−5.79637	−0.428480
$184$	−17.8105	−1.31301
$185$	0	0
$186$	4.13334	0.303071
$187$	−2.88415	−0.210910
$188$	−3.86529	−0.281906
$189$	−3.52370	−0.256312
$190$	0	0
$191$	12.7997	0.926151	0.463076	−	0.886319i	$-0.346746\pi$
$191$	12.7997	0.926151	0.463076	+	0.886319i	$0.346746\pi$
$192$	−2.26736	−0.163633
$193$	−11.8924	−0.856034	−0.428017	−	0.903771i	$-0.640788\pi$
$193$	−11.8924	−0.856034	−0.428017	+	0.903771i	$0.640788\pi$
$194$	13.6669	0.981224
$195$	0	0
$196$	7.33961	0.524258
$197$	−22.7415	−1.62027	−0.810134	−	0.586245i	$-0.800605\pi$
$197$	−22.7415	−1.62027	−0.810134	+	0.586245i	$0.800605\pi$
$198$	−1.87808	−0.133469
$199$	−11.6794	−0.827934	−0.413967	−	0.910292i	$-0.635857\pi$
$199$	−11.6794	−0.827934	−0.413967	+	0.910292i	$0.635857\pi$
$200$	0	0
$201$	7.87182	0.555235
$202$	−16.0146	−1.12679
$203$	−5.90276	−0.414293
$204$	2.41182	0.168861
$205$	0	0
$206$	−11.1893	−0.779595
$207$	17.5159	1.21744
$208$	0	0
$209$	2.25716	0.156131
$210$	0	0
$211$	−3.68652	−0.253790	−0.126895	−	0.991916i	$-0.540501\pi$
$211$	−3.68652	−0.253790	−0.126895	+	0.991916i	$0.540501\pi$
$212$	6.18778	0.424979
$213$	−0.286545	−0.0196337
$214$	−12.0382	−0.822918
$215$	0	0
$216$	8.39577	0.571260
$217$	−10.7664	−0.730869
$218$	3.46187	0.234468
$219$	0.0199322	0.00134690
$220$	0	0
$221$	0	0
$222$	2.69622	0.180958
$223$	1.62742	0.108980	0.0544899	−	0.998514i	$-0.482647\pi$
$223$	1.62742	0.108980	0.0544899	+	0.998514i	$0.482647\pi$
$224$	6.68220	0.446473
$225$	0	0
$226$	−0.894279	−0.0594865
$227$	−10.9557	−0.727156	−0.363578	−	0.931564i	$-0.618445\pi$
$227$	−10.9557	−0.727156	−0.363578	+	0.931564i	$0.618445\pi$
$228$	−1.88751	−0.125004
$229$	17.3128	1.14407	0.572033	−	0.820231i	$-0.306155\pi$
$229$	17.3128	1.14407	0.572033	+	0.820231i	$0.306155\pi$
$230$	0	0
$231$	−0.508290	−0.0334430
$232$	14.0642	0.923363
$233$	−25.7739	−1.68851	−0.844253	−	0.535944i	$-0.819956\pi$
$233$	−25.7739	−1.68851	−0.844253	+	0.535944i	$0.819956\pi$
$234$	0	0
$235$	0	0
$236$	9.72004	0.632720
$237$	−4.66187	−0.302821
$238$	3.39827	0.220277
$239$	−10.7002	−0.692138	−0.346069	−	0.938209i	$-0.612484\pi$
$239$	−10.7002	−0.692138	−0.346069	+	0.938209i	$0.612484\pi$
$240$	0	0
$241$	13.8833	0.894299	0.447149	−	0.894459i	$-0.352439\pi$
$241$	13.8833	0.894299	0.447149	+	0.894459i	$0.352439\pi$
$242$	8.64699	0.555850
$243$	−12.5892	−0.807598
$244$	−14.1577	−0.906352
$245$	0	0
$246$	0.538055	0.0343051
$247$	0	0
$248$	25.6525	1.62894
$249$	4.95783	0.314190
$250$	0	0
$251$	4.84837	0.306026	0.153013	−	0.988224i	$-0.451102\pi$
$251$	4.84837	0.306026	0.153013	+	0.988224i	$0.451102\pi$
$252$	−4.09081	−0.257697
$253$	5.31582	0.334203
$254$	16.5827	1.04049
$255$	0	0
$256$	−15.1935	−0.949592
$257$	−19.3126	−1.20469	−0.602343	−	0.798237i	$-0.705766\pi$
$257$	−19.3126	−1.20469	−0.602343	+	0.798237i	$0.705766\pi$
$258$	0.409648	0.0255036
$259$	−7.02301	−0.436389
$260$	0	0
$261$	−13.8316	−0.856154
$262$	6.01752	0.371764
$263$	8.55625	0.527601	0.263800	−	0.964577i	$-0.415024\pi$
$263$	8.55625	0.527601	0.263800	+	0.964577i	$0.415024\pi$
$264$	1.21108	0.0745368
$265$	0	0
$266$	−2.65951	−0.163065
$267$	−8.94592	−0.547481
$268$	19.2270	1.17448
$269$	−8.76168	−0.534209	−0.267105	−	0.963668i	$-0.586067\pi$
$269$	−8.76168	−0.534209	−0.267105	+	0.963668i	$0.586067\pi$
$270$	0	0
$271$	−12.5573	−0.762805	−0.381402	−	0.924409i	$-0.624559\pi$
$271$	−12.5573	−0.762805	−0.381402	+	0.924409i	$0.624559\pi$
$272$	0.980587	0.0594568
$273$	0	0
$274$	11.2778	0.681317
$275$	0	0
$276$	−4.44527	−0.267574
$277$	23.2314	1.39584	0.697919	−	0.716177i	$-0.254110\pi$
$277$	23.2314	1.39584	0.697919	+	0.716177i	$0.254110\pi$
$278$	−4.67573	−0.280432
$279$	−25.2282	−1.51037
$280$	0	0
$281$	4.36929	0.260650	0.130325	−	0.991471i	$-0.458398\pi$
$281$	4.36929	0.260650	0.130325	+	0.991471i	$0.458398\pi$
$282$	1.32599	0.0789618
$283$	−10.7175	−0.637087	−0.318543	−	0.947908i	$-0.603194\pi$
$283$	−10.7175	−0.637087	−0.318543	+	0.947908i	$0.603194\pi$
$284$	−0.699888	−0.0415307
$285$	0	0
$286$	0	0
$287$	−1.40151	−0.0827283
$288$	15.6580	0.922656
$289$	−4.77134	−0.280667
$290$	0	0
$291$	8.66731	0.508087
$292$	0.0486847	0.00284906
$293$	−19.8711	−1.16088	−0.580442	−	0.814302i	$-0.697120\pi$
$293$	−19.8711	−1.16088	−0.580442	+	0.814302i	$0.697120\pi$
$294$	−2.51786	−0.146845
$295$	0	0
$296$	16.7334	0.972610
$297$	−2.50584	−0.145404
$298$	6.38611	0.369937
$299$	0	0
$300$	0	0
$301$	−1.06704	−0.0615029
$302$	11.3454	0.652854
$303$	−10.1562	−0.583460
$304$	−0.767416	−0.0440143
$305$	0	0
$306$	7.96296	0.455212
$307$	−25.3032	−1.44413	−0.722064	−	0.691826i	$-0.756806\pi$
$307$	−25.3032	−1.44413	−0.722064	+	0.691826i	$0.756806\pi$
$308$	−1.24150	−0.0707412
$309$	−7.09607	−0.403681
$310$	0	0
$311$	−0.0484816	−0.00274914	−0.00137457	−	0.999999i	$-0.500438\pi$
$311$	−0.0484816	−0.00274914	−0.00137457	+	0.999999i	$0.500438\pi$
$312$	0	0
$313$	−5.01890	−0.283685	−0.141843	−	0.989889i	$-0.545303\pi$
$313$	−5.01890	−0.283685	−0.141843	+	0.989889i	$0.545303\pi$
$314$	−11.0246	−0.622156
$315$	0	0
$316$	−11.3867	−0.640550
$317$	−8.54572	−0.479975	−0.239988	−	0.970776i	$-0.577143\pi$
$317$	−8.54572	−0.479975	−0.239988	+	0.970776i	$0.577143\pi$
$318$	−2.12273	−0.119037
$319$	−4.19769	−0.235025
$320$	0	0
$321$	−7.63446	−0.426114
$322$	−6.26340	−0.349046
$323$	−9.57026	−0.532503
$324$	−8.48628	−0.471460
$325$	0	0
$326$	12.3483	0.683911
$327$	2.19546	0.121409
$328$	3.33931	0.184382
$329$	−3.45390	−0.190420
$330$	0	0
$331$	29.3486	1.61314	0.806571	−	0.591137i	$-0.201321\pi$
$331$	29.3486	1.61314	0.806571	+	0.591137i	$0.201321\pi$
$332$	12.1096	0.664598
$333$	−16.4566	−0.901817
$334$	−7.30984	−0.399976
$335$	0	0
$336$	0.172814	0.00942780
$337$	14.0811	0.767046	0.383523	−	0.923531i	$-0.374711\pi$
$337$	14.0811	0.767046	0.383523	+	0.923531i	$0.374711\pi$
$338$	0	0
$339$	−0.567137	−0.0308026
$340$	0	0
$341$	−7.65639	−0.414617
$342$	−6.23188	−0.336982
$343$	14.6768	0.792475
$344$	2.54238	0.137076
$345$	0	0
$346$	−13.8803	−0.746211
$347$	−8.31065	−0.446139	−0.223070	−	0.974803i	$-0.571608\pi$
$347$	−8.31065	−0.446139	−0.223070	+	0.974803i	$0.571608\pi$
$348$	3.51025	0.188169
$349$	−7.17924	−0.384296	−0.192148	−	0.981366i	$-0.561545\pi$
$349$	−7.17924	−0.384296	−0.192148	+	0.981366i	$0.561545\pi$
$350$	0	0
$351$	0	0
$352$	4.75198	0.253281
$353$	−33.1129	−1.76242	−0.881211	−	0.472723i	$-0.843271\pi$
$353$	−33.1129	−1.76242	−0.881211	+	0.472723i	$0.843271\pi$
$354$	−3.33447	−0.177225
$355$	0	0
$356$	−21.8505	−1.15807
$357$	2.15513	0.114061
$358$	11.5125	0.608453
$359$	5.26114	0.277673	0.138836	−	0.990315i	$-0.455664\pi$
$359$	5.26114	0.277673	0.138836	+	0.990315i	$0.455664\pi$
$360$	0	0
$361$	−11.5102	−0.605802
$362$	10.6548	0.560002
$363$	5.48378	0.287824
$364$	0	0
$365$	0	0
$366$	4.85681	0.253869
$367$	−18.6625	−0.974175	−0.487087	−	0.873353i	$-0.661941\pi$
$367$	−18.6625	−0.974175	−0.487087	+	0.873353i	$0.661941\pi$
$368$	−1.80733	−0.0942138
$369$	−3.28407	−0.170962
$370$	0	0
$371$	5.52920	0.287062
$372$	6.40254	0.331956
$373$	23.3956	1.21138	0.605690	−	0.795701i	$-0.292897\pi$
$373$	23.3956	1.21138	0.605690	+	0.795701i	$0.292897\pi$
$374$	2.41664	0.124962
$375$	0	0
$376$	8.22945	0.424401
$377$	0	0
$378$	2.95253	0.151862
$379$	−9.81011	−0.503911	−0.251956	−	0.967739i	$-0.581074\pi$
$379$	−9.81011	−0.503911	−0.251956	+	0.967739i	$0.581074\pi$
$380$	0	0
$381$	10.5165	0.538775
$382$	−10.7249	−0.548734
$383$	−19.4317	−0.992912	−0.496456	−	0.868062i	$-0.665366\pi$
$383$	−19.4317	−0.992912	−0.496456	+	0.868062i	$0.665366\pi$
$384$	−4.22347	−0.215528
$385$	0	0
$386$	9.96471	0.507191
$387$	−2.50032	−0.127099
$388$	21.1700	1.07474
$389$	3.77736	0.191520	0.0957598	−	0.995404i	$-0.469472\pi$
$389$	3.77736	0.191520	0.0957598	+	0.995404i	$0.469472\pi$
$390$	0	0
$391$	−22.5388	−1.13984
$392$	−15.6265	−0.789257
$393$	3.81621	0.192503
$394$	19.0553	0.959990
$395$	0	0
$396$	−2.90914	−0.146190
$397$	−0.952863	−0.0478228	−0.0239114	−	0.999714i	$-0.507612\pi$
$397$	−0.952863	−0.0478228	−0.0239114	+	0.999714i	$0.507612\pi$
$398$	9.78627	0.490541
$399$	−1.68662	−0.0844366
$400$	0	0
$401$	−22.6005	−1.12862	−0.564309	−	0.825564i	$-0.690857\pi$
$401$	−22.6005	−1.12862	−0.564309	+	0.825564i	$0.690857\pi$
$402$	−6.59584	−0.328971
$403$	0	0
$404$	−24.8067	−1.23418
$405$	0	0
$406$	4.94595	0.245464
$407$	−4.99434	−0.247560
$408$	−5.13492	−0.254217
$409$	27.6223	1.36583	0.682916	−	0.730496i	$-0.260711\pi$
$409$	27.6223	1.36583	0.682916	+	0.730496i	$0.260711\pi$
$410$	0	0
$411$	7.15220	0.352792
$412$	−17.3322	−0.853897
$413$	8.68550	0.427386
$414$	−14.6767	−0.721318
$415$	0	0
$416$	0	0
$417$	−2.96527	−0.145210
$418$	−1.89129	−0.0925058
$419$	−15.5881	−0.761527	−0.380763	−	0.924673i	$-0.624339\pi$
$419$	−15.5881	−0.761527	−0.380763	+	0.924673i	$0.624339\pi$
$420$	0	0
$421$	−24.8347	−1.21037	−0.605185	−	0.796085i	$-0.706901\pi$
$421$	−24.8347	−1.21037	−0.605185	+	0.796085i	$0.706901\pi$
$422$	3.08895	0.150368
$423$	−8.09332	−0.393511
$424$	−13.1742	−0.639794
$425$	0	0
$426$	0.240098	0.0116328
$427$	−12.6508	−0.612217
$428$	−18.6472	−0.901348
$429$	0	0
$430$	0	0
$431$	−7.10019	−0.342004	−0.171002	−	0.985271i	$-0.554700\pi$
$431$	−7.10019	−0.342004	−0.171002	+	0.985271i	$0.554700\pi$
$432$	0.851967	0.0409903
$433$	−4.90507	−0.235722	−0.117861	−	0.993030i	$-0.537604\pi$
$433$	−4.90507	−0.235722	−0.117861	+	0.993030i	$0.537604\pi$
$434$	9.02120	0.433031
$435$	0	0
$436$	5.36244	0.256814
$437$	17.6391	0.843792
$438$	−0.0167013	−0.000798021 0
$439$	3.21827	0.153600	0.0767998	−	0.997047i	$-0.475530\pi$
$439$	3.21827	0.153600	0.0767998	+	0.997047i	$0.475530\pi$
$440$	0	0
$441$	15.3680	0.731809
$442$	0	0
$443$	−24.2947	−1.15428	−0.577139	−	0.816646i	$-0.695831\pi$
$443$	−24.2947	−1.15428	−0.577139	+	0.816646i	$0.695831\pi$
$444$	4.17644	0.198205
$445$	0	0
$446$	−1.36362	−0.0645693
$447$	4.04996	0.191557
$448$	−4.94862	−0.233800
$449$	−18.8973	−0.891818	−0.445909	−	0.895078i	$-0.647119\pi$
$449$	−18.8973	−0.891818	−0.445909	+	0.895078i	$0.647119\pi$
$450$	0	0
$451$	−0.996667	−0.0469312
$452$	−1.38524	−0.0651561
$453$	7.19507	0.338054
$454$	9.17984	0.430831
$455$	0	0
$456$	4.01863	0.188190
$457$	19.4172	0.908299	0.454150	−	0.890925i	$-0.349943\pi$
$457$	19.4172	0.908299	0.454150	+	0.890925i	$0.349943\pi$
$458$	−14.5065	−0.677846
$459$	10.6247	0.495917
$460$	0	0
$461$	−24.8591	−1.15780	−0.578901	−	0.815398i	$-0.696518\pi$
$461$	−24.8591	−1.15780	−0.578901	+	0.815398i	$0.696518\pi$
$462$	0.425899	0.0198146
$463$	−14.9891	−0.696601	−0.348301	−	0.937383i	$-0.613241\pi$
$463$	−14.9891	−0.696601	−0.348301	+	0.937383i	$0.613241\pi$
$464$	1.42718	0.0662552
$465$	0	0
$466$	21.5961	1.00042
$467$	0.412333	0.0190805	0.00954025	−	0.999954i	$-0.496963\pi$
$467$	0.412333	0.0190805	0.00954025	+	0.999954i	$0.496963\pi$
$468$	0	0
$469$	17.1806	0.793326
$470$	0	0
$471$	−6.99164	−0.322158
$472$	−20.6946	−0.952544
$473$	−0.758811	−0.0348902
$474$	3.90621	0.179418
$475$	0	0
$476$	5.26391	0.241271
$477$	12.9562	0.593225
$478$	8.96574	0.410084
$479$	38.9699	1.78058	0.890291	−	0.455393i	$-0.150501\pi$
$479$	38.9699	1.78058	0.890291	+	0.455393i	$0.150501\pi$
$480$	0	0
$481$	0	0
$482$	−11.6329	−0.529862
$483$	−3.97215	−0.180739
$484$	13.3942	0.608826
$485$	0	0
$486$	10.5486	0.478493
$487$	−20.1399	−0.912627	−0.456314	−	0.889819i	$-0.650830\pi$
$487$	−20.1399	−0.912627	−0.456314	+	0.889819i	$0.650830\pi$
$488$	30.1426	1.36449
$489$	7.83111	0.354135
$490$	0	0
$491$	−26.1154	−1.17857	−0.589285	−	0.807925i	$-0.700590\pi$
$491$	−26.1154	−1.17857	−0.589285	+	0.807925i	$0.700590\pi$
$492$	0.833447	0.0375747
$493$	17.7980	0.801582
$494$	0	0
$495$	0	0
$496$	2.60311	0.116883
$497$	−0.625397	−0.0280529
$498$	−4.15420	−0.186154
$499$	−36.9100	−1.65232	−0.826160	−	0.563435i	$-0.809479\pi$
$499$	−36.9100	−1.65232	−0.826160	+	0.563435i	$0.809479\pi$
$500$	0	0
$501$	−4.63578	−0.207111
$502$	−4.06247	−0.181317
$503$	21.4909	0.958233	0.479117	−	0.877751i	$-0.340957\pi$
$503$	21.4909	0.958233	0.479117	+	0.877751i	$0.340957\pi$
$504$	8.70959	0.387956
$505$	0	0
$506$	−4.45415	−0.198011
$507$	0	0
$508$	25.6866	1.13966
$509$	36.7639	1.62953	0.814766	−	0.579790i	$-0.196865\pi$
$509$	36.7639	1.62953	0.814766	+	0.579790i	$0.196865\pi$
$510$	0	0
$511$	0.0435030	0.00192446
$512$	−3.16538	−0.139891
$513$	−8.31496	−0.367114
$514$	16.1821	0.713763
$515$	0	0
$516$	0.634544	0.0279343
$517$	−2.45620	−0.108024
$518$	5.88462	0.258555
$519$	−8.80268	−0.386395
$520$	0	0
$521$	−34.0226	−1.49056	−0.745279	−	0.666753i	$-0.767684\pi$
$521$	−34.0226	−1.49056	−0.745279	+	0.666753i	$0.767684\pi$
$522$	11.5896	0.507261
$523$	27.1269	1.18617	0.593087	−	0.805138i	$-0.297909\pi$
$523$	27.1269	1.18617	0.593087	+	0.805138i	$0.297909\pi$
$524$	9.32114	0.407196
$525$	0	0
$526$	−7.16932	−0.312597
$527$	32.4628	1.41410
$528$	0.122895	0.00534833
$529$	18.5417	0.806159
$530$	0	0
$531$	20.3522	0.883211
$532$	−4.11958	−0.178607
$533$	0	0
$534$	7.49583	0.324376
$535$	0	0
$536$	−40.9354	−1.76814
$537$	7.30102	0.315062
$538$	7.34146	0.316513
$539$	4.66396	0.200891
$540$	0	0
$541$	8.63199	0.371118	0.185559	−	0.982633i	$-0.440590\pi$
$541$	8.63199	0.371118	0.185559	+	0.982633i	$0.440590\pi$
$542$	10.5219	0.451953
$543$	6.75708	0.289974
$544$	−20.1482	−0.863845
$545$	0	0
$546$	0	0
$547$	−37.3901	−1.59868	−0.799342	−	0.600876i	$-0.794819\pi$
$547$	−37.3901	−1.59868	−0.799342	+	0.600876i	$0.794819\pi$
$548$	17.4693	0.746252
$549$	−29.6439	−1.26517
$550$	0	0
$551$	−13.9289	−0.593390
$552$	9.46425	0.402825
$553$	−10.1747	−0.432674
$554$	−19.4657	−0.827017
$555$	0	0
$556$	−7.24271	−0.307159
$557$	−32.5930	−1.38101	−0.690504	−	0.723328i	$-0.742611\pi$
$557$	−32.5930	−1.38101	−0.690504	+	0.723328i	$0.742611\pi$
$558$	21.1388	0.894879
$559$	0	0
$560$	0	0
$561$	1.53260	0.0647062
$562$	−3.66105	−0.154432
$563$	20.9787	0.884146	0.442073	−	0.896979i	$-0.354243\pi$
$563$	20.9787	0.884146	0.442073	+	0.896979i	$0.354243\pi$
$564$	2.05396	0.0864874
$565$	0	0
$566$	8.98021	0.377466
$567$	−7.58306	−0.318458
$568$	1.49011	0.0625234
$569$	17.9874	0.754069	0.377035	−	0.926199i	$-0.376944\pi$
$569$	17.9874	0.754069	0.377035	+	0.926199i	$0.376944\pi$
$570$	0	0
$571$	18.7821	0.786007	0.393003	−	0.919537i	$-0.371436\pi$
$571$	18.7821	0.786007	0.393003	+	0.919537i	$0.371436\pi$
$572$	0	0
$573$	−6.80156	−0.284139
$574$	1.17433	0.0490156
$575$	0	0
$576$	−11.5958	−0.483159
$577$	20.0135	0.833174	0.416587	−	0.909096i	$-0.363226\pi$
$577$	20.0135	0.833174	0.416587	+	0.909096i	$0.363226\pi$
$578$	3.99793	0.166292
$579$	6.31946	0.262628
$580$	0	0
$581$	10.8207	0.448918
$582$	−7.26238	−0.301036
$583$	3.93203	0.162848
$584$	−0.103653	−0.00428918
$585$	0	0
$586$	16.6501	0.687810
$587$	1.52744	0.0630443	0.0315222	−	0.999503i	$-0.489965\pi$
$587$	1.52744	0.0630443	0.0315222	+	0.999503i	$0.489965\pi$
$588$	−3.90017	−0.160840
$589$	−25.4056	−1.04682
$590$	0	0
$591$	12.0845	0.497091
$592$	1.69804	0.0697888
$593$	20.7422	0.851781	0.425891	−	0.904775i	$-0.359961\pi$
$593$	20.7422	0.851781	0.425891	+	0.904775i	$0.359961\pi$
$594$	2.09966	0.0861501
$595$	0	0
$596$	9.89207	0.405195
$597$	6.20629	0.254007
$598$	0	0
$599$	−13.0318	−0.532464	−0.266232	−	0.963909i	$-0.585779\pi$
$599$	−13.0318	−0.532464	−0.266232	+	0.963909i	$0.585779\pi$
$600$	0	0
$601$	14.4204	0.588220	0.294110	−	0.955772i	$-0.404977\pi$
$601$	14.4204	0.588220	0.294110	+	0.955772i	$0.404977\pi$
$602$	0.894075	0.0364398
$603$	40.2583	1.63944
$604$	17.5740	0.715077
$605$	0	0
$606$	8.50995	0.345693
$607$	23.0260	0.934595	0.467298	−	0.884100i	$-0.345228\pi$
$607$	23.0260	0.934595	0.467298	+	0.884100i	$0.345228\pi$
$608$	15.7681	0.639482
$609$	3.13664	0.127103
$610$	0	0
$611$	0	0
$612$	12.3346	0.498597
$613$	7.61356	0.307509	0.153754	−	0.988109i	$-0.450864\pi$
$613$	7.61356	0.307509	0.153754	+	0.988109i	$0.450864\pi$
$614$	21.2017	0.855629
$615$	0	0
$616$	2.64323	0.106499
$617$	48.0213	1.93326	0.966632	−	0.256171i	$-0.0824610\pi$
$617$	48.0213	1.93326	0.966632	+	0.256171i	$0.0824610\pi$
$618$	5.94583	0.239176
$619$	7.57517	0.304472	0.152236	−	0.988344i	$-0.451353\pi$
$619$	7.57517	0.304472	0.152236	+	0.988344i	$0.451353\pi$
$620$	0	0
$621$	−19.5825	−0.785818
$622$	0.0406230	0.00162883
$623$	−19.5249	−0.782248
$624$	0	0
$625$	0	0
$626$	4.20537	0.168080
$627$	−1.19942	−0.0479003
$628$	−17.0771	−0.681452
$629$	21.1758	0.844334
$630$	0	0
$631$	−18.8372	−0.749898	−0.374949	−	0.927045i	$-0.622340\pi$
$631$	−18.8372	−0.749898	−0.374949	+	0.927045i	$0.622340\pi$
$632$	24.2429	0.964331
$633$	1.95896	0.0778618
$634$	7.16050	0.284380
$635$	0	0
$636$	−3.28810	−0.130382
$637$	0	0
$638$	3.51726	0.139250
$639$	−1.46546	−0.0579726
$640$	0	0
$641$	6.21575	0.245508	0.122754	−	0.992437i	$-0.460827\pi$
$641$	6.21575	0.245508	0.122754	+	0.992437i	$0.460827\pi$
$642$	6.39695	0.252468
$643$	38.9992	1.53798	0.768989	−	0.639262i	$-0.220760\pi$
$643$	38.9992	1.53798	0.768989	+	0.639262i	$0.220760\pi$
$644$	−9.70200	−0.382312
$645$	0	0
$646$	8.01897	0.315502
$647$	43.2740	1.70127	0.850637	−	0.525753i	$-0.176216\pi$
$647$	43.2740	1.70127	0.850637	+	0.525753i	$0.176216\pi$
$648$	18.0678	0.709771
$649$	6.17661	0.242453
$650$	0	0
$651$	5.72110	0.224227
$652$	19.1276	0.749093
$653$	−26.8601	−1.05112	−0.525559	−	0.850757i	$-0.676144\pi$
$653$	−26.8601	−1.05112	−0.525559	+	0.850757i	$0.676144\pi$
$654$	−1.83959	−0.0719337
$655$	0	0
$656$	0.338859	0.0132302
$657$	0.101938	0.00397698
$658$	2.89404	0.112821
$659$	−38.4595	−1.49817	−0.749084	−	0.662475i	$-0.769506\pi$
$659$	−38.4595	−1.49817	−0.749084	+	0.662475i	$0.769506\pi$
$660$	0	0
$661$	−13.4978	−0.525004	−0.262502	−	0.964931i	$-0.584548\pi$
$661$	−13.4978	−0.525004	−0.262502	+	0.964931i	$0.584548\pi$
$662$	−24.5913	−0.955768
$663$	0	0
$664$	−25.7820	−1.00054
$665$	0	0
$666$	13.7891	0.534316
$667$	−32.8038	−1.27017
$668$	−11.3229	−0.438097
$669$	−0.864786	−0.0334346
$670$	0	0
$671$	−8.99651	−0.347306
$672$	−3.55083	−0.136976
$673$	−21.3368	−0.822473	−0.411236	−	0.911529i	$-0.634903\pi$
$673$	−21.3368	−0.822473	−0.411236	+	0.911529i	$0.634903\pi$
$674$	−11.7986	−0.454466
$675$	0	0
$676$	0	0
$677$	−2.03363	−0.0781587	−0.0390794	−	0.999236i	$-0.512443\pi$
$677$	−2.03363	−0.0781587	−0.0390794	+	0.999236i	$0.512443\pi$
$678$	0.475207	0.0182502
$679$	18.9168	0.725960
$680$	0	0
$681$	5.82171	0.223088
$682$	6.41533	0.245656
$683$	7.26359	0.277934	0.138967	−	0.990297i	$-0.455622\pi$
$683$	7.26359	0.277934	0.138967	+	0.990297i	$0.455622\pi$
$684$	−9.65318	−0.369099
$685$	0	0
$686$	−12.2978	−0.469532
$687$	−9.19980	−0.350994
$688$	0.257990	0.00983577
$689$	0	0
$690$	0	0
$691$	−16.5837	−0.630874	−0.315437	−	0.948946i	$-0.602151\pi$
$691$	−16.5837	−0.630874	−0.315437	+	0.948946i	$0.602151\pi$
$692$	−21.5006	−0.817331
$693$	−2.59951	−0.0987472
$694$	6.96354	0.264332
$695$	0	0
$696$	−7.47354	−0.283284
$697$	4.22582	0.160064
$698$	6.01552	0.227691
$699$	13.6959	0.518027
$700$	0	0
$701$	−7.00926	−0.264736	−0.132368	−	0.991201i	$-0.542258\pi$
$701$	−7.00926	−0.264736	−0.132368	+	0.991201i	$0.542258\pi$
$702$	0	0
$703$	−16.5724	−0.625038
$704$	−3.51916	−0.132633
$705$	0	0
$706$	27.7455	1.04421
$707$	−22.1664	−0.833653
$708$	−5.16509	−0.194116
$709$	15.7366	0.591001	0.295501	−	0.955343i	$-0.404514\pi$
$709$	15.7366	0.591001	0.295501	+	0.955343i	$0.404514\pi$
$710$	0	0
$711$	−23.8419	−0.894140
$712$	46.5210	1.74345
$713$	−59.8326	−2.24075
$714$	−1.80579	−0.0675800
$715$	0	0
$716$	17.8328	0.666443
$717$	5.68593	0.212345
$718$	−4.40834	−0.164518
$719$	−43.0849	−1.60680	−0.803398	−	0.595443i	$-0.796977\pi$
$719$	−43.0849	−1.60680	−0.803398	+	0.595443i	$0.796977\pi$
$720$	0	0
$721$	−15.4875	−0.576784
$722$	9.64449	0.358931
$723$	−7.37736	−0.274367
$724$	16.5042	0.613375
$725$	0	0
$726$	−4.59489	−0.170532
$727$	−13.8952	−0.515344	−0.257672	−	0.966232i	$-0.582955\pi$
$727$	−13.8952	−0.515344	−0.257672	+	0.966232i	$0.582955\pi$
$728$	0	0
$729$	−12.9255	−0.478720
$730$	0	0
$731$	3.21733	0.118997
$732$	7.52319	0.278065
$733$	37.7741	1.39522	0.697610	−	0.716478i	$-0.254247\pi$
$733$	37.7741	1.39522	0.697610	+	0.716478i	$0.254247\pi$
$734$	15.6374	0.577187
$735$	0	0
$736$	37.1354	1.36883
$737$	12.2178	0.450049
$738$	2.75174	0.101293
$739$	−21.2237	−0.780725	−0.390362	−	0.920661i	$-0.627650\pi$
$739$	−21.2237	−0.780725	−0.390362	+	0.920661i	$0.627650\pi$
$740$	0	0
$741$	0	0
$742$	−4.63294	−0.170081
$743$	−23.7248	−0.870377	−0.435189	−	0.900339i	$-0.643318\pi$
$743$	−23.7248	−0.870377	−0.435189	+	0.900339i	$0.643318\pi$
$744$	−13.6314	−0.499751
$745$	0	0
$746$	−19.6033	−0.717729
$747$	25.3555	0.927709
$748$	3.74338	0.136871
$749$	−16.6626	−0.608836
$750$	0	0
$751$	−23.3632	−0.852534	−0.426267	−	0.904597i	$-0.640172\pi$
$751$	−23.3632	−0.852534	−0.426267	+	0.904597i	$0.640172\pi$
$752$	0.835090	0.0304526
$753$	−2.57635	−0.0938876
$754$	0	0
$755$	0	0
$756$	4.57346	0.166335
$757$	1.02041	0.0370875	0.0185437	−	0.999828i	$-0.494097\pi$
$757$	1.02041	0.0370875	0.0185437	+	0.999828i	$0.494097\pi$
$758$	8.21994	0.298562
$759$	−2.82475	−0.102532
$760$	0	0
$761$	−27.4016	−0.993307	−0.496654	−	0.867949i	$-0.665438\pi$
$761$	−27.4016	−0.993307	−0.496654	+	0.867949i	$0.665438\pi$
$762$	−8.81181	−0.319218
$763$	4.79170	0.173471
$764$	−16.6129	−0.601033
$765$	0	0
$766$	16.2819	0.588289
$767$	0	0
$768$	8.07359	0.291331
$769$	25.7737	0.929425	0.464712	−	0.885462i	$-0.346158\pi$
$769$	25.7737	0.929425	0.464712	+	0.885462i	$0.346158\pi$
$770$	0	0
$771$	10.2624	0.369593
$772$	15.4353	0.555530
$773$	11.5788	0.416461	0.208230	−	0.978080i	$-0.433230\pi$
$773$	11.5788	0.416461	0.208230	+	0.978080i	$0.433230\pi$
$774$	2.09503	0.0753044
$775$	0	0
$776$	−45.0722	−1.61800
$777$	3.73193	0.133882
$778$	−3.16507	−0.113473
$779$	−3.30717	−0.118491
$780$	0	0
$781$	−0.444745	−0.0159142
$782$	18.8854	0.675341
$783$	15.4635	0.552621
$784$	−1.58571	−0.0566325
$785$	0	0
$786$	−3.19763	−0.114056
$787$	34.1619	1.21774	0.608870	−	0.793270i	$-0.291623\pi$
$787$	34.1619	1.21774	0.608870	+	0.793270i	$0.291623\pi$
$788$	29.5166	1.05148
$789$	−4.54667	−0.161866
$790$	0	0
$791$	−1.23780	−0.0440112
$792$	6.19374	0.220085
$793$	0	0
$794$	0.798409	0.0283345
$795$	0	0
$796$	15.1589	0.537294
$797$	31.5188	1.11645	0.558226	−	0.829689i	$-0.311482\pi$
$797$	31.5188	1.11645	0.558226	+	0.829689i	$0.311482\pi$
$798$	1.41323	0.0500277
$799$	10.4142	0.368428
$800$	0	0
$801$	−45.7515	−1.61655
$802$	18.9371	0.668693
$803$	0.0309367	0.00109173
$804$	−10.2169	−0.360324
$805$	0	0
$806$	0	0
$807$	4.65583	0.163893
$808$	52.8149	1.85802
$809$	21.7313	0.764031	0.382016	−	0.924156i	$-0.375230\pi$
$809$	21.7313	0.764031	0.382016	+	0.924156i	$0.375230\pi$
$810$	0	0
$811$	30.0916	1.05666	0.528330	−	0.849039i	$-0.322818\pi$
$811$	30.0916	1.05666	0.528330	+	0.849039i	$0.322818\pi$
$812$	7.66128	0.268858
$813$	6.67280	0.234025
$814$	4.18479	0.146677
$815$	0	0
$816$	−0.521070	−0.0182411
$817$	−2.51791	−0.0880904
$818$	−23.1448	−0.809240
$819$	0	0
$820$	0	0
$821$	−51.5157	−1.79791	−0.898955	−	0.438041i	$-0.855672\pi$
$821$	−51.5157	−1.79791	−0.898955	+	0.438041i	$0.855672\pi$
$822$	−5.99287	−0.209025
$823$	−21.1230	−0.736301	−0.368150	−	0.929766i	$-0.620009\pi$
$823$	−21.1230	−0.736301	−0.368150	+	0.929766i	$0.620009\pi$
$824$	36.9013	1.28552
$825$	0	0
$826$	−7.27763	−0.253221
$827$	−48.5138	−1.68699	−0.843494	−	0.537138i	$-0.819505\pi$
$827$	−48.5138	−1.68699	−0.843494	+	0.537138i	$0.819505\pi$
$828$	−22.7341	−0.790065
$829$	28.0093	0.972802	0.486401	−	0.873736i	$-0.338309\pi$
$829$	28.0093	0.972802	0.486401	+	0.873736i	$0.338309\pi$
$830$	0	0
$831$	−12.3448	−0.428237
$832$	0	0
$833$	−19.7750	−0.685163
$834$	2.48462	0.0860353
$835$	0	0
$836$	−2.92960	−0.101322
$837$	28.2047	0.974898
$838$	13.0613	0.451196
$839$	55.8214	1.92717	0.963585	−	0.267404i	$-0.0861658\pi$
$839$	55.8214	1.92717	0.963585	+	0.267404i	$0.0861658\pi$
$840$	0	0
$841$	−3.09617	−0.106764
$842$	20.8091	0.717130
$843$	−2.32178	−0.0799662
$844$	4.78478	0.164699
$845$	0	0
$846$	6.78143	0.233151
$847$	11.9686	0.411246
$848$	−1.33686	−0.0459079
$849$	5.69510	0.195455
$850$	0	0
$851$	−39.0294	−1.33791
$852$	0.371911	0.0127415
$853$	−37.0872	−1.26984	−0.634921	−	0.772577i	$-0.718968\pi$
$853$	−37.0872	−1.26984	−0.634921	+	0.772577i	$0.718968\pi$
$854$	10.6002	0.362731
$855$	0	0
$856$	39.7011	1.35696
$857$	−50.7725	−1.73436	−0.867178	−	0.497998i	$-0.834069\pi$
$857$	−50.7725	−1.73436	−0.867178	+	0.497998i	$0.834069\pi$
$858$	0	0
$859$	24.8213	0.846891	0.423446	−	0.905921i	$-0.360820\pi$
$859$	24.8213	0.846891	0.423446	+	0.905921i	$0.360820\pi$
$860$	0	0
$861$	0.744741	0.0253807
$862$	5.94929	0.202634
$863$	−30.1931	−1.02778	−0.513892	−	0.857855i	$-0.671797\pi$
$863$	−30.1931	−1.02778	−0.513892	+	0.857855i	$0.671797\pi$
$864$	−17.5054	−0.595546
$865$	0	0
$866$	4.10998	0.139663
$867$	2.53542	0.0861075
$868$	13.9738	0.474303
$869$	−7.23567	−0.245453
$870$	0	0
$871$	0	0
$872$	−11.4170	−0.386627
$873$	44.3266	1.50023
$874$	−14.7799	−0.499937
$875$	0	0
$876$	−0.0258704	−0.000874078 0
$877$	22.1079	0.746532	0.373266	−	0.927724i	$-0.378238\pi$
$877$	22.1079	0.746532	0.373266	+	0.927724i	$0.378238\pi$
$878$	−2.69661	−0.0910061
$879$	10.5592	0.356154
$880$	0	0
$881$	5.48395	0.184759	0.0923794	−	0.995724i	$-0.470553\pi$
$881$	5.48395	0.184759	0.0923794	+	0.995724i	$0.470553\pi$
$882$	−12.8769	−0.433588
$883$	13.9041	0.467909	0.233955	−	0.972248i	$-0.424833\pi$
$883$	13.9041	0.467909	0.233955	+	0.972248i	$0.424833\pi$
$884$	0	0
$885$	0	0
$886$	20.3567	0.683896
$887$	19.7523	0.663216	0.331608	−	0.943417i	$-0.392409\pi$
$887$	19.7523	0.663216	0.331608	+	0.943417i	$0.392409\pi$
$888$	−8.89190	−0.298393
$889$	22.9527	0.769808
$890$	0	0
$891$	−5.39261	−0.180659
$892$	−2.11225	−0.0707233
$893$	−8.15024	−0.272737
$894$	−3.39349	−0.113495
$895$	0	0
$896$	−9.21792	−0.307949
$897$	0	0
$898$	15.8341	0.528392
$899$	47.2474	1.57579
$900$	0	0
$901$	−16.6716	−0.555413
$902$	0.835112	0.0278062
$903$	0.567008	0.0188688
$904$	2.94925	0.0980907
$905$	0	0
$906$	−6.02879	−0.200293
$907$	50.4025	1.67359	0.836794	−	0.547518i	$-0.184427\pi$
$907$	50.4025	1.67359	0.836794	+	0.547518i	$0.184427\pi$
$908$	14.2196	0.471893
$909$	−51.9412	−1.72278
$910$	0	0
$911$	55.5002	1.83880	0.919402	−	0.393319i	$-0.128673\pi$
$911$	55.5002	1.83880	0.919402	+	0.393319i	$0.128673\pi$
$912$	0.407794	0.0135034
$913$	7.69503	0.254668
$914$	−16.2698	−0.538157
$915$	0	0
$916$	−22.4706	−0.742450
$917$	8.32906	0.275050
$918$	−8.90247	−0.293825
$919$	40.6408	1.34062	0.670309	−	0.742082i	$-0.266162\pi$
$919$	40.6408	1.34062	0.670309	+	0.742082i	$0.266162\pi$
$920$	0	0
$921$	13.4457	0.443052
$922$	20.8295	0.685984
$923$	0	0
$924$	0.659717	0.0217031
$925$	0	0
$926$	12.5594	0.412728
$927$	−36.2909	−1.19195
$928$	−29.3243	−0.962618
$929$	29.5963	0.971024	0.485512	−	0.874230i	$-0.338633\pi$
$929$	29.5963	0.971024	0.485512	+	0.874230i	$0.338633\pi$
$930$	0	0
$931$	15.4761	0.507208
$932$	33.4524	1.09577
$933$	0.0257625	0.000843425 0
$934$	−0.345496	−0.0113050
$935$	0	0
$936$	0	0
$937$	−38.2185	−1.24854	−0.624272	−	0.781207i	$-0.714604\pi$
$937$	−38.2185	−1.24854	−0.624272	+	0.781207i	$0.714604\pi$
$938$	−14.3957	−0.470037
$939$	2.66698	0.0870334
$940$	0	0
$941$	−6.20726	−0.202351	−0.101175	−	0.994869i	$-0.532260\pi$
$941$	−6.20726	−0.202351	−0.101175	+	0.994869i	$0.532260\pi$
$942$	5.85833	0.190875
$943$	−7.78868	−0.253634
$944$	−2.10000	−0.0683490
$945$	0	0
$946$	0.635812	0.0206720
$947$	−20.3360	−0.660831	−0.330415	−	0.943836i	$-0.607189\pi$
$947$	−20.3360	−0.660831	−0.330415	+	0.943836i	$0.607189\pi$
$948$	6.05071	0.196518
$949$	0	0
$950$	0	0
$951$	4.54107	0.147254
$952$	−11.2072	−0.363227
$953$	13.6309	0.441548	0.220774	−	0.975325i	$-0.429142\pi$
$953$	13.6309	0.441548	0.220774	+	0.975325i	$0.429142\pi$
$954$	−10.8561	−0.351479
$955$	0	0
$956$	13.8879	0.449168
$957$	2.23059	0.0721048
$958$	−32.6531	−1.05497
$959$	15.6100	0.504073
$960$	0	0
$961$	55.1772	1.77991
$962$	0	0
$963$	−39.0444	−1.25819
$964$	−18.0193	−0.580362
$965$	0	0
$966$	3.32828	0.107086
$967$	32.5909	1.04805	0.524026	−	0.851702i	$-0.324429\pi$
$967$	32.5909	1.04805	0.524026	+	0.851702i	$0.324429\pi$
$968$	−28.5170	−0.916572
$969$	5.08550	0.163370
$970$	0	0
$971$	23.4891	0.753801	0.376900	−	0.926254i	$-0.376990\pi$
$971$	23.4891	0.753801	0.376900	+	0.926254i	$0.376990\pi$
$972$	16.3397	0.524097
$973$	−6.47185	−0.207478
$974$	16.8754	0.540721
$975$	0	0
$976$	3.05874	0.0979079
$977$	−5.55803	−0.177817	−0.0889085	−	0.996040i	$-0.528338\pi$
$977$	−5.55803	−0.177817	−0.0889085	+	0.996040i	$0.528338\pi$
$978$	−6.56173	−0.209821
$979$	−13.8849	−0.443764
$980$	0	0
$981$	11.2281	0.358486
$982$	21.8822	0.698289
$983$	−20.8723	−0.665723	−0.332861	−	0.942976i	$-0.608014\pi$
$983$	−20.8723	−0.665723	−0.332861	+	0.942976i	$0.608014\pi$
$984$	−1.77446	−0.0565677
$985$	0	0
$986$	−14.9130	−0.474928
$987$	1.83535	0.0584199
$988$	0	0
$989$	−5.92990	−0.188560
$990$	0	0
$991$	37.4195	1.18867	0.594335	−	0.804218i	$-0.297416\pi$
$991$	37.4195	1.18867	0.594335	+	0.804218i	$0.297416\pi$
$992$	−53.4862	−1.69819
$993$	−15.5954	−0.494905
$994$	0.524024	0.0166210
$995$	0	0
$996$	−6.43485	−0.203896
$997$	−25.0667	−0.793869	−0.396935	−	0.917847i	$-0.629926\pi$
$997$	−25.0667	−0.793869	−0.396935	+	0.917847i	$0.629926\pi$
$998$	30.9271	0.978981
$999$	18.3982	0.582094

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4225.2.a.bx.1.7	yes	12
5.4	even	2		4225.2.a.by.1.6	yes	12
13.12	even	2		4225.2.a.bz.1.6	yes	12
65.64	even	2		4225.2.a.bw.1.7	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4225.2.a.bw.1.7	✓	12	65.64	even	2
4225.2.a.bx.1.7	yes	12	1.1	even	1	trivial
4225.2.a.by.1.6	yes	12	5.4	even	2
4225.2.a.bz.1.6	yes	12	13.12	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:	\( N \)	\(=\)	\( 4225 = 5^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4225.a (trivial)

\(f(q)\)	\(=\)	\(q-0.837905 q^{2} -0.531386 q^{3} -1.29791 q^{4} +0.445251 q^{6} -1.15977 q^{7} +2.76334 q^{8} -2.71763 q^{9} +O(q^{10})\)	\(q-0.837905 q^{2} -0.531386 q^{3} -1.29791 q^{4} +0.445251 q^{6} -1.15977 q^{7} +2.76334 q^{8} -2.71763 q^{9} -0.824761 q^{11} +0.689694 q^{12} +0.971780 q^{14} +0.280412 q^{16} +3.49695 q^{17} +2.27712 q^{18} -2.73674 q^{19} +0.616287 q^{21} +0.691072 q^{22} -6.44528 q^{23} -1.46840 q^{24} +3.03827 q^{27} +1.50529 q^{28} +5.08958 q^{29} +9.28317 q^{31} -5.76164 q^{32} +0.438266 q^{33} -2.93011 q^{34} +3.52725 q^{36} +6.05550 q^{37} +2.29313 q^{38} +1.20843 q^{41} -0.516390 q^{42} +0.920038 q^{43} +1.07047 q^{44} +5.40054 q^{46} +2.97808 q^{47} -0.149007 q^{48} -5.65493 q^{49} -1.85823 q^{51} -4.76748 q^{53} -2.54578 q^{54} -3.20485 q^{56} +1.45427 q^{57} -4.26459 q^{58} -7.48896 q^{59} +10.9080 q^{61} -7.77841 q^{62} +3.15183 q^{63} +4.26688 q^{64} -0.367226 q^{66} -14.8138 q^{67} -4.53874 q^{68} +3.42493 q^{69} +0.539241 q^{71} -7.50973 q^{72} -0.0375099 q^{73} -5.07394 q^{74} +3.55206 q^{76} +0.956536 q^{77} +8.77305 q^{79} +6.53839 q^{81} -1.01255 q^{82} -9.33001 q^{83} -0.799888 q^{84} -0.770905 q^{86} -2.70453 q^{87} -2.27910 q^{88} +16.8351 q^{89} +8.36543 q^{92} -4.93294 q^{93} -2.49535 q^{94} +3.06165 q^{96} -16.3108 q^{97} +4.73829 q^{98} +2.24139 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 7 q^{2} + q^{3} + 13 q^{4} - 3 q^{6} - 12 q^{7} - 18 q^{8} + 11 q^{9}+O(q^{10}) \) 12 * q - 7 * q^2 + q^3 + 13 * q^4 - 3 * q^6 - 12 * q^7 - 18 * q^8 + 11 * q^9	\( 12 q - 7 q^{2} + q^{3} + 13 q^{4} - 3 q^{6} - 12 q^{7} - 18 q^{8} + 11 q^{9} + 3 q^{11} + 8 q^{12} + 2 q^{14} + 11 q^{16} - 2 q^{17} - 14 q^{18} - 5 q^{21} - 12 q^{22} + 3 q^{23} - 4 q^{24} - 5 q^{27} - 29 q^{28} + 9 q^{29} - 4 q^{31} - 48 q^{32} - 30 q^{33} - 3 q^{34} + 44 q^{36} - 17 q^{37} - 15 q^{38} + 9 q^{41} + 80 q^{42} - q^{43} + 10 q^{44} + 3 q^{46} - 61 q^{47} - 35 q^{48} + 8 q^{49} - 26 q^{51} - 23 q^{53} - 48 q^{54} + 51 q^{56} - 10 q^{57} + 17 q^{58} - q^{59} + 13 q^{61} - 29 q^{62} - 46 q^{63} + 50 q^{64} + 29 q^{66} - 43 q^{67} + 26 q^{68} - 31 q^{69} + 19 q^{71} - 48 q^{72} - 21 q^{73} + 7 q^{74} + 46 q^{76} + 42 q^{77} - 7 q^{79} - 16 q^{81} - 15 q^{82} - 75 q^{83} - 83 q^{84} + 17 q^{86} - 5 q^{87} - 38 q^{88} - 19 q^{89} + 35 q^{92} - 11 q^{93} + 26 q^{94} + 90 q^{96} - 17 q^{97} - 11 q^{98} - 6 q^{99}+O(q^{100}) \) 12 * q - 7 * q^2 + q^3 + 13 * q^4 - 3 * q^6 - 12 * q^7 - 18 * q^8 + 11 * q^9 + 3 * q^11 + 8 * q^12 + 2 * q^14 + 11 * q^16 - 2 * q^17 - 14 * q^18 - 5 * q^21 - 12 * q^22 + 3 * q^23 - 4 * q^24 - 5 * q^27 - 29 * q^28 + 9 * q^29 - 4 * q^31 - 48 * q^32 - 30 * q^33 - 3 * q^34 + 44 * q^36 - 17 * q^37 - 15 * q^38 + 9 * q^41 + 80 * q^42 - q^43 + 10 * q^44 + 3 * q^46 - 61 * q^47 - 35 * q^48 + 8 * q^49 - 26 * q^51 - 23 * q^53 - 48 * q^54 + 51 * q^56 - 10 * q^57 + 17 * q^58 - q^59 + 13 * q^61 - 29 * q^62 - 46 * q^63 + 50 * q^64 + 29 * q^66 - 43 * q^67 + 26 * q^68 - 31 * q^69 + 19 * q^71 - 48 * q^72 - 21 * q^73 + 7 * q^74 + 46 * q^76 + 42 * q^77 - 7 * q^79 - 16 * q^81 - 15 * q^82 - 75 * q^83 - 83 * q^84 + 17 * q^86 - 5 * q^87 - 38 * q^88 - 19 * q^89 + 35 * q^92 - 11 * q^93 + 26 * q^94 + 90 * q^96 - 17 * q^97 - 11 * q^98 - 6 * q^99

Introduction

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