LMFDB - Embedded newform 6025.2.a.o.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("6025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$48.1098672178$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Character	$\chi$	$=$	6025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.30206 q^{2} +0.621403 q^{3} +3.29948 q^{4} -1.43051 q^{6} +0.210935 q^{7} -2.99147 q^{8} -2.61386 q^{9} +O(q^{10})$	\(q-2.30206 q^{2} +0.621403 q^{3} +3.29948 q^{4} -1.43051 q^{6} +0.210935 q^{7} -2.99147 q^{8} -2.61386 q^{9} +2.40298 q^{11} +2.05030 q^{12} +0.974615 q^{13} -0.485586 q^{14} +0.287590 q^{16} -0.931440 q^{17} +6.01726 q^{18} -4.45706 q^{19} +0.131076 q^{21} -5.53180 q^{22} -3.73038 q^{23} -1.85891 q^{24} -2.24362 q^{26} -3.48847 q^{27} +0.695976 q^{28} -6.80679 q^{29} -10.0607 q^{31} +5.32089 q^{32} +1.49322 q^{33} +2.14423 q^{34} -8.62436 q^{36} +6.37525 q^{37} +10.2604 q^{38} +0.605629 q^{39} +2.10120 q^{41} -0.301744 q^{42} +4.46976 q^{43} +7.92857 q^{44} +8.58756 q^{46} +3.60254 q^{47} +0.178709 q^{48} -6.95551 q^{49} -0.578799 q^{51} +3.21572 q^{52} -9.36731 q^{53} +8.03066 q^{54} -0.631007 q^{56} -2.76963 q^{57} +15.6696 q^{58} +0.136888 q^{59} +14.9090 q^{61} +23.1604 q^{62} -0.551355 q^{63} -12.8242 q^{64} -3.43747 q^{66} -1.71602 q^{67} -3.07326 q^{68} -2.31807 q^{69} +8.78360 q^{71} +7.81928 q^{72} +11.8818 q^{73} -14.6762 q^{74} -14.7060 q^{76} +0.506873 q^{77} -1.39419 q^{78} -2.51865 q^{79} +5.67383 q^{81} -4.83709 q^{82} +12.6167 q^{83} +0.432481 q^{84} -10.2897 q^{86} -4.22976 q^{87} -7.18844 q^{88} -0.0938543 q^{89} +0.205581 q^{91} -12.3083 q^{92} -6.25176 q^{93} -8.29326 q^{94} +3.30642 q^{96} -5.72436 q^{97} +16.0120 q^{98} -6.28105 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 11 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 16 q^{7} + 33 q^{8} + 38 q^{9}+O(q^{10}) $ 40 * q + 11 * q^2 + 8 * q^3 + 41 * q^4 + 3 * q^6 + 16 * q^7 + 33 * q^8 + 38 * q^9	$ 40 q + 11 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 16 q^{7} + 33 q^{8} + 38 q^{9} + q^{11} + 14 q^{12} + 9 q^{13} - q^{14} + 43 q^{16} + 12 q^{17} + 42 q^{18} + 2 q^{21} + 5 q^{22} + 77 q^{23} - 2 q^{24} + 2 q^{26} + 38 q^{27} + 42 q^{28} + 2 q^{29} + q^{31} + 72 q^{32} + 20 q^{33} + 5 q^{34} + 32 q^{36} + 28 q^{37} + 23 q^{38} - 2 q^{39} - 2 q^{41} + 37 q^{42} + 31 q^{43} + 3 q^{44} + 14 q^{46} + 96 q^{47} + 13 q^{48} + 40 q^{49} - 10 q^{51} + 42 q^{52} + 54 q^{53} + 4 q^{54} - 15 q^{56} + 37 q^{57} + 27 q^{58} + q^{59} + 5 q^{61} + 39 q^{62} + 70 q^{63} + 65 q^{64} - 52 q^{66} + 34 q^{67} + 52 q^{68} + 21 q^{69} - 9 q^{71} + 70 q^{72} + 25 q^{73} + 22 q^{74} - 47 q^{76} + 54 q^{77} + 58 q^{78} + 13 q^{79} + 12 q^{81} - 5 q^{82} + 63 q^{83} + 95 q^{84} - 18 q^{86} + 47 q^{87} + 13 q^{88} + 19 q^{89} - 31 q^{91} + 137 q^{92} + 52 q^{93} + 120 q^{94} - 49 q^{96} + 36 q^{97} + 64 q^{98} + 16 q^{99}+O(q^{100}) $ 40 * q + 11 * q^2 + 8 * q^3 + 41 * q^4 + 3 * q^6 + 16 * q^7 + 33 * q^8 + 38 * q^9 + q^11 + 14 * q^12 + 9 * q^13 - q^14 + 43 * q^16 + 12 * q^17 + 42 * q^18 + 2 * q^21 + 5 * q^22 + 77 * q^23 - 2 * q^24 + 2 * q^26 + 38 * q^27 + 42 * q^28 + 2 * q^29 + q^31 + 72 * q^32 + 20 * q^33 + 5 * q^34 + 32 * q^36 + 28 * q^37 + 23 * q^38 - 2 * q^39 - 2 * q^41 + 37 * q^42 + 31 * q^43 + 3 * q^44 + 14 * q^46 + 96 * q^47 + 13 * q^48 + 40 * q^49 - 10 * q^51 + 42 * q^52 + 54 * q^53 + 4 * q^54 - 15 * q^56 + 37 * q^57 + 27 * q^58 + q^59 + 5 * q^61 + 39 * q^62 + 70 * q^63 + 65 * q^64 - 52 * q^66 + 34 * q^67 + 52 * q^68 + 21 * q^69 - 9 * q^71 + 70 * q^72 + 25 * q^73 + 22 * q^74 - 47 * q^76 + 54 * q^77 + 58 * q^78 + 13 * q^79 + 12 * q^81 - 5 * q^82 + 63 * q^83 + 95 * q^84 - 18 * q^86 + 47 * q^87 + 13 * q^88 + 19 * q^89 - 31 * q^91 + 137 * q^92 + 52 * q^93 + 120 * q^94 - 49 * q^96 + 36 * q^97 + 64 * q^98 + 16 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.30206	−1.62780	−0.813901	−	0.581004i	$-0.802660\pi$
$2$	−2.30206	−1.62780	−0.813901	+	0.581004i	$0.802660\pi$
$3$	0.621403	0.358767	0.179384	−	0.983779i	$-0.442590\pi$
$3$	0.621403	0.358767	0.179384	+	0.983779i	$0.442590\pi$
$4$	3.29948	1.64974
$5$	0	0
$5$	0	0
$6$	−1.43051	−0.584002
$7$	0.210935	0.0797261	0.0398630	−	0.999205i	$-0.487308\pi$
$7$	0.210935	0.0797261	0.0398630	+	0.999205i	$0.487308\pi$
$8$	−2.99147	−1.05764
$9$	−2.61386	−0.871286
$10$	0	0
$11$	2.40298	0.724525	0.362263	−	0.932076i	$-0.382004\pi$
$11$	2.40298	0.724525	0.362263	+	0.932076i	$0.382004\pi$
$12$	2.05030	0.591872
$13$	0.974615	0.270310	0.135155	−	0.990824i	$-0.456847\pi$
$13$	0.974615	0.270310	0.135155	+	0.990824i	$0.456847\pi$
$14$	−0.485586	−0.129778
$15$	0	0
$16$	0.287590	0.0718975
$17$	−0.931440	−0.225907	−0.112954	−	0.993600i	$-0.536031\pi$
$17$	−0.931440	−0.225907	−0.112954	+	0.993600i	$0.536031\pi$
$18$	6.01726	1.41828
$19$	−4.45706	−1.02252	−0.511259	−	0.859426i	$-0.670821\pi$
$19$	−4.45706	−1.02252	−0.511259	+	0.859426i	$0.670821\pi$
$20$	0	0
$21$	0.131076	0.0286031
$22$	−5.53180	−1.17938
$23$	−3.73038	−0.777839	−0.388919	−	0.921272i	$-0.627152\pi$
$23$	−3.73038	−0.777839	−0.388919	+	0.921272i	$0.627152\pi$
$24$	−1.85891	−0.379448
$25$	0	0
$26$	−2.24362	−0.440011
$27$	−3.48847	−0.671356
$28$	0.695976	0.131527
$29$	−6.80679	−1.26399	−0.631995	−	0.774973i	$-0.717764\pi$
$29$	−6.80679	−1.26399	−0.631995	+	0.774973i	$0.717764\pi$
$30$	0	0
$31$	−10.0607	−1.80696	−0.903480	−	0.428631i	$-0.858996\pi$
$31$	−10.0607	−1.80696	−0.903480	+	0.428631i	$0.858996\pi$
$32$	5.32089	0.940610
$33$	1.49322	0.259936
$34$	2.14423	0.367732
$35$	0	0
$36$	−8.62436	−1.43739
$37$	6.37525	1.04809	0.524043	−	0.851692i	$-0.324423\pi$
$37$	6.37525	1.04809	0.524043	+	0.851692i	$0.324423\pi$
$38$	10.2604	1.66446
$39$	0.605629	0.0969782
$40$	0	0
$41$	2.10120	0.328153	0.164076	−	0.986448i	$-0.447536\pi$
$41$	2.10120	0.328153	0.164076	+	0.986448i	$0.447536\pi$
$42$	−0.301744	−0.0465602
$43$	4.46976	0.681633	0.340816	−	0.940130i	$-0.389297\pi$
$43$	4.46976	0.681633	0.340816	+	0.940130i	$0.389297\pi$
$44$	7.92857	1.19528
$45$	0	0
$46$	8.58756	1.26617
$47$	3.60254	0.525484	0.262742	−	0.964866i	$-0.415373\pi$
$47$	3.60254	0.525484	0.262742	+	0.964866i	$0.415373\pi$
$48$	0.178709	0.0257945
$49$	−6.95551	−0.993644
$50$	0	0
$51$	−0.578799	−0.0810481
$52$	3.21572	0.445940
$53$	−9.36731	−1.28670	−0.643350	−	0.765572i	$-0.722456\pi$
$53$	−9.36731	−1.28670	−0.643350	+	0.765572i	$0.722456\pi$
$54$	8.03066	1.09283
$55$	0	0
$56$	−0.631007	−0.0843219
$57$	−2.76963	−0.366846
$58$	15.6696	2.05752
$59$	0.136888	0.0178213	0.00891067	−	0.999960i	$-0.497164\pi$
$59$	0.136888	0.0178213	0.00891067	+	0.999960i	$0.497164\pi$
$60$	0	0
$61$	14.9090	1.90890	0.954450	−	0.298371i	$-0.0964433\pi$
$61$	14.9090	1.90890	0.954450	+	0.298371i	$0.0964433\pi$
$62$	23.1604	2.94137
$63$	−0.551355	−0.0694642
$64$	−12.8242	−1.60302
$65$	0	0
$66$	−3.43747	−0.423124
$67$	−1.71602	−0.209646	−0.104823	−	0.994491i	$-0.533428\pi$
$67$	−1.71602	−0.209646	−0.104823	+	0.994491i	$0.533428\pi$
$68$	−3.07326	−0.372688
$69$	−2.31807	−0.279063
$70$	0	0
$71$	8.78360	1.04242	0.521211	−	0.853428i	$-0.325481\pi$
$71$	8.78360	1.04242	0.521211	+	0.853428i	$0.325481\pi$
$72$	7.81928	0.921511
$73$	11.8818	1.39066	0.695330	−	0.718690i	$-0.255258\pi$
$73$	11.8818	1.39066	0.695330	+	0.718690i	$0.255258\pi$
$74$	−14.6762	−1.70607
$75$	0	0
$76$	−14.7060	−1.68689
$77$	0.506873	0.0577636
$78$	−1.39419	−0.157861
$79$	−2.51865	−0.283370	−0.141685	−	0.989912i	$-0.545252\pi$
$79$	−2.51865	−0.283370	−0.141685	+	0.989912i	$0.545252\pi$
$80$	0	0
$81$	5.67383	0.630426
$82$	−4.83709	−0.534167
$83$	12.6167	1.38486	0.692431	−	0.721485i	$-0.256540\pi$
$83$	12.6167	1.38486	0.692431	+	0.721485i	$0.256540\pi$
$84$	0.432481	0.0471876
$85$	0	0
$86$	−10.2897	−1.10956
$87$	−4.22976	−0.453478
$88$	−7.18844	−0.766290
$89$	−0.0938543	−0.00994853	−0.00497427	−	0.999988i	$-0.501583\pi$
$89$	−0.0938543	−0.00994853	−0.00497427	+	0.999988i	$0.501583\pi$
$90$	0	0
$91$	0.205581	0.0215507
$92$	−12.3083	−1.28323
$93$	−6.25176	−0.648277
$94$	−8.29326	−0.855384
$95$	0	0
$96$	3.30642	0.337460
$97$	−5.72436	−0.581221	−0.290611	−	0.956841i	$-0.593858\pi$
$97$	−5.72436	−0.581221	−0.290611	+	0.956841i	$0.593858\pi$
$98$	16.0120	1.61745
$99$	−6.28105	−0.631269
$100$	0	0
$101$	5.99970	0.596993	0.298496	−	0.954411i	$-0.403515\pi$
$101$	5.99970	0.596993	0.298496	+	0.954411i	$0.403515\pi$
$102$	1.33243	0.131930
$103$	0.771897	0.0760572	0.0380286	−	0.999277i	$-0.487892\pi$
$103$	0.771897	0.0760572	0.0380286	+	0.999277i	$0.487892\pi$
$104$	−2.91553	−0.285892
$105$	0	0
$106$	21.5641	2.09449
$107$	12.9356	1.25054	0.625268	−	0.780410i	$-0.284990\pi$
$107$	12.9356	1.25054	0.625268	+	0.780410i	$0.284990\pi$
$108$	−11.5101	−1.10756
$109$	−16.2915	−1.56045	−0.780224	−	0.625501i	$-0.784895\pi$
$109$	−16.2915	−1.56045	−0.780224	+	0.625501i	$0.784895\pi$
$110$	0	0
$111$	3.96160	0.376018
$112$	0.0606629	0.00573211
$113$	1.67287	0.157370	0.0786851	−	0.996900i	$-0.474928\pi$
$113$	1.67287	0.157370	0.0786851	+	0.996900i	$0.474928\pi$
$114$	6.37585	0.597153
$115$	0	0
$116$	−22.4589	−2.08525
$117$	−2.54751	−0.235517
$118$	−0.315125	−0.0290096
$119$	−0.196474	−0.0180107
$120$	0	0
$121$	−5.22569	−0.475063
$122$	−34.3214	−3.10731
$123$	1.30569	0.117730
$124$	−33.1951	−2.98101
$125$	0	0
$126$	1.26925	0.113074
$127$	5.38313	0.477675	0.238838	−	0.971060i	$-0.423234\pi$
$127$	5.38313	0.477675	0.238838	+	0.971060i	$0.423234\pi$
$128$	18.8803	1.66879
$129$	2.77752	0.244547
$130$	0	0
$131$	12.1425	1.06090	0.530449	−	0.847717i	$-0.322023\pi$
$131$	12.1425	1.06090	0.530449	+	0.847717i	$0.322023\pi$
$132$	4.92683	0.428826
$133$	−0.940151	−0.0815214
$134$	3.95039	0.341262
$135$	0	0
$136$	2.78638	0.238930
$137$	−8.93069	−0.763000	−0.381500	−	0.924369i	$-0.624592\pi$
$137$	−8.93069	−0.763000	−0.381500	+	0.924369i	$0.624592\pi$
$138$	5.33634	0.454259
$139$	17.0420	1.44548	0.722740	−	0.691120i	$-0.242882\pi$
$139$	17.0420	1.44548	0.722740	+	0.691120i	$0.242882\pi$
$140$	0	0
$141$	2.23863	0.188527
$142$	−20.2204	−1.69686
$143$	2.34198	0.195846
$144$	−0.751720	−0.0626433
$145$	0	0
$146$	−27.3526	−2.26372
$147$	−4.32217	−0.356487
$148$	21.0350	1.72907
$149$	13.6099	1.11497	0.557483	−	0.830189i	$-0.311767\pi$
$149$	13.6099	1.11497	0.557483	+	0.830189i	$0.311767\pi$
$150$	0	0
$151$	−22.2322	−1.80923	−0.904616	−	0.426227i	$-0.859843\pi$
$151$	−22.2322	−1.80923	−0.904616	+	0.426227i	$0.859843\pi$
$152$	13.3332	1.08146
$153$	2.43465	0.196830
$154$	−1.16685	−0.0940276
$155$	0	0
$156$	1.99826	0.159989
$157$	23.2505	1.85559	0.927795	−	0.373090i	$-0.121702\pi$
$157$	23.2505	1.85559	0.927795	+	0.373090i	$0.121702\pi$
$158$	5.79808	0.461270
$159$	−5.82087	−0.461625
$160$	0	0
$161$	−0.786870	−0.0620140
$162$	−13.0615	−1.02621
$163$	20.4199	1.59941	0.799706	−	0.600392i	$-0.204989\pi$
$163$	20.4199	1.59941	0.799706	+	0.600392i	$0.204989\pi$
$164$	6.93287	0.541366
$165$	0	0
$166$	−29.0444	−2.25428
$167$	−5.28146	−0.408692	−0.204346	−	0.978899i	$-0.565507\pi$
$167$	−5.28146	−0.408692	−0.204346	+	0.978899i	$0.565507\pi$
$168$	−0.392109	−0.0302519
$169$	−12.0501	−0.926933
$170$	0	0
$171$	11.6501	0.890907
$172$	14.7479	1.12452
$173$	22.6464	1.72177	0.860887	−	0.508795i	$-0.169909\pi$
$173$	22.6464	1.72177	0.860887	+	0.508795i	$0.169909\pi$
$174$	9.73716	0.738172
$175$	0	0
$176$	0.691073	0.0520916
$177$	0.0850627	0.00639371
$178$	0.216058	0.0161942
$179$	−14.7098	−1.09946	−0.549731	−	0.835342i	$-0.685270\pi$
$179$	−14.7098	−1.09946	−0.549731	+	0.835342i	$0.685270\pi$
$180$	0	0
$181$	6.46739	0.480718	0.240359	−	0.970684i	$-0.422735\pi$
$181$	6.46739	0.480718	0.240359	+	0.970684i	$0.422735\pi$
$182$	−0.473259	−0.0350803
$183$	9.26448	0.684850
$184$	11.1593	0.822677
$185$	0	0
$186$	14.3919	1.05527
$187$	−2.23823	−0.163676
$188$	11.8865	0.866912
$189$	−0.735841	−0.0535246
$190$	0	0
$191$	8.88339	0.642779	0.321390	−	0.946947i	$-0.395850\pi$
$191$	8.88339	0.642779	0.321390	+	0.946947i	$0.395850\pi$
$192$	−7.96899	−0.575112
$193$	5.31061	0.382266	0.191133	−	0.981564i	$-0.438784\pi$
$193$	5.31061	0.382266	0.191133	+	0.981564i	$0.438784\pi$
$194$	13.1778	0.946112
$195$	0	0
$196$	−22.9495	−1.63925
$197$	21.6383	1.54167	0.770834	−	0.637036i	$-0.219840\pi$
$197$	21.6383	1.54167	0.770834	+	0.637036i	$0.219840\pi$
$198$	14.4593	1.02758
$199$	15.5901	1.10515	0.552577	−	0.833462i	$-0.313644\pi$
$199$	15.5901	1.10515	0.552577	+	0.833462i	$0.313644\pi$
$200$	0	0
$201$	−1.06634	−0.0752140
$202$	−13.8117	−0.971785
$203$	−1.43579	−0.100773
$204$	−1.90973	−0.133708
$205$	0	0
$206$	−1.77695	−0.123806
$207$	9.75070	0.677720
$208$	0.280290	0.0194346
$209$	−10.7102	−0.740841
$210$	0	0
$211$	6.51783	0.448706	0.224353	−	0.974508i	$-0.427973\pi$
$211$	6.51783	0.448706	0.224353	+	0.974508i	$0.427973\pi$
$212$	−30.9072	−2.12272
$213$	5.45815	0.373987
$214$	−29.7786	−2.03562
$215$	0	0
$216$	10.4356	0.710056
$217$	−2.12216	−0.144062
$218$	37.5041	2.54010
$219$	7.38339	0.498923
$220$	0	0
$221$	−0.907796	−0.0610650
$222$	−9.11984	−0.612083
$223$	−15.7100	−1.05202	−0.526010	−	0.850478i	$-0.676313\pi$
$223$	−15.7100	−1.05202	−0.526010	+	0.850478i	$0.676313\pi$
$224$	1.12236	0.0749911
$225$	0	0
$226$	−3.85104	−0.256168
$227$	−4.47811	−0.297223	−0.148611	−	0.988896i	$-0.547480\pi$
$227$	−4.47811	−0.297223	−0.148611	+	0.988896i	$0.547480\pi$
$228$	−9.13832	−0.605200
$229$	−15.7525	−1.04096	−0.520478	−	0.853875i	$-0.674246\pi$
$229$	−15.7525	−1.04096	−0.520478	+	0.853875i	$0.674246\pi$
$230$	0	0
$231$	0.314972	0.0207237
$232$	20.3623	1.33685
$233$	−17.7331	−1.16173	−0.580867	−	0.813998i	$-0.697287\pi$
$233$	−17.7331	−1.16173	−0.580867	+	0.813998i	$0.697287\pi$
$234$	5.86451	0.383375
$235$	0	0
$236$	0.451659	0.0294005
$237$	−1.56509	−0.101664
$238$	0.452294	0.0293179
$239$	21.9067	1.41703	0.708514	−	0.705696i	$-0.249366\pi$
$239$	21.9067	1.41703	0.708514	+	0.705696i	$0.249366\pi$
$240$	0	0
$241$	−1.00000	−0.0644157
$241$	−1.00000	−0.0644157
$242$	12.0299	0.773309
$243$	13.9911	0.897532
$244$	49.1918	3.14918
$245$	0	0
$246$	−3.00578	−0.191642
$247$	−4.34392	−0.276397
$248$	30.0964	1.91112
$249$	7.84004	0.496843
$250$	0	0
$251$	14.5327	0.917293	0.458647	−	0.888619i	$-0.348334\pi$
$251$	14.5327	0.917293	0.458647	+	0.888619i	$0.348334\pi$
$252$	−1.81918	−0.114598
$253$	−8.96403	−0.563564
$254$	−12.3923	−0.777561
$255$	0	0
$256$	−17.8151	−1.11344
$257$	−7.60643	−0.474476	−0.237238	−	0.971452i	$-0.576242\pi$
$257$	−7.60643	−0.474476	−0.237238	+	0.971452i	$0.576242\pi$
$258$	−6.39402	−0.398075
$259$	1.34477	0.0835597
$260$	0	0
$261$	17.7920	1.10130
$262$	−27.9528	−1.72693
$263$	−11.7074	−0.721910	−0.360955	−	0.932583i	$-0.617549\pi$
$263$	−11.7074	−0.721910	−0.360955	+	0.932583i	$0.617549\pi$
$264$	−4.46692	−0.274920
$265$	0	0
$266$	2.16428	0.132701
$267$	−0.0583213	−0.00356921
$268$	−5.66198	−0.345860
$269$	28.7925	1.75551	0.877754	−	0.479112i	$-0.159041\pi$
$269$	28.7925	1.75551	0.877754	+	0.479112i	$0.159041\pi$
$270$	0	0
$271$	−17.5105	−1.06369	−0.531843	−	0.846843i	$-0.678500\pi$
$271$	−17.5105	−1.06369	−0.531843	+	0.846843i	$0.678500\pi$
$272$	−0.267873	−0.0162422
$273$	0.127749	0.00773169
$274$	20.5590	1.24201
$275$	0	0
$276$	−7.64842	−0.460381
$277$	−3.47856	−0.209007	−0.104503	−	0.994525i	$-0.533325\pi$
$277$	−3.47856	−0.209007	−0.104503	+	0.994525i	$0.533325\pi$
$278$	−39.2316	−2.35296
$279$	26.2973	1.57438
$280$	0	0
$281$	−25.5910	−1.52663	−0.763317	−	0.646024i	$-0.776431\pi$
$281$	−25.5910	−1.52663	−0.763317	+	0.646024i	$0.776431\pi$
$282$	−5.15345	−0.306884
$283$	−12.7295	−0.756688	−0.378344	−	0.925665i	$-0.623506\pi$
$283$	−12.7295	−0.756688	−0.378344	+	0.925665i	$0.623506\pi$
$284$	28.9813	1.71972
$285$	0	0
$286$	−5.39138	−0.318799
$287$	0.443218	0.0261623
$288$	−13.9081	−0.819540
$289$	−16.1324	−0.948966
$290$	0	0
$291$	−3.55713	−0.208523
$292$	39.2038	2.29423
$293$	31.1009	1.81693	0.908466	−	0.417958i	$-0.137254\pi$
$293$	31.1009	1.81693	0.908466	+	0.417958i	$0.137254\pi$
$294$	9.94989	0.580289
$295$	0	0
$296$	−19.0714	−1.10850
$297$	−8.38271	−0.486414
$298$	−31.3308	−1.81494
$299$	−3.63569	−0.210257
$300$	0	0
$301$	0.942831	0.0543439
$302$	51.1799	2.94507
$303$	3.72823	0.214181
$304$	−1.28181	−0.0735166
$305$	0	0
$306$	−5.60471	−0.320400
$307$	−30.9837	−1.76834	−0.884168	−	0.467170i	$-0.845274\pi$
$307$	−30.9837	−1.76834	−0.884168	+	0.467170i	$0.845274\pi$
$308$	1.67242	0.0952947
$309$	0.479659	0.0272868
$310$	0	0
$311$	20.2422	1.14783	0.573915	−	0.818915i	$-0.305424\pi$
$311$	20.2422	1.14783	0.573915	+	0.818915i	$0.305424\pi$
$312$	−1.81172	−0.102568
$313$	19.2756	1.08952	0.544762	−	0.838591i	$-0.316620\pi$
$313$	19.2756	1.08952	0.544762	+	0.838591i	$0.316620\pi$
$314$	−53.5240	−3.02053
$315$	0	0
$316$	−8.31022	−0.467486
$317$	−28.7017	−1.61205	−0.806023	−	0.591884i	$-0.798384\pi$
$317$	−28.7017	−1.61205	−0.806023	+	0.591884i	$0.798384\pi$
$318$	13.4000	0.751435
$319$	−16.3566	−0.915793
$320$	0	0
$321$	8.03825	0.448651
$322$	1.81142	0.100947
$323$	4.15148	0.230995
$324$	18.7207	1.04004
$325$	0	0
$326$	−47.0079	−2.60353
$327$	−10.1236	−0.559837
$328$	−6.28568	−0.347069
$329$	0.759903	0.0418948
$330$	0	0
$331$	3.66766	0.201593	0.100796	−	0.994907i	$-0.467861\pi$
$331$	3.66766	0.201593	0.100796	+	0.994907i	$0.467861\pi$
$332$	41.6285	2.28466
$333$	−16.6640	−0.913182
$334$	12.1582	0.665269
$335$	0	0
$336$	0.0376961	0.00205649
$337$	5.65473	0.308033	0.154016	−	0.988068i	$-0.450779\pi$
$337$	5.65473	0.308033	0.154016	+	0.988068i	$0.450779\pi$
$338$	27.7401	1.50886
$339$	1.03953	0.0564593
$340$	0	0
$341$	−24.1757	−1.30919
$342$	−26.8193	−1.45022
$343$	−2.94371	−0.158945
$344$	−13.3712	−0.720925
$345$	0	0
$346$	−52.1334	−2.80271
$347$	−9.82236	−0.527292	−0.263646	−	0.964620i	$-0.584925\pi$
$347$	−9.82236	−0.527292	−0.263646	+	0.964620i	$0.584925\pi$
$348$	−13.9560	−0.748120
$349$	9.14553	0.489549	0.244775	−	0.969580i	$-0.421286\pi$
$349$	9.14553	0.489549	0.244775	+	0.969580i	$0.421286\pi$
$350$	0	0
$351$	−3.39991	−0.181474
$352$	12.7860	0.681495
$353$	−12.2849	−0.653858	−0.326929	−	0.945049i	$-0.606014\pi$
$353$	−12.2849	−0.653858	−0.326929	+	0.945049i	$0.606014\pi$
$354$	−0.195819	−0.0104077
$355$	0	0
$356$	−0.309670	−0.0164125
$357$	−0.122089	−0.00646165
$358$	33.8628	1.78971
$359$	9.35891	0.493944	0.246972	−	0.969023i	$-0.420564\pi$
$359$	9.35891	0.493944	0.246972	+	0.969023i	$0.420564\pi$
$360$	0	0
$361$	0.865352	0.0455449
$362$	−14.8883	−0.782513
$363$	−3.24726	−0.170437
$364$	0.678309	0.0355531
$365$	0	0
$366$	−21.3274	−1.11480
$367$	28.3939	1.48215	0.741076	−	0.671421i	$-0.234316\pi$
$367$	28.3939	1.48215	0.741076	+	0.671421i	$0.234316\pi$
$368$	−1.07282	−0.0559247
$369$	−5.49225	−0.285915
$370$	0	0
$371$	−1.97590	−0.102584
$372$	−20.6275	−1.06949
$373$	−32.8172	−1.69921	−0.849605	−	0.527419i	$-0.823160\pi$
$373$	−32.8172	−1.69921	−0.849605	+	0.527419i	$0.823160\pi$
$374$	5.15254	0.266431
$375$	0	0
$376$	−10.7769	−0.555776
$377$	−6.63401	−0.341669
$378$	1.69395	0.0871274
$379$	−36.5564	−1.87778	−0.938888	−	0.344222i	$-0.888143\pi$
$379$	−36.5564	−1.87778	−0.938888	+	0.344222i	$0.888143\pi$
$380$	0	0
$381$	3.34509	0.171374
$382$	−20.4501	−1.04632
$383$	2.86409	0.146348	0.0731741	−	0.997319i	$-0.476687\pi$
$383$	2.86409	0.146348	0.0731741	+	0.997319i	$0.476687\pi$
$384$	11.7322	0.598708
$385$	0	0
$386$	−12.2253	−0.622254
$387$	−11.6833	−0.593897
$388$	−18.8874	−0.958862
$389$	33.6956	1.70844	0.854218	−	0.519915i	$-0.174036\pi$
$389$	33.6956	1.70844	0.854218	+	0.519915i	$0.174036\pi$
$390$	0	0
$391$	3.47463	0.175720
$392$	20.8072	1.05092
$393$	7.54540	0.380615
$394$	−49.8127	−2.50953
$395$	0	0
$396$	−20.7242	−1.04143
$397$	−17.2778	−0.867147	−0.433574	−	0.901118i	$-0.642748\pi$
$397$	−17.2778	−0.867147	−0.433574	+	0.901118i	$0.642748\pi$
$398$	−35.8894	−1.79897
$399$	−0.584212	−0.0292472
$400$	0	0
$401$	−1.82076	−0.0909246	−0.0454623	−	0.998966i	$-0.514476\pi$
$401$	−1.82076	−0.0909246	−0.0454623	+	0.998966i	$0.514476\pi$
$402$	2.45478	0.122433
$403$	−9.80534	−0.488439
$404$	19.7959	0.984881
$405$	0	0
$406$	3.30528	0.164038
$407$	15.3196	0.759364
$408$	1.73146	0.0857201
$409$	28.2861	1.39866	0.699329	−	0.714800i	$-0.253482\pi$
$409$	28.2861	1.39866	0.699329	+	0.714800i	$0.253482\pi$
$410$	0	0
$411$	−5.54955	−0.273739
$412$	2.54685	0.125475
$413$	0.0288746	0.00142082
$414$	−22.4467	−1.10319
$415$	0	0
$416$	5.18582	0.254256
$417$	10.5899	0.518591
$418$	24.6555	1.20594
$419$	20.8218	1.01721	0.508606	−	0.860999i	$-0.330161\pi$
$419$	20.8218	1.01721	0.508606	+	0.860999i	$0.330161\pi$
$420$	0	0
$421$	1.24934	0.0608889	0.0304444	−	0.999536i	$-0.490308\pi$
$421$	1.24934	0.0608889	0.0304444	+	0.999536i	$0.490308\pi$
$422$	−15.0044	−0.730405
$423$	−9.41653	−0.457847
$424$	28.0220	1.36087
$425$	0	0
$426$	−12.5650	−0.608776
$427$	3.14483	0.152189
$428$	42.6809	2.06306
$429$	1.45531	0.0702632
$430$	0	0
$431$	14.6970	0.707931	0.353966	−	0.935258i	$-0.384833\pi$
$431$	14.6970	0.707931	0.353966	+	0.935258i	$0.384833\pi$
$432$	−1.00325	−0.0482688
$433$	14.5749	0.700424	0.350212	−	0.936670i	$-0.386109\pi$
$433$	14.5749	0.700424	0.350212	+	0.936670i	$0.386109\pi$
$434$	4.88534	0.234504
$435$	0	0
$436$	−53.7536	−2.57433
$437$	16.6265	0.795355
$438$	−16.9970	−0.812148
$439$	−6.56204	−0.313189	−0.156594	−	0.987663i	$-0.550052\pi$
$439$	−6.56204	−0.313189	−0.156594	+	0.987663i	$0.550052\pi$
$440$	0	0
$441$	18.1807	0.865748
$442$	2.08980	0.0994016
$443$	−5.86656	−0.278729	−0.139364	−	0.990241i	$-0.544506\pi$
$443$	−5.86656	−0.278729	−0.139364	+	0.990241i	$0.544506\pi$
$444$	13.0712	0.620332
$445$	0	0
$446$	36.1654	1.71248
$447$	8.45722	0.400013
$448$	−2.70507	−0.127803
$449$	41.6763	1.96683	0.983413	−	0.181380i	$-0.0580565\pi$
$449$	41.6763	1.96683	0.983413	+	0.181380i	$0.0580565\pi$
$450$	0	0
$451$	5.04914	0.237755
$452$	5.51959	0.259620
$453$	−13.8152	−0.649093
$454$	10.3089	0.483819
$455$	0	0
$456$	8.28526	0.387993
$457$	17.3603	0.812079	0.406039	−	0.913856i	$-0.366910\pi$
$457$	17.3603	0.812079	0.406039	+	0.913856i	$0.366910\pi$
$458$	36.2632	1.69447
$459$	3.24930	0.151664
$460$	0	0
$461$	−15.8619	−0.738761	−0.369381	−	0.929278i	$-0.620430\pi$
$461$	−15.8619	−0.738761	−0.369381	+	0.929278i	$0.620430\pi$
$462$	−0.725085	−0.0337340
$463$	38.1251	1.77182	0.885912	−	0.463853i	$-0.153533\pi$
$463$	38.1251	1.77182	0.885912	+	0.463853i	$0.153533\pi$
$464$	−1.95757	−0.0908777
$465$	0	0
$466$	40.8227	1.89107
$467$	3.00097	0.138868	0.0694341	−	0.997587i	$-0.477881\pi$
$467$	3.00097	0.138868	0.0694341	+	0.997587i	$0.477881\pi$
$468$	−8.40544	−0.388542
$469$	−0.361970	−0.0167142
$470$	0	0
$471$	14.4479	0.665725
$472$	−0.409497	−0.0188486
$473$	10.7407	0.493860
$474$	3.60294	0.165489
$475$	0	0
$476$	−0.648260	−0.0297130
$477$	24.4848	1.12108
$478$	−50.4306	−2.30664
$479$	9.13452	0.417367	0.208683	−	0.977983i	$-0.433082\pi$
$479$	9.13452	0.417367	0.208683	+	0.977983i	$0.433082\pi$
$480$	0	0
$481$	6.21342	0.283308
$482$	2.30206	0.104856
$483$	−0.488963	−0.0222486
$484$	−17.2421	−0.783730
$485$	0	0
$486$	−32.2084	−1.46100
$487$	−33.1282	−1.50118	−0.750590	−	0.660768i	$-0.770231\pi$
$487$	−33.1282	−1.50118	−0.750590	+	0.660768i	$0.770231\pi$
$488$	−44.5998	−2.01894
$489$	12.6890	0.573816
$490$	0	0
$491$	−9.79550	−0.442065	−0.221032	−	0.975266i	$-0.570943\pi$
$491$	−9.79550	−0.442065	−0.221032	+	0.975266i	$0.570943\pi$
$492$	4.30810	0.194224
$493$	6.34012	0.285545
$494$	9.99995	0.449919
$495$	0	0
$496$	−2.89336	−0.129916
$497$	1.85277	0.0831082
$498$	−18.0482	−0.808761
$499$	−33.9595	−1.52023	−0.760117	−	0.649786i	$-0.774858\pi$
$499$	−33.9595	−1.52023	−0.760117	+	0.649786i	$0.774858\pi$
$500$	0	0
$501$	−3.28191	−0.146625
$502$	−33.4550	−1.49317
$503$	5.66981	0.252804	0.126402	−	0.991979i	$-0.459657\pi$
$503$	5.66981	0.252804	0.126402	+	0.991979i	$0.459657\pi$
$504$	1.64936	0.0734685
$505$	0	0
$506$	20.6357	0.917370
$507$	−7.48798	−0.332553
$508$	17.7615	0.788039
$509$	−35.5544	−1.57592	−0.787960	−	0.615727i	$-0.788863\pi$
$509$	−35.5544	−1.57592	−0.787960	+	0.615727i	$0.788863\pi$
$510$	0	0
$511$	2.50629	0.110872
$512$	3.25087	0.143670
$513$	15.5483	0.686474
$514$	17.5104	0.772352
$515$	0	0
$516$	9.16437	0.403439
$517$	8.65683	0.380727
$518$	−3.09573	−0.136019
$519$	14.0725	0.617716
$520$	0	0
$521$	42.5505	1.86417	0.932085	−	0.362239i	$-0.117988\pi$
$521$	42.5505	1.86417	0.932085	+	0.362239i	$0.117988\pi$
$522$	−40.9582	−1.79269
$523$	3.91190	0.171055	0.0855277	−	0.996336i	$-0.472742\pi$
$523$	3.91190	0.171055	0.0855277	+	0.996336i	$0.472742\pi$
$524$	40.0640	1.75020
$525$	0	0
$526$	26.9512	1.17513
$527$	9.37096	0.408205
$528$	0.429435	0.0186887
$529$	−9.08424	−0.394967
$530$	0	0
$531$	−0.357806	−0.0155275
$532$	−3.10201	−0.134489
$533$	2.04786	0.0887028
$534$	0.134259	0.00580996
$535$	0	0
$536$	5.13343	0.221731
$537$	−9.14071	−0.394451
$538$	−66.2819	−2.85762
$539$	−16.7139	−0.719920
$540$	0	0
$541$	12.2335	0.525960	0.262980	−	0.964801i	$-0.415295\pi$
$541$	12.2335	0.525960	0.262980	+	0.964801i	$0.415295\pi$
$542$	40.3102	1.73147
$543$	4.01886	0.172466
$544$	−4.95609	−0.212491
$545$	0	0
$546$	−0.294085	−0.0125857
$547$	−21.1242	−0.903207	−0.451603	−	0.892219i	$-0.649148\pi$
$547$	−21.1242	−0.903207	−0.451603	+	0.892219i	$0.649148\pi$
$548$	−29.4666	−1.25875
$549$	−38.9700	−1.66320
$550$	0	0
$551$	30.3383	1.29245
$552$	6.93444	0.295149
$553$	−0.531272	−0.0225920
$554$	8.00786	0.340221
$555$	0	0
$556$	56.2296	2.38466
$557$	21.6076	0.915543	0.457772	−	0.889070i	$-0.348648\pi$
$557$	21.6076	0.915543	0.457772	+	0.889070i	$0.348648\pi$
$558$	−60.5380	−2.56278
$559$	4.35630	0.184252
$560$	0	0
$561$	−1.39084	−0.0587214
$562$	58.9121	2.48506
$563$	5.61389	0.236597	0.118299	−	0.992978i	$-0.462256\pi$
$563$	5.61389	0.236597	0.118299	+	0.992978i	$0.462256\pi$
$564$	7.38630	0.311019
$565$	0	0
$566$	29.3040	1.23174
$567$	1.19681	0.0502614
$568$	−26.2759	−1.10251
$569$	20.8782	0.875260	0.437630	−	0.899155i	$-0.355818\pi$
$569$	20.8782	0.875260	0.437630	+	0.899155i	$0.355818\pi$
$570$	0	0
$571$	−21.6611	−0.906490	−0.453245	−	0.891386i	$-0.649734\pi$
$571$	−21.6611	−0.906490	−0.453245	+	0.891386i	$0.649734\pi$
$572$	7.72731	0.323095
$573$	5.52016	0.230608
$574$	−1.02031	−0.0425871
$575$	0	0
$576$	33.5206	1.39669
$577$	27.6119	1.14950	0.574749	−	0.818330i	$-0.305100\pi$
$577$	27.6119	1.14950	0.574749	+	0.818330i	$0.305100\pi$
$578$	37.1378	1.54473
$579$	3.30003	0.137145
$580$	0	0
$581$	2.66131	0.110410
$582$	8.18873	0.339434
$583$	−22.5095	−0.932246
$584$	−35.5441	−1.47082
$585$	0	0
$586$	−71.5960	−2.95761
$587$	33.5649	1.38537	0.692685	−	0.721240i	$-0.256428\pi$
$587$	33.5649	1.38537	0.692685	+	0.721240i	$0.256428\pi$
$588$	−14.2609	−0.588110
$589$	44.8412	1.84765
$590$	0	0
$591$	13.4461	0.553099
$592$	1.83346	0.0753547
$593$	16.8437	0.691689	0.345844	−	0.938292i	$-0.387593\pi$
$593$	16.8437	0.691689	0.345844	+	0.938292i	$0.387593\pi$
$594$	19.2975	0.791786
$595$	0	0
$596$	44.9055	1.83940
$597$	9.68775	0.396493
$598$	8.36957	0.342257
$599$	−15.8531	−0.647741	−0.323871	−	0.946101i	$-0.604984\pi$
$599$	−15.8531	−0.647741	−0.323871	+	0.946101i	$0.604984\pi$
$600$	0	0
$601$	−29.8027	−1.21568	−0.607839	−	0.794060i	$-0.707964\pi$
$601$	−29.8027	−1.21568	−0.607839	+	0.794060i	$0.707964\pi$
$602$	−2.17045	−0.0884611
$603$	4.48544	0.182661
$604$	−73.3547	−2.98476
$605$	0	0
$606$	−8.58261	−0.348645
$607$	8.19562	0.332650	0.166325	−	0.986071i	$-0.446810\pi$
$607$	8.19562	0.332650	0.166325	+	0.986071i	$0.446810\pi$
$608$	−23.7155	−0.961791
$609$	−0.892206	−0.0361540
$610$	0	0
$611$	3.51109	0.142044
$612$	8.03308	0.324718
$613$	−3.11827	−0.125946	−0.0629730	−	0.998015i	$-0.520058\pi$
$613$	−3.11827	−0.125946	−0.0629730	+	0.998015i	$0.520058\pi$
$614$	71.3264	2.87850
$615$	0	0
$616$	−1.51630	−0.0610933
$617$	−22.7055	−0.914089	−0.457044	−	0.889444i	$-0.651092\pi$
$617$	−22.7055	−0.914089	−0.457044	+	0.889444i	$0.651092\pi$
$618$	−1.10420	−0.0444175
$619$	24.0523	0.966745	0.483373	−	0.875415i	$-0.339412\pi$
$619$	24.0523	0.966745	0.483373	+	0.875415i	$0.339412\pi$
$620$	0	0
$621$	13.0133	0.522207
$622$	−46.5988	−1.86844
$623$	−0.0197972	−0.000793157 0
$624$	0.174173	0.00697249
$625$	0	0
$626$	−44.3737	−1.77353
$627$	−6.65535	−0.265789
$628$	76.7144	3.06124
$629$	−5.93817	−0.236770
$630$	0	0
$631$	−18.8649	−0.750999	−0.375500	−	0.926823i	$-0.622529\pi$
$631$	−18.8649	−0.750999	−0.375500	+	0.926823i	$0.622529\pi$
$632$	7.53446	0.299705
$633$	4.05020	0.160981
$634$	66.0729	2.62409
$635$	0	0
$636$	−19.2058	−0.761561
$637$	−6.77894	−0.268592
$638$	37.6538	1.49073
$639$	−22.9591	−0.908248
$640$	0	0
$641$	24.1790	0.955012	0.477506	−	0.878628i	$-0.341541\pi$
$641$	24.1790	0.955012	0.477506	+	0.878628i	$0.341541\pi$
$642$	−18.5045	−0.730315
$643$	−13.3248	−0.525480	−0.262740	−	0.964867i	$-0.584626\pi$
$643$	−13.3248	−0.525480	−0.262740	+	0.964867i	$0.584626\pi$
$644$	−2.59626	−0.102307
$645$	0	0
$646$	−9.55695	−0.376013
$647$	−4.31010	−0.169448	−0.0847238	−	0.996404i	$-0.527001\pi$
$647$	−4.31010	−0.169448	−0.0847238	+	0.996404i	$0.527001\pi$
$648$	−16.9731	−0.666767
$649$	0.328939	0.0129120
$650$	0	0
$651$	−1.31872	−0.0516846
$652$	67.3750	2.63861
$653$	−9.01003	−0.352590	−0.176295	−	0.984337i	$-0.556411\pi$
$653$	−9.01003	−0.352590	−0.176295	+	0.984337i	$0.556411\pi$
$654$	23.3052	0.911304
$655$	0	0
$656$	0.604285	0.0235934
$657$	−31.0574	−1.21166
$658$	−1.74934	−0.0681964
$659$	24.4469	0.952315	0.476158	−	0.879360i	$-0.342029\pi$
$659$	24.4469	0.952315	0.476158	+	0.879360i	$0.342029\pi$
$660$	0	0
$661$	27.3702	1.06458	0.532288	−	0.846563i	$-0.321332\pi$
$661$	27.3702	1.06458	0.532288	+	0.846563i	$0.321332\pi$
$662$	−8.44316	−0.328153
$663$	−0.564107	−0.0219081
$664$	−37.7424	−1.46469
$665$	0	0
$666$	38.3615	1.48648
$667$	25.3920	0.983180
$668$	−17.4261	−0.674234
$669$	−9.76225	−0.377430
$670$	0	0
$671$	35.8260	1.38305
$672$	0.697440	0.0269043
$673$	−26.8965	−1.03678	−0.518392	−	0.855143i	$-0.673469\pi$
$673$	−26.8965	−1.03678	−0.518392	+	0.855143i	$0.673469\pi$
$674$	−13.0175	−0.501416
$675$	0	0
$676$	−39.7591	−1.52920
$677$	−6.02201	−0.231445	−0.115722	−	0.993282i	$-0.536918\pi$
$677$	−6.02201	−0.231445	−0.115722	+	0.993282i	$0.536918\pi$
$678$	−2.39305	−0.0919045
$679$	−1.20747	−0.0463385
$680$	0	0
$681$	−2.78271	−0.106634
$682$	55.6539	2.13110
$683$	39.2858	1.50323	0.751614	−	0.659603i	$-0.229276\pi$
$683$	39.2858	1.50323	0.751614	+	0.659603i	$0.229276\pi$
$684$	38.4393	1.46976
$685$	0	0
$686$	6.77659	0.258732
$687$	−9.78865	−0.373460
$688$	1.28546	0.0490077
$689$	−9.12953	−0.347807
$690$	0	0
$691$	14.7694	0.561856	0.280928	−	0.959729i	$-0.409358\pi$
$691$	14.7694	0.561856	0.280928	+	0.959729i	$0.409358\pi$
$692$	74.7213	2.84048
$693$	−1.32489	−0.0503286
$694$	22.6116	0.858326
$695$	0	0
$696$	12.6532	0.479618
$697$	−1.95714	−0.0741321
$698$	−21.0536	−0.796889
$699$	−11.0194	−0.416792
$700$	0	0
$701$	43.5098	1.64334	0.821672	−	0.569961i	$-0.193042\pi$
$701$	43.5098	1.64334	0.821672	+	0.569961i	$0.193042\pi$
$702$	7.82680	0.295404
$703$	−28.4149	−1.07169
$704$	−30.8162	−1.16143
$705$	0	0
$706$	28.2805	1.06435
$707$	1.26555	0.0475959
$708$	0.280662	0.0105479
$709$	−10.4609	−0.392866	−0.196433	−	0.980517i	$-0.562936\pi$
$709$	−10.4609	−0.392866	−0.196433	+	0.980517i	$0.562936\pi$
$710$	0	0
$711$	6.58339	0.246896
$712$	0.280762	0.0105220
$713$	37.5304	1.40552
$714$	0.281057	0.0105183
$715$	0	0
$716$	−48.5346	−1.81382
$717$	13.6129	0.508383
$718$	−21.5448	−0.804043
$719$	−39.7332	−1.48180	−0.740900	−	0.671615i	$-0.765601\pi$
$719$	−39.7332	−1.48180	−0.740900	+	0.671615i	$0.765601\pi$
$720$	0	0
$721$	0.162820	0.00606375
$722$	−1.99209	−0.0741380
$723$	−0.621403	−0.0231102
$724$	21.3390	0.793058
$725$	0	0
$726$	7.47539	0.277438
$727$	5.02605	0.186406	0.0932029	−	0.995647i	$-0.470289\pi$
$727$	5.02605	0.186406	0.0932029	+	0.995647i	$0.470289\pi$
$728$	−0.614989	−0.0227930
$729$	−8.32737	−0.308421
$730$	0	0
$731$	−4.16332	−0.153986
$732$	30.5679	1.12982
$733$	−2.25167	−0.0831675	−0.0415837	−	0.999135i	$-0.513240\pi$
$733$	−2.25167	−0.0831675	−0.0415837	+	0.999135i	$0.513240\pi$
$734$	−65.3645	−2.41265
$735$	0	0
$736$	−19.8490	−0.731643
$737$	−4.12357	−0.151894
$738$	12.6435	0.465413
$739$	12.6976	0.467088	0.233544	−	0.972346i	$-0.424968\pi$
$739$	12.6976	0.467088	0.233544	+	0.972346i	$0.424968\pi$
$740$	0	0
$741$	−2.69932	−0.0991620
$742$	4.54863	0.166986
$743$	15.3764	0.564105	0.282052	−	0.959399i	$-0.408985\pi$
$743$	15.3764	0.564105	0.282052	+	0.959399i	$0.408985\pi$
$744$	18.7020	0.685647
$745$	0	0
$746$	75.5471	2.76598
$747$	−32.9782	−1.20661
$748$	−7.38499	−0.270022
$749$	2.72858	0.0997003
$750$	0	0
$751$	−23.2707	−0.849159	−0.424580	−	0.905391i	$-0.639578\pi$
$751$	−23.2707	−0.849159	−0.424580	+	0.905391i	$0.639578\pi$
$752$	1.03605	0.0377810
$753$	9.03063	0.329095
$754$	15.2719	0.556169
$755$	0	0
$756$	−2.42789	−0.0883015
$757$	33.5311	1.21871	0.609354	−	0.792898i	$-0.291429\pi$
$757$	33.5311	1.21871	0.609354	+	0.792898i	$0.291429\pi$
$758$	84.1550	3.05665
$759$	−5.57027	−0.202188
$760$	0	0
$761$	−2.94990	−0.106934	−0.0534669	−	0.998570i	$-0.517027\pi$
$761$	−2.94990	−0.106934	−0.0534669	+	0.998570i	$0.517027\pi$
$762$	−7.70060	−0.278963
$763$	−3.43646	−0.124408
$764$	29.3105	1.06042
$765$	0	0
$766$	−6.59330	−0.238226
$767$	0.133413	0.00481728
$768$	−11.0703	−0.399467
$769$	24.1690	0.871556	0.435778	−	0.900054i	$-0.356473\pi$
$769$	24.1690	0.871556	0.435778	+	0.900054i	$0.356473\pi$
$770$	0	0
$771$	−4.72665	−0.170226
$772$	17.5222	0.630639
$773$	11.7946	0.424223	0.212111	−	0.977246i	$-0.431966\pi$
$773$	11.7946	0.424223	0.212111	+	0.977246i	$0.431966\pi$
$774$	26.8957	0.966747
$775$	0	0
$776$	17.1243	0.614725
$777$	0.835642	0.0299785
$778$	−77.5693	−2.78099
$779$	−9.36518	−0.335542
$780$	0	0
$781$	21.1068	0.755261
$782$	−7.99880	−0.286037
$783$	23.7453	0.848587
$784$	−2.00033	−0.0714405
$785$	0	0
$786$	−17.3700	−0.619566
$787$	34.2291	1.22013	0.610067	−	0.792349i	$-0.291142\pi$
$787$	34.2291	1.22013	0.610067	+	0.792349i	$0.291142\pi$
$788$	71.3952	2.54335
$789$	−7.27502	−0.258998
$790$	0	0
$791$	0.352867	0.0125465
$792$	18.7896	0.667658
$793$	14.5305	0.515994
$794$	39.7745	1.41154
$795$	0	0
$796$	51.4393	1.82322
$797$	4.91446	0.174079	0.0870395	−	0.996205i	$-0.472259\pi$
$797$	4.91446	0.174079	0.0870395	+	0.996205i	$0.472259\pi$
$798$	1.34489	0.0476086
$799$	−3.35555	−0.118711
$800$	0	0
$801$	0.245322	0.00866802
$802$	4.19151	0.148007
$803$	28.5517	1.00757
$804$	−3.51837	−0.124083
$805$	0	0
$806$	22.5725	0.795081
$807$	17.8917	0.629818
$808$	−17.9479	−0.631406
$809$	8.69376	0.305656	0.152828	−	0.988253i	$-0.451162\pi$
$809$	8.69376	0.305656	0.152828	+	0.988253i	$0.451162\pi$
$810$	0	0
$811$	45.8146	1.60877	0.804384	−	0.594110i	$-0.202496\pi$
$811$	45.8146	1.60877	0.804384	+	0.594110i	$0.202496\pi$
$812$	−4.73737	−0.166249
$813$	−10.8811	−0.381616
$814$	−35.2666	−1.23609
$815$	0	0
$816$	−0.166457	−0.00582716
$817$	−19.9220	−0.696982
$818$	−65.1163	−2.27674
$819$	−0.537359	−0.0187769
$820$	0	0
$821$	37.5369	1.31005	0.655023	−	0.755609i	$-0.272659\pi$
$821$	37.5369	1.31005	0.655023	+	0.755609i	$0.272659\pi$
$822$	12.7754	0.445593
$823$	−31.1636	−1.08630	−0.543148	−	0.839637i	$-0.682768\pi$
$823$	−31.1636	−1.08630	−0.543148	+	0.839637i	$0.682768\pi$
$824$	−2.30911	−0.0804415
$825$	0	0
$826$	−0.0664710	−0.00231282
$827$	44.5635	1.54962	0.774812	−	0.632191i	$-0.217844\pi$
$827$	44.5635	1.54962	0.774812	+	0.632191i	$0.217844\pi$
$828$	32.1722	1.11806
$829$	21.1083	0.733122	0.366561	−	0.930394i	$-0.380535\pi$
$829$	21.1083	0.733122	0.366561	+	0.930394i	$0.380535\pi$
$830$	0	0
$831$	−2.16159	−0.0749847
$832$	−12.4987	−0.433313
$833$	6.47864	0.224471
$834$	−24.3786	−0.844163
$835$	0	0
$836$	−35.3381	−1.22219
$837$	35.0965	1.21311
$838$	−47.9330	−1.65582
$839$	−3.65940	−0.126337	−0.0631683	−	0.998003i	$-0.520120\pi$
$839$	−3.65940	−0.126337	−0.0631683	+	0.998003i	$0.520120\pi$
$840$	0	0
$841$	17.3324	0.597670
$842$	−2.87604	−0.0991150
$843$	−15.9023	−0.547706
$844$	21.5054	0.740248
$845$	0	0
$846$	21.6774	0.745285
$847$	−1.10228	−0.0378749
$848$	−2.69395	−0.0925105
$849$	−7.91012	−0.271475
$850$	0	0
$851$	−23.7821	−0.815241
$852$	18.0090	0.616980
$853$	2.01691	0.0690575	0.0345288	−	0.999404i	$-0.489007\pi$
$853$	2.01691	0.0690575	0.0345288	+	0.999404i	$0.489007\pi$
$854$	−7.23959	−0.247734
$855$	0	0
$856$	−38.6966	−1.32262
$857$	−39.8911	−1.36266	−0.681328	−	0.731979i	$-0.738597\pi$
$857$	−39.8911	−1.36266	−0.681328	+	0.731979i	$0.738597\pi$
$858$	−3.35022	−0.114374
$859$	32.6232	1.11309	0.556544	−	0.830818i	$-0.312127\pi$
$859$	32.6232	1.11309	0.556544	+	0.830818i	$0.312127\pi$
$860$	0	0
$861$	0.275417	0.00938618
$862$	−33.8334	−1.15237
$863$	11.5451	0.393000	0.196500	−	0.980504i	$-0.437042\pi$
$863$	11.5451	0.393000	0.196500	+	0.980504i	$0.437042\pi$
$864$	−18.5618	−0.631484
$865$	0	0
$866$	−33.5522	−1.14015
$867$	−10.0247	−0.340458
$868$	−7.00202	−0.237664
$869$	−6.05226	−0.205309
$870$	0	0
$871$	−1.67246	−0.0566693
$872$	48.7357	1.65040
$873$	14.9627	0.506410
$874$	−38.2753	−1.29468
$875$	0	0
$876$	24.3613	0.823093
$877$	−23.1859	−0.782934	−0.391467	−	0.920192i	$-0.628032\pi$
$877$	−23.1859	−0.782934	−0.391467	+	0.920192i	$0.628032\pi$
$878$	15.1062	0.509810
$879$	19.3262	0.651855
$880$	0	0
$881$	−2.05103	−0.0691010	−0.0345505	−	0.999403i	$-0.511000\pi$
$881$	−2.05103	−0.0691010	−0.0345505	+	0.999403i	$0.511000\pi$
$882$	−41.8531	−1.40927
$883$	29.5602	0.994780	0.497390	−	0.867527i	$-0.334292\pi$
$883$	29.5602	0.994780	0.497390	+	0.867527i	$0.334292\pi$
$884$	−2.99525	−0.100741
$885$	0	0
$886$	13.5052	0.453715
$887$	−52.0973	−1.74926	−0.874628	−	0.484795i	$-0.838894\pi$
$887$	−52.0973	−1.74926	−0.874628	+	0.484795i	$0.838894\pi$
$888$	−11.8510	−0.397694
$889$	1.13549	0.0380832
$890$	0	0
$891$	13.6341	0.456759
$892$	−51.8348	−1.73556
$893$	−16.0567	−0.537318
$894$	−19.4690	−0.651141
$895$	0	0
$896$	3.98251	0.133046
$897$	−2.25923	−0.0754334
$898$	−95.9413	−3.20160
$899$	68.4813	2.28398
$900$	0	0
$901$	8.72509	0.290675
$902$	−11.6234	−0.387018
$903$	0.585878	0.0194968
$904$	−5.00434	−0.166442
$905$	0	0
$906$	31.8033	1.05659
$907$	57.7679	1.91815	0.959077	−	0.283147i	$-0.0913784\pi$
$907$	57.7679	1.91815	0.959077	+	0.283147i	$0.0913784\pi$
$908$	−14.7754	−0.490339
$909$	−15.6824	−0.520151
$910$	0	0
$911$	25.9833	0.860864	0.430432	−	0.902623i	$-0.358361\pi$
$911$	25.9833	0.860864	0.430432	+	0.902623i	$0.358361\pi$
$912$	−0.796517	−0.0263753
$913$	30.3176	1.00337
$914$	−39.9643	−1.32190
$915$	0	0
$916$	−51.9750	−1.71730
$917$	2.56129	0.0845812
$918$	−7.48008	−0.246879
$919$	−8.27163	−0.272856	−0.136428	−	0.990650i	$-0.543562\pi$
$919$	−8.27163	−0.272856	−0.136428	+	0.990650i	$0.543562\pi$
$920$	0	0
$921$	−19.2534	−0.634420
$922$	36.5150	1.20256
$923$	8.56063	0.281777
$924$	1.03924	0.0341886
$925$	0	0
$926$	−87.7663	−2.88418
$927$	−2.01763	−0.0662676
$928$	−36.2182	−1.18892
$929$	−23.1255	−0.758724	−0.379362	−	0.925248i	$-0.623857\pi$
$929$	−23.1255	−0.758724	−0.379362	+	0.925248i	$0.623857\pi$
$930$	0	0
$931$	31.0011	1.01602
$932$	−58.5100	−1.91656
$933$	12.5786	0.411804
$934$	−6.90841	−0.226050
$935$	0	0
$936$	7.62079	0.249093
$937$	9.69548	0.316737	0.158369	−	0.987380i	$-0.449377\pi$
$937$	9.69548	0.316737	0.158369	+	0.987380i	$0.449377\pi$
$938$	0.833276	0.0272074
$939$	11.9779	0.390885
$940$	0	0
$941$	−46.8304	−1.52663	−0.763314	−	0.646028i	$-0.776429\pi$
$941$	−46.8304	−1.52663	−0.763314	+	0.646028i	$0.776429\pi$
$942$	−33.2599	−1.08367
$943$	−7.83829	−0.255250
$944$	0.0393677	0.00128131
$945$	0	0
$946$	−24.7258	−0.803906
$947$	18.3025	0.594753	0.297376	−	0.954760i	$-0.403888\pi$
$947$	18.3025	0.594753	0.297376	+	0.954760i	$0.403888\pi$
$948$	−5.16399	−0.167719
$949$	11.5802	0.375909
$950$	0	0
$951$	−17.8353	−0.578349
$952$	0.587745	0.0190489
$953$	−21.6178	−0.700269	−0.350135	−	0.936699i	$-0.613864\pi$
$953$	−21.6178	−0.700269	−0.350135	+	0.936699i	$0.613864\pi$
$954$	−56.3655	−1.82490
$955$	0	0
$956$	72.2807	2.33773
$957$	−10.1640	−0.328556
$958$	−21.0282	−0.679390
$959$	−1.88380	−0.0608310
$960$	0	0
$961$	70.2182	2.26510
$962$	−14.3037	−0.461169
$963$	−33.8119	−1.08957
$964$	−3.29948	−0.106269
$965$	0	0
$966$	1.12562	0.0362163
$967$	14.0708	0.452485	0.226242	−	0.974071i	$-0.427356\pi$
$967$	14.0708	0.452485	0.226242	+	0.974071i	$0.427356\pi$
$968$	15.6325	0.502448
$969$	2.57974	0.0828732
$970$	0	0
$971$	−9.84488	−0.315937	−0.157969	−	0.987444i	$-0.550495\pi$
$971$	−9.84488	−0.315937	−0.157969	+	0.987444i	$0.550495\pi$
$972$	46.1634	1.48069
$973$	3.59475	0.115243
$974$	76.2630	2.44362
$975$	0	0
$976$	4.28768	0.137245
$977$	43.8289	1.40221	0.701105	−	0.713058i	$-0.252690\pi$
$977$	43.8289	1.40221	0.701105	+	0.713058i	$0.252690\pi$
$978$	−29.2108	−0.934059
$979$	−0.225530	−0.00720796
$980$	0	0
$981$	42.5838	1.35960
$982$	22.5498	0.719594
$983$	51.3060	1.63641	0.818203	−	0.574930i	$-0.194971\pi$
$983$	51.3060	1.63641	0.818203	+	0.574930i	$0.194971\pi$
$984$	−3.90594	−0.124517
$985$	0	0
$986$	−14.5953	−0.464810
$987$	0.472206	0.0150305
$988$	−14.3326	−0.455982
$989$	−16.6739	−0.530200
$990$	0	0
$991$	−38.8108	−1.23287	−0.616433	−	0.787407i	$-0.711423\pi$
$991$	−38.8108	−1.23287	−0.616433	+	0.787407i	$0.711423\pi$
$992$	−53.5320	−1.69964
$993$	2.27909	0.0723248
$994$	−4.26519	−0.135284
$995$	0	0
$996$	25.8680	0.819660
$997$	49.9861	1.58307	0.791537	−	0.611121i	$-0.209281\pi$
$997$	49.9861	1.58307	0.791537	+	0.611121i	$0.209281\pi$
$998$	78.1767	2.47464
$999$	−22.2399	−0.703638

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.o.1.3	yes	40
5.4	even	2		6025.2.a.l.1.38	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6025.2.a.l.1.38	✓	40	5.4	even	2
6025.2.a.o.1.3	yes	40	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:	\( N \)	\(=\)	\( 6025 = 5^{2} \cdot 241 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6025.a (trivial)

\(f(q)\)	\(=\)	\(q-2.30206 q^{2} +0.621403 q^{3} +3.29948 q^{4} -1.43051 q^{6} +0.210935 q^{7} -2.99147 q^{8} -2.61386 q^{9} +O(q^{10})\)	\(q-2.30206 q^{2} +0.621403 q^{3} +3.29948 q^{4} -1.43051 q^{6} +0.210935 q^{7} -2.99147 q^{8} -2.61386 q^{9} +2.40298 q^{11} +2.05030 q^{12} +0.974615 q^{13} -0.485586 q^{14} +0.287590 q^{16} -0.931440 q^{17} +6.01726 q^{18} -4.45706 q^{19} +0.131076 q^{21} -5.53180 q^{22} -3.73038 q^{23} -1.85891 q^{24} -2.24362 q^{26} -3.48847 q^{27} +0.695976 q^{28} -6.80679 q^{29} -10.0607 q^{31} +5.32089 q^{32} +1.49322 q^{33} +2.14423 q^{34} -8.62436 q^{36} +6.37525 q^{37} +10.2604 q^{38} +0.605629 q^{39} +2.10120 q^{41} -0.301744 q^{42} +4.46976 q^{43} +7.92857 q^{44} +8.58756 q^{46} +3.60254 q^{47} +0.178709 q^{48} -6.95551 q^{49} -0.578799 q^{51} +3.21572 q^{52} -9.36731 q^{53} +8.03066 q^{54} -0.631007 q^{56} -2.76963 q^{57} +15.6696 q^{58} +0.136888 q^{59} +14.9090 q^{61} +23.1604 q^{62} -0.551355 q^{63} -12.8242 q^{64} -3.43747 q^{66} -1.71602 q^{67} -3.07326 q^{68} -2.31807 q^{69} +8.78360 q^{71} +7.81928 q^{72} +11.8818 q^{73} -14.6762 q^{74} -14.7060 q^{76} +0.506873 q^{77} -1.39419 q^{78} -2.51865 q^{79} +5.67383 q^{81} -4.83709 q^{82} +12.6167 q^{83} +0.432481 q^{84} -10.2897 q^{86} -4.22976 q^{87} -7.18844 q^{88} -0.0938543 q^{89} +0.205581 q^{91} -12.3083 q^{92} -6.25176 q^{93} -8.29326 q^{94} +3.30642 q^{96} -5.72436 q^{97} +16.0120 q^{98} -6.28105 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 11 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 16 q^{7} + 33 q^{8} + 38 q^{9}+O(q^{10}) \) 40 * q + 11 * q^2 + 8 * q^3 + 41 * q^4 + 3 * q^6 + 16 * q^7 + 33 * q^8 + 38 * q^9	\( 40 q + 11 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 16 q^{7} + 33 q^{8} + 38 q^{9} + q^{11} + 14 q^{12} + 9 q^{13} - q^{14} + 43 q^{16} + 12 q^{17} + 42 q^{18} + 2 q^{21} + 5 q^{22} + 77 q^{23} - 2 q^{24} + 2 q^{26} + 38 q^{27} + 42 q^{28} + 2 q^{29} + q^{31} + 72 q^{32} + 20 q^{33} + 5 q^{34} + 32 q^{36} + 28 q^{37} + 23 q^{38} - 2 q^{39} - 2 q^{41} + 37 q^{42} + 31 q^{43} + 3 q^{44} + 14 q^{46} + 96 q^{47} + 13 q^{48} + 40 q^{49} - 10 q^{51} + 42 q^{52} + 54 q^{53} + 4 q^{54} - 15 q^{56} + 37 q^{57} + 27 q^{58} + q^{59} + 5 q^{61} + 39 q^{62} + 70 q^{63} + 65 q^{64} - 52 q^{66} + 34 q^{67} + 52 q^{68} + 21 q^{69} - 9 q^{71} + 70 q^{72} + 25 q^{73} + 22 q^{74} - 47 q^{76} + 54 q^{77} + 58 q^{78} + 13 q^{79} + 12 q^{81} - 5 q^{82} + 63 q^{83} + 95 q^{84} - 18 q^{86} + 47 q^{87} + 13 q^{88} + 19 q^{89} - 31 q^{91} + 137 q^{92} + 52 q^{93} + 120 q^{94} - 49 q^{96} + 36 q^{97} + 64 q^{98} + 16 q^{99}+O(q^{100}) \) 40 * q + 11 * q^2 + 8 * q^3 + 41 * q^4 + 3 * q^6 + 16 * q^7 + 33 * q^8 + 38 * q^9 + q^11 + 14 * q^12 + 9 * q^13 - q^14 + 43 * q^16 + 12 * q^17 + 42 * q^18 + 2 * q^21 + 5 * q^22 + 77 * q^23 - 2 * q^24 + 2 * q^26 + 38 * q^27 + 42 * q^28 + 2 * q^29 + q^31 + 72 * q^32 + 20 * q^33 + 5 * q^34 + 32 * q^36 + 28 * q^37 + 23 * q^38 - 2 * q^39 - 2 * q^41 + 37 * q^42 + 31 * q^43 + 3 * q^44 + 14 * q^46 + 96 * q^47 + 13 * q^48 + 40 * q^49 - 10 * q^51 + 42 * q^52 + 54 * q^53 + 4 * q^54 - 15 * q^56 + 37 * q^57 + 27 * q^58 + q^59 + 5 * q^61 + 39 * q^62 + 70 * q^63 + 65 * q^64 - 52 * q^66 + 34 * q^67 + 52 * q^68 + 21 * q^69 - 9 * q^71 + 70 * q^72 + 25 * q^73 + 22 * q^74 - 47 * q^76 + 54 * q^77 + 58 * q^78 + 13 * q^79 + 12 * q^81 - 5 * q^82 + 63 * q^83 + 95 * q^84 - 18 * q^86 + 47 * q^87 + 13 * q^88 + 19 * q^89 - 31 * q^91 + 137 * q^92 + 52 * q^93 + 120 * q^94 - 49 * q^96 + 36 * q^97 + 64 * q^98 + 16 * q^99

Introduction

L-functions

Modular forms

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Database

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