LMFDB - Embedded newform 3025.2.a.bj.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3025,2,Mod(1,3025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3025, 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("3025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3025 = 5^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3025.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:	$24.1547466114$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 7x^{6} + 18x^{5} + 16x^{4} - 30x^{3} - 12x^{2} + 12x + 4 $ x^8 - 3x^7 - 7x^6 + 18x^5 + 16x^4 - 30x^3 - 12x^2 + 12*x + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 275)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$3.29384$ of defining polynomial
Character	$\chi$	$=$	3025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.29384 q^{2} -0.0424059 q^{3} +3.26171 q^{4} -0.0972724 q^{6} -1.13968 q^{7} +2.89416 q^{8} -2.99820 q^{9} +O(q^{10})$	\(q+2.29384 q^{2} -0.0424059 q^{3} +3.26171 q^{4} -0.0972724 q^{6} -1.13968 q^{7} +2.89416 q^{8} -2.99820 q^{9} -0.138316 q^{12} -2.80970 q^{13} -2.61424 q^{14} +0.115334 q^{16} -7.73764 q^{17} -6.87740 q^{18} -3.28300 q^{19} +0.0483291 q^{21} +5.54471 q^{23} -0.122730 q^{24} -6.44501 q^{26} +0.254359 q^{27} -3.71730 q^{28} +6.96286 q^{29} -0.746958 q^{31} -5.52377 q^{32} -17.7489 q^{34} -9.77927 q^{36} -4.81446 q^{37} -7.53069 q^{38} +0.119148 q^{39} -6.66398 q^{41} +0.110859 q^{42} +0.698596 q^{43} +12.7187 q^{46} -8.64868 q^{47} -0.00489083 q^{48} -5.70113 q^{49} +0.328121 q^{51} -9.16443 q^{52} +8.76226 q^{53} +0.583459 q^{54} -3.29842 q^{56} +0.139219 q^{57} +15.9717 q^{58} +10.6321 q^{59} -9.25529 q^{61} -1.71340 q^{62} +3.41699 q^{63} -12.9013 q^{64} -6.69671 q^{67} -25.2379 q^{68} -0.235128 q^{69} +7.46035 q^{71} -8.67729 q^{72} -1.52785 q^{73} -11.0436 q^{74} -10.7082 q^{76} +0.273306 q^{78} +10.0289 q^{79} +8.98382 q^{81} -15.2861 q^{82} -4.89694 q^{83} +0.157635 q^{84} +1.60247 q^{86} -0.295266 q^{87} -9.00636 q^{89} +3.20216 q^{91} +18.0852 q^{92} +0.0316754 q^{93} -19.8387 q^{94} +0.234240 q^{96} -5.92237 q^{97} -13.0775 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 5 q^{2} + q^{3} + 9 q^{4} - q^{6} - 8 q^{7} - 12 q^{8} + 9 q^{9}+O(q^{10}) $ 8 * q - 5 * q^2 + q^3 + 9 * q^4 - q^6 - 8 * q^7 - 12 * q^8 + 9 * q^9	$ 8 q - 5 q^{2} + q^{3} + 9 q^{4} - q^{6} - 8 q^{7} - 12 q^{8} + 9 q^{9} + 3 q^{12} - 9 q^{13} + 9 q^{14} + 23 q^{16} - 19 q^{17} - 22 q^{18} - q^{19} - 5 q^{21} + 2 q^{23} - q^{24} - 2 q^{26} - 2 q^{27} - 9 q^{28} - 7 q^{29} - 5 q^{31} - 29 q^{32} + 10 q^{34} - 16 q^{36} - 8 q^{37} - 37 q^{38} + q^{39} + 41 q^{42} - 14 q^{43} + 20 q^{46} + 11 q^{47} - 27 q^{48} - 12 q^{49} + 25 q^{51} + 7 q^{52} + 11 q^{53} - 30 q^{54} - 10 q^{56} + 2 q^{57} + 27 q^{58} + 17 q^{59} + 2 q^{61} - 25 q^{62} - 41 q^{63} + 30 q^{64} + 7 q^{67} - 66 q^{68} + 17 q^{71} + 19 q^{72} - 34 q^{73} + 6 q^{74} + 31 q^{76} - 17 q^{78} - 23 q^{79} - 4 q^{81} - 17 q^{82} - 41 q^{83} - 83 q^{84} + q^{86} - 25 q^{87} - 11 q^{89} - 7 q^{91} + 33 q^{92} - 59 q^{93} - 50 q^{94} + 61 q^{96} + 2 q^{97} - 26 q^{98}+O(q^{100}) $ 8 * q - 5 * q^2 + q^3 + 9 * q^4 - q^6 - 8 * q^7 - 12 * q^8 + 9 * q^9 + 3 * q^12 - 9 * q^13 + 9 * q^14 + 23 * q^16 - 19 * q^17 - 22 * q^18 - q^19 - 5 * q^21 + 2 * q^23 - q^24 - 2 * q^26 - 2 * q^27 - 9 * q^28 - 7 * q^29 - 5 * q^31 - 29 * q^32 + 10 * q^34 - 16 * q^36 - 8 * q^37 - 37 * q^38 + q^39 + 41 * q^42 - 14 * q^43 + 20 * q^46 + 11 * q^47 - 27 * q^48 - 12 * q^49 + 25 * q^51 + 7 * q^52 + 11 * q^53 - 30 * q^54 - 10 * q^56 + 2 * q^57 + 27 * q^58 + 17 * q^59 + 2 * q^61 - 25 * q^62 - 41 * q^63 + 30 * q^64 + 7 * q^67 - 66 * q^68 + 17 * q^71 + 19 * q^72 - 34 * q^73 + 6 * q^74 + 31 * q^76 - 17 * q^78 - 23 * q^79 - 4 * q^81 - 17 * q^82 - 41 * q^83 - 83 * q^84 + q^86 - 25 * q^87 - 11 * q^89 - 7 * q^91 + 33 * q^92 - 59 * q^93 - 50 * q^94 + 61 * q^96 + 2 * q^97 - 26 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.29384	1.62199	0.810996	−	0.585052i	$-0.198926\pi$
$2$	2.29384	1.62199	0.810996	+	0.585052i	$0.198926\pi$
$3$	−0.0424059	−0.0244830	−0.0122415	−	0.999925i	$-0.503897\pi$
$3$	−0.0424059	−0.0244830	−0.0122415	+	0.999925i	$0.503897\pi$
$4$	3.26171	1.63086
$5$	0	0
$5$	0	0
$6$	−0.0972724	−0.0397113
$7$	−1.13968	−0.430758	−0.215379	−	0.976531i	$-0.569099\pi$
$7$	−1.13968	−0.430758	−0.215379	+	0.976531i	$0.569099\pi$
$8$	2.89416	1.02324
$9$	−2.99820	−0.999401
$10$	0	0
$11$	0	0
$11$	0	0
$12$	−0.138316	−0.0399283
$13$	−2.80970	−0.779271	−0.389636	−	0.920969i	$-0.627399\pi$
$13$	−2.80970	−0.779271	−0.389636	+	0.920969i	$0.627399\pi$
$14$	−2.61424	−0.698685
$15$	0	0
$16$	0.115334	0.0288335
$17$	−7.73764	−1.87665	−0.938326	−	0.345751i	$-0.887624\pi$
$17$	−7.73764	−1.87665	−0.938326	+	0.345751i	$0.887624\pi$
$18$	−6.87740	−1.62102
$19$	−3.28300	−0.753172	−0.376586	−	0.926382i	$-0.622902\pi$
$19$	−3.28300	−0.753172	−0.376586	+	0.926382i	$0.622902\pi$
$20$	0	0
$21$	0.0483291	0.0105463
$22$	0	0
$23$	5.54471	1.15615	0.578076	−	0.815983i	$-0.303804\pi$
$23$	5.54471	1.15615	0.578076	+	0.815983i	$0.303804\pi$
$24$	−0.122730	−0.0250521
$25$	0	0
$26$	−6.44501	−1.26397
$27$	0.254359	0.0489514
$28$	−3.71730	−0.702504
$29$	6.96286	1.29297	0.646486	−	0.762926i	$-0.276238\pi$
$29$	6.96286	1.29297	0.646486	+	0.762926i	$0.276238\pi$
$30$	0	0
$31$	−0.746958	−0.134158	−0.0670788	−	0.997748i	$-0.521368\pi$
$31$	−0.746958	−0.134158	−0.0670788	+	0.997748i	$0.521368\pi$
$32$	−5.52377	−0.976474
$33$	0	0
$34$	−17.7489	−3.04391
$35$	0	0
$36$	−9.77927	−1.62988
$37$	−4.81446	−0.791493	−0.395747	−	0.918360i	$-0.629514\pi$
$37$	−4.81446	−0.791493	−0.395747	+	0.918360i	$0.629514\pi$
$38$	−7.53069	−1.22164
$39$	0.119148	0.0190789
$40$	0	0
$41$	−6.66398	−1.04074	−0.520369	−	0.853941i	$-0.674206\pi$
$41$	−6.66398	−1.04074	−0.520369	+	0.853941i	$0.674206\pi$
$42$	0.110859	0.0171059
$43$	0.698596	0.106535	0.0532674	−	0.998580i	$-0.483036\pi$
$43$	0.698596	0.106535	0.0532674	+	0.998580i	$0.483036\pi$
$44$	0	0
$45$	0	0
$46$	12.7187	1.87527
$47$	−8.64868	−1.26154	−0.630770	−	0.775970i	$-0.717261\pi$
$47$	−8.64868	−1.26154	−0.630770	+	0.775970i	$0.717261\pi$
$48$	−0.00489083	−0.000705931 0
$49$	−5.70113	−0.814448
$50$	0	0
$51$	0.328121	0.0459462
$52$	−9.16443	−1.27088
$53$	8.76226	1.20359	0.601795	−	0.798651i	$-0.294453\pi$
$53$	8.76226	1.20359	0.601795	+	0.798651i	$0.294453\pi$
$54$	0.583459	0.0793988
$55$	0	0
$56$	−3.29842	−0.440769
$57$	0.139219	0.0184399
$58$	15.9717	2.09719
$59$	10.6321	1.38419	0.692093	−	0.721809i	$-0.256689\pi$
$59$	10.6321	1.38419	0.692093	+	0.721809i	$0.256689\pi$
$60$	0	0
$61$	−9.25529	−1.18502	−0.592509	−	0.805564i	$-0.701863\pi$
$61$	−9.25529	−1.18502	−0.592509	+	0.805564i	$0.701863\pi$
$62$	−1.71340	−0.217602
$63$	3.41699	0.430500
$64$	−12.9013	−1.61267
$65$	0	0
$66$	0	0
$67$	−6.69671	−0.818133	−0.409066	−	0.912505i	$-0.634146\pi$
$67$	−6.69671	−0.818133	−0.409066	+	0.912505i	$0.634146\pi$
$68$	−25.2379	−3.06055
$69$	−0.235128	−0.0283061
$70$	0	0
$71$	7.46035	0.885380	0.442690	−	0.896675i	$-0.354024\pi$
$71$	7.46035	0.885380	0.442690	+	0.896675i	$0.354024\pi$
$72$	−8.67729	−1.02263
$73$	−1.52785	−0.178821	−0.0894107	−	0.995995i	$-0.528498\pi$
$73$	−1.52785	−0.178821	−0.0894107	+	0.995995i	$0.528498\pi$
$74$	−11.0436	−1.28379
$75$	0	0
$76$	−10.7082	−1.22831
$77$	0	0
$78$	0.273306	0.0309459
$79$	10.0289	1.12834	0.564171	−	0.825658i	$-0.309196\pi$
$79$	10.0289	1.12834	0.564171	+	0.825658i	$0.309196\pi$
$80$	0	0
$81$	8.98382	0.998202
$82$	−15.2861	−1.68807
$83$	−4.89694	−0.537509	−0.268755	−	0.963209i	$-0.586612\pi$
$83$	−4.89694	−0.537509	−0.268755	+	0.963209i	$0.586612\pi$
$84$	0.157635	0.0171994
$85$	0	0
$86$	1.60247	0.172799
$87$	−0.295266	−0.0316559
$88$	0	0
$89$	−9.00636	−0.954672	−0.477336	−	0.878721i	$-0.658398\pi$
$89$	−9.00636	−0.954672	−0.477336	+	0.878721i	$0.658398\pi$
$90$	0	0
$91$	3.20216	0.335677
$92$	18.0852	1.88552
$93$	0.0316754	0.00328459
$94$	−19.8387	−2.04621
$95$	0	0
$96$	0.234240	0.0239071
$97$	−5.92237	−0.601325	−0.300663	−	0.953731i	$-0.597208\pi$
$97$	−5.92237	−0.601325	−0.300663	+	0.953731i	$0.597208\pi$
$98$	−13.0775	−1.32103
$99$	0	0
$100$	0	0
$101$	9.34119	0.929483	0.464742	−	0.885446i	$-0.346147\pi$
$101$	9.34119	0.929483	0.464742	+	0.885446i	$0.346147\pi$
$102$	0.752658	0.0745243
$103$	10.4100	1.02572	0.512862	−	0.858471i	$-0.328585\pi$
$103$	10.4100	1.02572	0.512862	+	0.858471i	$0.328585\pi$
$104$	−8.13174	−0.797383
$105$	0	0
$106$	20.0992	1.95221
$107$	−4.42524	−0.427804	−0.213902	−	0.976855i	$-0.568617\pi$
$107$	−4.42524	−0.427804	−0.213902	+	0.976855i	$0.568617\pi$
$108$	0.829645	0.0798327
$109$	3.22523	0.308921	0.154460	−	0.987999i	$-0.450636\pi$
$109$	3.22523	0.308921	0.154460	+	0.987999i	$0.450636\pi$
$110$	0	0
$111$	0.204162	0.0193782
$112$	−0.131443	−0.0124202
$113$	4.85509	0.456728	0.228364	−	0.973576i	$-0.426662\pi$
$113$	4.85509	0.456728	0.228364	+	0.973576i	$0.426662\pi$
$114$	0.319345	0.0299094
$115$	0	0
$116$	22.7108	2.10865
$117$	8.42405	0.778804
$118$	24.3884	2.24514
$119$	8.81842	0.808383
$120$	0	0
$121$	0	0
$122$	−21.2302	−1.92209
$123$	0.282592	0.0254804
$124$	−2.43636	−0.218792
$125$	0	0
$126$	7.83802	0.698267
$127$	15.6539	1.38906	0.694528	−	0.719466i	$-0.255613\pi$
$127$	15.6539	1.38906	0.694528	+	0.719466i	$0.255613\pi$
$128$	−18.5461	−1.63926
$129$	−0.0296246	−0.00260830
$130$	0	0
$131$	21.8905	1.91258	0.956291	−	0.292415i	$-0.0944589\pi$
$131$	21.8905	1.91258	0.956291	+	0.292415i	$0.0944589\pi$
$132$	0	0
$133$	3.74157	0.324435
$134$	−15.3612	−1.32700
$135$	0	0
$136$	−22.3940	−1.92027
$137$	9.57907	0.818395	0.409197	−	0.912446i	$-0.365809\pi$
$137$	9.57907	0.818395	0.409197	+	0.912446i	$0.365809\pi$
$138$	−0.539347	−0.0459123
$139$	−12.5047	−1.06064	−0.530318	−	0.847799i	$-0.677927\pi$
$139$	−12.5047	−1.06064	−0.530318	+	0.847799i	$0.677927\pi$
$140$	0	0
$141$	0.366755	0.0308863
$142$	17.1129	1.43608
$143$	0	0
$144$	−0.345794	−0.0288162
$145$	0	0
$146$	−3.50465	−0.290047
$147$	0.241762	0.0199402
$148$	−15.7034	−1.29081
$149$	−8.30659	−0.680502	−0.340251	−	0.940335i	$-0.610512\pi$
$149$	−8.30659	−0.680502	−0.340251	+	0.940335i	$0.610512\pi$
$150$	0	0
$151$	−12.9295	−1.05218	−0.526092	−	0.850427i	$-0.676343\pi$
$151$	−12.9295	−1.05218	−0.526092	+	0.850427i	$0.676343\pi$
$152$	−9.50155	−0.770677
$153$	23.1990	1.87553
$154$	0	0
$155$	0	0
$156$	0.388626	0.0311150
$157$	7.70468	0.614900	0.307450	−	0.951564i	$-0.400524\pi$
$157$	7.70468	0.614900	0.307450	+	0.951564i	$0.400524\pi$
$158$	23.0047	1.83016
$159$	−0.371571	−0.0294675
$160$	0	0
$161$	−6.31919	−0.498022
$162$	20.6075	1.61907
$163$	−21.6793	−1.69806	−0.849029	−	0.528347i	$-0.822812\pi$
$163$	−21.6793	−1.69806	−0.849029	+	0.528347i	$0.822812\pi$
$164$	−21.7360	−1.69729
$165$	0	0
$166$	−11.2328	−0.871835
$167$	−14.2142	−1.09993	−0.549964	−	0.835188i	$-0.685359\pi$
$167$	−14.2142	−1.09993	−0.549964	+	0.835188i	$0.685359\pi$
$168$	0.139872	0.0107914
$169$	−5.10557	−0.392736
$170$	0	0
$171$	9.84310	0.752721
$172$	2.27862	0.173743
$173$	−19.0351	−1.44721	−0.723606	−	0.690214i	$-0.757517\pi$
$173$	−19.0351	−1.44721	−0.723606	+	0.690214i	$0.757517\pi$
$174$	−0.677294	−0.0513456
$175$	0	0
$176$	0	0
$177$	−0.450865	−0.0338891
$178$	−20.6592	−1.54847
$179$	−10.6379	−0.795118	−0.397559	−	0.917577i	$-0.630143\pi$
$179$	−10.6379	−0.795118	−0.397559	+	0.917577i	$0.630143\pi$
$180$	0	0
$181$	−15.3635	−1.14196	−0.570981	−	0.820964i	$-0.693437\pi$
$181$	−15.3635	−1.14196	−0.570981	+	0.820964i	$0.693437\pi$
$182$	7.34524	0.544465
$183$	0.392479	0.0290129
$184$	16.0473	1.18302
$185$	0	0
$186$	0.0726584	0.00532757
$187$	0	0
$188$	−28.2095	−2.05739
$189$	−0.289887	−0.0210862
$190$	0	0
$191$	14.9133	1.07909	0.539543	−	0.841958i	$-0.318597\pi$
$191$	14.9133	1.07909	0.539543	+	0.841958i	$0.318597\pi$
$192$	0.547092	0.0394830
$193$	−6.68480	−0.481182	−0.240591	−	0.970627i	$-0.577341\pi$
$193$	−6.68480	−0.481182	−0.240591	+	0.970627i	$0.577341\pi$
$194$	−13.5850	−0.975344
$195$	0	0
$196$	−18.5954	−1.32825
$197$	−8.96183	−0.638504	−0.319252	−	0.947670i	$-0.603432\pi$
$197$	−8.96183	−0.638504	−0.319252	+	0.947670i	$0.603432\pi$
$198$	0	0
$199$	13.7830	0.977053	0.488527	−	0.872549i	$-0.337534\pi$
$199$	13.7830	0.977053	0.488527	+	0.872549i	$0.337534\pi$
$200$	0	0
$201$	0.283980	0.0200304
$202$	21.4272	1.50761
$203$	−7.93542	−0.556958
$204$	1.07024	0.0749315
$205$	0	0
$206$	23.8788	1.66372
$207$	−16.6242	−1.15546
$208$	−0.324054	−0.0224691
$209$	0	0
$210$	0	0
$211$	−8.78478	−0.604769	−0.302385	−	0.953186i	$-0.597783\pi$
$211$	−8.78478	−0.604769	−0.302385	+	0.953186i	$0.597783\pi$
$212$	28.5800	1.96288
$213$	−0.316363	−0.0216768
$214$	−10.1508	−0.693895
$215$	0	0
$216$	0.736157	0.0500891
$217$	0.851292	0.0577894
$218$	7.39817	0.501067
$219$	0.0647899	0.00437809
$220$	0	0
$221$	21.7405	1.46242
$222$	0.468314	0.0314312
$223$	9.03661	0.605136	0.302568	−	0.953128i	$-0.402156\pi$
$223$	9.03661	0.605136	0.302568	+	0.953128i	$0.402156\pi$
$224$	6.29532	0.420624
$225$	0	0
$226$	11.1368	0.740809
$227$	−5.68198	−0.377127	−0.188563	−	0.982061i	$-0.560383\pi$
$227$	−5.68198	−0.377127	−0.188563	+	0.982061i	$0.560383\pi$
$228$	0.454091	0.0300729
$229$	5.89416	0.389497	0.194749	−	0.980853i	$-0.437611\pi$
$229$	5.89416	0.389497	0.194749	+	0.980853i	$0.437611\pi$
$230$	0	0
$231$	0	0
$232$	20.1517	1.32302
$233$	0.0409351	0.00268175	0.00134087	−	0.999999i	$-0.499573\pi$
$233$	0.0409351	0.00268175	0.00134087	+	0.999999i	$0.499573\pi$
$234$	19.3234	1.26321
$235$	0	0
$236$	34.6789	2.25741
$237$	−0.425285	−0.0276252
$238$	20.2281	1.31119
$239$	7.53414	0.487343	0.243672	−	0.969858i	$-0.421648\pi$
$239$	7.53414	0.487343	0.243672	+	0.969858i	$0.421648\pi$
$240$	0	0
$241$	0.718212	0.0462641	0.0231321	−	0.999732i	$-0.492636\pi$
$241$	0.718212	0.0462641	0.0231321	+	0.999732i	$0.492636\pi$
$242$	0	0
$243$	−1.14404	−0.0733904
$244$	−30.1881	−1.93259
$245$	0	0
$246$	0.648221	0.0413290
$247$	9.22426	0.586925
$248$	−2.16182	−0.137276
$249$	0.207659	0.0131599
$250$	0	0
$251$	−6.45420	−0.407386	−0.203693	−	0.979035i	$-0.565294\pi$
$251$	−6.45420	−0.407386	−0.203693	+	0.979035i	$0.565294\pi$
$252$	11.1452	0.702083
$253$	0	0
$254$	35.9075	2.25304
$255$	0	0
$256$	−16.7391	−1.04619
$257$	−23.4402	−1.46216	−0.731080	−	0.682292i	$-0.760983\pi$
$257$	−23.4402	−1.46216	−0.731080	+	0.682292i	$0.760983\pi$
$258$	−0.0679541	−0.00423064
$259$	5.48694	0.340942
$260$	0	0
$261$	−20.8761	−1.29220
$262$	50.2134	3.10219
$263$	−20.1226	−1.24081	−0.620405	−	0.784281i	$-0.713032\pi$
$263$	−20.1226	−1.24081	−0.620405	+	0.784281i	$0.713032\pi$
$264$	0	0
$265$	0	0
$266$	8.58256	0.526230
$267$	0.381923	0.0233733
$268$	−21.8427	−1.33426
$269$	0.704825	0.0429739	0.0214870	−	0.999769i	$-0.493160\pi$
$269$	0.704825	0.0429739	0.0214870	+	0.999769i	$0.493160\pi$
$270$	0	0
$271$	20.4338	1.24126	0.620631	−	0.784103i	$-0.286876\pi$
$271$	20.4338	1.24126	0.620631	+	0.784103i	$0.286876\pi$
$272$	−0.892412	−0.0541104
$273$	−0.135790	−0.00821840
$274$	21.9729	1.32743
$275$	0	0
$276$	−0.766921	−0.0461632
$277$	−15.3420	−0.921812	−0.460906	−	0.887449i	$-0.652475\pi$
$277$	−15.3420	−0.921812	−0.460906	+	0.887449i	$0.652475\pi$
$278$	−28.6838	−1.72034
$279$	2.23953	0.134077
$280$	0	0
$281$	−13.8322	−0.825157	−0.412579	−	0.910922i	$-0.635372\pi$
$281$	−13.8322	−0.825157	−0.412579	+	0.910922i	$0.635372\pi$
$282$	0.841278	0.0500974
$283$	−16.9958	−1.01030	−0.505148	−	0.863033i	$-0.668562\pi$
$283$	−16.9958	−1.01030	−0.505148	+	0.863033i	$0.668562\pi$
$284$	24.3335	1.44393
$285$	0	0
$286$	0	0
$287$	7.59479	0.448306
$288$	16.5614	0.975889
$289$	42.8710	2.52182
$290$	0	0
$291$	0.251143	0.0147223
$292$	−4.98341	−0.291632
$293$	−0.839450	−0.0490412	−0.0245206	−	0.999699i	$-0.507806\pi$
$293$	−0.839450	−0.0490412	−0.0245206	+	0.999699i	$0.507806\pi$
$294$	0.554563	0.0323428
$295$	0	0
$296$	−13.9338	−0.809889
$297$	0	0
$298$	−19.0540	−1.10377
$299$	−15.5790	−0.900956
$300$	0	0
$301$	−0.796175	−0.0458907
$302$	−29.6581	−1.70663
$303$	−0.396121	−0.0227566
$304$	−0.378641	−0.0217166
$305$	0	0
$306$	53.2148	3.04209
$307$	−6.41496	−0.366121	−0.183060	−	0.983102i	$-0.558600\pi$
$307$	−6.41496	−0.366121	−0.183060	+	0.983102i	$0.558600\pi$
$308$	0	0
$309$	−0.441444	−0.0251129
$310$	0	0
$311$	30.3839	1.72291	0.861457	−	0.507831i	$-0.169553\pi$
$311$	30.3839	1.72291	0.861457	+	0.507831i	$0.169553\pi$
$312$	0.344834	0.0195224
$313$	−2.35148	−0.132914	−0.0664568	−	0.997789i	$-0.521169\pi$
$313$	−2.35148	−0.132914	−0.0664568	+	0.997789i	$0.521169\pi$
$314$	17.6733	0.997363
$315$	0	0
$316$	32.7114	1.84016
$317$	−0.803582	−0.0451337	−0.0225668	−	0.999745i	$-0.507184\pi$
$317$	−0.803582	−0.0451337	−0.0225668	+	0.999745i	$0.507184\pi$
$318$	−0.852326	−0.0477961
$319$	0	0
$320$	0	0
$321$	0.187656	0.0104740
$322$	−14.4952	−0.807787
$323$	25.4027	1.41344
$324$	29.3026	1.62792
$325$	0	0
$326$	−49.7290	−2.75423
$327$	−0.136769	−0.00756333
$328$	−19.2866	−1.06493
$329$	9.85672	0.543418
$330$	0	0
$331$	29.5735	1.62551	0.812753	−	0.582608i	$-0.197968\pi$
$331$	29.5735	1.62551	0.812753	+	0.582608i	$0.197968\pi$
$332$	−15.9724	−0.876600
$333$	14.4347	0.791019
$334$	−32.6052	−1.78407
$335$	0	0
$336$	0.00557398	0.000304085 0
$337$	−28.4781	−1.55130	−0.775651	−	0.631163i	$-0.782578\pi$
$337$	−28.4781	−1.55130	−0.775651	+	0.631163i	$0.782578\pi$
$338$	−11.7114	−0.637015
$339$	−0.205884	−0.0111821
$340$	0	0
$341$	0	0
$342$	22.5785	1.22091
$343$	14.4752	0.781588
$344$	2.02185	0.109011
$345$	0	0
$346$	−43.6635	−2.34736
$347$	−8.07245	−0.433352	−0.216676	−	0.976244i	$-0.569522\pi$
$347$	−8.07245	−0.433352	−0.216676	+	0.976244i	$0.569522\pi$
$348$	−0.963073	−0.0516262
$349$	30.0062	1.60619	0.803097	−	0.595848i	$-0.203184\pi$
$349$	30.0062	1.60619	0.803097	+	0.595848i	$0.203184\pi$
$350$	0	0
$351$	−0.714673	−0.0381464
$352$	0	0
$353$	3.17893	0.169197	0.0845987	−	0.996415i	$-0.473039\pi$
$353$	3.17893	0.169197	0.0845987	+	0.996415i	$0.473039\pi$
$354$	−1.03421	−0.0549678
$355$	0	0
$356$	−29.3761	−1.55693
$357$	−0.373953	−0.0197917
$358$	−24.4018	−1.28967
$359$	2.93152	0.154720	0.0773598	−	0.997003i	$-0.475351\pi$
$359$	2.93152	0.154720	0.0773598	+	0.997003i	$0.475351\pi$
$360$	0	0
$361$	−8.22190	−0.432732
$362$	−35.2415	−1.85225
$363$	0	0
$364$	10.4445	0.547441
$365$	0	0
$366$	0.900284	0.0470586
$367$	−18.2980	−0.955148	−0.477574	−	0.878592i	$-0.658484\pi$
$367$	−18.2980	−0.955148	−0.477574	+	0.878592i	$0.658484\pi$
$368$	0.639493	0.0333359
$369$	19.9799	1.04011
$370$	0	0
$371$	−9.98616	−0.518456
$372$	0.103316	0.00535669
$373$	8.37860	0.433827	0.216914	−	0.976191i	$-0.430401\pi$
$373$	8.37860	0.433827	0.216914	+	0.976191i	$0.430401\pi$
$374$	0	0
$375$	0	0
$376$	−25.0307	−1.29086
$377$	−19.5636	−1.00758
$378$	−0.664956	−0.0342016
$379$	−14.3396	−0.736576	−0.368288	−	0.929712i	$-0.620056\pi$
$379$	−14.3396	−0.736576	−0.368288	+	0.929712i	$0.620056\pi$
$380$	0	0
$381$	−0.663816	−0.0340083
$382$	34.2087	1.75027
$383$	27.1972	1.38971	0.694856	−	0.719149i	$-0.255468\pi$
$383$	27.1972	1.38971	0.694856	+	0.719149i	$0.255468\pi$
$384$	0.786462	0.0401340
$385$	0	0
$386$	−15.3339	−0.780473
$387$	−2.09453	−0.106471
$388$	−19.3170	−0.980674
$389$	20.0526	1.01671	0.508353	−	0.861149i	$-0.330254\pi$
$389$	20.0526	1.01671	0.508353	+	0.861149i	$0.330254\pi$
$390$	0	0
$391$	−42.9030	−2.16970
$392$	−16.5000	−0.833377
$393$	−0.928286	−0.0468259
$394$	−20.5570	−1.03565
$395$	0	0
$396$	0	0
$397$	29.4343	1.47727	0.738633	−	0.674108i	$-0.235472\pi$
$397$	29.4343	1.47727	0.738633	+	0.674108i	$0.235472\pi$
$398$	31.6161	1.58477
$399$	−0.158664	−0.00794315
$400$	0	0
$401$	19.2235	0.959976	0.479988	−	0.877275i	$-0.340641\pi$
$401$	19.2235	0.959976	0.479988	+	0.877275i	$0.340641\pi$
$402$	0.651405	0.0324891
$403$	2.09873	0.104545
$404$	30.4683	1.51585
$405$	0	0
$406$	−18.2026	−0.903380
$407$	0	0
$408$	0.949637	0.0470140
$409$	12.7912	0.632482	0.316241	−	0.948679i	$-0.397579\pi$
$409$	12.7912	0.632482	0.316241	+	0.948679i	$0.397579\pi$
$410$	0	0
$411$	−0.406209	−0.0200368
$412$	33.9543	1.67281
$413$	−12.1172	−0.596249
$414$	−38.1332	−1.87414
$415$	0	0
$416$	15.5201	0.760938
$417$	0.530273	0.0259676
$418$	0	0
$419$	4.22237	0.206276	0.103138	−	0.994667i	$-0.467112\pi$
$419$	4.22237	0.206276	0.103138	+	0.994667i	$0.467112\pi$
$420$	0	0
$421$	0.818215	0.0398773	0.0199387	−	0.999801i	$-0.493653\pi$
$421$	0.818215	0.0398773	0.0199387	+	0.999801i	$0.493653\pi$
$422$	−20.1509	−0.980931
$423$	25.9305	1.26078
$424$	25.3594	1.23156
$425$	0	0
$426$	−0.725686	−0.0351596
$427$	10.5481	0.510456
$428$	−14.4339	−0.697687
$429$	0	0
$430$	0	0
$431$	5.80964	0.279841	0.139920	−	0.990163i	$-0.455315\pi$
$431$	5.80964	0.279841	0.139920	+	0.990163i	$0.455315\pi$
$432$	0.0293362	0.00141144
$433$	−15.6176	−0.750535	−0.375268	−	0.926917i	$-0.622449\pi$
$433$	−15.6176	−0.750535	−0.375268	+	0.926917i	$0.622449\pi$
$434$	1.95273	0.0937340
$435$	0	0
$436$	10.5198	0.503805
$437$	−18.2033	−0.870782
$438$	0.148618	0.00710123
$439$	−21.6614	−1.03384	−0.516922	−	0.856032i	$-0.672922\pi$
$439$	−21.6614	−1.03384	−0.516922	+	0.856032i	$0.672922\pi$
$440$	0	0
$441$	17.0931	0.813959
$442$	49.8692	2.37203
$443$	−28.7252	−1.36478	−0.682389	−	0.730989i	$-0.739059\pi$
$443$	−28.7252	−1.36478	−0.682389	+	0.730989i	$0.739059\pi$
$444$	0.665916	0.0316030
$445$	0	0
$446$	20.7285	0.981525
$447$	0.352248	0.0166608
$448$	14.7034	0.694668
$449$	17.5325	0.827410	0.413705	−	0.910411i	$-0.364234\pi$
$449$	17.5325	0.827410	0.413705	+	0.910411i	$0.364234\pi$
$450$	0	0
$451$	0	0
$452$	15.8359	0.744858
$453$	0.548285	0.0257607
$454$	−13.0336	−0.611696
$455$	0	0
$456$	0.402921	0.0188685
$457$	−19.7587	−0.924274	−0.462137	−	0.886809i	$-0.652917\pi$
$457$	−19.7587	−0.924274	−0.462137	+	0.886809i	$0.652917\pi$
$458$	13.5203	0.631761
$459$	−1.96814	−0.0918648
$460$	0	0
$461$	12.9859	0.604812	0.302406	−	0.953179i	$-0.402210\pi$
$461$	12.9859	0.604812	0.302406	+	0.953179i	$0.402210\pi$
$462$	0	0
$463$	9.66418	0.449133	0.224566	−	0.974459i	$-0.427903\pi$
$463$	9.66418	0.449133	0.224566	+	0.974459i	$0.427903\pi$
$464$	0.803054	0.0372808
$465$	0	0
$466$	0.0938986	0.00434977
$467$	9.69415	0.448592	0.224296	−	0.974521i	$-0.427992\pi$
$467$	9.69415	0.448592	0.224296	+	0.974521i	$0.427992\pi$
$468$	27.4768	1.27012
$469$	7.63209	0.352417
$470$	0	0
$471$	−0.326724	−0.0150546
$472$	30.7711	1.41636
$473$	0	0
$474$	−0.975537	−0.0448079
$475$	0	0
$476$	28.7631	1.31836
$477$	−26.2710	−1.20287
$478$	17.2821	0.790466
$479$	−5.22980	−0.238955	−0.119478	−	0.992837i	$-0.538122\pi$
$479$	−5.22980	−0.238955	−0.119478	+	0.992837i	$0.538122\pi$
$480$	0	0
$481$	13.5272	0.616788
$482$	1.64747	0.0750400
$483$	0.267971	0.0121931
$484$	0	0
$485$	0	0
$486$	−2.62426	−0.119039
$487$	5.45313	0.247105	0.123553	−	0.992338i	$-0.460571\pi$
$487$	5.45313	0.247105	0.123553	+	0.992338i	$0.460571\pi$
$488$	−26.7863	−1.21256
$489$	0.919332	0.0415736
$490$	0	0
$491$	−16.1990	−0.731052	−0.365526	−	0.930801i	$-0.619111\pi$
$491$	−16.1990	−0.731052	−0.365526	+	0.930801i	$0.619111\pi$
$492$	0.921732	0.0415549
$493$	−53.8761	−2.42646
$494$	21.1590	0.951988
$495$	0	0
$496$	−0.0861495	−0.00386823
$497$	−8.50240	−0.381385
$498$	0.476337	0.0213452
$499$	11.7560	0.526270	0.263135	−	0.964759i	$-0.415244\pi$
$499$	11.7560	0.526270	0.263135	+	0.964759i	$0.415244\pi$
$500$	0	0
$501$	0.602766	0.0269296
$502$	−14.8049	−0.660776
$503$	−32.5241	−1.45018	−0.725089	−	0.688655i	$-0.758201\pi$
$503$	−32.5241	−1.45018	−0.725089	+	0.688655i	$0.758201\pi$
$504$	9.88932	0.440505
$505$	0	0
$506$	0	0
$507$	0.216506	0.00961539
$508$	51.0584	2.26535
$509$	−12.7537	−0.565296	−0.282648	−	0.959224i	$-0.591213\pi$
$509$	−12.7537	−0.565296	−0.282648	+	0.959224i	$0.591213\pi$
$510$	0	0
$511$	1.74126	0.0770287
$512$	−1.30467	−0.0576587
$513$	−0.835061	−0.0368688
$514$	−53.7681	−2.37161
$515$	0	0
$516$	−0.0966268	−0.00425376
$517$	0	0
$518$	12.5862	0.553005
$519$	0.807200	0.0354321
$520$	0	0
$521$	19.3491	0.847699	0.423850	−	0.905733i	$-0.360678\pi$
$521$	19.3491	0.847699	0.423850	+	0.905733i	$0.360678\pi$
$522$	−47.8864	−2.09593
$523$	−15.7423	−0.688364	−0.344182	−	0.938903i	$-0.611844\pi$
$523$	−15.7423	−0.688364	−0.344182	+	0.938903i	$0.611844\pi$
$524$	71.4005	3.11915
$525$	0	0
$526$	−46.1580	−2.01258
$527$	5.77969	0.251767
$528$	0	0
$529$	7.74383	0.336688
$530$	0	0
$531$	−31.8773	−1.38336
$532$	12.2039	0.529106
$533$	18.7238	0.811017
$534$	0.876070	0.0379112
$535$	0	0
$536$	−19.3814	−0.837148
$537$	0.451112	0.0194669
$538$	1.61676	0.0697033
$539$	0	0
$540$	0	0
$541$	−17.0579	−0.733375	−0.366687	−	0.930344i	$-0.619508\pi$
$541$	−17.0579	−0.733375	−0.366687	+	0.930344i	$0.619508\pi$
$542$	46.8718	2.01332
$543$	0.651503	0.0279587
$544$	42.7409	1.83250
$545$	0	0
$546$	−0.311481	−0.0133302
$547$	−15.2548	−0.652249	−0.326124	−	0.945327i	$-0.605743\pi$
$547$	−15.2548	−0.652249	−0.326124	+	0.945327i	$0.605743\pi$
$548$	31.2441	1.33468
$549$	27.7492	1.18431
$550$	0	0
$551$	−22.8591	−0.973830
$552$	−0.680500	−0.0289640
$553$	−11.4297	−0.486042
$554$	−35.1921	−1.49517
$555$	0	0
$556$	−40.7868	−1.72974
$557$	10.1477	0.429971	0.214986	−	0.976617i	$-0.431030\pi$
$557$	10.1477	0.429971	0.214986	+	0.976617i	$0.431030\pi$
$558$	5.13713	0.217472
$559$	−1.96285	−0.0830196
$560$	0	0
$561$	0	0
$562$	−31.7288	−1.33840
$563$	−32.8329	−1.38374	−0.691871	−	0.722021i	$-0.743213\pi$
$563$	−32.8329	−1.38374	−0.691871	+	0.722021i	$0.743213\pi$
$564$	1.19625	0.0503712
$565$	0	0
$566$	−38.9857	−1.63869
$567$	−10.2387	−0.429983
$568$	21.5915	0.905958
$569$	−22.5290	−0.944464	−0.472232	−	0.881474i	$-0.656552\pi$
$569$	−22.5290	−0.944464	−0.472232	+	0.881474i	$0.656552\pi$
$570$	0	0
$571$	25.5317	1.06847	0.534234	−	0.845336i	$-0.320600\pi$
$571$	25.5317	1.06847	0.534234	+	0.845336i	$0.320600\pi$
$572$	0	0
$573$	−0.632410	−0.0264193
$574$	17.4212	0.727149
$575$	0	0
$576$	38.6808	1.61170
$577$	42.4719	1.76813	0.884063	−	0.467367i	$-0.154797\pi$
$577$	42.4719	1.76813	0.884063	+	0.467367i	$0.154797\pi$
$578$	98.3393	4.09038
$579$	0.283475	0.0117808
$580$	0	0
$581$	5.58094	0.231536
$582$	0.576083	0.0238794
$583$	0	0
$584$	−4.42185	−0.182977
$585$	0	0
$586$	−1.92557	−0.0795444
$587$	17.6468	0.728361	0.364180	−	0.931328i	$-0.381349\pi$
$587$	17.6468	0.728361	0.364180	+	0.931328i	$0.381349\pi$
$588$	0.788556	0.0325195
$589$	2.45226	0.101044
$590$	0	0
$591$	0.380034	0.0156325
$592$	−0.555271	−0.0228215
$593$	16.3570	0.671700	0.335850	−	0.941915i	$-0.390976\pi$
$593$	16.3570	0.671700	0.335850	+	0.941915i	$0.390976\pi$
$594$	0	0
$595$	0	0
$596$	−27.0937	−1.10980
$597$	−0.584482	−0.0239212
$598$	−35.7357	−1.46134
$599$	−32.7516	−1.33819	−0.669097	−	0.743175i	$-0.733319\pi$
$599$	−32.7516	−1.33819	−0.669097	+	0.743175i	$0.733319\pi$
$600$	0	0
$601$	−4.98097	−0.203178	−0.101589	−	0.994826i	$-0.532393\pi$
$601$	−4.98097	−0.203178	−0.101589	+	0.994826i	$0.532393\pi$
$602$	−1.82630	−0.0744344
$603$	20.0781	0.817643
$604$	−42.1722	−1.71596
$605$	0	0
$606$	−0.908640	−0.0369110
$607$	−9.97769	−0.404982	−0.202491	−	0.979284i	$-0.564904\pi$
$607$	−9.97769	−0.404982	−0.202491	+	0.979284i	$0.564904\pi$
$608$	18.1345	0.735453
$609$	0.336509	0.0136360
$610$	0	0
$611$	24.3002	0.983082
$612$	75.6684	3.05871
$613$	34.5107	1.39387	0.696936	−	0.717133i	$-0.254546\pi$
$613$	34.5107	1.39387	0.696936	+	0.717133i	$0.254546\pi$
$614$	−14.7149	−0.593845
$615$	0	0
$616$	0	0
$617$	−34.5830	−1.39226	−0.696129	−	0.717916i	$-0.745096\pi$
$617$	−34.5830	−1.39226	−0.696129	+	0.717916i	$0.745096\pi$
$618$	−1.01260	−0.0407328
$619$	−4.06860	−0.163531	−0.0817654	−	0.996652i	$-0.526056\pi$
$619$	−4.06860	−0.163531	−0.0817654	+	0.996652i	$0.526056\pi$
$620$	0	0
$621$	1.41035	0.0565953
$622$	69.6959	2.79455
$623$	10.2643	0.411232
$624$	0.0137418	0.000550112 0
$625$	0	0
$626$	−5.39393	−0.215585
$627$	0	0
$628$	25.1304	1.00281
$629$	37.2526	1.48536
$630$	0	0
$631$	−6.56111	−0.261194	−0.130597	−	0.991436i	$-0.541689\pi$
$631$	−6.56111	−0.261194	−0.130597	+	0.991436i	$0.541689\pi$
$632$	29.0253	1.15457
$633$	0.372526	0.0148066
$634$	−1.84329	−0.0732064
$635$	0	0
$636$	−1.21196	−0.0480573
$637$	16.0185	0.634676
$638$	0	0
$639$	−22.3676	−0.884850
$640$	0	0
$641$	−6.62453	−0.261653	−0.130827	−	0.991405i	$-0.541763\pi$
$641$	−6.62453	−0.261653	−0.130827	+	0.991405i	$0.541763\pi$
$642$	0.430454	0.0169887
$643$	−46.7123	−1.84215	−0.921077	−	0.389380i	$-0.872689\pi$
$643$	−46.7123	−1.84215	−0.921077	+	0.389380i	$0.872689\pi$
$644$	−20.6114	−0.812201
$645$	0	0
$646$	58.2697	2.29259
$647$	−42.9962	−1.69035	−0.845177	−	0.534486i	$-0.820505\pi$
$647$	−42.9962	−1.69035	−0.845177	+	0.534486i	$0.820505\pi$
$648$	26.0006	1.02140
$649$	0	0
$650$	0	0
$651$	−0.0360998	−0.00141486
$652$	−70.7117	−2.76929
$653$	−18.3866	−0.719522	−0.359761	−	0.933044i	$-0.617142\pi$
$653$	−18.3866	−0.719522	−0.359761	+	0.933044i	$0.617142\pi$
$654$	−0.313726	−0.0122676
$655$	0	0
$656$	−0.768582	−0.0300081
$657$	4.58080	0.178714
$658$	22.6097	0.881420
$659$	−13.6816	−0.532959	−0.266480	−	0.963841i	$-0.585860\pi$
$659$	−13.6816	−0.532959	−0.266480	+	0.963841i	$0.585860\pi$
$660$	0	0
$661$	−35.5389	−1.38230	−0.691151	−	0.722710i	$-0.742896\pi$
$661$	−35.5389	−1.38230	−0.691151	+	0.722710i	$0.742896\pi$
$662$	67.8369	2.63656
$663$	−0.921923	−0.0358045
$664$	−14.1726	−0.550002
$665$	0	0
$666$	33.1110	1.28303
$667$	38.6071	1.49487
$668$	−46.3626	−1.79382
$669$	−0.383205	−0.0148156
$670$	0	0
$671$	0	0
$672$	−0.266959	−0.0102982
$673$	16.5236	0.636939	0.318470	−	0.947933i	$-0.396831\pi$
$673$	16.5236	0.636939	0.318470	+	0.947933i	$0.396831\pi$
$674$	−65.3243	−2.51620
$675$	0	0
$676$	−16.6529	−0.640496
$677$	28.4913	1.09501	0.547505	−	0.836802i	$-0.315578\pi$
$677$	28.4913	1.09501	0.547505	+	0.836802i	$0.315578\pi$
$678$	−0.472266	−0.0181373
$679$	6.74959	0.259026
$680$	0	0
$681$	0.240950	0.00923321
$682$	0	0
$683$	−3.48712	−0.133431	−0.0667156	−	0.997772i	$-0.521252\pi$
$683$	−3.48712	−0.133431	−0.0667156	+	0.997772i	$0.521252\pi$
$684$	32.1053	1.22758
$685$	0	0
$686$	33.2038	1.26773
$687$	−0.249947	−0.00953608
$688$	0.0805718	0.00307177
$689$	−24.6193	−0.937922
$690$	0	0
$691$	−36.0080	−1.36981	−0.684904	−	0.728633i	$-0.740156\pi$
$691$	−36.0080	−1.36981	−0.684904	+	0.728633i	$0.740156\pi$
$692$	−62.0870	−2.36019
$693$	0	0
$694$	−18.5169	−0.702893
$695$	0	0
$696$	−0.854549	−0.0323916
$697$	51.5634	1.95310
$698$	68.8295	2.60523
$699$	−0.00173589	−6.56573e−5 0
$700$	0	0
$701$	−50.5415	−1.90893	−0.954464	−	0.298327i	$-0.903571\pi$
$701$	−50.5415	−1.90893	−0.954464	+	0.298327i	$0.903571\pi$
$702$	−1.63935	−0.0618732
$703$	15.8059	0.596131
$704$	0	0
$705$	0	0
$706$	7.29196	0.274437
$707$	−10.6460	−0.400382
$708$	−1.47059	−0.0552682
$709$	25.7847	0.968364	0.484182	−	0.874967i	$-0.339117\pi$
$709$	25.7847	0.968364	0.484182	+	0.874967i	$0.339117\pi$
$710$	0	0
$711$	−30.0687	−1.12766
$712$	−26.0659	−0.976860
$713$	−4.14167	−0.155107
$714$	−0.857788	−0.0321019
$715$	0	0
$716$	−34.6979	−1.29672
$717$	−0.319492	−0.0119316
$718$	6.72444	0.250954
$719$	−26.8799	−1.00245	−0.501226	−	0.865316i	$-0.667118\pi$
$719$	−26.8799	−1.00245	−0.501226	+	0.865316i	$0.667118\pi$
$720$	0	0
$721$	−11.8640	−0.441839
$722$	−18.8597	−0.701887
$723$	−0.0304564	−0.00113269
$724$	−50.1113	−1.86237
$725$	0	0
$726$	0	0
$727$	−6.40984	−0.237728	−0.118864	−	0.992911i	$-0.537925\pi$
$727$	−6.40984	−0.237728	−0.118864	+	0.992911i	$0.537925\pi$
$728$	9.26757	0.343479
$729$	−26.9029	−0.996405
$730$	0	0
$731$	−5.40548	−0.199929
$732$	1.28015	0.0473158
$733$	10.2910	0.380107	0.190053	−	0.981774i	$-0.439134\pi$
$733$	10.2910	0.380107	0.190053	+	0.981774i	$0.439134\pi$
$734$	−41.9727	−1.54924
$735$	0	0
$736$	−30.6277	−1.12895
$737$	0	0
$738$	45.8308	1.68706
$739$	23.8032	0.875614	0.437807	−	0.899069i	$-0.355755\pi$
$739$	23.8032	0.875614	0.437807	+	0.899069i	$0.355755\pi$
$740$	0	0
$741$	−0.391163	−0.0143697
$742$	−22.9067	−0.840930
$743$	11.8906	0.436224	0.218112	−	0.975924i	$-0.430010\pi$
$743$	11.8906	0.436224	0.218112	+	0.975924i	$0.430010\pi$
$744$	0.0916738	0.00336093
$745$	0	0
$746$	19.2192	0.703664
$747$	14.6820	0.537187
$748$	0	0
$749$	5.04335	0.184280
$750$	0	0
$751$	43.4322	1.58486	0.792432	−	0.609960i	$-0.208815\pi$
$751$	43.4322	1.58486	0.792432	+	0.609960i	$0.208815\pi$
$752$	−0.997486	−0.0363746
$753$	0.273696	0.00997404
$754$	−44.8757	−1.63428
$755$	0	0
$756$	−0.945529	−0.0343886
$757$	−12.0047	−0.436320	−0.218160	−	0.975913i	$-0.570005\pi$
$757$	−12.0047	−0.436320	−0.218160	+	0.975913i	$0.570005\pi$
$758$	−32.8928	−1.19472
$759$	0	0
$760$	0	0
$761$	−27.0583	−0.980864	−0.490432	−	0.871479i	$-0.663161\pi$
$761$	−27.0583	−0.980864	−0.490432	+	0.871479i	$0.663161\pi$
$762$	−1.52269	−0.0551612
$763$	−3.67572	−0.133070
$764$	48.6427	1.75983
$765$	0	0
$766$	62.3861	2.25410
$767$	−29.8731	−1.07866
$768$	0.709835	0.0256140
$769$	−22.0504	−0.795158	−0.397579	−	0.917568i	$-0.630150\pi$
$769$	−22.0504	−0.795158	−0.397579	+	0.917568i	$0.630150\pi$
$770$	0	0
$771$	0.994003	0.0357981
$772$	−21.8039	−0.784739
$773$	39.9016	1.43516	0.717581	−	0.696476i	$-0.245250\pi$
$773$	39.9016	1.43516	0.717581	+	0.696476i	$0.245250\pi$
$774$	−4.80452	−0.172695
$775$	0	0
$776$	−17.1403	−0.615301
$777$	−0.232679	−0.00834730
$778$	45.9975	1.64909
$779$	21.8778	0.783855
$780$	0	0
$781$	0	0
$782$	−98.4126	−3.51923
$783$	1.77107	0.0632928
$784$	−0.657534	−0.0234833
$785$	0	0
$786$	−2.12934	−0.0759511
$787$	−23.4937	−0.837461	−0.418731	−	0.908110i	$-0.637525\pi$
$787$	−23.4937	−0.837461	−0.418731	+	0.908110i	$0.637525\pi$
$788$	−29.2309	−1.04131
$789$	0.853315	0.0303788
$790$	0	0
$791$	−5.53324	−0.196739
$792$	0	0
$793$	26.0046	0.923451
$794$	67.5177	2.39611
$795$	0	0
$796$	44.9563	1.59343
$797$	−5.92754	−0.209964	−0.104982	−	0.994474i	$-0.533479\pi$
$797$	−5.92754	−0.209964	−0.104982	+	0.994474i	$0.533479\pi$
$798$	−0.363951	−0.0128837
$799$	66.9204	2.36747
$800$	0	0
$801$	27.0029	0.954100
$802$	44.0957	1.55707
$803$	0	0
$804$	0.926260	0.0326667
$805$	0	0
$806$	4.81415	0.169571
$807$	−0.0298887	−0.00105213
$808$	27.0349	0.951086
$809$	8.43318	0.296495	0.148247	−	0.988950i	$-0.452637\pi$
$809$	8.43318	0.296495	0.148247	+	0.988950i	$0.452637\pi$
$810$	0	0
$811$	−7.04102	−0.247243	−0.123622	−	0.992329i	$-0.539451\pi$
$811$	−7.04102	−0.247243	−0.123622	+	0.992329i	$0.539451\pi$
$812$	−25.8831	−0.908317
$813$	−0.866511	−0.0303899
$814$	0	0
$815$	0	0
$816$	0.0378435	0.00132479
$817$	−2.29349	−0.0802391
$818$	29.3409	1.02588
$819$	−9.60071	−0.335476
$820$	0	0
$821$	41.3164	1.44195	0.720976	−	0.692960i	$-0.243694\pi$
$821$	41.3164	1.44195	0.720976	+	0.692960i	$0.243694\pi$
$822$	−0.931779	−0.0324995
$823$	31.0563	1.08255	0.541277	−	0.840845i	$-0.317941\pi$
$823$	31.0563	1.08255	0.541277	+	0.840845i	$0.317941\pi$
$824$	30.1282	1.04956
$825$	0	0
$826$	−27.7949	−0.967110
$827$	−25.9473	−0.902276	−0.451138	−	0.892454i	$-0.648982\pi$
$827$	−25.9473	−0.902276	−0.451138	+	0.892454i	$0.648982\pi$
$828$	−54.2232	−1.88439
$829$	−0.0809572	−0.00281176	−0.00140588	−	0.999999i	$-0.500448\pi$
$829$	−0.0809572	−0.00281176	−0.00140588	+	0.999999i	$0.500448\pi$
$830$	0	0
$831$	0.650591	0.0225688
$832$	36.2489	1.25670
$833$	44.1133	1.52844
$834$	1.21636	0.0421192
$835$	0	0
$836$	0	0
$837$	−0.189995	−0.00656720
$838$	9.68546	0.334579
$839$	−14.7008	−0.507528	−0.253764	−	0.967266i	$-0.581669\pi$
$839$	−14.7008	−0.507528	−0.253764	+	0.967266i	$0.581669\pi$
$840$	0	0
$841$	19.4815	0.671775
$842$	1.87686	0.0646807
$843$	0.586565	0.0202024
$844$	−28.6534	−0.986291
$845$	0	0
$846$	59.4805	2.04498
$847$	0	0
$848$	1.01059	0.0347036
$849$	0.720722	0.0247351
$850$	0	0
$851$	−26.6948	−0.915087
$852$	−1.03188	−0.0353517
$853$	44.0374	1.50781	0.753906	−	0.656982i	$-0.228167\pi$
$853$	44.0374	1.50781	0.753906	+	0.656982i	$0.228167\pi$
$854$	24.1956	0.827955
$855$	0	0
$856$	−12.8074	−0.437747
$857$	29.9206	1.02207	0.511035	−	0.859560i	$-0.329262\pi$
$857$	29.9206	1.02207	0.511035	+	0.859560i	$0.329262\pi$
$858$	0	0
$859$	−52.2985	−1.78440	−0.892200	−	0.451640i	$-0.850839\pi$
$859$	−52.2985	−1.78440	−0.892200	+	0.451640i	$0.850839\pi$
$860$	0	0
$861$	−0.322064	−0.0109759
$862$	13.3264	0.453899
$863$	37.0318	1.26058	0.630289	−	0.776360i	$-0.282936\pi$
$863$	37.0318	1.26058	0.630289	+	0.776360i	$0.282936\pi$
$864$	−1.40502	−0.0477998
$865$	0	0
$866$	−35.8244	−1.21736
$867$	−1.81798	−0.0617419
$868$	2.77667	0.0942462
$869$	0	0
$870$	0	0
$871$	18.8158	0.637547
$872$	9.33434	0.316101
$873$	17.7564	0.600965
$874$	−41.7555	−1.41240
$875$	0	0
$876$	0.211326	0.00714003
$877$	−21.6606	−0.731427	−0.365714	−	0.930727i	$-0.619175\pi$
$877$	−21.6606	−0.731427	−0.365714	+	0.930727i	$0.619175\pi$
$878$	−49.6879	−1.67689
$879$	0.0355976	0.00120068
$880$	0	0
$881$	−5.86710	−0.197668	−0.0988339	−	0.995104i	$-0.531511\pi$
$881$	−5.86710	−0.197668	−0.0988339	+	0.995104i	$0.531511\pi$
$882$	39.2090	1.32023
$883$	−0.304363	−0.0102426	−0.00512132	−	0.999987i	$-0.501630\pi$
$883$	−0.304363	−0.0102426	−0.00512132	+	0.999987i	$0.501630\pi$
$884$	70.9111	2.38500
$885$	0	0
$886$	−65.8912	−2.21366
$887$	−14.5603	−0.488886	−0.244443	−	0.969664i	$-0.578605\pi$
$887$	−14.5603	−0.488886	−0.244443	+	0.969664i	$0.578605\pi$
$888$	0.590877	0.0198285
$889$	−17.8404	−0.598347
$890$	0	0
$891$	0	0
$892$	29.4748	0.986889
$893$	28.3936	0.950157
$894$	0.808002	0.0270236
$895$	0	0
$896$	21.1365	0.706122
$897$	0.660641	0.0220582
$898$	40.2168	1.34205
$899$	−5.20097	−0.173462
$900$	0	0
$901$	−67.7992	−2.25872
$902$	0	0
$903$	0.0337625	0.00112355
$904$	14.0514	0.467343
$905$	0	0
$906$	1.25768	0.0417836
$907$	0.343092	0.0113922	0.00569609	−	0.999984i	$-0.498187\pi$
$907$	0.343092	0.0113922	0.00569609	+	0.999984i	$0.498187\pi$
$908$	−18.5330	−0.615039
$909$	−28.0068	−0.928926
$910$	0	0
$911$	−29.5389	−0.978668	−0.489334	−	0.872097i	$-0.662760\pi$
$911$	−29.5389	−0.978668	−0.489334	+	0.872097i	$0.662760\pi$
$912$	0.0160566	0.000531688 0
$913$	0	0
$914$	−45.3234	−1.49916
$915$	0	0
$916$	19.2251	0.635214
$917$	−24.9481	−0.823860
$918$	−4.51460	−0.149004
$919$	−26.7701	−0.883063	−0.441531	−	0.897246i	$-0.645565\pi$
$919$	−26.7701	−0.883063	−0.441531	+	0.897246i	$0.645565\pi$
$920$	0	0
$921$	0.272032	0.00896376
$922$	29.7875	0.980999
$923$	−20.9614	−0.689951
$924$	0	0
$925$	0	0
$926$	22.1681	0.728489
$927$	−31.2112	−1.02511
$928$	−38.4613	−1.26255
$929$	−15.1478	−0.496983	−0.248491	−	0.968634i	$-0.579935\pi$
$929$	−15.1478	−0.496983	−0.248491	+	0.968634i	$0.579935\pi$
$930$	0	0
$931$	18.7168	0.613419
$932$	0.133518	0.00437354
$933$	−1.28846	−0.0421822
$934$	22.2369	0.727612
$935$	0	0
$936$	24.3806	0.796905
$937$	52.1467	1.70356	0.851779	−	0.523901i	$-0.175524\pi$
$937$	52.1467	1.70356	0.851779	+	0.523901i	$0.175524\pi$
$938$	17.5068	0.571618
$939$	0.0997167	0.00325413
$940$	0	0
$941$	38.5195	1.25570	0.627850	−	0.778334i	$-0.283935\pi$
$941$	38.5195	1.25570	0.627850	+	0.778334i	$0.283935\pi$
$942$	−0.749452	−0.0244185
$943$	−36.9498	−1.20325
$944$	1.22624	0.0399109
$945$	0	0
$946$	0	0
$947$	48.1131	1.56347	0.781733	−	0.623613i	$-0.214336\pi$
$947$	48.1131	1.56347	0.781733	+	0.623613i	$0.214336\pi$
$948$	−1.38716	−0.0450528
$949$	4.29281	0.139350
$950$	0	0
$951$	0.0340766	0.00110501
$952$	25.5219	0.827171
$953$	−39.2379	−1.27104	−0.635521	−	0.772084i	$-0.719215\pi$
$953$	−39.2379	−1.27104	−0.635521	+	0.772084i	$0.719215\pi$
$954$	−60.2616	−1.95104
$955$	0	0
$956$	24.5742	0.794786
$957$	0	0
$958$	−11.9963	−0.387584
$959$	−10.9171	−0.352530
$960$	0	0
$961$	−30.4421	−0.982002
$962$	31.0293	1.00042
$963$	13.2678	0.427548
$964$	2.34260	0.0754501
$965$	0	0
$966$	0.614682	0.0197771
$967$	−38.7795	−1.24706	−0.623532	−	0.781798i	$-0.714303\pi$
$967$	−38.7795	−1.24706	−0.623532	+	0.781798i	$0.714303\pi$
$968$	0	0
$969$	−1.07722	−0.0346054
$970$	0	0
$971$	−42.7333	−1.37138	−0.685688	−	0.727895i	$-0.740499\pi$
$971$	−42.7333	−1.37138	−0.685688	+	0.727895i	$0.740499\pi$
$972$	−3.73154	−0.119689
$973$	14.2514	0.456877
$974$	12.5086	0.400802
$975$	0	0
$976$	−1.06745	−0.0341682
$977$	−48.0953	−1.53870	−0.769352	−	0.638825i	$-0.779421\pi$
$977$	−48.0953	−1.53870	−0.769352	+	0.638825i	$0.779421\pi$
$978$	2.10880	0.0674320
$979$	0	0
$980$	0	0
$981$	−9.66989	−0.308736
$982$	−37.1580	−1.18576
$983$	−54.5875	−1.74107	−0.870536	−	0.492104i	$-0.836228\pi$
$983$	−54.5875	−1.74107	−0.870536	+	0.492104i	$0.836228\pi$
$984$	0.817867	0.0260726
$985$	0	0
$986$	−123.583	−3.93569
$987$	−0.417983	−0.0133045
$988$	30.0869	0.957190
$989$	3.87351	0.123171
$990$	0	0
$991$	1.26477	0.0401766	0.0200883	−	0.999798i	$-0.493605\pi$
$991$	1.26477	0.0401766	0.0200883	+	0.999798i	$0.493605\pi$
$992$	4.12602	0.131001
$993$	−1.25409	−0.0397974
$994$	−19.5032	−0.618602
$995$	0	0
$996$	0.677324	0.0214618
$997$	−5.65195	−0.178999	−0.0894995	−	0.995987i	$-0.528527\pi$
$997$	−5.65195	−0.178999	−0.0894995	+	0.995987i	$0.528527\pi$
$998$	26.9664	0.853606
$999$	−1.22460	−0.0387447

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3025.2.a.bj.1.8		8
5.4	even	2		3025.2.a.bm.1.1		8
11.2	odd	10		275.2.h.e.26.4	yes	16
11.6	odd	10		275.2.h.e.201.4	yes	16
11.10	odd	2		3025.2.a.bn.1.1		8
55.2	even	20		275.2.z.c.224.2		32
55.13	even	20		275.2.z.c.224.7		32
55.17	even	20		275.2.z.c.124.7		32
55.24	odd	10		275.2.h.c.26.1	✓	16
55.28	even	20		275.2.z.c.124.2		32
55.39	odd	10		275.2.h.c.201.1	yes	16
55.54	odd	2		3025.2.a.bi.1.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
275.2.h.c.26.1	✓	16	55.24	odd	10
275.2.h.c.201.1	yes	16	55.39	odd	10
275.2.h.e.26.4	yes	16	11.2	odd	10
275.2.h.e.201.4	yes	16	11.6	odd	10
275.2.z.c.124.2		32	55.28	even	20
275.2.z.c.124.7		32	55.17	even	20
275.2.z.c.224.2		32	55.2	even	20
275.2.z.c.224.7		32	55.13	even	20
3025.2.a.bi.1.8		8	55.54	odd	2
3025.2.a.bj.1.8		8	1.1	even	1	trivial
3025.2.a.bm.1.1		8	5.4	even	2
3025.2.a.bn.1.1		8	11.10	odd	2

Overview	Random
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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:	\( N \)	\(=\)	\( 3025 = 5^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3025.a (trivial)

\(f(q)\)	\(=\)	\(q+2.29384 q^{2} -0.0424059 q^{3} +3.26171 q^{4} -0.0972724 q^{6} -1.13968 q^{7} +2.89416 q^{8} -2.99820 q^{9} +O(q^{10})\)	\(q+2.29384 q^{2} -0.0424059 q^{3} +3.26171 q^{4} -0.0972724 q^{6} -1.13968 q^{7} +2.89416 q^{8} -2.99820 q^{9} -0.138316 q^{12} -2.80970 q^{13} -2.61424 q^{14} +0.115334 q^{16} -7.73764 q^{17} -6.87740 q^{18} -3.28300 q^{19} +0.0483291 q^{21} +5.54471 q^{23} -0.122730 q^{24} -6.44501 q^{26} +0.254359 q^{27} -3.71730 q^{28} +6.96286 q^{29} -0.746958 q^{31} -5.52377 q^{32} -17.7489 q^{34} -9.77927 q^{36} -4.81446 q^{37} -7.53069 q^{38} +0.119148 q^{39} -6.66398 q^{41} +0.110859 q^{42} +0.698596 q^{43} +12.7187 q^{46} -8.64868 q^{47} -0.00489083 q^{48} -5.70113 q^{49} +0.328121 q^{51} -9.16443 q^{52} +8.76226 q^{53} +0.583459 q^{54} -3.29842 q^{56} +0.139219 q^{57} +15.9717 q^{58} +10.6321 q^{59} -9.25529 q^{61} -1.71340 q^{62} +3.41699 q^{63} -12.9013 q^{64} -6.69671 q^{67} -25.2379 q^{68} -0.235128 q^{69} +7.46035 q^{71} -8.67729 q^{72} -1.52785 q^{73} -11.0436 q^{74} -10.7082 q^{76} +0.273306 q^{78} +10.0289 q^{79} +8.98382 q^{81} -15.2861 q^{82} -4.89694 q^{83} +0.157635 q^{84} +1.60247 q^{86} -0.295266 q^{87} -9.00636 q^{89} +3.20216 q^{91} +18.0852 q^{92} +0.0316754 q^{93} -19.8387 q^{94} +0.234240 q^{96} -5.92237 q^{97} -13.0775 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 5 q^{2} + q^{3} + 9 q^{4} - q^{6} - 8 q^{7} - 12 q^{8} + 9 q^{9}+O(q^{10}) \) 8 * q - 5 * q^2 + q^3 + 9 * q^4 - q^6 - 8 * q^7 - 12 * q^8 + 9 * q^9	\( 8 q - 5 q^{2} + q^{3} + 9 q^{4} - q^{6} - 8 q^{7} - 12 q^{8} + 9 q^{9} + 3 q^{12} - 9 q^{13} + 9 q^{14} + 23 q^{16} - 19 q^{17} - 22 q^{18} - q^{19} - 5 q^{21} + 2 q^{23} - q^{24} - 2 q^{26} - 2 q^{27} - 9 q^{28} - 7 q^{29} - 5 q^{31} - 29 q^{32} + 10 q^{34} - 16 q^{36} - 8 q^{37} - 37 q^{38} + q^{39} + 41 q^{42} - 14 q^{43} + 20 q^{46} + 11 q^{47} - 27 q^{48} - 12 q^{49} + 25 q^{51} + 7 q^{52} + 11 q^{53} - 30 q^{54} - 10 q^{56} + 2 q^{57} + 27 q^{58} + 17 q^{59} + 2 q^{61} - 25 q^{62} - 41 q^{63} + 30 q^{64} + 7 q^{67} - 66 q^{68} + 17 q^{71} + 19 q^{72} - 34 q^{73} + 6 q^{74} + 31 q^{76} - 17 q^{78} - 23 q^{79} - 4 q^{81} - 17 q^{82} - 41 q^{83} - 83 q^{84} + q^{86} - 25 q^{87} - 11 q^{89} - 7 q^{91} + 33 q^{92} - 59 q^{93} - 50 q^{94} + 61 q^{96} + 2 q^{97} - 26 q^{98}+O(q^{100}) \) 8 * q - 5 * q^2 + q^3 + 9 * q^4 - q^6 - 8 * q^7 - 12 * q^8 + 9 * q^9 + 3 * q^12 - 9 * q^13 + 9 * q^14 + 23 * q^16 - 19 * q^17 - 22 * q^18 - q^19 - 5 * q^21 + 2 * q^23 - q^24 - 2 * q^26 - 2 * q^27 - 9 * q^28 - 7 * q^29 - 5 * q^31 - 29 * q^32 + 10 * q^34 - 16 * q^36 - 8 * q^37 - 37 * q^38 + q^39 + 41 * q^42 - 14 * q^43 + 20 * q^46 + 11 * q^47 - 27 * q^48 - 12 * q^49 + 25 * q^51 + 7 * q^52 + 11 * q^53 - 30 * q^54 - 10 * q^56 + 2 * q^57 + 27 * q^58 + 17 * q^59 + 2 * q^61 - 25 * q^62 - 41 * q^63 + 30 * q^64 + 7 * q^67 - 66 * q^68 + 17 * q^71 + 19 * q^72 - 34 * q^73 + 6 * q^74 + 31 * q^76 - 17 * q^78 - 23 * q^79 - 4 * q^81 - 17 * q^82 - 41 * q^83 - 83 * q^84 + q^86 - 25 * q^87 - 11 * q^89 - 7 * q^91 + 33 * q^92 - 59 * q^93 - 50 * q^94 + 61 * q^96 + 2 * q^97 - 26 * q^98

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