LMFDB - Embedded newform 7225.2.a.bw.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7225,2,Mod(1,7225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7225, 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("7225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7225 = 5^{2} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7225.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:	$57.6919154604$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 21 x^{13} - 2 x^{12} + 171 x^{11} + 30 x^{10} - 678 x^{9} - 153 x^{8} + 1350 x^{7} + 301 x^{6} + \cdots + 3 $ x^15 - 21x^13 - 2x^12 + 171x^11 + 30x^10 - 678x^9 - 153x^8 + 1350x^7 + 301x^6 - 1284x^5 - 183x^4 + 555x^3 + 18x^2 - 81*x + 3
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			1.5
Root			$-1.10615$ of defining polynomial
Character	$\chi$	$=$	7225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.10615 q^{2} +3.09602 q^{3} -0.776422 q^{4} -3.42468 q^{6} +2.90465 q^{7} +3.07115 q^{8} +6.58534 q^{9} +O(q^{10})$	\(q-1.10615 q^{2} +3.09602 q^{3} -0.776422 q^{4} -3.42468 q^{6} +2.90465 q^{7} +3.07115 q^{8} +6.58534 q^{9} -2.75521 q^{11} -2.40382 q^{12} +4.63304 q^{13} -3.21299 q^{14} -1.84432 q^{16} -7.28440 q^{18} +8.35290 q^{19} +8.99286 q^{21} +3.04769 q^{22} +6.17263 q^{23} +9.50835 q^{24} -5.12486 q^{26} +11.1003 q^{27} -2.25524 q^{28} +1.37656 q^{29} +2.25976 q^{31} -4.10220 q^{32} -8.53019 q^{33} -5.11300 q^{36} -1.24362 q^{37} -9.23960 q^{38} +14.3440 q^{39} +9.73468 q^{41} -9.94749 q^{42} -0.0582420 q^{43} +2.13921 q^{44} -6.82788 q^{46} +3.78140 q^{47} -5.71006 q^{48} +1.43700 q^{49} -3.59720 q^{52} -8.41519 q^{53} -12.2786 q^{54} +8.92063 q^{56} +25.8607 q^{57} -1.52269 q^{58} -4.40059 q^{59} -10.5516 q^{61} -2.49965 q^{62} +19.1281 q^{63} +8.22631 q^{64} +9.43570 q^{66} -9.25722 q^{67} +19.1106 q^{69} -1.48552 q^{71} +20.2246 q^{72} -15.9514 q^{73} +1.37563 q^{74} -6.48538 q^{76} -8.00293 q^{77} -15.8667 q^{78} -3.18167 q^{79} +14.6107 q^{81} -10.7681 q^{82} -7.45601 q^{83} -6.98226 q^{84} +0.0644246 q^{86} +4.26185 q^{87} -8.46167 q^{88} +10.4873 q^{89} +13.4574 q^{91} -4.79257 q^{92} +6.99627 q^{93} -4.18281 q^{94} -12.7005 q^{96} -9.17821 q^{97} -1.58955 q^{98} -18.1440 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 9 q^{3} + 12 q^{4} + 9 q^{6} + 12 q^{7} + 6 q^{8} + 12 q^{9}+O(q^{10}) $ 15 * q + 9 * q^3 + 12 * q^4 + 9 * q^6 + 12 * q^7 + 6 * q^8 + 12 * q^9	$ 15 q + 9 q^{3} + 12 q^{4} + 9 q^{6} + 12 q^{7} + 6 q^{8} + 12 q^{9} + 6 q^{11} + 24 q^{12} + 6 q^{16} + 12 q^{18} + 6 q^{19} + 30 q^{21} + 12 q^{22} + 36 q^{23} + 18 q^{24} + 36 q^{26} + 36 q^{27} + 24 q^{28} - 18 q^{29} + 12 q^{32} + 12 q^{33} - 9 q^{36} + 12 q^{37} - 6 q^{38} + 9 q^{39} - 18 q^{41} + 36 q^{42} - 3 q^{43} - 12 q^{44} + 21 q^{46} - 3 q^{47} - 12 q^{48} + 15 q^{49} - 27 q^{52} + 21 q^{54} - 6 q^{56} + 39 q^{57} + 18 q^{58} - 12 q^{59} - 15 q^{61} + 54 q^{62} + 60 q^{63} - 36 q^{64} + 18 q^{66} - 24 q^{67} + 42 q^{69} + 6 q^{71} + 66 q^{72} - 9 q^{73} - 36 q^{74} - 18 q^{76} + 30 q^{77} + 30 q^{78} - 9 q^{79} + 51 q^{81} - 36 q^{82} + 15 q^{83} + 9 q^{84} - 36 q^{86} - 51 q^{87} + 30 q^{88} - 24 q^{89} + 27 q^{91} + 15 q^{92} - 42 q^{93} - 57 q^{94} + 42 q^{96} + 48 q^{97} + 12 q^{99}+O(q^{100}) $ 15 * q + 9 * q^3 + 12 * q^4 + 9 * q^6 + 12 * q^7 + 6 * q^8 + 12 * q^9 + 6 * q^11 + 24 * q^12 + 6 * q^16 + 12 * q^18 + 6 * q^19 + 30 * q^21 + 12 * q^22 + 36 * q^23 + 18 * q^24 + 36 * q^26 + 36 * q^27 + 24 * q^28 - 18 * q^29 + 12 * q^32 + 12 * q^33 - 9 * q^36 + 12 * q^37 - 6 * q^38 + 9 * q^39 - 18 * q^41 + 36 * q^42 - 3 * q^43 - 12 * q^44 + 21 * q^46 - 3 * q^47 - 12 * q^48 + 15 * q^49 - 27 * q^52 + 21 * q^54 - 6 * q^56 + 39 * q^57 + 18 * q^58 - 12 * q^59 - 15 * q^61 + 54 * q^62 + 60 * q^63 - 36 * q^64 + 18 * q^66 - 24 * q^67 + 42 * q^69 + 6 * q^71 + 66 * q^72 - 9 * q^73 - 36 * q^74 - 18 * q^76 + 30 * q^77 + 30 * q^78 - 9 * q^79 + 51 * q^81 - 36 * q^82 + 15 * q^83 + 9 * q^84 - 36 * q^86 - 51 * q^87 + 30 * q^88 - 24 * q^89 + 27 * q^91 + 15 * q^92 - 42 * q^93 - 57 * q^94 + 42 * q^96 + 48 * q^97 + 12 * 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 $

$2$	−1.10615	−0.782169	−0.391085	−	0.920355i	$-0.627900\pi$
$2$	−1.10615	−0.782169	−0.391085	+	0.920355i	$0.627900\pi$
$3$	3.09602	1.78749	0.893744	−	0.448578i	$-0.148069\pi$
$3$	3.09602	1.78749	0.893744	+	0.448578i	$0.148069\pi$
$4$	−0.776422	−0.388211
$5$	0	0
$5$	0	0
$6$	−3.42468	−1.39812
$7$	2.90465	1.09786	0.548928	−	0.835870i	$-0.315036\pi$
$7$	2.90465	1.09786	0.548928	+	0.835870i	$0.315036\pi$
$8$	3.07115	1.08582
$9$	6.58534	2.19511
$10$	0	0
$11$	−2.75521	−0.830727	−0.415364	−	0.909655i	$-0.636346\pi$
$11$	−2.75521	−0.830727	−0.415364	+	0.909655i	$0.636346\pi$
$12$	−2.40382	−0.693923
$13$	4.63304	1.28498	0.642488	−	0.766296i	$-0.277902\pi$
$13$	4.63304	1.28498	0.642488	+	0.766296i	$0.277902\pi$
$14$	−3.21299	−0.858709
$15$	0	0
$16$	−1.84432	−0.461081
$17$	0	0
$17$	0	0
$18$	−7.28440	−1.71695
$19$	8.35290	1.91629	0.958144	−	0.286288i	$-0.0924215\pi$
$19$	8.35290	1.91629	0.958144	+	0.286288i	$0.0924215\pi$
$20$	0	0
$21$	8.99286	1.96240
$22$	3.04769	0.649769
$23$	6.17263	1.28708	0.643541	−	0.765411i	$-0.277464\pi$
$23$	6.17263	1.28708	0.643541	+	0.765411i	$0.277464\pi$
$24$	9.50835	1.94088
$25$	0	0
$26$	−5.12486	−1.00507
$27$	11.1003	2.13625
$28$	−2.25524	−0.426200
$29$	1.37656	0.255621	0.127810	−	0.991799i	$-0.459205\pi$
$29$	1.37656	0.255621	0.127810	+	0.991799i	$0.459205\pi$
$30$	0	0
$31$	2.25976	0.405865	0.202933	−	0.979193i	$-0.434953\pi$
$31$	2.25976	0.405865	0.202933	+	0.979193i	$0.434953\pi$
$32$	−4.10220	−0.725173
$33$	−8.53019	−1.48491
$34$	0	0
$35$	0	0
$36$	−5.11300	−0.852167
$37$	−1.24362	−0.204449	−0.102225	−	0.994761i	$-0.532596\pi$
$37$	−1.24362	−0.204449	−0.102225	+	0.994761i	$0.532596\pi$
$38$	−9.23960	−1.49886
$39$	14.3440	2.29688
$40$	0	0
$41$	9.73468	1.52030	0.760151	−	0.649746i	$-0.225125\pi$
$41$	9.73468	1.52030	0.760151	+	0.649746i	$0.225125\pi$
$42$	−9.94749	−1.53493
$43$	−0.0582420	−0.00888182	−0.00444091	−	0.999990i	$-0.501414\pi$
$43$	−0.0582420	−0.00888182	−0.00444091	+	0.999990i	$0.501414\pi$
$44$	2.13921	0.322498
$45$	0	0
$46$	−6.82788	−1.00672
$47$	3.78140	0.551574	0.275787	−	0.961219i	$-0.411062\pi$
$47$	3.78140	0.551574	0.275787	+	0.961219i	$0.411062\pi$
$48$	−5.71006	−0.824176
$49$	1.43700	0.205286
$50$	0	0
$51$	0	0
$52$	−3.59720	−0.498842
$53$	−8.41519	−1.15591	−0.577957	−	0.816067i	$-0.696150\pi$
$53$	−8.41519	−1.15591	−0.577957	+	0.816067i	$0.696150\pi$
$54$	−12.2786	−1.67091
$55$	0	0
$56$	8.92063	1.19207
$57$	25.8607	3.42534
$58$	−1.52269	−0.199939
$59$	−4.40059	−0.572907	−0.286454	−	0.958094i	$-0.592476\pi$
$59$	−4.40059	−0.572907	−0.286454	+	0.958094i	$0.592476\pi$
$60$	0	0
$61$	−10.5516	−1.35099	−0.675497	−	0.737363i	$-0.736071\pi$
$61$	−10.5516	−1.35099	−0.675497	+	0.737363i	$0.736071\pi$
$62$	−2.49965	−0.317455
$63$	19.1281	2.40992
$64$	8.22631	1.02829
$65$	0	0
$66$	9.43570	1.16145
$67$	−9.25722	−1.13095	−0.565475	−	0.824766i	$-0.691307\pi$
$67$	−9.25722	−1.13095	−0.565475	+	0.824766i	$0.691307\pi$
$68$	0	0
$69$	19.1106	2.30064
$70$	0	0
$71$	−1.48552	−0.176299	−0.0881493	−	0.996107i	$-0.528095\pi$
$71$	−1.48552	−0.176299	−0.0881493	+	0.996107i	$0.528095\pi$
$72$	20.2246	2.38349
$73$	−15.9514	−1.86697	−0.933487	−	0.358611i	$-0.883250\pi$
$73$	−15.9514	−1.86697	−0.933487	+	0.358611i	$0.883250\pi$
$74$	1.37563	0.159914
$75$	0	0
$76$	−6.48538	−0.743924
$77$	−8.00293	−0.912018
$78$	−15.8667	−1.79655
$79$	−3.18167	−0.357966	−0.178983	−	0.983852i	$-0.557281\pi$
$79$	−3.18167	−0.357966	−0.178983	+	0.983852i	$0.557281\pi$
$80$	0	0
$81$	14.6107	1.62341
$82$	−10.7681	−1.18913
$83$	−7.45601	−0.818403	−0.409202	−	0.912444i	$-0.634193\pi$
$83$	−7.45601	−0.818403	−0.409202	+	0.912444i	$0.634193\pi$
$84$	−6.98226	−0.761827
$85$	0	0
$86$	0.0644246	0.00694709
$87$	4.26185	0.456919
$88$	−8.46167	−0.902017
$89$	10.4873	1.11166	0.555828	−	0.831297i	$-0.312401\pi$
$89$	10.4873	1.11166	0.555828	+	0.831297i	$0.312401\pi$
$90$	0	0
$91$	13.4574	1.41072
$92$	−4.79257	−0.499660
$93$	6.99627	0.725479
$94$	−4.18281	−0.431424
$95$	0	0
$96$	−12.7005	−1.29624
$97$	−9.17821	−0.931906	−0.465953	−	0.884809i	$-0.654288\pi$
$97$	−9.17821	−0.931906	−0.465953	+	0.884809i	$0.654288\pi$
$98$	−1.58955	−0.160569
$99$	−18.1440	−1.82354
$100$	0	0
$101$	−11.8730	−1.18141	−0.590703	−	0.806889i	$-0.701150\pi$
$101$	−11.8730	−1.18141	−0.590703	+	0.806889i	$0.701150\pi$
$102$	0	0
$103$	1.60590	0.158234	0.0791170	−	0.996865i	$-0.474790\pi$
$103$	1.60590	0.158234	0.0791170	+	0.996865i	$0.474790\pi$
$104$	14.2288	1.39525
$105$	0	0
$106$	9.30849	0.904121
$107$	13.3427	1.28989	0.644946	−	0.764228i	$-0.276880\pi$
$107$	13.3427	1.28989	0.644946	+	0.764228i	$0.276880\pi$
$108$	−8.61850	−0.829316
$109$	1.94165	0.185976	0.0929880	−	0.995667i	$-0.470358\pi$
$109$	1.94165	0.185976	0.0929880	+	0.995667i	$0.470358\pi$
$110$	0	0
$111$	−3.85026	−0.365451
$112$	−5.35712	−0.506200
$113$	−5.05740	−0.475761	−0.237880	−	0.971294i	$-0.576453\pi$
$113$	−5.05740	−0.475761	−0.237880	+	0.971294i	$0.576453\pi$
$114$	−28.6060	−2.67920
$115$	0	0
$116$	−1.06879	−0.0992348
$117$	30.5102	2.82067
$118$	4.86773	0.448111
$119$	0	0
$120$	0	0
$121$	−3.40882	−0.309892
$122$	11.6717	1.05671
$123$	30.1388	2.71752
$124$	−1.75453	−0.157561
$125$	0	0
$126$	−21.1587	−1.88496
$127$	−15.6761	−1.39103	−0.695513	−	0.718514i	$-0.744823\pi$
$127$	−15.6761	−1.39103	−0.695513	+	0.718514i	$0.744823\pi$
$128$	−0.895174	−0.0791229
$129$	−0.180318	−0.0158761
$130$	0	0
$131$	10.8233	0.945638	0.472819	−	0.881160i	$-0.343237\pi$
$131$	10.8233	0.945638	0.472819	+	0.881160i	$0.343237\pi$
$132$	6.62303	0.576461
$133$	24.2623	2.10381
$134$	10.2399	0.884594
$135$	0	0
$136$	0	0
$137$	17.0936	1.46040	0.730202	−	0.683231i	$-0.239426\pi$
$137$	17.0936	1.46040	0.730202	+	0.683231i	$0.239426\pi$
$138$	−21.1393	−1.79949
$139$	−14.6432	−1.24202	−0.621011	−	0.783802i	$-0.713278\pi$
$139$	−14.6432	−1.24202	−0.621011	+	0.783802i	$0.713278\pi$
$140$	0	0
$141$	11.7073	0.985931
$142$	1.64321	0.137895
$143$	−12.7650	−1.06746
$144$	−12.1455	−1.01212
$145$	0	0
$146$	17.6447	1.46029
$147$	4.44900	0.366947
$148$	0.965572	0.0793695
$149$	17.1627	1.40603	0.703013	−	0.711177i	$-0.251838\pi$
$149$	17.1627	1.40603	0.703013	+	0.711177i	$0.251838\pi$
$150$	0	0
$151$	−13.7967	−1.12276	−0.561378	−	0.827559i	$-0.689729\pi$
$151$	−13.7967	−1.12276	−0.561378	+	0.827559i	$0.689729\pi$
$152$	25.6530	2.08074
$153$	0	0
$154$	8.85247	0.713353
$155$	0	0
$156$	−11.1370	−0.891674
$157$	−11.5721	−0.923554	−0.461777	−	0.886996i	$-0.652788\pi$
$157$	−11.5721	−0.923554	−0.461777	+	0.886996i	$0.652788\pi$
$158$	3.51942	0.279990
$159$	−26.0536	−2.06618
$160$	0	0
$161$	17.9294	1.41303
$162$	−16.1616	−1.26978
$163$	−5.00786	−0.392246	−0.196123	−	0.980579i	$-0.562835\pi$
$163$	−5.00786	−0.392246	−0.196123	+	0.980579i	$0.562835\pi$
$164$	−7.55823	−0.590198
$165$	0	0
$166$	8.24750	0.640130
$167$	18.1140	1.40170	0.700851	−	0.713307i	$-0.252804\pi$
$167$	18.1140	1.40170	0.700851	+	0.713307i	$0.252804\pi$
$168$	27.6184	2.13081
$169$	8.46510	0.651161
$170$	0	0
$171$	55.0067	4.20647
$172$	0.0452204	0.00344802
$173$	14.9338	1.13540	0.567699	−	0.823236i	$-0.307834\pi$
$173$	14.9338	1.13540	0.567699	+	0.823236i	$0.307834\pi$
$174$	−4.71427	−0.357388
$175$	0	0
$176$	5.08150	0.383032
$177$	−13.6243	−1.02406
$178$	−11.6006	−0.869503
$179$	4.21621	0.315134	0.157567	−	0.987508i	$-0.449635\pi$
$179$	4.21621	0.315134	0.157567	+	0.987508i	$0.449635\pi$
$180$	0	0
$181$	15.9686	1.18694	0.593468	−	0.804857i	$-0.297758\pi$
$181$	15.9686	1.18694	0.593468	+	0.804857i	$0.297758\pi$
$182$	−14.8859	−1.10342
$183$	−32.6680	−2.41488
$184$	18.9571	1.39754
$185$	0	0
$186$	−7.73895	−0.567448
$187$	0	0
$188$	−2.93596	−0.214127
$189$	32.2424	2.34529
$190$	0	0
$191$	−9.11831	−0.659778	−0.329889	−	0.944020i	$-0.607011\pi$
$191$	−9.11831	−0.659778	−0.329889	+	0.944020i	$0.607011\pi$
$192$	25.4688	1.83805
$193$	−5.55204	−0.399645	−0.199822	−	0.979832i	$-0.564037\pi$
$193$	−5.55204	−0.399645	−0.199822	+	0.979832i	$0.564037\pi$
$194$	10.1525	0.728908
$195$	0	0
$196$	−1.11572	−0.0796945
$197$	6.60139	0.470330	0.235165	−	0.971956i	$-0.424437\pi$
$197$	6.60139	0.470330	0.235165	+	0.971956i	$0.424437\pi$
$198$	20.0701	1.42632
$199$	−17.3665	−1.23108	−0.615539	−	0.788106i	$-0.711062\pi$
$199$	−17.3665	−1.23108	−0.615539	+	0.788106i	$0.711062\pi$
$200$	0	0
$201$	−28.6605	−2.02156
$202$	13.1334	0.924060
$203$	3.99843	0.280634
$204$	0	0
$205$	0	0
$206$	−1.77637	−0.123766
$207$	40.6489	2.82529
$208$	−8.54483	−0.592477
$209$	−23.0140	−1.59191
$210$	0	0
$211$	0.628498	0.0432676	0.0216338	−	0.999766i	$-0.493113\pi$
$211$	0.628498	0.0432676	0.0216338	+	0.999766i	$0.493113\pi$
$212$	6.53374	0.448739
$213$	−4.59920	−0.315132
$214$	−14.7591	−1.00891
$215$	0	0
$216$	34.0906	2.31957
$217$	6.56382	0.445581
$218$	−2.14776	−0.145465
$219$	−49.3859	−3.33719
$220$	0	0
$221$	0	0
$222$	4.25898	0.285844
$223$	−13.2121	−0.884747	−0.442373	−	0.896831i	$-0.645863\pi$
$223$	−13.2121	−0.884747	−0.442373	+	0.896831i	$0.645863\pi$
$224$	−11.9155	−0.796135
$225$	0	0
$226$	5.59427	0.372125
$227$	−14.2526	−0.945977	−0.472989	−	0.881069i	$-0.656825\pi$
$227$	−14.2526	−0.945977	−0.472989	+	0.881069i	$0.656825\pi$
$228$	−20.0789	−1.32976
$229$	−7.03989	−0.465209	−0.232604	−	0.972571i	$-0.574725\pi$
$229$	−7.03989	−0.465209	−0.232604	+	0.972571i	$0.574725\pi$
$230$	0	0
$231$	−24.7772	−1.63022
$232$	4.22762	0.277557
$233$	−12.8646	−0.842786	−0.421393	−	0.906878i	$-0.638459\pi$
$233$	−12.8646	−0.842786	−0.421393	+	0.906878i	$0.638459\pi$
$234$	−33.7489	−2.20624
$235$	0	0
$236$	3.41671	0.222409
$237$	−9.85051	−0.639859
$238$	0	0
$239$	−4.31255	−0.278956	−0.139478	−	0.990225i	$-0.544542\pi$
$239$	−4.31255	−0.278956	−0.139478	+	0.990225i	$0.544542\pi$
$240$	0	0
$241$	7.25418	0.467283	0.233642	−	0.972323i	$-0.424936\pi$
$241$	7.25418	0.467283	0.233642	+	0.972323i	$0.424936\pi$
$242$	3.77068	0.242388
$243$	11.9341	0.765571
$244$	8.19250	0.524471
$245$	0	0
$246$	−33.3381	−2.12556
$247$	38.6994	2.46238
$248$	6.94007	0.440695
$249$	−23.0840	−1.46289
$250$	0	0
$251$	5.65828	0.357147	0.178574	−	0.983927i	$-0.442852\pi$
$251$	5.65828	0.357147	0.178574	+	0.983927i	$0.442852\pi$
$252$	−14.8515	−0.935557
$253$	−17.0069	−1.06921
$254$	17.3401	1.08802
$255$	0	0
$256$	−15.4624	−0.966401
$257$	1.56762	0.0977856	0.0488928	−	0.998804i	$-0.484431\pi$
$257$	1.56762	0.0977856	0.0488928	+	0.998804i	$0.484431\pi$
$258$	0.199460	0.0124178
$259$	−3.61227	−0.224456
$260$	0	0
$261$	9.06511	0.561116
$262$	−11.9723	−0.739649
$263$	1.06686	0.0657854	0.0328927	−	0.999459i	$-0.489528\pi$
$263$	1.06686	0.0657854	0.0328927	+	0.999459i	$0.489528\pi$
$264$	−26.1975	−1.61234
$265$	0	0
$266$	−26.8378	−1.64553
$267$	32.4690	1.98707
$268$	7.18752	0.439047
$269$	−4.91216	−0.299500	−0.149750	−	0.988724i	$-0.547847\pi$
$269$	−4.91216	−0.299500	−0.149750	+	0.988724i	$0.547847\pi$
$270$	0	0
$271$	−4.77840	−0.290267	−0.145134	−	0.989412i	$-0.546361\pi$
$271$	−4.77840	−0.290267	−0.145134	+	0.989412i	$0.546361\pi$
$272$	0	0
$273$	41.6643	2.52164
$274$	−18.9082	−1.14228
$275$	0	0
$276$	−14.8379	−0.893136
$277$	−6.52598	−0.392108	−0.196054	−	0.980593i	$-0.562813\pi$
$277$	−6.52598	−0.392108	−0.196054	+	0.980593i	$0.562813\pi$
$278$	16.1977	0.971471
$279$	14.8813	0.890920
$280$	0	0
$281$	−17.1957	−1.02581	−0.512905	−	0.858445i	$-0.671431\pi$
$281$	−17.1957	−1.02581	−0.512905	+	0.858445i	$0.671431\pi$
$282$	−12.9501	−0.771165
$283$	8.02818	0.477226	0.238613	−	0.971115i	$-0.423307\pi$
$283$	8.02818	0.477226	0.238613	+	0.971115i	$0.423307\pi$
$284$	1.15339	0.0684411
$285$	0	0
$286$	14.1201	0.834937
$287$	28.2759	1.66907
$288$	−27.0144	−1.59184
$289$	0	0
$290$	0	0
$291$	−28.4159	−1.66577
$292$	12.3851	0.724780
$293$	4.31155	0.251883	0.125942	−	0.992038i	$-0.459805\pi$
$293$	4.31155	0.251883	0.125942	+	0.992038i	$0.459805\pi$
$294$	−4.92128	−0.287015
$295$	0	0
$296$	−3.81934	−0.221994
$297$	−30.5836	−1.77464
$298$	−18.9846	−1.09975
$299$	28.5981	1.65387
$300$	0	0
$301$	−0.169173	−0.00975096
$302$	15.2612	0.878186
$303$	−36.7590	−2.11175
$304$	−15.4054	−0.883563
$305$	0	0
$306$	0	0
$307$	7.89436	0.450555	0.225277	−	0.974295i	$-0.427671\pi$
$307$	7.89436	0.450555	0.225277	+	0.974295i	$0.427671\pi$
$308$	6.21365	0.354056
$309$	4.97189	0.282841
$310$	0	0
$311$	−15.6322	−0.886422	−0.443211	−	0.896417i	$-0.646161\pi$
$311$	−15.6322	−0.886422	−0.443211	+	0.896417i	$0.646161\pi$
$312$	44.0526	2.49399
$313$	−4.33994	−0.245308	−0.122654	−	0.992449i	$-0.539141\pi$
$313$	−4.33994	−0.245308	−0.122654	+	0.992449i	$0.539141\pi$
$314$	12.8005	0.722375
$315$	0	0
$316$	2.47032	0.138966
$317$	17.9720	1.00941	0.504705	−	0.863292i	$-0.331601\pi$
$317$	17.9720	1.00941	0.504705	+	0.863292i	$0.331601\pi$
$318$	28.8193	1.61611
$319$	−3.79271	−0.212351
$320$	0	0
$321$	41.3094	2.30567
$322$	−19.8326	−1.10523
$323$	0	0
$324$	−11.3440	−0.630225
$325$	0	0
$326$	5.53946	0.306802
$327$	6.01138	0.332430
$328$	29.8967	1.65077
$329$	10.9836	0.605548
$330$	0	0
$331$	−6.35660	−0.349391	−0.174695	−	0.984623i	$-0.555894\pi$
$331$	−6.35660	−0.349391	−0.174695	+	0.984623i	$0.555894\pi$
$332$	5.78901	0.317713
$333$	−8.18964	−0.448789
$334$	−20.0369	−1.09637
$335$	0	0
$336$	−16.5857	−0.904826
$337$	1.77707	0.0968034	0.0484017	−	0.998828i	$-0.484587\pi$
$337$	1.77707	0.0968034	0.0484017	+	0.998828i	$0.484587\pi$
$338$	−9.36371	−0.509318
$339$	−15.6578	−0.850416
$340$	0	0
$341$	−6.22612	−0.337163
$342$	−60.8459	−3.29017
$343$	−16.1586	−0.872481
$344$	−0.178870	−0.00964403
$345$	0	0
$346$	−16.5191	−0.888073
$347$	−13.4893	−0.724145	−0.362073	−	0.932150i	$-0.617931\pi$
$347$	−13.4893	−0.724145	−0.362073	+	0.932150i	$0.617931\pi$
$348$	−3.30900	−0.177381
$349$	25.9021	1.38651	0.693255	−	0.720693i	$-0.256176\pi$
$349$	25.9021	1.38651	0.693255	+	0.720693i	$0.256176\pi$
$350$	0	0
$351$	51.4281	2.74503
$352$	11.3024	0.602421
$353$	32.1662	1.71203	0.856016	−	0.516949i	$-0.172932\pi$
$353$	32.1662	1.71203	0.856016	+	0.516949i	$0.172932\pi$
$354$	15.0706	0.800992
$355$	0	0
$356$	−8.14261	−0.431558
$357$	0	0
$358$	−4.66377	−0.246488
$359$	−33.9431	−1.79145	−0.895724	−	0.444610i	$-0.853342\pi$
$359$	−33.9431	−1.79145	−0.895724	+	0.444610i	$0.853342\pi$
$360$	0	0
$361$	50.7710	2.67216
$362$	−17.6637	−0.928385
$363$	−10.5538	−0.553929
$364$	−10.4486	−0.547656
$365$	0	0
$366$	36.1358	1.88885
$367$	−0.783787	−0.0409133	−0.0204567	−	0.999791i	$-0.506512\pi$
$367$	−0.783787	−0.0409133	−0.0204567	+	0.999791i	$0.506512\pi$
$368$	−11.3843	−0.593449
$369$	64.1062	3.33723
$370$	0	0
$371$	−24.4432	−1.26903
$372$	−5.43206	−0.281639
$373$	−13.9714	−0.723411	−0.361706	−	0.932292i	$-0.617805\pi$
$373$	−13.9714	−0.723411	−0.361706	+	0.932292i	$0.617805\pi$
$374$	0	0
$375$	0	0
$376$	11.6132	0.598908
$377$	6.37766	0.328466
$378$	−35.6651	−1.83442
$379$	4.95724	0.254636	0.127318	−	0.991862i	$-0.459363\pi$
$379$	4.95724	0.254636	0.127318	+	0.991862i	$0.459363\pi$
$380$	0	0
$381$	−48.5334	−2.48644
$382$	10.0863	0.516058
$383$	30.6897	1.56817	0.784086	−	0.620652i	$-0.213132\pi$
$383$	30.6897	1.56817	0.784086	+	0.620652i	$0.213132\pi$
$384$	−2.77148	−0.141431
$385$	0	0
$386$	6.14142	0.312590
$387$	−0.383543	−0.0194966
$388$	7.12617	0.361776
$389$	8.42482	0.427155	0.213578	−	0.976926i	$-0.431488\pi$
$389$	8.42482	0.427155	0.213578	+	0.976926i	$0.431488\pi$
$390$	0	0
$391$	0	0
$392$	4.41326	0.222903
$393$	33.5092	1.69032
$394$	−7.30216	−0.367877
$395$	0	0
$396$	14.0874	0.707919
$397$	−1.58289	−0.0794430	−0.0397215	−	0.999211i	$-0.512647\pi$
$397$	−1.58289	−0.0794430	−0.0397215	+	0.999211i	$0.512647\pi$
$398$	19.2100	0.962911
$399$	75.1165	3.76053
$400$	0	0
$401$	−3.73161	−0.186348	−0.0931739	−	0.995650i	$-0.529701\pi$
$401$	−3.73161	−0.186348	−0.0931739	+	0.995650i	$0.529701\pi$
$402$	31.7030	1.58120
$403$	10.4696	0.521527
$404$	9.21845	0.458635
$405$	0	0
$406$	−4.42288	−0.219504
$407$	3.42643	0.169842
$408$	0	0
$409$	−33.6009	−1.66146	−0.830728	−	0.556678i	$-0.812076\pi$
$409$	−33.6009	−1.66146	−0.830728	+	0.556678i	$0.812076\pi$
$410$	0	0
$411$	52.9221	2.61045
$412$	−1.24686	−0.0614282
$413$	−12.7822	−0.628969
$414$	−44.9639	−2.20986
$415$	0	0
$416$	−19.0057	−0.931829
$417$	−45.3357	−2.22010
$418$	25.4570	1.24514
$419$	33.1973	1.62180	0.810898	−	0.585188i	$-0.198979\pi$
$419$	33.1973	1.62180	0.810898	+	0.585188i	$0.198979\pi$
$420$	0	0
$421$	−25.7067	−1.25287	−0.626434	−	0.779475i	$-0.715486\pi$
$421$	−25.7067	−1.25287	−0.626434	+	0.779475i	$0.715486\pi$
$422$	−0.695216	−0.0338426
$423$	24.9018	1.21077
$424$	−25.8443	−1.25511
$425$	0	0
$426$	5.08742	0.246486
$427$	−30.6487	−1.48320
$428$	−10.3596	−0.500750
$429$	−39.5207	−1.90808
$430$	0	0
$431$	9.45697	0.455526	0.227763	−	0.973717i	$-0.426859\pi$
$431$	9.45697	0.455526	0.227763	+	0.973717i	$0.426859\pi$
$432$	−20.4725	−0.984983
$433$	9.51726	0.457370	0.228685	−	0.973500i	$-0.426557\pi$
$433$	9.51726	0.457370	0.228685	+	0.973500i	$0.426557\pi$
$434$	−7.26060	−0.348520
$435$	0	0
$436$	−1.50754	−0.0721980
$437$	51.5594	2.46642
$438$	54.6285	2.61025
$439$	−0.503647	−0.0240378	−0.0120189	−	0.999928i	$-0.503826\pi$
$439$	−0.503647	−0.0240378	−0.0120189	+	0.999928i	$0.503826\pi$
$440$	0	0
$441$	9.46316	0.450627
$442$	0	0
$443$	−13.5880	−0.645584	−0.322792	−	0.946470i	$-0.604621\pi$
$443$	−13.5880	−0.645584	−0.322792	+	0.946470i	$0.604621\pi$
$444$	2.98943	0.141872
$445$	0	0
$446$	14.6146	0.692022
$447$	53.1361	2.51325
$448$	23.8946	1.12891
$449$	20.3910	0.962312	0.481156	−	0.876635i	$-0.340217\pi$
$449$	20.3910	0.962312	0.481156	+	0.876635i	$0.340217\pi$
$450$	0	0
$451$	−26.8211	−1.26296
$452$	3.92668	0.184696
$453$	−42.7147	−2.00691
$454$	15.7656	0.739914
$455$	0	0
$456$	79.4223	3.71929
$457$	6.39312	0.299058	0.149529	−	0.988757i	$-0.452224\pi$
$457$	6.39312	0.299058	0.149529	+	0.988757i	$0.452224\pi$
$458$	7.78720	0.363872
$459$	0	0
$460$	0	0
$461$	−23.3149	−1.08589	−0.542943	−	0.839770i	$-0.682690\pi$
$461$	−23.3149	−1.08589	−0.542943	+	0.839770i	$0.682690\pi$
$462$	27.4074	1.27511
$463$	39.4556	1.83366	0.916828	−	0.399282i	$-0.130740\pi$
$463$	39.4556	1.83366	0.916828	+	0.399282i	$0.130740\pi$
$464$	−2.53882	−0.117862
$465$	0	0
$466$	14.2302	0.659201
$467$	8.87512	0.410691	0.205346	−	0.978689i	$-0.434168\pi$
$467$	8.87512	0.410691	0.205346	+	0.978689i	$0.434168\pi$
$468$	−23.6888	−1.09501
$469$	−26.8890	−1.24162
$470$	0	0
$471$	−35.8274	−1.65084
$472$	−13.5149	−0.622072
$473$	0.160469	0.00737837
$474$	10.8962	0.500478
$475$	0	0
$476$	0	0
$477$	−55.4168	−2.53736
$478$	4.77035	0.218191
$479$	7.13146	0.325845	0.162922	−	0.986639i	$-0.447908\pi$
$479$	7.13146	0.325845	0.162922	+	0.986639i	$0.447908\pi$
$480$	0	0
$481$	−5.76173	−0.262712
$482$	−8.02425	−0.365494
$483$	55.5096	2.52578
$484$	2.64668	0.120304
$485$	0	0
$486$	−13.2009	−0.598806
$487$	−6.98977	−0.316737	−0.158368	−	0.987380i	$-0.550623\pi$
$487$	−6.98977	−0.316737	−0.158368	+	0.987380i	$0.550623\pi$
$488$	−32.4056	−1.46693
$489$	−15.5044	−0.701134
$490$	0	0
$491$	−10.9267	−0.493113	−0.246556	−	0.969128i	$-0.579299\pi$
$491$	−10.9267	−0.493113	−0.246556	+	0.969128i	$0.579299\pi$
$492$	−23.4004	−1.05497
$493$	0	0
$494$	−42.8075	−1.92600
$495$	0	0
$496$	−4.16773	−0.187137
$497$	−4.31492	−0.193550
$498$	25.5344	1.14422
$499$	26.2455	1.17491	0.587456	−	0.809256i	$-0.300130\pi$
$499$	26.2455	1.17491	0.587456	+	0.809256i	$0.300130\pi$
$500$	0	0
$501$	56.0813	2.50553
$502$	−6.25893	−0.279350
$503$	31.1917	1.39077	0.695384	−	0.718638i	$-0.255234\pi$
$503$	31.1917	1.39077	0.695384	+	0.718638i	$0.255234\pi$
$504$	58.7454	2.61673
$505$	0	0
$506$	18.8123	0.836307
$507$	26.2081	1.16394
$508$	12.1712	0.540012
$509$	31.4174	1.39255	0.696275	−	0.717775i	$-0.254839\pi$
$509$	31.4174	1.39255	0.696275	+	0.717775i	$0.254839\pi$
$510$	0	0
$511$	−46.3334	−2.04967
$512$	18.8942	0.835012
$513$	92.7195	4.09367
$514$	−1.73403	−0.0764849
$515$	0	0
$516$	0.140003	0.00616330
$517$	−10.4185	−0.458207
$518$	3.99573	0.175562
$519$	46.2354	2.02951
$520$	0	0
$521$	−27.3391	−1.19775	−0.598874	−	0.800843i	$-0.704385\pi$
$521$	−27.3391	−1.19775	−0.598874	+	0.800843i	$0.704385\pi$
$522$	−10.0274	−0.438888
$523$	32.5448	1.42309	0.711543	−	0.702642i	$-0.247997\pi$
$523$	32.5448	1.42309	0.711543	+	0.702642i	$0.247997\pi$
$524$	−8.40347	−0.367107
$525$	0	0
$526$	−1.18011	−0.0514553
$527$	0	0
$528$	15.7324	0.684666
$529$	15.1014	0.656582
$530$	0	0
$531$	−28.9793	−1.25760
$532$	−18.8378	−0.816721
$533$	45.1012	1.95355
$534$	−35.9158	−1.55423
$535$	0	0
$536$	−28.4303	−1.22800
$537$	13.0535	0.563298
$538$	5.43361	0.234259
$539$	−3.95925	−0.170537
$540$	0	0
$541$	−19.2060	−0.825729	−0.412864	−	0.910793i	$-0.635472\pi$
$541$	−19.2060	−0.825729	−0.412864	+	0.910793i	$0.635472\pi$
$542$	5.28565	0.227038
$543$	49.4391	2.12163
$544$	0	0
$545$	0	0
$546$	−46.0872	−1.97235
$547$	−28.3358	−1.21155	−0.605776	−	0.795636i	$-0.707137\pi$
$547$	−28.3358	−1.21155	−0.605776	+	0.795636i	$0.707137\pi$
$548$	−13.2718	−0.566945
$549$	−69.4858	−2.96558
$550$	0	0
$551$	11.4983	0.489843
$552$	58.6915	2.49808
$553$	−9.24164	−0.392995
$554$	7.21874	0.306695
$555$	0	0
$556$	11.3693	0.482167
$557$	−3.73907	−0.158429	−0.0792147	−	0.996858i	$-0.525241\pi$
$557$	−3.73907	−0.158429	−0.0792147	+	0.996858i	$0.525241\pi$
$558$	−16.4610	−0.696850
$559$	−0.269838	−0.0114129
$560$	0	0
$561$	0	0
$562$	19.0211	0.802357
$563$	−22.7786	−0.960002	−0.480001	−	0.877268i	$-0.659364\pi$
$563$	−22.7786	−0.960002	−0.480001	+	0.877268i	$0.659364\pi$
$564$	−9.08980	−0.382750
$565$	0	0
$566$	−8.88040	−0.373271
$567$	42.4389	1.78227
$568$	−4.56225	−0.191428
$569$	32.6405	1.36836	0.684181	−	0.729313i	$-0.260160\pi$
$569$	32.6405	1.36836	0.684181	+	0.729313i	$0.260160\pi$
$570$	0	0
$571$	−12.7492	−0.533537	−0.266768	−	0.963761i	$-0.585956\pi$
$571$	−12.7492	−0.533537	−0.266768	+	0.963761i	$0.585956\pi$
$572$	9.91104	0.414401
$573$	−28.2305	−1.17934
$574$	−31.2775	−1.30550
$575$	0	0
$576$	54.1730	2.25721
$577$	4.53542	0.188812	0.0944060	−	0.995534i	$-0.469905\pi$
$577$	4.53542	0.188812	0.0944060	+	0.995534i	$0.469905\pi$
$578$	0	0
$579$	−17.1892	−0.714360
$580$	0	0
$581$	−21.6571	−0.898489
$582$	31.4324	1.30291
$583$	23.1856	0.960250
$584$	−48.9893	−2.02719
$585$	0	0
$586$	−4.76924	−0.197015
$587$	−20.6362	−0.851746	−0.425873	−	0.904783i	$-0.640033\pi$
$587$	−20.6362	−0.851746	−0.425873	+	0.904783i	$0.640033\pi$
$588$	−3.45430	−0.142453
$589$	18.8756	0.777754
$590$	0	0
$591$	20.4380	0.840708
$592$	2.29363	0.0942677
$593$	0.527710	0.0216704	0.0108352	−	0.999941i	$-0.496551\pi$
$593$	0.527710	0.0216704	0.0108352	+	0.999941i	$0.496551\pi$
$594$	33.8302	1.38807
$595$	0	0
$596$	−13.3255	−0.545835
$597$	−53.7670	−2.20054
$598$	−31.6339	−1.29361
$599$	−19.9890	−0.816729	−0.408364	−	0.912819i	$-0.633901\pi$
$599$	−19.9890	−0.816729	−0.408364	+	0.912819i	$0.633901\pi$
$600$	0	0
$601$	36.2410	1.47830	0.739151	−	0.673540i	$-0.235227\pi$
$601$	36.2410	1.47830	0.739151	+	0.673540i	$0.235227\pi$
$602$	0.187131	0.00762690
$603$	−60.9619	−2.48256
$604$	10.7120	0.435867
$605$	0	0
$606$	40.6611	1.65175
$607$	19.0565	0.773480	0.386740	−	0.922189i	$-0.373601\pi$
$607$	19.0565	0.773480	0.386740	+	0.922189i	$0.373601\pi$
$608$	−34.2653	−1.38964
$609$	12.3792	0.501631
$610$	0	0
$611$	17.5194	0.708758
$612$	0	0
$613$	12.8132	0.517521	0.258760	−	0.965942i	$-0.416686\pi$
$613$	12.8132	0.517521	0.258760	+	0.965942i	$0.416686\pi$
$614$	−8.73238	−0.352410
$615$	0	0
$616$	−24.5782	−0.990284
$617$	−35.3589	−1.42350	−0.711749	−	0.702434i	$-0.752096\pi$
$617$	−35.3589	−1.42350	−0.711749	+	0.702434i	$0.752096\pi$
$618$	−5.49968	−0.221230
$619$	8.25731	0.331889	0.165945	−	0.986135i	$-0.446933\pi$
$619$	8.25731	0.331889	0.165945	+	0.986135i	$0.446933\pi$
$620$	0	0
$621$	68.5179	2.74953
$622$	17.2917	0.693332
$623$	30.4621	1.22044
$624$	−26.4550	−1.05905
$625$	0	0
$626$	4.80065	0.191872
$627$	−71.2518	−2.84552
$628$	8.98484	0.358534
$629$	0	0
$630$	0	0
$631$	19.5842	0.779633	0.389816	−	0.920893i	$-0.372538\pi$
$631$	19.5842	0.779633	0.389816	+	0.920893i	$0.372538\pi$
$632$	−9.77139	−0.388685
$633$	1.94584	0.0773403
$634$	−19.8798	−0.789529
$635$	0	0
$636$	20.2286	0.802116
$637$	6.65771	0.263788
$638$	4.19532	0.166094
$639$	−9.78264	−0.386995
$640$	0	0
$641$	−4.82277	−0.190488	−0.0952440	−	0.995454i	$-0.530363\pi$
$641$	−4.82277	−0.190488	−0.0952440	+	0.995454i	$0.530363\pi$
$642$	−45.6946	−1.80342
$643$	21.7833	0.859048	0.429524	−	0.903055i	$-0.358681\pi$
$643$	21.7833	0.859048	0.429524	+	0.903055i	$0.358681\pi$
$644$	−13.9208	−0.548554
$645$	0	0
$646$	0	0
$647$	−36.7458	−1.44463	−0.722313	−	0.691566i	$-0.756921\pi$
$647$	−36.7458	−1.44463	−0.722313	+	0.691566i	$0.756921\pi$
$648$	44.8716	1.76272
$649$	12.1245	0.475930
$650$	0	0
$651$	20.3217	0.796471
$652$	3.88821	0.152274
$653$	−20.5441	−0.803953	−0.401977	−	0.915650i	$-0.631677\pi$
$653$	−20.5441	−0.803953	−0.401977	+	0.915650i	$0.631677\pi$
$654$	−6.64951	−0.260017
$655$	0	0
$656$	−17.9539	−0.700982
$657$	−105.046	−4.09822
$658$	−12.1496	−0.473641
$659$	−48.1223	−1.87458	−0.937289	−	0.348552i	$-0.886673\pi$
$659$	−48.1223	−1.87458	−0.937289	+	0.348552i	$0.886673\pi$
$660$	0	0
$661$	−36.6815	−1.42674	−0.713372	−	0.700786i	$-0.752833\pi$
$661$	−36.6815	−1.42674	−0.713372	+	0.700786i	$0.752833\pi$
$662$	7.03139	0.273283
$663$	0	0
$664$	−22.8985	−0.888636
$665$	0	0
$666$	9.05900	0.351029
$667$	8.49699	0.329005
$668$	−14.0641	−0.544157
$669$	−40.9049	−1.58147
$670$	0	0
$671$	29.0719	1.12231
$672$	−36.8905	−1.42308
$673$	2.50043	0.0963845	0.0481923	−	0.998838i	$-0.484654\pi$
$673$	2.50043	0.0963845	0.0481923	+	0.998838i	$0.484654\pi$
$674$	−1.96572	−0.0757166
$675$	0	0
$676$	−6.57249	−0.252788
$677$	−35.6682	−1.37084	−0.685421	−	0.728147i	$-0.740382\pi$
$677$	−35.6682	−1.37084	−0.685421	+	0.728147i	$0.740382\pi$
$678$	17.3200	0.665169
$679$	−26.6595	−1.02310
$680$	0	0
$681$	−44.1263	−1.69092
$682$	6.88705	0.263719
$683$	−7.74927	−0.296517	−0.148259	−	0.988949i	$-0.547367\pi$
$683$	−7.74927	−0.296517	−0.148259	+	0.988949i	$0.547367\pi$
$684$	−42.7084	−1.63300
$685$	0	0
$686$	17.8739	0.682427
$687$	−21.7956	−0.831555
$688$	0.107417	0.00409524
$689$	−38.9879	−1.48532
$690$	0	0
$691$	−45.3235	−1.72419	−0.862093	−	0.506750i	$-0.830847\pi$
$691$	−45.3235	−1.72419	−0.862093	+	0.506750i	$0.830847\pi$
$692$	−11.5950	−0.440774
$693$	−52.7020	−2.00198
$694$	14.9213	0.566404
$695$	0	0
$696$	13.0888	0.496130
$697$	0	0
$698$	−28.6518	−1.08448
$699$	−39.8290	−1.50647
$700$	0	0
$701$	39.3992	1.48809	0.744044	−	0.668130i	$-0.232905\pi$
$701$	39.3992	1.48809	0.744044	+	0.668130i	$0.232905\pi$
$702$	−56.8874	−2.14708
$703$	−10.3878	−0.391784
$704$	−22.6652	−0.854227
$705$	0	0
$706$	−35.5807	−1.33910
$707$	−34.4869	−1.29701
$708$	10.5782	0.397554
$709$	11.5879	0.435193	0.217596	−	0.976039i	$-0.430178\pi$
$709$	11.5879	0.435193	0.217596	+	0.976039i	$0.430178\pi$
$710$	0	0
$711$	−20.9524	−0.785775
$712$	32.2082	1.20705
$713$	13.9487	0.522382
$714$	0	0
$715$	0	0
$716$	−3.27356	−0.122339
$717$	−13.3518	−0.498630
$718$	37.5463	1.40122
$719$	28.8585	1.07624	0.538121	−	0.842868i	$-0.319135\pi$
$719$	28.8585	1.07624	0.538121	+	0.842868i	$0.319135\pi$
$720$	0	0
$721$	4.66458	0.173718
$722$	−56.1605	−2.09008
$723$	22.4591	0.835263
$724$	−12.3984	−0.460782
$725$	0	0
$726$	11.6741	0.433266
$727$	−0.112849	−0.00418535	−0.00209267	−	0.999998i	$-0.500666\pi$
$727$	−0.112849	−0.00418535	−0.00209267	+	0.999998i	$0.500666\pi$
$728$	41.3297	1.53178
$729$	−6.88388	−0.254959
$730$	0	0
$731$	0	0
$732$	25.3641	0.937485
$733$	−33.4520	−1.23558	−0.617789	−	0.786344i	$-0.711971\pi$
$733$	−33.4520	−1.23558	−0.617789	+	0.786344i	$0.711971\pi$
$734$	0.866989	0.0320012
$735$	0	0
$736$	−25.3214	−0.933358
$737$	25.5056	0.939511
$738$	−70.9113	−2.61028
$739$	−42.1942	−1.55214	−0.776069	−	0.630648i	$-0.782789\pi$
$739$	−42.1942	−1.55214	−0.776069	+	0.630648i	$0.782789\pi$
$740$	0	0
$741$	119.814	4.40148
$742$	27.0379	0.992594
$743$	−43.4721	−1.59484	−0.797418	−	0.603428i	$-0.793801\pi$
$743$	−43.4721	−1.59484	−0.797418	+	0.603428i	$0.793801\pi$
$744$	21.4866	0.787737
$745$	0	0
$746$	15.4545	0.565830
$747$	−49.1003	−1.79649
$748$	0	0
$749$	38.7560	1.41611
$750$	0	0
$751$	50.8183	1.85438	0.927192	−	0.374586i	$-0.122215\pi$
$751$	50.8183	1.85438	0.927192	+	0.374586i	$0.122215\pi$
$752$	−6.97412	−0.254320
$753$	17.5181	0.638397
$754$	−7.05468	−0.256916
$755$	0	0
$756$	−25.0338	−0.910469
$757$	20.8735	0.758659	0.379330	−	0.925262i	$-0.376155\pi$
$757$	20.8735	0.758659	0.379330	+	0.925262i	$0.376155\pi$
$758$	−5.48347	−0.199169
$759$	−52.6537	−1.91121
$760$	0	0
$761$	3.97049	0.143930	0.0719651	−	0.997407i	$-0.477073\pi$
$761$	3.97049	0.143930	0.0719651	+	0.997407i	$0.477073\pi$
$762$	53.6854	1.94482
$763$	5.63981	0.204175
$764$	7.07966	0.256133
$765$	0	0
$766$	−33.9476	−1.22658
$767$	−20.3881	−0.736172
$768$	−47.8720	−1.72743
$769$	44.2142	1.59440	0.797202	−	0.603713i	$-0.206313\pi$
$769$	44.2142	1.59440	0.797202	+	0.603713i	$0.206313\pi$
$770$	0	0
$771$	4.85339	0.174791
$772$	4.31073	0.155147
$773$	−48.0188	−1.72711	−0.863557	−	0.504251i	$-0.831769\pi$
$773$	−48.0188	−1.72711	−0.863557	+	0.504251i	$0.831769\pi$
$774$	0.424258	0.0152496
$775$	0	0
$776$	−28.1877	−1.01188
$777$	−11.1837	−0.401212
$778$	−9.31915	−0.334108
$779$	81.3129	2.91334
$780$	0	0
$781$	4.09292	0.146456
$782$	0	0
$783$	15.2802	0.546069
$784$	−2.65030	−0.0946536
$785$	0	0
$786$	−37.0664	−1.32211
$787$	37.2679	1.32846	0.664228	−	0.747530i	$-0.268761\pi$
$787$	37.2679	1.32846	0.664228	+	0.747530i	$0.268761\pi$
$788$	−5.12547	−0.182587
$789$	3.30302	0.117591
$790$	0	0
$791$	−14.6900	−0.522316
$792$	−55.7230	−1.98003
$793$	−48.8860	−1.73599
$794$	1.75092	0.0621379
$795$	0	0
$796$	13.4837	0.477918
$797$	9.72080	0.344328	0.172164	−	0.985068i	$-0.444924\pi$
$797$	9.72080	0.344328	0.172164	+	0.985068i	$0.444924\pi$
$798$	−83.0904	−2.94137
$799$	0	0
$800$	0	0
$801$	69.0627	2.44021
$802$	4.12774	0.145755
$803$	43.9495	1.55095
$804$	22.2527	0.784792
$805$	0	0
$806$	−11.5810	−0.407922
$807$	−15.2081	−0.535352
$808$	−36.4637	−1.28279
$809$	33.6621	1.18350	0.591748	−	0.806123i	$-0.298438\pi$
$809$	33.6621	1.18350	0.591748	+	0.806123i	$0.298438\pi$
$810$	0	0
$811$	−2.43879	−0.0856375	−0.0428188	−	0.999083i	$-0.513634\pi$
$811$	−2.43879	−0.0856375	−0.0428188	+	0.999083i	$0.513634\pi$
$812$	−3.10447	−0.108945
$813$	−14.7940	−0.518849
$814$	−3.79016	−0.132845
$815$	0	0
$816$	0	0
$817$	−0.486490	−0.0170201
$818$	37.1678	1.29954
$819$	88.6214	3.09668
$820$	0	0
$821$	−55.9146	−1.95143	−0.975716	−	0.219038i	$-0.929708\pi$
$821$	−55.9146	−1.95143	−0.975716	+	0.219038i	$0.929708\pi$
$822$	−58.5400	−2.04182
$823$	54.2832	1.89219	0.946097	−	0.323882i	$-0.104988\pi$
$823$	54.2832	1.89219	0.946097	+	0.323882i	$0.104988\pi$
$824$	4.93196	0.171813
$825$	0	0
$826$	14.1391	0.491961
$827$	21.8616	0.760203	0.380102	−	0.924945i	$-0.375889\pi$
$827$	21.8616	0.760203	0.380102	+	0.924945i	$0.375889\pi$
$828$	−31.5607	−1.09681
$829$	46.7084	1.62225	0.811126	−	0.584872i	$-0.198855\pi$
$829$	46.7084	1.62225	0.811126	+	0.584872i	$0.198855\pi$
$830$	0	0
$831$	−20.2046	−0.700889
$832$	38.1129	1.32133
$833$	0	0
$834$	50.1483	1.73649
$835$	0	0
$836$	17.8686	0.617998
$837$	25.0840	0.867029
$838$	−36.7214	−1.26852
$839$	50.4053	1.74018	0.870092	−	0.492889i	$-0.164059\pi$
$839$	50.4053	1.74018	0.870092	+	0.492889i	$0.164059\pi$
$840$	0	0
$841$	−27.1051	−0.934658
$842$	28.4356	0.979954
$843$	−53.2383	−1.83362
$844$	−0.487980	−0.0167970
$845$	0	0
$846$	−27.5452	−0.947024
$847$	−9.90143	−0.340217
$848$	15.5203	0.532970
$849$	24.8554	0.853035
$850$	0	0
$851$	−7.67639	−0.263143
$852$	3.57092	0.122338
$853$	18.5407	0.634821	0.317410	−	0.948288i	$-0.397187\pi$
$853$	18.5407	0.634821	0.317410	+	0.948288i	$0.397187\pi$
$854$	33.9022	1.16011
$855$	0	0
$856$	40.9776	1.40059
$857$	31.7575	1.08482	0.542408	−	0.840115i	$-0.317513\pi$
$857$	31.7575	1.08482	0.542408	+	0.840115i	$0.317513\pi$
$858$	43.7160	1.49244
$859$	18.6880	0.637627	0.318814	−	0.947817i	$-0.396716\pi$
$859$	18.6880	0.637627	0.318814	+	0.947817i	$0.396716\pi$
$860$	0	0
$861$	87.5427	2.98345
$862$	−10.4609	−0.356299
$863$	−38.0188	−1.29418	−0.647088	−	0.762415i	$-0.724013\pi$
$863$	−38.0188	−1.29418	−0.647088	+	0.762415i	$0.724013\pi$
$864$	−45.5355	−1.54915
$865$	0	0
$866$	−10.5276	−0.357741
$867$	0	0
$868$	−5.09630	−0.172980
$869$	8.76617	0.297372
$870$	0	0
$871$	−42.8891	−1.45324
$872$	5.96309	0.201936
$873$	−60.4416	−2.04564
$874$	−57.0326	−1.92916
$875$	0	0
$876$	38.3444	1.29554
$877$	−44.1406	−1.49052	−0.745261	−	0.666773i	$-0.767675\pi$
$877$	−44.1406	−1.49052	−0.745261	+	0.666773i	$0.767675\pi$
$878$	0.557112	0.0188016
$879$	13.3486	0.450238
$880$	0	0
$881$	15.8070	0.532552	0.266276	−	0.963897i	$-0.414207\pi$
$881$	15.8070	0.532552	0.266276	+	0.963897i	$0.414207\pi$
$882$	−10.4677	−0.352466
$883$	43.9965	1.48060	0.740300	−	0.672277i	$-0.234684\pi$
$883$	43.9965	1.48060	0.740300	+	0.672277i	$0.234684\pi$
$884$	0	0
$885$	0	0
$886$	15.0304	0.504956
$887$	9.71566	0.326220	0.163110	−	0.986608i	$-0.447847\pi$
$887$	9.71566	0.326220	0.163110	+	0.986608i	$0.447847\pi$
$888$	−11.8247	−0.396812
$889$	−45.5335	−1.52714
$890$	0	0
$891$	−40.2554	−1.34861
$892$	10.2582	0.343469
$893$	31.5856	1.05697
$894$	−58.7768	−1.96579
$895$	0	0
$896$	−2.60017	−0.0868655
$897$	88.5402	2.95627
$898$	−22.5556	−0.752691
$899$	3.11070	0.103748
$900$	0	0
$901$	0	0
$902$	29.6683	0.987846
$903$	−0.523762	−0.0174297
$904$	−15.5321	−0.516589
$905$	0	0
$906$	47.2491	1.56975
$907$	−24.7263	−0.821023	−0.410512	−	0.911855i	$-0.634650\pi$
$907$	−24.7263	−0.821023	−0.410512	+	0.911855i	$0.634650\pi$
$908$	11.0660	0.367239
$909$	−78.1876	−2.59332
$910$	0	0
$911$	−9.02165	−0.298901	−0.149450	−	0.988769i	$-0.547750\pi$
$911$	−9.02165	−0.298901	−0.149450	+	0.988769i	$0.547750\pi$
$912$	−47.6956	−1.57936
$913$	20.5429	0.679870
$914$	−7.07178	−0.233914
$915$	0	0
$916$	5.46593	0.180599
$917$	31.4380	1.03817
$918$	0	0
$919$	11.9804	0.395197	0.197599	−	0.980283i	$-0.436686\pi$
$919$	11.9804	0.395197	0.197599	+	0.980283i	$0.436686\pi$
$920$	0	0
$921$	24.4411	0.805361
$922$	25.7899	0.849346
$923$	−6.88247	−0.226539
$924$	19.2376	0.632870
$925$	0	0
$926$	−43.6440	−1.43423
$927$	10.5754	0.347341
$928$	−5.64692	−0.185369
$929$	11.1426	0.365577	0.182789	−	0.983152i	$-0.441488\pi$
$929$	11.1426	0.365577	0.182789	+	0.983152i	$0.441488\pi$
$930$	0	0
$931$	12.0032	0.393388
$932$	9.98834	0.327179
$933$	−48.3977	−1.58447
$934$	−9.81725	−0.321230
$935$	0	0
$936$	93.7013	3.06272
$937$	−0.0466154	−0.00152286	−0.000761429	−	1.00000i	$-0.500242\pi$
$937$	−0.0466154	−0.00152286	−0.000761429		1.00000i	$0.500242\pi$
$938$	29.7434	0.971156
$939$	−13.4365	−0.438485
$940$	0	0
$941$	−21.2527	−0.692818	−0.346409	−	0.938084i	$-0.612599\pi$
$941$	−21.2527	−0.692818	−0.346409	+	0.938084i	$0.612599\pi$
$942$	39.6307	1.29124
$943$	60.0886	1.95675
$944$	8.11610	0.264157
$945$	0	0
$946$	−0.177503	−0.00577114
$947$	1.89235	0.0614932	0.0307466	−	0.999527i	$-0.490212\pi$
$947$	1.89235	0.0614932	0.0307466	+	0.999527i	$0.490212\pi$
$948$	7.64816	0.248401
$949$	−73.9037	−2.39902
$950$	0	0
$951$	55.6417	1.80431
$952$	0	0
$953$	−27.3568	−0.886172	−0.443086	−	0.896479i	$-0.646116\pi$
$953$	−27.3568	−0.886172	−0.443086	+	0.896479i	$0.646116\pi$
$954$	61.2996	1.98465
$955$	0	0
$956$	3.34836	0.108294
$957$	−11.7423	−0.379575
$958$	−7.88849	−0.254866
$959$	49.6509	1.60331
$960$	0	0
$961$	−25.8935	−0.835273
$962$	6.37336	0.205485
$963$	87.8665	2.83146
$964$	−5.63231	−0.181405
$965$	0	0
$966$	−61.4022	−1.97558
$967$	−11.4902	−0.369500	−0.184750	−	0.982786i	$-0.559147\pi$
$967$	−11.4902	−0.369500	−0.184750	+	0.982786i	$0.559147\pi$
$968$	−10.4690	−0.336486
$969$	0	0
$970$	0	0
$971$	−17.4146	−0.558862	−0.279431	−	0.960166i	$-0.590146\pi$
$971$	−17.4146	−0.558862	−0.279431	+	0.960166i	$0.590146\pi$
$972$	−9.26588	−0.297203
$973$	−42.5335	−1.36356
$974$	7.73177	0.247742
$975$	0	0
$976$	19.4606	0.622917
$977$	−0.658377	−0.0210633	−0.0105317	−	0.999945i	$-0.503352\pi$
$977$	−0.658377	−0.0210633	−0.0105317	+	0.999945i	$0.503352\pi$
$978$	17.1503	0.548406
$979$	−28.8948	−0.923483
$980$	0	0
$981$	12.7864	0.408238
$982$	12.0866	0.385698
$983$	13.6618	0.435742	0.217871	−	0.975978i	$-0.430089\pi$
$983$	13.6618	0.435742	0.217871	+	0.975978i	$0.430089\pi$
$984$	92.5607	2.95073
$985$	0	0
$986$	0	0
$987$	34.0056	1.08241
$988$	−30.0471	−0.955924
$989$	−0.359506	−0.0114316
$990$	0	0
$991$	35.6573	1.13269	0.566346	−	0.824168i	$-0.308357\pi$
$991$	35.6573	1.13269	0.566346	+	0.824168i	$0.308357\pi$
$992$	−9.26999	−0.294323
$993$	−19.6802	−0.624531
$994$	4.77296	0.151389
$995$	0	0
$996$	17.9229	0.567909
$997$	20.0746	0.635768	0.317884	−	0.948130i	$-0.397028\pi$
$997$	20.0746	0.635768	0.317884	+	0.948130i	$0.397028\pi$
$998$	−29.0316	−0.918980
$999$	−13.8045	−0.436755

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7225.2.a.bw.1.5	yes	15
5.4	even	2		7225.2.a.bu.1.11	yes	15
17.16	even	2		7225.2.a.bt.1.5	✓	15
85.84	even	2		7225.2.a.bv.1.11	yes	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7225.2.a.bt.1.5	✓	15	17.16	even	2
7225.2.a.bu.1.11	yes	15	5.4	even	2
7225.2.a.bv.1.11	yes	15	85.84	even	2
7225.2.a.bw.1.5	yes	15	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7225 = 5^{2} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7225.a (trivial)

\(f(q)\)	\(=\)	\(q-1.10615 q^{2} +3.09602 q^{3} -0.776422 q^{4} -3.42468 q^{6} +2.90465 q^{7} +3.07115 q^{8} +6.58534 q^{9} +O(q^{10})\)	\(q-1.10615 q^{2} +3.09602 q^{3} -0.776422 q^{4} -3.42468 q^{6} +2.90465 q^{7} +3.07115 q^{8} +6.58534 q^{9} -2.75521 q^{11} -2.40382 q^{12} +4.63304 q^{13} -3.21299 q^{14} -1.84432 q^{16} -7.28440 q^{18} +8.35290 q^{19} +8.99286 q^{21} +3.04769 q^{22} +6.17263 q^{23} +9.50835 q^{24} -5.12486 q^{26} +11.1003 q^{27} -2.25524 q^{28} +1.37656 q^{29} +2.25976 q^{31} -4.10220 q^{32} -8.53019 q^{33} -5.11300 q^{36} -1.24362 q^{37} -9.23960 q^{38} +14.3440 q^{39} +9.73468 q^{41} -9.94749 q^{42} -0.0582420 q^{43} +2.13921 q^{44} -6.82788 q^{46} +3.78140 q^{47} -5.71006 q^{48} +1.43700 q^{49} -3.59720 q^{52} -8.41519 q^{53} -12.2786 q^{54} +8.92063 q^{56} +25.8607 q^{57} -1.52269 q^{58} -4.40059 q^{59} -10.5516 q^{61} -2.49965 q^{62} +19.1281 q^{63} +8.22631 q^{64} +9.43570 q^{66} -9.25722 q^{67} +19.1106 q^{69} -1.48552 q^{71} +20.2246 q^{72} -15.9514 q^{73} +1.37563 q^{74} -6.48538 q^{76} -8.00293 q^{77} -15.8667 q^{78} -3.18167 q^{79} +14.6107 q^{81} -10.7681 q^{82} -7.45601 q^{83} -6.98226 q^{84} +0.0644246 q^{86} +4.26185 q^{87} -8.46167 q^{88} +10.4873 q^{89} +13.4574 q^{91} -4.79257 q^{92} +6.99627 q^{93} -4.18281 q^{94} -12.7005 q^{96} -9.17821 q^{97} -1.58955 q^{98} -18.1440 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 9 q^{3} + 12 q^{4} + 9 q^{6} + 12 q^{7} + 6 q^{8} + 12 q^{9}+O(q^{10}) \) 15 * q + 9 * q^3 + 12 * q^4 + 9 * q^6 + 12 * q^7 + 6 * q^8 + 12 * q^9	\( 15 q + 9 q^{3} + 12 q^{4} + 9 q^{6} + 12 q^{7} + 6 q^{8} + 12 q^{9} + 6 q^{11} + 24 q^{12} + 6 q^{16} + 12 q^{18} + 6 q^{19} + 30 q^{21} + 12 q^{22} + 36 q^{23} + 18 q^{24} + 36 q^{26} + 36 q^{27} + 24 q^{28} - 18 q^{29} + 12 q^{32} + 12 q^{33} - 9 q^{36} + 12 q^{37} - 6 q^{38} + 9 q^{39} - 18 q^{41} + 36 q^{42} - 3 q^{43} - 12 q^{44} + 21 q^{46} - 3 q^{47} - 12 q^{48} + 15 q^{49} - 27 q^{52} + 21 q^{54} - 6 q^{56} + 39 q^{57} + 18 q^{58} - 12 q^{59} - 15 q^{61} + 54 q^{62} + 60 q^{63} - 36 q^{64} + 18 q^{66} - 24 q^{67} + 42 q^{69} + 6 q^{71} + 66 q^{72} - 9 q^{73} - 36 q^{74} - 18 q^{76} + 30 q^{77} + 30 q^{78} - 9 q^{79} + 51 q^{81} - 36 q^{82} + 15 q^{83} + 9 q^{84} - 36 q^{86} - 51 q^{87} + 30 q^{88} - 24 q^{89} + 27 q^{91} + 15 q^{92} - 42 q^{93} - 57 q^{94} + 42 q^{96} + 48 q^{97} + 12 q^{99}+O(q^{100}) \) 15 * q + 9 * q^3 + 12 * q^4 + 9 * q^6 + 12 * q^7 + 6 * q^8 + 12 * q^9 + 6 * q^11 + 24 * q^12 + 6 * q^16 + 12 * q^18 + 6 * q^19 + 30 * q^21 + 12 * q^22 + 36 * q^23 + 18 * q^24 + 36 * q^26 + 36 * q^27 + 24 * q^28 - 18 * q^29 + 12 * q^32 + 12 * q^33 - 9 * q^36 + 12 * q^37 - 6 * q^38 + 9 * q^39 - 18 * q^41 + 36 * q^42 - 3 * q^43 - 12 * q^44 + 21 * q^46 - 3 * q^47 - 12 * q^48 + 15 * q^49 - 27 * q^52 + 21 * q^54 - 6 * q^56 + 39 * q^57 + 18 * q^58 - 12 * q^59 - 15 * q^61 + 54 * q^62 + 60 * q^63 - 36 * q^64 + 18 * q^66 - 24 * q^67 + 42 * q^69 + 6 * q^71 + 66 * q^72 - 9 * q^73 - 36 * q^74 - 18 * q^76 + 30 * q^77 + 30 * q^78 - 9 * q^79 + 51 * q^81 - 36 * q^82 + 15 * q^83 + 9 * q^84 - 36 * q^86 - 51 * q^87 + 30 * q^88 - 24 * q^89 + 27 * q^91 + 15 * q^92 - 42 * q^93 - 57 * q^94 + 42 * q^96 + 48 * q^97 + 12 * q^99

Introduction

L-functions

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Database

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