LMFDB - Embedded newform 5225.2.a.j.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5225,2,Mod(1,5225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("5225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5225 = 5^{2} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5225.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:	$41.7218350561$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\Q(\zeta_{22})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1 $ x^5 - x^4 - 4x^3 + 3x^2 + 3*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1045)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.91899$ of defining polynomial
Character	$\chi$	$=$	5225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.39788 q^{2} +2.59435 q^{3} +3.74982 q^{4} +6.22094 q^{6} +2.89389 q^{7} +4.19584 q^{8} +3.73066 q^{9} +O(q^{10})$	\(q+2.39788 q^{2} +2.59435 q^{3} +3.74982 q^{4} +6.22094 q^{6} +2.89389 q^{7} +4.19584 q^{8} +3.73066 q^{9} -1.00000 q^{11} +9.72834 q^{12} -4.73780 q^{13} +6.93920 q^{14} +2.56149 q^{16} +5.65389 q^{17} +8.94566 q^{18} +1.00000 q^{19} +7.50778 q^{21} -2.39788 q^{22} +4.00714 q^{23} +10.8855 q^{24} -11.3607 q^{26} +1.89558 q^{27} +10.8516 q^{28} +9.32825 q^{29} -6.60270 q^{31} -2.24955 q^{32} -2.59435 q^{33} +13.5573 q^{34} +13.9893 q^{36} -6.07686 q^{37} +2.39788 q^{38} -12.2915 q^{39} +5.47333 q^{41} +18.0027 q^{42} -10.9515 q^{43} -3.74982 q^{44} +9.60863 q^{46} +0.295438 q^{47} +6.64540 q^{48} +1.37462 q^{49} +14.6682 q^{51} -17.7659 q^{52} +3.81149 q^{53} +4.54538 q^{54} +12.1423 q^{56} +2.59435 q^{57} +22.3680 q^{58} -5.54910 q^{59} -1.01018 q^{61} -15.8325 q^{62} +10.7961 q^{63} -10.5171 q^{64} -6.22094 q^{66} -6.98807 q^{67} +21.2010 q^{68} +10.3959 q^{69} -1.02085 q^{71} +15.6533 q^{72} +0.202033 q^{73} -14.5716 q^{74} +3.74982 q^{76} -2.89389 q^{77} -29.4735 q^{78} +7.28690 q^{79} -6.27417 q^{81} +13.1244 q^{82} +13.7041 q^{83} +28.1528 q^{84} -26.2603 q^{86} +24.2008 q^{87} -4.19584 q^{88} -15.8152 q^{89} -13.7107 q^{91} +15.0260 q^{92} -17.1297 q^{93} +0.708423 q^{94} -5.83613 q^{96} -4.81308 q^{97} +3.29618 q^{98} -3.73066 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 3 q^{2} + 7 q^{3} + 5 q^{4} + 2 q^{6} + 11 q^{7} - 3 q^{8} + 8 q^{9}+O(q^{10}) $ 5 * q + 3 * q^2 + 7 * q^3 + 5 * q^4 + 2 * q^6 + 11 * q^7 - 3 * q^8 + 8 * q^9	$ 5 q + 3 q^{2} + 7 q^{3} + 5 q^{4} + 2 q^{6} + 11 q^{7} - 3 q^{8} + 8 q^{9} - 5 q^{11} + 7 q^{12} - q^{13} - 3 q^{16} + 3 q^{17} + 7 q^{18} + 5 q^{19} + 11 q^{21} - 3 q^{22} + 8 q^{23} + 9 q^{24} - 16 q^{26} + 10 q^{27} + 22 q^{28} + 11 q^{29} - 5 q^{31} + 2 q^{32} - 7 q^{33} + 4 q^{34} - 3 q^{36} + 9 q^{37} + 3 q^{38} - 8 q^{39} + 15 q^{41} - 11 q^{42} + 13 q^{43} - 5 q^{44} + 18 q^{46} + 20 q^{47} + 20 q^{48} + 20 q^{49} + 24 q^{51} - q^{52} + 5 q^{53} + 17 q^{54} + 7 q^{57} + 33 q^{58} - 17 q^{59} + 3 q^{61} - 14 q^{62} + 22 q^{63} - 17 q^{64} - 2 q^{66} + 28 q^{67} + 25 q^{68} - 2 q^{69} - 6 q^{71} + 26 q^{72} + 16 q^{73} - 21 q^{74} + 5 q^{76} - 11 q^{77} - 29 q^{78} + 3 q^{79} + q^{81} - 2 q^{82} + 33 q^{83} + 33 q^{84} + 10 q^{86} + 3 q^{88} - 16 q^{89} - 22 q^{91} + 19 q^{92} + 26 q^{93} - 10 q^{94} + 5 q^{96} + 14 q^{97} - 10 q^{98} - 8 q^{99}+O(q^{100}) $ 5 * q + 3 * q^2 + 7 * q^3 + 5 * q^4 + 2 * q^6 + 11 * q^7 - 3 * q^8 + 8 * q^9 - 5 * q^11 + 7 * q^12 - q^13 - 3 * q^16 + 3 * q^17 + 7 * q^18 + 5 * q^19 + 11 * q^21 - 3 * q^22 + 8 * q^23 + 9 * q^24 - 16 * q^26 + 10 * q^27 + 22 * q^28 + 11 * q^29 - 5 * q^31 + 2 * q^32 - 7 * q^33 + 4 * q^34 - 3 * q^36 + 9 * q^37 + 3 * q^38 - 8 * q^39 + 15 * q^41 - 11 * q^42 + 13 * q^43 - 5 * q^44 + 18 * q^46 + 20 * q^47 + 20 * q^48 + 20 * q^49 + 24 * q^51 - q^52 + 5 * q^53 + 17 * q^54 + 7 * q^57 + 33 * q^58 - 17 * q^59 + 3 * q^61 - 14 * q^62 + 22 * q^63 - 17 * q^64 - 2 * q^66 + 28 * q^67 + 25 * q^68 - 2 * q^69 - 6 * q^71 + 26 * q^72 + 16 * q^73 - 21 * q^74 + 5 * q^76 - 11 * q^77 - 29 * q^78 + 3 * q^79 + q^81 - 2 * q^82 + 33 * q^83 + 33 * q^84 + 10 * q^86 + 3 * q^88 - 16 * q^89 - 22 * q^91 + 19 * q^92 + 26 * q^93 - 10 * q^94 + 5 * q^96 + 14 * q^97 - 10 * q^98 - 8 * 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 $

$2$	2.39788	1.69556	0.847778	−	0.530352i	$-0.177940\pi$
$2$	2.39788	1.69556	0.847778	+	0.530352i	$0.177940\pi$
$3$	2.59435	1.49785	0.748925	−	0.662655i	$-0.230570\pi$
$3$	2.59435	1.49785	0.748925	+	0.662655i	$0.230570\pi$
$4$	3.74982	1.87491
$5$	0	0
$5$	0	0
$6$	6.22094	2.53969
$7$	2.89389	1.09379	0.546895	−	0.837201i	$-0.315810\pi$
$7$	2.89389	1.09379	0.546895	+	0.837201i	$0.315810\pi$
$8$	4.19584	1.48345
$9$	3.73066	1.24355
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	9.72834	2.80833
$13$	−4.73780	−1.31403	−0.657015	−	0.753878i	$-0.728181\pi$
$13$	−4.73780	−1.31403	−0.657015	+	0.753878i	$0.728181\pi$
$14$	6.93920	1.85458
$15$	0	0
$16$	2.56149	0.640372
$17$	5.65389	1.37127	0.685635	−	0.727946i	$-0.259525\pi$
$17$	5.65389	1.37127	0.685635	+	0.727946i	$0.259525\pi$
$18$	8.94566	2.10851
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	7.50778	1.63833
$22$	−2.39788	−0.511229
$23$	4.00714	0.835547	0.417773	−	0.908551i	$-0.362811\pi$
$23$	4.00714	0.835547	0.417773	+	0.908551i	$0.362811\pi$
$24$	10.8855	2.22199
$25$	0	0
$26$	−11.3607	−2.22801
$27$	1.89558	0.364805
$28$	10.8516	2.05075
$29$	9.32825	1.73221	0.866106	−	0.499860i	$-0.166615\pi$
$29$	9.32825	1.73221	0.866106	+	0.499860i	$0.166615\pi$
$30$	0	0
$31$	−6.60270	−1.18588	−0.592940	−	0.805247i	$-0.702033\pi$
$31$	−6.60270	−1.18588	−0.592940	+	0.805247i	$0.702033\pi$
$32$	−2.24955	−0.397669
$33$	−2.59435	−0.451619
$34$	13.5573	2.32506
$35$	0	0
$36$	13.9893	2.33155
$37$	−6.07686	−0.999029	−0.499515	−	0.866305i	$-0.666488\pi$
$37$	−6.07686	−0.999029	−0.499515	+	0.866305i	$0.666488\pi$
$38$	2.39788	0.388987
$39$	−12.2915	−1.96822
$40$	0	0
$41$	5.47333	0.854791	0.427395	−	0.904065i	$-0.359431\pi$
$41$	5.47333	0.854791	0.427395	+	0.904065i	$0.359431\pi$
$42$	18.0027	2.77788
$43$	−10.9515	−1.67009	−0.835043	−	0.550185i	$-0.814557\pi$
$43$	−10.9515	−1.67009	−0.835043	+	0.550185i	$0.814557\pi$
$44$	−3.74982	−0.565306
$45$	0	0
$46$	9.60863	1.41672
$47$	0.295438	0.0430940	0.0215470	−	0.999768i	$-0.493141\pi$
$47$	0.295438	0.0430940	0.0215470	+	0.999768i	$0.493141\pi$
$48$	6.64540	0.959181
$49$	1.37462	0.196375
$50$	0	0
$51$	14.6682	2.05395
$52$	−17.7659	−2.46368
$53$	3.81149	0.523549	0.261774	−	0.965129i	$-0.415692\pi$
$53$	3.81149	0.523549	0.261774	+	0.965129i	$0.415692\pi$
$54$	4.54538	0.618547
$55$	0	0
$56$	12.1423	1.62259
$57$	2.59435	0.343630
$58$	22.3680	2.93706
$59$	−5.54910	−0.722431	−0.361215	−	0.932482i	$-0.617638\pi$
$59$	−5.54910	−0.722431	−0.361215	+	0.932482i	$0.617638\pi$
$60$	0	0
$61$	−1.01018	−0.129340	−0.0646700	−	0.997907i	$-0.520599\pi$
$61$	−1.01018	−0.129340	−0.0646700	+	0.997907i	$0.520599\pi$
$62$	−15.8325	−2.01073
$63$	10.7961	1.36018
$64$	−10.5171	−1.31464
$65$	0	0
$66$	−6.22094	−0.765744
$67$	−6.98807	−0.853729	−0.426864	−	0.904316i	$-0.640382\pi$
$67$	−6.98807	−0.853729	−0.426864	+	0.904316i	$0.640382\pi$
$68$	21.2010	2.57100
$69$	10.3959	1.25152
$70$	0	0
$71$	−1.02085	−0.121152	−0.0605761	−	0.998164i	$-0.519294\pi$
$71$	−1.02085	−0.121152	−0.0605761	+	0.998164i	$0.519294\pi$
$72$	15.6533	1.84475
$73$	0.202033	0.0236462	0.0118231	−	0.999930i	$-0.496237\pi$
$73$	0.202033	0.0236462	0.0118231	+	0.999930i	$0.496237\pi$
$74$	−14.5716	−1.69391
$75$	0	0
$76$	3.74982	0.430133
$77$	−2.89389	−0.329790
$78$	−29.4735	−3.33722
$79$	7.28690	0.819840	0.409920	−	0.912121i	$-0.365557\pi$
$79$	7.28690	0.819840	0.409920	+	0.912121i	$0.365557\pi$
$80$	0	0
$81$	−6.27417	−0.697130
$82$	13.1244	1.44935
$83$	13.7041	1.50422	0.752109	−	0.659039i	$-0.229037\pi$
$83$	13.7041	1.50422	0.752109	+	0.659039i	$0.229037\pi$
$84$	28.1528	3.07172
$85$	0	0
$86$	−26.2603	−2.83172
$87$	24.2008	2.59459
$88$	−4.19584	−0.447278
$89$	−15.8152	−1.67641	−0.838203	−	0.545359i	$-0.816394\pi$
$89$	−15.8152	−1.67641	−0.838203	+	0.545359i	$0.816394\pi$
$90$	0	0
$91$	−13.7107	−1.43727
$92$	15.0260	1.56657
$93$	−17.1297	−1.77627
$94$	0.708423	0.0730683
$95$	0	0
$96$	−5.83613	−0.595648
$97$	−4.81308	−0.488694	−0.244347	−	0.969688i	$-0.578574\pi$
$97$	−4.81308	−0.488694	−0.244347	+	0.969688i	$0.578574\pi$
$98$	3.29618	0.332964
$99$	−3.73066	−0.374945
$100$	0	0
$101$	1.76221	0.175346	0.0876730	−	0.996149i	$-0.472057\pi$
$101$	1.76221	0.175346	0.0876730	+	0.996149i	$0.472057\pi$
$102$	35.1725	3.48259
$103$	5.53571	0.545450	0.272725	−	0.962092i	$-0.412075\pi$
$103$	5.53571	0.545450	0.272725	+	0.962092i	$0.412075\pi$
$104$	−19.8791	−1.94930
$105$	0	0
$106$	9.13949	0.887706
$107$	4.32073	0.417701	0.208851	−	0.977948i	$-0.433028\pi$
$107$	4.32073	0.417701	0.208851	+	0.977948i	$0.433028\pi$
$108$	7.10809	0.683976
$109$	−13.0039	−1.24555	−0.622776	−	0.782400i	$-0.713995\pi$
$109$	−13.0039	−1.24555	−0.622776	+	0.782400i	$0.713995\pi$
$110$	0	0
$111$	−15.7655	−1.49640
$112$	7.41267	0.700432
$113$	17.0047	1.59967	0.799833	−	0.600223i	$-0.204922\pi$
$113$	17.0047	1.59967	0.799833	+	0.600223i	$0.204922\pi$
$114$	6.22094	0.582644
$115$	0	0
$116$	34.9792	3.24774
$117$	−17.6751	−1.63406
$118$	−13.3061	−1.22492
$119$	16.3618	1.49988
$120$	0	0
$121$	1.00000	0.0909091
$122$	−2.42228	−0.219303
$123$	14.1997	1.28035
$124$	−24.7589	−2.22342
$125$	0	0
$126$	25.8878	2.30627
$127$	9.80245	0.869826	0.434913	−	0.900472i	$-0.356779\pi$
$127$	9.80245	0.869826	0.434913	+	0.900472i	$0.356779\pi$
$128$	−20.7197	−1.83138
$129$	−28.4120	−2.50154
$130$	0	0
$131$	−13.4383	−1.17411	−0.587053	−	0.809549i	$-0.699712\pi$
$131$	−13.4383	−1.17411	−0.587053	+	0.809549i	$0.699712\pi$
$132$	−9.72834	−0.846743
$133$	2.89389	0.250932
$134$	−16.7565	−1.44754
$135$	0	0
$136$	23.7228	2.03422
$137$	16.2122	1.38510	0.692549	−	0.721371i	$-0.256488\pi$
$137$	16.2122	1.38510	0.692549	+	0.721371i	$0.256488\pi$
$138$	24.9282	2.12203
$139$	19.0373	1.61472	0.807359	−	0.590060i	$-0.200896\pi$
$139$	19.0373	1.61472	0.807359	+	0.590060i	$0.200896\pi$
$140$	0	0
$141$	0.766469	0.0645484
$142$	−2.44787	−0.205420
$143$	4.73780	0.396195
$144$	9.55603	0.796336
$145$	0	0
$146$	0.484451	0.0400934
$147$	3.56626	0.294140
$148$	−22.7871	−1.87309
$149$	−15.3234	−1.25534	−0.627670	−	0.778480i	$-0.715991\pi$
$149$	−15.3234	−1.25534	−0.627670	+	0.778480i	$0.715991\pi$
$150$	0	0
$151$	−1.25936	−0.102485	−0.0512426	−	0.998686i	$-0.516318\pi$
$151$	−1.25936	−0.102485	−0.0512426	+	0.998686i	$0.516318\pi$
$152$	4.19584	0.340328
$153$	21.0927	1.70525
$154$	−6.93920	−0.559177
$155$	0	0
$156$	−46.0909	−3.69023
$157$	−12.1861	−0.972559	−0.486279	−	0.873803i	$-0.661646\pi$
$157$	−12.1861	−0.972559	−0.486279	+	0.873803i	$0.661646\pi$
$158$	17.4731	1.39008
$159$	9.88835	0.784197
$160$	0	0
$161$	11.5962	0.913912
$162$	−15.0447	−1.18202
$163$	−6.09722	−0.477571	−0.238786	−	0.971072i	$-0.576749\pi$
$163$	−6.09722	−0.477571	−0.238786	+	0.971072i	$0.576749\pi$
$164$	20.5240	1.60265
$165$	0	0
$166$	32.8607	2.55048
$167$	−1.06720	−0.0825826	−0.0412913	−	0.999147i	$-0.513147\pi$
$167$	−1.06720	−0.0825826	−0.0412913	+	0.999147i	$0.513147\pi$
$168$	31.5015	2.43039
$169$	9.44675	0.726673
$170$	0	0
$171$	3.73066	0.285291
$172$	−41.0661	−3.13126
$173$	13.0289	0.990571	0.495286	−	0.868730i	$-0.335063\pi$
$173$	13.0289	0.990571	0.495286	+	0.868730i	$0.335063\pi$
$174$	58.0304	4.39928
$175$	0	0
$176$	−2.56149	−0.193079
$177$	−14.3963	−1.08209
$178$	−37.9229	−2.84244
$179$	20.6273	1.54176	0.770878	−	0.636983i	$-0.219818\pi$
$179$	20.6273	1.54176	0.770878	+	0.636983i	$0.219818\pi$
$180$	0	0
$181$	−20.5908	−1.53050	−0.765251	−	0.643732i	$-0.777385\pi$
$181$	−20.5908	−1.53050	−0.765251	+	0.643732i	$0.777385\pi$
$182$	−32.8766	−2.43697
$183$	−2.62076	−0.193732
$184$	16.8133	1.23950
$185$	0	0
$186$	−41.0750	−3.01176
$187$	−5.65389	−0.413453
$188$	1.10784	0.0807973
$189$	5.48562	0.399020
$190$	0	0
$191$	8.40047	0.607837	0.303918	−	0.952698i	$-0.401705\pi$
$191$	8.40047	0.607837	0.303918	+	0.952698i	$0.401705\pi$
$192$	−27.2851	−1.96913
$193$	−11.4982	−0.827661	−0.413830	−	0.910354i	$-0.635809\pi$
$193$	−11.4982	−0.827661	−0.413830	+	0.910354i	$0.635809\pi$
$194$	−11.5412	−0.828607
$195$	0	0
$196$	5.15458	0.368185
$197$	−27.0376	−1.92635	−0.963173	−	0.268883i	$-0.913345\pi$
$197$	−27.0376	−1.92635	−0.963173	+	0.268883i	$0.913345\pi$
$198$	−8.94566	−0.635740
$199$	−5.18460	−0.367526	−0.183763	−	0.982971i	$-0.558828\pi$
$199$	−5.18460	−0.367526	−0.183763	+	0.982971i	$0.558828\pi$
$200$	0	0
$201$	−18.1295	−1.27876
$202$	4.22555	0.297309
$203$	26.9950	1.89468
$204$	55.0029	3.85098
$205$	0	0
$206$	13.2740	0.924840
$207$	14.9493	1.03905
$208$	−12.1358	−0.841467
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	−21.5427	−1.48306	−0.741529	−	0.670921i	$-0.765899\pi$
$211$	−21.5427	−1.48306	−0.741529	+	0.670921i	$0.765899\pi$
$212$	14.2924	0.981606
$213$	−2.64844	−0.181468
$214$	10.3606	0.708235
$215$	0	0
$216$	7.95357	0.541172
$217$	−19.1075	−1.29710
$218$	−31.1818	−2.11190
$219$	0.524145	0.0354184
$220$	0	0
$221$	−26.7870	−1.80189
$222$	−37.8037	−2.53722
$223$	26.9600	1.80537	0.902687	−	0.430298i	$-0.141592\pi$
$223$	26.9600	1.80537	0.902687	+	0.430298i	$0.141592\pi$
$224$	−6.50997	−0.434966
$225$	0	0
$226$	40.7751	2.71232
$227$	−9.02570	−0.599057	−0.299528	−	0.954087i	$-0.596829\pi$
$227$	−9.02570	−0.599057	−0.299528	+	0.954087i	$0.596829\pi$
$228$	9.72834	0.644275
$229$	−13.7697	−0.909926	−0.454963	−	0.890510i	$-0.650347\pi$
$229$	−13.7697	−0.909926	−0.454963	+	0.890510i	$0.650347\pi$
$230$	0	0
$231$	−7.50778	−0.493976
$232$	39.1399	2.56966
$233$	2.45587	0.160890	0.0804448	−	0.996759i	$-0.474366\pi$
$233$	2.45587	0.160890	0.0804448	+	0.996759i	$0.474366\pi$
$234$	−42.3827	−2.77065
$235$	0	0
$236$	−20.8081	−1.35449
$237$	18.9048	1.22800
$238$	39.2335	2.54313
$239$	−22.8439	−1.47765	−0.738825	−	0.673897i	$-0.764619\pi$
$239$	−22.8439	−1.47765	−0.738825	+	0.673897i	$0.764619\pi$
$240$	0	0
$241$	16.1181	1.03826	0.519129	−	0.854696i	$-0.326256\pi$
$241$	16.1181	1.03826	0.519129	+	0.854696i	$0.326256\pi$
$242$	2.39788	0.154141
$243$	−21.9641	−1.40900
$244$	−3.78798	−0.242501
$245$	0	0
$246$	34.0492	2.17090
$247$	−4.73780	−0.301459
$248$	−27.7039	−1.75920
$249$	35.5532	2.25309
$250$	0	0
$251$	−18.8758	−1.19143	−0.595716	−	0.803195i	$-0.703132\pi$
$251$	−18.8758	−1.19143	−0.595716	+	0.803195i	$0.703132\pi$
$252$	40.4835	2.55022
$253$	−4.00714	−0.251927
$254$	23.5051	1.47484
$255$	0	0
$256$	−28.6490	−1.79056
$257$	−7.48029	−0.466608	−0.233304	−	0.972404i	$-0.574954\pi$
$257$	−7.48029	−0.466608	−0.233304	+	0.972404i	$0.574954\pi$
$258$	−68.1285	−4.24150
$259$	−17.5858	−1.09273
$260$	0	0
$261$	34.8005	2.15410
$262$	−32.2233	−1.99076
$263$	17.7703	1.09577	0.547883	−	0.836555i	$-0.315434\pi$
$263$	17.7703	1.09577	0.547883	+	0.836555i	$0.315434\pi$
$264$	−10.8855	−0.669956
$265$	0	0
$266$	6.93920	0.425470
$267$	−41.0301	−2.51100
$268$	−26.2040	−1.60066
$269$	25.9320	1.58110	0.790550	−	0.612398i	$-0.209795\pi$
$269$	25.9320	1.58110	0.790550	+	0.612398i	$0.209795\pi$
$270$	0	0
$271$	−16.2686	−0.988248	−0.494124	−	0.869391i	$-0.664511\pi$
$271$	−16.2686	−0.988248	−0.494124	+	0.869391i	$0.664511\pi$
$272$	14.4824	0.878122
$273$	−35.5703	−2.15282
$274$	38.8748	2.34851
$275$	0	0
$276$	38.9828	2.34649
$277$	17.9217	1.07681	0.538405	−	0.842686i	$-0.319027\pi$
$277$	17.9217	1.07681	0.538405	+	0.842686i	$0.319027\pi$
$278$	45.6490	2.73784
$279$	−24.6324	−1.47470
$280$	0	0
$281$	−28.8563	−1.72142	−0.860712	−	0.509093i	$-0.829981\pi$
$281$	−28.8563	−1.72142	−0.860712	+	0.509093i	$0.829981\pi$
$282$	1.83790	0.109445
$283$	10.7620	0.639736	0.319868	−	0.947462i	$-0.396361\pi$
$283$	10.7620	0.639736	0.319868	+	0.947462i	$0.396361\pi$
$284$	−3.82799	−0.227149
$285$	0	0
$286$	11.3607	0.671770
$287$	15.8392	0.934961
$288$	−8.39232	−0.494522
$289$	14.9665	0.880380
$290$	0	0
$291$	−12.4868	−0.731990
$292$	0.757587	0.0443345
$293$	−5.14307	−0.300462	−0.150231	−	0.988651i	$-0.548002\pi$
$293$	−5.14307	−0.300462	−0.150231	+	0.988651i	$0.548002\pi$
$294$	8.55144	0.498730
$295$	0	0
$296$	−25.4975	−1.48201
$297$	−1.89558	−0.109993
$298$	−36.7436	−2.12850
$299$	−18.9850	−1.09793
$300$	0	0
$301$	−31.6924	−1.82672
$302$	−3.01979	−0.173769
$303$	4.57178	0.262642
$304$	2.56149	0.146911
$305$	0	0
$306$	50.5778	2.89134
$307$	−28.1017	−1.60385	−0.801925	−	0.597425i	$-0.796191\pi$
$307$	−28.1017	−1.60385	−0.801925	+	0.597425i	$0.796191\pi$
$308$	−10.8516	−0.618326
$309$	14.3616	0.817002
$310$	0	0
$311$	−14.1909	−0.804692	−0.402346	−	0.915488i	$-0.631805\pi$
$311$	−14.1909	−0.804692	−0.402346	+	0.915488i	$0.631805\pi$
$312$	−51.5733	−2.91976
$313$	9.46083	0.534758	0.267379	−	0.963591i	$-0.413842\pi$
$313$	9.46083	0.534758	0.267379	+	0.963591i	$0.413842\pi$
$314$	−29.2208	−1.64903
$315$	0	0
$316$	27.3245	1.53712
$317$	23.8673	1.34052	0.670260	−	0.742126i	$-0.266182\pi$
$317$	23.8673	1.34052	0.670260	+	0.742126i	$0.266182\pi$
$318$	23.7111	1.32965
$319$	−9.32825	−0.522282
$320$	0	0
$321$	11.2095	0.625653
$322$	27.8064	1.54959
$323$	5.65389	0.314591
$324$	−23.5270	−1.30705
$325$	0	0
$326$	−14.6204	−0.809748
$327$	−33.7368	−1.86565
$328$	22.9652	1.26804
$329$	0.854966	0.0471358
$330$	0	0
$331$	−13.1843	−0.724674	−0.362337	−	0.932047i	$-0.618021\pi$
$331$	−13.1843	−0.724674	−0.362337	+	0.932047i	$0.618021\pi$
$332$	51.3877	2.82027
$333$	−22.6707	−1.24235
$334$	−2.55902	−0.140023
$335$	0	0
$336$	19.2311	1.04914
$337$	−17.0148	−0.926854	−0.463427	−	0.886135i	$-0.653380\pi$
$337$	−17.0148	−0.926854	−0.463427	+	0.886135i	$0.653380\pi$
$338$	22.6521	1.23211
$339$	44.1161	2.39606
$340$	0	0
$341$	6.60270	0.357556
$342$	8.94566	0.483726
$343$	−16.2792	−0.878997
$344$	−45.9507	−2.47750
$345$	0	0
$346$	31.2418	1.67957
$347$	30.2606	1.62447	0.812236	−	0.583329i	$-0.198250\pi$
$347$	30.2606	1.62447	0.812236	+	0.583329i	$0.198250\pi$
$348$	90.7484	4.86462
$349$	5.13463	0.274851	0.137425	−	0.990512i	$-0.456117\pi$
$349$	5.13463	0.274851	0.137425	+	0.990512i	$0.456117\pi$
$350$	0	0
$351$	−8.98089	−0.479365
$352$	2.24955	0.119902
$353$	10.5661	0.562376	0.281188	−	0.959653i	$-0.409272\pi$
$353$	10.5661	0.562376	0.281188	+	0.959653i	$0.409272\pi$
$354$	−34.5206	−1.83475
$355$	0	0
$356$	−59.3040	−3.14311
$357$	42.4481	2.24659
$358$	49.4617	2.61413
$359$	16.5178	0.871778	0.435889	−	0.900000i	$-0.356434\pi$
$359$	16.5178	0.871778	0.435889	+	0.900000i	$0.356434\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−49.3742	−2.59505
$363$	2.59435	0.136168
$364$	−51.4126	−2.69475
$365$	0	0
$366$	−6.28426	−0.328483
$367$	−19.8761	−1.03752	−0.518762	−	0.854919i	$-0.673607\pi$
$367$	−19.8761	−1.03752	−0.518762	+	0.854919i	$0.673607\pi$
$368$	10.2642	0.535061
$369$	20.4191	1.06298
$370$	0	0
$371$	11.0301	0.572652
$372$	−64.2333	−3.33034
$373$	11.4151	0.591052	0.295526	−	0.955335i	$-0.404505\pi$
$373$	11.4151	0.591052	0.295526	+	0.955335i	$0.404505\pi$
$374$	−13.5573	−0.701033
$375$	0	0
$376$	1.23961	0.0639280
$377$	−44.1954	−2.27618
$378$	13.1538	0.676560
$379$	1.99925	0.102695	0.0513474	−	0.998681i	$-0.483648\pi$
$379$	1.99925	0.102695	0.0513474	+	0.998681i	$0.483648\pi$
$380$	0	0
$381$	25.4310	1.30287
$382$	20.1433	1.03062
$383$	7.07007	0.361264	0.180632	−	0.983551i	$-0.442186\pi$
$383$	7.07007	0.361264	0.180632	+	0.983551i	$0.442186\pi$
$384$	−53.7541	−2.74313
$385$	0	0
$386$	−27.5713	−1.40334
$387$	−40.8563	−2.07684
$388$	−18.0481	−0.916256
$389$	17.0237	0.863135	0.431568	−	0.902081i	$-0.357961\pi$
$389$	17.0237	0.863135	0.431568	+	0.902081i	$0.357961\pi$
$390$	0	0
$391$	22.6559	1.14576
$392$	5.76771	0.291313
$393$	−34.8636	−1.75863
$394$	−64.8327	−3.26623
$395$	0	0
$396$	−13.9893	−0.702988
$397$	25.3590	1.27273	0.636366	−	0.771387i	$-0.280437\pi$
$397$	25.3590	1.27273	0.636366	+	0.771387i	$0.280437\pi$
$398$	−12.4320	−0.623161
$399$	7.50778	0.375859
$400$	0	0
$401$	19.6449	0.981019	0.490509	−	0.871436i	$-0.336811\pi$
$401$	19.6449	0.981019	0.490509	+	0.871436i	$0.336811\pi$
$402$	−43.4723	−2.16820
$403$	31.2823	1.55828
$404$	6.60795	0.328758
$405$	0	0
$406$	64.7306	3.21253
$407$	6.07686	0.301219
$408$	61.5454	3.04695
$409$	−28.6527	−1.41678	−0.708392	−	0.705819i	$-0.750579\pi$
$409$	−28.6527	−1.41678	−0.708392	+	0.705819i	$0.750579\pi$
$410$	0	0
$411$	42.0600	2.07467
$412$	20.7579	1.02267
$413$	−16.0585	−0.790187
$414$	35.8465	1.76176
$415$	0	0
$416$	10.6579	0.522548
$417$	49.3893	2.41861
$418$	−2.39788	−0.117284
$419$	−2.80518	−0.137042	−0.0685211	−	0.997650i	$-0.521828\pi$
$419$	−2.80518	−0.137042	−0.0685211	+	0.997650i	$0.521828\pi$
$420$	0	0
$421$	3.26165	0.158963	0.0794815	−	0.996836i	$-0.474674\pi$
$421$	3.26165	0.158963	0.0794815	+	0.996836i	$0.474674\pi$
$422$	−51.6566	−2.51461
$423$	1.10218	0.0535897
$424$	15.9924	0.776661
$425$	0	0
$426$	−6.35062	−0.307689
$427$	−2.92335	−0.141471
$428$	16.2020	0.783151
$429$	12.2915	0.593440
$430$	0	0
$431$	32.7606	1.57802	0.789012	−	0.614378i	$-0.210593\pi$
$431$	32.7606	1.57802	0.789012	+	0.614378i	$0.210593\pi$
$432$	4.85551	0.233611
$433$	21.8656	1.05079	0.525396	−	0.850858i	$-0.323917\pi$
$433$	21.8656	1.05079	0.525396	+	0.850858i	$0.323917\pi$
$434$	−45.8175	−2.19931
$435$	0	0
$436$	−48.7624	−2.33529
$437$	4.00714	0.191688
$438$	1.25684	0.0600539
$439$	30.6280	1.46180	0.730898	−	0.682486i	$-0.239101\pi$
$439$	30.6280	1.46180	0.730898	+	0.682486i	$0.239101\pi$
$440$	0	0
$441$	5.12825	0.244202
$442$	−64.2319	−3.05520
$443$	−4.54914	−0.216136	−0.108068	−	0.994144i	$-0.534466\pi$
$443$	−4.54914	−0.216136	−0.108068	+	0.994144i	$0.534466\pi$
$444$	−59.1177	−2.80560
$445$	0	0
$446$	64.6467	3.06111
$447$	−39.7542	−1.88031
$448$	−30.4355	−1.43794
$449$	11.5308	0.544172	0.272086	−	0.962273i	$-0.412286\pi$
$449$	11.5308	0.544172	0.272086	+	0.962273i	$0.412286\pi$
$450$	0	0
$451$	−5.47333	−0.257729
$452$	63.7644	2.99923
$453$	−3.26722	−0.153507
$454$	−21.6425	−1.01573
$455$	0	0
$456$	10.8855	0.509760
$457$	−11.3778	−0.532231	−0.266115	−	0.963941i	$-0.585740\pi$
$457$	−11.3778	−0.532231	−0.266115	+	0.963941i	$0.585740\pi$
$458$	−33.0180	−1.54283
$459$	10.7174	0.500246
$460$	0	0
$461$	4.37265	0.203655	0.101827	−	0.994802i	$-0.467531\pi$
$461$	4.37265	0.203655	0.101827	+	0.994802i	$0.467531\pi$
$462$	−18.0027	−0.837563
$463$	9.31982	0.433129	0.216564	−	0.976268i	$-0.430515\pi$
$463$	9.31982	0.433129	0.216564	+	0.976268i	$0.430515\pi$
$464$	23.8942	1.10926
$465$	0	0
$466$	5.88888	0.272797
$467$	28.0711	1.29897	0.649487	−	0.760372i	$-0.274984\pi$
$467$	28.0711	1.29897	0.649487	+	0.760372i	$0.274984\pi$
$468$	−66.2784	−3.06372
$469$	−20.2227	−0.933799
$470$	0	0
$471$	−31.6151	−1.45675
$472$	−23.2831	−1.07169
$473$	10.9515	0.503550
$474$	45.3313	2.08214
$475$	0	0
$476$	61.3536	2.81214
$477$	14.2194	0.651060
$478$	−54.7769	−2.50544
$479$	−5.65184	−0.258239	−0.129120	−	0.991629i	$-0.541215\pi$
$479$	−5.65184	−0.258239	−0.129120	+	0.991629i	$0.541215\pi$
$480$	0	0
$481$	28.7909	1.31275
$482$	38.6492	1.76042
$483$	30.0847	1.36890
$484$	3.74982	0.170446
$485$	0	0
$486$	−52.6673	−2.38904
$487$	6.09349	0.276122	0.138061	−	0.990424i	$-0.455913\pi$
$487$	6.09349	0.276122	0.138061	+	0.990424i	$0.455913\pi$
$488$	−4.23855	−0.191870
$489$	−15.8183	−0.715330
$490$	0	0
$491$	8.56286	0.386436	0.193218	−	0.981156i	$-0.438107\pi$
$491$	8.56286	0.386436	0.193218	+	0.981156i	$0.438107\pi$
$492$	53.2464	2.40053
$493$	52.7409	2.37533
$494$	−11.3607	−0.511140
$495$	0	0
$496$	−16.9127	−0.759404
$497$	−2.95422	−0.132515
$498$	85.2522	3.82024
$499$	−1.35537	−0.0606749	−0.0303374	−	0.999540i	$-0.509658\pi$
$499$	−1.35537	−0.0606749	−0.0303374	+	0.999540i	$0.509658\pi$
$500$	0	0
$501$	−2.76870	−0.123696
$502$	−45.2619	−2.02014
$503$	−13.9030	−0.619903	−0.309952	−	0.950752i	$-0.600313\pi$
$503$	−13.9030	−0.619903	−0.309952	+	0.950752i	$0.600313\pi$
$504$	45.2989	2.01777
$505$	0	0
$506$	−9.60863	−0.427156
$507$	24.5082	1.08845
$508$	36.7574	1.63084
$509$	−3.21978	−0.142714	−0.0713572	−	0.997451i	$-0.522733\pi$
$509$	−3.21978	−0.142714	−0.0713572	+	0.997451i	$0.522733\pi$
$510$	0	0
$511$	0.584663	0.0258640
$512$	−27.2574	−1.20462
$513$	1.89558	0.0836920
$514$	−17.9368	−0.791160
$515$	0	0
$516$	−106.540	−4.69015
$517$	−0.295438	−0.0129933
$518$	−42.1686	−1.85278
$519$	33.8016	1.48373
$520$	0	0
$521$	3.09750	0.135704	0.0678520	−	0.997695i	$-0.478385\pi$
$521$	3.09750	0.135704	0.0678520	+	0.997695i	$0.478385\pi$
$522$	83.4474	3.65239
$523$	−13.2007	−0.577228	−0.288614	−	0.957446i	$-0.593194\pi$
$523$	−13.2007	−0.577228	−0.288614	+	0.957446i	$0.593194\pi$
$524$	−50.3910	−2.20134
$525$	0	0
$526$	42.6111	1.85793
$527$	−37.3309	−1.62616
$528$	−6.64540	−0.289204
$529$	−6.94281	−0.301861
$530$	0	0
$531$	−20.7018	−0.898381
$532$	10.8516	0.470475
$533$	−25.9316	−1.12322
$534$	−98.3852	−4.25755
$535$	0	0
$536$	−29.3209	−1.26647
$537$	53.5144	2.30932
$538$	62.1817	2.68084
$539$	−1.37462	−0.0592092
$540$	0	0
$541$	−6.57608	−0.282728	−0.141364	−	0.989958i	$-0.545149\pi$
$541$	−6.57608	−0.282728	−0.141364	+	0.989958i	$0.545149\pi$
$542$	−39.0102	−1.67563
$543$	−53.4198	−2.29246
$544$	−12.7187	−0.545311
$545$	0	0
$546$	−85.2933	−3.65022
$547$	23.9559	1.02428	0.512139	−	0.858902i	$-0.328853\pi$
$547$	23.9559	1.02428	0.512139	+	0.858902i	$0.328853\pi$
$548$	60.7926	2.59693
$549$	−3.76863	−0.160841
$550$	0	0
$551$	9.32825	0.397397
$552$	43.6197	1.85658
$553$	21.0875	0.896732
$554$	42.9740	1.82579
$555$	0	0
$556$	71.3862	3.02745
$557$	11.2404	0.476271	0.238135	−	0.971232i	$-0.423464\pi$
$557$	11.2404	0.476271	0.238135	+	0.971232i	$0.423464\pi$
$558$	−59.0655	−2.50044
$559$	51.8860	2.19454
$560$	0	0
$561$	−14.6682	−0.619291
$562$	−69.1939	−2.91877
$563$	−44.9880	−1.89602	−0.948009	−	0.318242i	$-0.896907\pi$
$563$	−44.9880	−1.89602	−0.948009	+	0.318242i	$0.896907\pi$
$564$	2.87412	0.121022
$565$	0	0
$566$	25.8060	1.08471
$567$	−18.1568	−0.762513
$568$	−4.28331	−0.179724
$569$	33.2099	1.39223	0.696115	−	0.717930i	$-0.254910\pi$
$569$	33.2099	1.39223	0.696115	+	0.717930i	$0.254910\pi$
$570$	0	0
$571$	−44.6440	−1.86829	−0.934146	−	0.356891i	$-0.883837\pi$
$571$	−44.6440	−1.86829	−0.934146	+	0.356891i	$0.883837\pi$
$572$	17.7659	0.742829
$573$	21.7938	0.910448
$574$	37.9806	1.58528
$575$	0	0
$576$	−39.2358	−1.63483
$577$	45.6776	1.90158	0.950792	−	0.309831i	$-0.100272\pi$
$577$	45.6776	1.90158	0.950792	+	0.309831i	$0.100272\pi$
$578$	35.8877	1.49273
$579$	−29.8304	−1.23971
$580$	0	0
$581$	39.6581	1.64530
$582$	−29.9418	−1.24113
$583$	−3.81149	−0.157856
$584$	0.847700	0.0350781
$585$	0	0
$586$	−12.3325	−0.509449
$587$	17.5933	0.726155	0.363078	−	0.931759i	$-0.381726\pi$
$587$	17.5933	0.726155	0.363078	+	0.931759i	$0.381726\pi$
$588$	13.3728	0.551485
$589$	−6.60270	−0.272060
$590$	0	0
$591$	−70.1449	−2.88538
$592$	−15.5658	−0.639750
$593$	−32.4709	−1.33342	−0.666710	−	0.745317i	$-0.732298\pi$
$593$	−32.4709	−1.33342	−0.666710	+	0.745317i	$0.732298\pi$
$594$	−4.54538	−0.186499
$595$	0	0
$596$	−57.4598	−2.35365
$597$	−13.4507	−0.550499
$598$	−45.5238	−1.86161
$599$	1.31594	0.0537679	0.0268839	−	0.999639i	$-0.491442\pi$
$599$	1.31594	0.0537679	0.0268839	+	0.999639i	$0.491442\pi$
$600$	0	0
$601$	−4.53331	−0.184918	−0.0924588	−	0.995717i	$-0.529473\pi$
$601$	−4.53331	−0.184918	−0.0924588	+	0.995717i	$0.529473\pi$
$602$	−75.9946	−3.09731
$603$	−26.0701	−1.06166
$604$	−4.72236	−0.192150
$605$	0	0
$606$	10.9626	0.445324
$607$	14.2654	0.579016	0.289508	−	0.957176i	$-0.406508\pi$
$607$	14.2654	0.579016	0.289508	+	0.957176i	$0.406508\pi$
$608$	−2.24955	−0.0912315
$609$	70.0344	2.83794
$610$	0	0
$611$	−1.39972	−0.0566268
$612$	79.0938	3.19718
$613$	−5.06662	−0.204639	−0.102319	−	0.994752i	$-0.532626\pi$
$613$	−5.06662	−0.204639	−0.102319	+	0.994752i	$0.532626\pi$
$614$	−67.3845	−2.71942
$615$	0	0
$616$	−12.1423	−0.489228
$617$	−19.5460	−0.786893	−0.393446	−	0.919348i	$-0.628717\pi$
$617$	−19.5460	−0.786893	−0.393446	+	0.919348i	$0.628717\pi$
$618$	34.4373	1.38527
$619$	8.78654	0.353161	0.176580	−	0.984286i	$-0.443496\pi$
$619$	8.78654	0.353161	0.176580	+	0.984286i	$0.443496\pi$
$620$	0	0
$621$	7.59587	0.304812
$622$	−34.0280	−1.36440
$623$	−45.7675	−1.83363
$624$	−31.4846	−1.26039
$625$	0	0
$626$	22.6859	0.906712
$627$	−2.59435	−0.103608
$628$	−45.6957	−1.82346
$629$	−34.3579	−1.36994
$630$	0	0
$631$	−0.0559751	−0.00222833	−0.00111417	−	0.999999i	$-0.500355\pi$
$631$	−0.0559751	−0.00222833	−0.00111417	+	0.999999i	$0.500355\pi$
$632$	30.5747	1.21620
$633$	−55.8892	−2.22140
$634$	57.2308	2.27293
$635$	0	0
$636$	37.0795	1.47030
$637$	−6.51269	−0.258042
$638$	−22.3680	−0.885558
$639$	−3.80843	−0.150659
$640$	0	0
$641$	4.49262	0.177448	0.0887239	−	0.996056i	$-0.471721\pi$
$641$	4.49262	0.177448	0.0887239	+	0.996056i	$0.471721\pi$
$642$	26.8790	1.06083
$643$	5.48669	0.216374	0.108187	−	0.994131i	$-0.465495\pi$
$643$	5.48669	0.216374	0.108187	+	0.994131i	$0.465495\pi$
$644$	43.4838	1.71350
$645$	0	0
$646$	13.5573	0.533406
$647$	34.5798	1.35947	0.679736	−	0.733456i	$-0.262094\pi$
$647$	34.5798	1.35947	0.679736	+	0.733456i	$0.262094\pi$
$648$	−26.3254	−1.03416
$649$	5.54910	0.217821
$650$	0	0
$651$	−49.5716	−1.94286
$652$	−22.8635	−0.895402
$653$	−11.3537	−0.444304	−0.222152	−	0.975012i	$-0.571308\pi$
$653$	−11.3537	−0.444304	−0.222152	+	0.975012i	$0.571308\pi$
$654$	−80.8967	−3.16331
$655$	0	0
$656$	14.0199	0.547384
$657$	0.753717	0.0294053
$658$	2.05010	0.0799213
$659$	27.1448	1.05741	0.528706	−	0.848805i	$-0.322677\pi$
$659$	27.1448	1.05741	0.528706	+	0.848805i	$0.322677\pi$
$660$	0	0
$661$	5.01901	0.195217	0.0976084	−	0.995225i	$-0.468881\pi$
$661$	5.01901	0.195217	0.0976084	+	0.995225i	$0.468881\pi$
$662$	−31.6143	−1.22873
$663$	−69.4949	−2.69896
$664$	57.5002	2.23144
$665$	0	0
$666$	−54.3615	−2.10647
$667$	37.3796	1.44734
$668$	−4.00181	−0.154835
$669$	69.9436	2.70418
$670$	0	0
$671$	1.01018	0.0389975
$672$	−16.8892	−0.651513
$673$	38.5035	1.48420	0.742100	−	0.670289i	$-0.233830\pi$
$673$	38.5035	1.48420	0.742100	+	0.670289i	$0.233830\pi$
$674$	−40.7994	−1.57153
$675$	0	0
$676$	35.4236	1.36244
$677$	−17.8660	−0.686647	−0.343324	−	0.939217i	$-0.611553\pi$
$677$	−17.8660	−0.686647	−0.343324	+	0.939217i	$0.611553\pi$
$678$	105.785	4.06265
$679$	−13.9285	−0.534528
$680$	0	0
$681$	−23.4158	−0.897297
$682$	15.8325	0.606257
$683$	−44.9512	−1.72001	−0.860005	−	0.510286i	$-0.829540\pi$
$683$	−44.9512	−1.72001	−0.860005	+	0.510286i	$0.829540\pi$
$684$	13.9893	0.534894
$685$	0	0
$686$	−39.0356	−1.49039
$687$	−35.7234	−1.36293
$688$	−28.0521	−1.06948
$689$	−18.0581	−0.687958
$690$	0	0
$691$	24.1309	0.917983	0.458991	−	0.888441i	$-0.348211\pi$
$691$	24.1309	0.917983	0.458991	+	0.888441i	$0.348211\pi$
$692$	48.8561	1.85723
$693$	−10.7961	−0.410111
$694$	72.5611	2.75438
$695$	0	0
$696$	101.543	3.84896
$697$	30.9456	1.17215
$698$	12.3122	0.466024
$699$	6.37139	0.240988
$700$	0	0
$701$	−1.53772	−0.0580789	−0.0290395	−	0.999578i	$-0.509245\pi$
$701$	−1.53772	−0.0580789	−0.0290395	+	0.999578i	$0.509245\pi$
$702$	−21.5351	−0.812789
$703$	−6.07686	−0.229193
$704$	10.5171	0.396379
$705$	0	0
$706$	25.3362	0.953540
$707$	5.09964	0.191792
$708$	−53.9835	−2.02882
$709$	5.19348	0.195045	0.0975226	−	0.995233i	$-0.468908\pi$
$709$	5.19348	0.195045	0.0975226	+	0.995233i	$0.468908\pi$
$710$	0	0
$711$	27.1849	1.01951
$712$	−66.3580	−2.48687
$713$	−26.4580	−0.990858
$714$	101.785	3.80922
$715$	0	0
$716$	77.3485	2.89065
$717$	−59.2652	−2.21330
$718$	39.6077	1.47815
$719$	−37.4841	−1.39792	−0.698960	−	0.715160i	$-0.746354\pi$
$719$	−37.4841	−1.39792	−0.698960	+	0.715160i	$0.746354\pi$
$720$	0	0
$721$	16.0198	0.596607
$722$	2.39788	0.0892398
$723$	41.8160	1.55515
$724$	−77.2117	−2.86955
$725$	0	0
$726$	6.22094	0.230881
$727$	−28.1916	−1.04557	−0.522785	−	0.852464i	$-0.675107\pi$
$727$	−28.1916	−1.04557	−0.522785	+	0.852464i	$0.675107\pi$
$728$	−57.5279	−2.13213
$729$	−38.1602	−1.41334
$730$	0	0
$731$	−61.9185	−2.29014
$732$	−9.82736	−0.363230
$733$	18.1906	0.671884	0.335942	−	0.941883i	$-0.390945\pi$
$733$	18.1906	0.671884	0.335942	+	0.941883i	$0.390945\pi$
$734$	−47.6604	−1.75918
$735$	0	0
$736$	−9.01428	−0.332271
$737$	6.98807	0.257409
$738$	48.9626	1.80234
$739$	5.65484	0.208017	0.104008	−	0.994576i	$-0.466833\pi$
$739$	5.65484	0.208017	0.104008	+	0.994576i	$0.466833\pi$
$740$	0	0
$741$	−12.2915	−0.451540
$742$	26.4487	0.970963
$743$	44.5565	1.63462	0.817310	−	0.576198i	$-0.195464\pi$
$743$	44.5565	1.63462	0.817310	+	0.576198i	$0.195464\pi$
$744$	−71.8737	−2.63502
$745$	0	0
$746$	27.3720	1.00216
$747$	51.1252	1.87057
$748$	−21.2010	−0.775187
$749$	12.5037	0.456877
$750$	0	0
$751$	−18.8205	−0.686769	−0.343384	−	0.939195i	$-0.611573\pi$
$751$	−18.8205	−0.686769	−0.343384	+	0.939195i	$0.611573\pi$
$752$	0.756760	0.0275962
$753$	−48.9705	−1.78459
$754$	−105.975	−3.85939
$755$	0	0
$756$	20.5701	0.748126
$757$	−2.97780	−0.108230	−0.0541150	−	0.998535i	$-0.517234\pi$
$757$	−2.97780	−0.108230	−0.0541150	+	0.998535i	$0.517234\pi$
$758$	4.79397	0.174125
$759$	−10.3959	−0.377348
$760$	0	0
$761$	−41.9642	−1.52120	−0.760601	−	0.649219i	$-0.775096\pi$
$761$	−41.9642	−1.52120	−0.760601	+	0.649219i	$0.775096\pi$
$762$	60.9804	2.20909
$763$	−37.6320	−1.36237
$764$	31.5002	1.13964
$765$	0	0
$766$	16.9532	0.612542
$767$	26.2905	0.949295
$768$	−74.3256	−2.68199
$769$	−2.33683	−0.0842683	−0.0421341	−	0.999112i	$-0.513416\pi$
$769$	−2.33683	−0.0842683	−0.0421341	+	0.999112i	$0.513416\pi$
$770$	0	0
$771$	−19.4065	−0.698908
$772$	−43.1162	−1.55179
$773$	−4.40125	−0.158302	−0.0791509	−	0.996863i	$-0.525221\pi$
$773$	−4.40125	−0.158302	−0.0791509	+	0.996863i	$0.525221\pi$
$774$	−97.9683	−3.52140
$775$	0	0
$776$	−20.1949	−0.724955
$777$	−45.6237	−1.63674
$778$	40.8207	1.46349
$779$	5.47333	0.196102
$780$	0	0
$781$	1.02085	0.0365288
$782$	54.3262	1.94270
$783$	17.6825	0.631920
$784$	3.52108	0.125753
$785$	0	0
$786$	−83.5985	−2.98186
$787$	13.1902	0.470180	0.235090	−	0.971974i	$-0.424461\pi$
$787$	13.1902	0.470180	0.235090	+	0.971974i	$0.424461\pi$
$788$	−101.386	−3.61172
$789$	46.1025	1.64129
$790$	0	0
$791$	49.2097	1.74970
$792$	−15.6533	−0.556214
$793$	4.78602	0.169957
$794$	60.8078	2.15799
$795$	0	0
$796$	−19.4413	−0.689078
$797$	39.1687	1.38743	0.693713	−	0.720252i	$-0.255974\pi$
$797$	39.1687	1.38743	0.693713	+	0.720252i	$0.255974\pi$
$798$	18.0027	0.637290
$799$	1.67037	0.0590935
$800$	0	0
$801$	−59.0010	−2.08470
$802$	47.1060	1.66337
$803$	−0.202033	−0.00712960
$804$	−67.9823	−2.39755
$805$	0	0
$806$	75.0111	2.64215
$807$	67.2766	2.36825
$808$	7.39394	0.260118
$809$	18.2003	0.639887	0.319944	−	0.947437i	$-0.396336\pi$
$809$	18.2003	0.639887	0.319944	+	0.947437i	$0.396336\pi$
$810$	0	0
$811$	−54.6434	−1.91879	−0.959395	−	0.282065i	$-0.908981\pi$
$811$	−54.6434	−1.91879	−0.959395	+	0.282065i	$0.908981\pi$
$812$	101.226	3.55234
$813$	−42.2065	−1.48025
$814$	14.5716	0.510733
$815$	0	0
$816$	37.5723	1.31530
$817$	−10.9515	−0.383144
$818$	−68.7056	−2.40224
$819$	−51.1499	−1.78732
$820$	0	0
$821$	18.3540	0.640560	0.320280	−	0.947323i	$-0.396223\pi$
$821$	18.3540	0.640560	0.320280	+	0.947323i	$0.396223\pi$
$822$	100.855	3.51772
$823$	−9.63416	−0.335826	−0.167913	−	0.985802i	$-0.553703\pi$
$823$	−9.63416	−0.335826	−0.167913	+	0.985802i	$0.553703\pi$
$824$	23.2270	0.809150
$825$	0	0
$826$	−38.5063	−1.33981
$827$	−26.2829	−0.913947	−0.456973	−	0.889480i	$-0.651067\pi$
$827$	−26.2829	−0.913947	−0.456973	+	0.889480i	$0.651067\pi$
$828$	56.0570	1.94812
$829$	−45.2690	−1.57226	−0.786128	−	0.618064i	$-0.787917\pi$
$829$	−45.2690	−1.57226	−0.786128	+	0.618064i	$0.787917\pi$
$830$	0	0
$831$	46.4952	1.61290
$832$	49.8281	1.72748
$833$	7.77197	0.269283
$834$	118.430	4.10088
$835$	0	0
$836$	−3.74982	−0.129690
$837$	−12.5160	−0.432615
$838$	−6.72648	−0.232362
$839$	23.0656	0.796315	0.398157	−	0.917317i	$-0.369650\pi$
$839$	23.0656	0.796315	0.398157	+	0.917317i	$0.369650\pi$
$840$	0	0
$841$	58.0162	2.00056
$842$	7.82104	0.269531
$843$	−74.8634	−2.57843
$844$	−80.7810	−2.78060
$845$	0	0
$846$	2.64289	0.0908643
$847$	2.89389	0.0994354
$848$	9.76309	0.335266
$849$	27.9205	0.958228
$850$	0	0
$851$	−24.3508	−0.834736
$852$	−9.93115	−0.340235
$853$	10.1175	0.346418	0.173209	−	0.984885i	$-0.444586\pi$
$853$	10.1175	0.346418	0.173209	+	0.984885i	$0.444586\pi$
$854$	−7.00983	−0.239872
$855$	0	0
$856$	18.1291	0.619641
$857$	−17.5975	−0.601119	−0.300559	−	0.953763i	$-0.597173\pi$
$857$	−17.5975	−0.601119	−0.300559	+	0.953763i	$0.597173\pi$
$858$	29.4735	1.00621
$859$	51.9873	1.77378	0.886891	−	0.461978i	$-0.152860\pi$
$859$	51.9873	1.77378	0.886891	+	0.461978i	$0.152860\pi$
$860$	0	0
$861$	41.0926	1.40043
$862$	78.5560	2.67563
$863$	42.2968	1.43980	0.719900	−	0.694078i	$-0.244188\pi$
$863$	42.2968	1.43980	0.719900	+	0.694078i	$0.244188\pi$
$864$	−4.26422	−0.145072
$865$	0	0
$866$	52.4310	1.78168
$867$	38.8282	1.31868
$868$	−71.6497	−2.43195
$869$	−7.28690	−0.247191
$870$	0	0
$871$	33.1081	1.12182
$872$	−54.5625	−1.84772
$873$	−17.9559	−0.607716
$874$	9.60863	0.325017
$875$	0	0
$876$	1.96545	0.0664063
$877$	4.60606	0.155535	0.0777677	−	0.996972i	$-0.475221\pi$
$877$	4.60606	0.155535	0.0777677	+	0.996972i	$0.475221\pi$
$878$	73.4423	2.47856
$879$	−13.3429	−0.450046
$880$	0	0
$881$	45.1124	1.51987	0.759937	−	0.649997i	$-0.225230\pi$
$881$	45.1124	1.51987	0.759937	+	0.649997i	$0.225230\pi$
$882$	12.2969	0.414059
$883$	40.3849	1.35906	0.679530	−	0.733648i	$-0.262184\pi$
$883$	40.3849	1.35906	0.679530	+	0.733648i	$0.262184\pi$
$884$	−100.446	−3.37837
$885$	0	0
$886$	−10.9083	−0.366471
$887$	−50.2462	−1.68710	−0.843551	−	0.537050i	$-0.819539\pi$
$887$	−50.2462	−1.68710	−0.843551	+	0.537050i	$0.819539\pi$
$888$	−66.1496	−2.21984
$889$	28.3672	0.951407
$890$	0	0
$891$	6.27417	0.210192
$892$	101.095	3.38491
$893$	0.295438	0.00988645
$894$	−95.3257	−3.18817
$895$	0	0
$896$	−59.9606	−2.00314
$897$	−49.2538	−1.64454
$898$	27.6495	0.922674
$899$	−61.5916	−2.05420
$900$	0	0
$901$	21.5498	0.717926
$902$	−13.1244	−0.436994
$903$	−82.2213	−2.73615
$904$	71.3490	2.37303
$905$	0	0
$906$	−7.83439	−0.260280
$907$	−58.4194	−1.93979	−0.969893	−	0.243532i	$-0.921694\pi$
$907$	−58.4194	−1.93979	−0.969893	+	0.243532i	$0.921694\pi$
$908$	−33.8447	−1.12318
$909$	6.57419	0.218052
$910$	0	0
$911$	−39.2409	−1.30011	−0.650054	−	0.759888i	$-0.725254\pi$
$911$	−39.2409	−1.30011	−0.650054	+	0.759888i	$0.725254\pi$
$912$	6.64540	0.220051
$913$	−13.7041	−0.453539
$914$	−27.2826	−0.902427
$915$	0	0
$916$	−51.6337	−1.70603
$917$	−38.8889	−1.28422
$918$	25.6990	0.848195
$919$	18.4113	0.607334	0.303667	−	0.952778i	$-0.401789\pi$
$919$	18.4113	0.607334	0.303667	+	0.952778i	$0.401789\pi$
$920$	0	0
$921$	−72.9057	−2.40233
$922$	10.4851	0.345308
$923$	4.83657	0.159198
$924$	−28.1528	−0.926159
$925$	0	0
$926$	22.3478	0.734393
$927$	20.6518	0.678296
$928$	−20.9844	−0.688847
$929$	−32.8679	−1.07836	−0.539180	−	0.842191i	$-0.681266\pi$
$929$	−32.8679	−1.07836	−0.539180	+	0.842191i	$0.681266\pi$
$930$	0	0
$931$	1.37462	0.0450515
$932$	9.20907	0.301653
$933$	−36.8162	−1.20531
$934$	67.3110	2.20248
$935$	0	0
$936$	−74.1620	−2.42406
$937$	−4.34616	−0.141983	−0.0709914	−	0.997477i	$-0.522616\pi$
$937$	−4.34616	−0.141983	−0.0709914	+	0.997477i	$0.522616\pi$
$938$	−48.4917	−1.58331
$939$	24.5447	0.800987
$940$	0	0
$941$	−55.1238	−1.79699	−0.898493	−	0.438989i	$-0.855337\pi$
$941$	−55.1238	−1.79699	−0.898493	+	0.438989i	$0.855337\pi$
$942$	−75.8091	−2.46999
$943$	21.9324	0.714218
$944$	−14.2139	−0.462625
$945$	0	0
$946$	26.2603	0.853797
$947$	−16.6794	−0.542007	−0.271004	−	0.962578i	$-0.587356\pi$
$947$	−16.6794	−0.542007	−0.271004	+	0.962578i	$0.587356\pi$
$948$	70.8894	2.30238
$949$	−0.957193	−0.0310718
$950$	0	0
$951$	61.9201	2.00790
$952$	68.6514	2.22500
$953$	−51.8274	−1.67885	−0.839427	−	0.543472i	$-0.817109\pi$
$953$	−51.8274	−1.67885	−0.839427	+	0.543472i	$0.817109\pi$
$954$	34.0963	1.10391
$955$	0	0
$956$	−85.6605	−2.77046
$957$	−24.2008	−0.782299
$958$	−13.5524	−0.437859
$959$	46.9163	1.51501
$960$	0	0
$961$	12.5957	0.406312
$962$	69.0371	2.22585
$963$	16.1192	0.519433
$964$	60.4399	1.94664
$965$	0	0
$966$	72.1395	2.32105
$967$	28.7146	0.923398	0.461699	−	0.887037i	$-0.347240\pi$
$967$	28.7146	0.923398	0.461699	+	0.887037i	$0.347240\pi$
$968$	4.19584	0.134860
$969$	14.6682	0.471210
$970$	0	0
$971$	43.9744	1.41121	0.705603	−	0.708607i	$-0.250676\pi$
$971$	43.9744	1.41121	0.705603	+	0.708607i	$0.250676\pi$
$972$	−82.3615	−2.64175
$973$	55.0918	1.76616
$974$	14.6114	0.468180
$975$	0	0
$976$	−2.58756	−0.0828258
$977$	−19.2496	−0.615848	−0.307924	−	0.951411i	$-0.599634\pi$
$977$	−19.2496	−0.615848	−0.307924	+	0.951411i	$0.599634\pi$
$978$	−37.9304	−1.21288
$979$	15.8152	0.505455
$980$	0	0
$981$	−48.5132	−1.54891
$982$	20.5327	0.655224
$983$	12.7410	0.406373	0.203187	−	0.979140i	$-0.434870\pi$
$983$	12.7410	0.406373	0.203187	+	0.979140i	$0.434870\pi$
$984$	59.5799	1.89934
$985$	0	0
$986$	126.466	4.02750
$987$	2.21808	0.0706023
$988$	−17.7659	−0.565208
$989$	−43.8842	−1.39544
$990$	0	0
$991$	27.9827	0.888900	0.444450	−	0.895804i	$-0.353399\pi$
$991$	27.9827	0.888900	0.444450	+	0.895804i	$0.353399\pi$
$992$	14.8531	0.471588
$993$	−34.2047	−1.08545
$994$	−7.08387	−0.224687
$995$	0	0
$996$	133.318	4.22434
$997$	47.7121	1.51106	0.755528	−	0.655116i	$-0.227380\pi$
$997$	47.7121	1.51106	0.755528	+	0.655116i	$0.227380\pi$
$998$	−3.25002	−0.102878
$999$	−11.5192	−0.364451

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5225.2.a.j.1.5		5
5.4	even	2		1045.2.a.d.1.1	✓	5
15.14	odd	2		9405.2.a.v.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.a.d.1.1	✓	5	5.4	even	2
5225.2.a.j.1.5		5	1.1	even	1	trivial
9405.2.a.v.1.5		5	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5225.a (trivial)

\(f(q)\)	\(=\)	\(q+2.39788 q^{2} +2.59435 q^{3} +3.74982 q^{4} +6.22094 q^{6} +2.89389 q^{7} +4.19584 q^{8} +3.73066 q^{9} +O(q^{10})\)	\(q+2.39788 q^{2} +2.59435 q^{3} +3.74982 q^{4} +6.22094 q^{6} +2.89389 q^{7} +4.19584 q^{8} +3.73066 q^{9} -1.00000 q^{11} +9.72834 q^{12} -4.73780 q^{13} +6.93920 q^{14} +2.56149 q^{16} +5.65389 q^{17} +8.94566 q^{18} +1.00000 q^{19} +7.50778 q^{21} -2.39788 q^{22} +4.00714 q^{23} +10.8855 q^{24} -11.3607 q^{26} +1.89558 q^{27} +10.8516 q^{28} +9.32825 q^{29} -6.60270 q^{31} -2.24955 q^{32} -2.59435 q^{33} +13.5573 q^{34} +13.9893 q^{36} -6.07686 q^{37} +2.39788 q^{38} -12.2915 q^{39} +5.47333 q^{41} +18.0027 q^{42} -10.9515 q^{43} -3.74982 q^{44} +9.60863 q^{46} +0.295438 q^{47} +6.64540 q^{48} +1.37462 q^{49} +14.6682 q^{51} -17.7659 q^{52} +3.81149 q^{53} +4.54538 q^{54} +12.1423 q^{56} +2.59435 q^{57} +22.3680 q^{58} -5.54910 q^{59} -1.01018 q^{61} -15.8325 q^{62} +10.7961 q^{63} -10.5171 q^{64} -6.22094 q^{66} -6.98807 q^{67} +21.2010 q^{68} +10.3959 q^{69} -1.02085 q^{71} +15.6533 q^{72} +0.202033 q^{73} -14.5716 q^{74} +3.74982 q^{76} -2.89389 q^{77} -29.4735 q^{78} +7.28690 q^{79} -6.27417 q^{81} +13.1244 q^{82} +13.7041 q^{83} +28.1528 q^{84} -26.2603 q^{86} +24.2008 q^{87} -4.19584 q^{88} -15.8152 q^{89} -13.7107 q^{91} +15.0260 q^{92} -17.1297 q^{93} +0.708423 q^{94} -5.83613 q^{96} -4.81308 q^{97} +3.29618 q^{98} -3.73066 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 3 q^{2} + 7 q^{3} + 5 q^{4} + 2 q^{6} + 11 q^{7} - 3 q^{8} + 8 q^{9}+O(q^{10}) \) 5 * q + 3 * q^2 + 7 * q^3 + 5 * q^4 + 2 * q^6 + 11 * q^7 - 3 * q^8 + 8 * q^9	\( 5 q + 3 q^{2} + 7 q^{3} + 5 q^{4} + 2 q^{6} + 11 q^{7} - 3 q^{8} + 8 q^{9} - 5 q^{11} + 7 q^{12} - q^{13} - 3 q^{16} + 3 q^{17} + 7 q^{18} + 5 q^{19} + 11 q^{21} - 3 q^{22} + 8 q^{23} + 9 q^{24} - 16 q^{26} + 10 q^{27} + 22 q^{28} + 11 q^{29} - 5 q^{31} + 2 q^{32} - 7 q^{33} + 4 q^{34} - 3 q^{36} + 9 q^{37} + 3 q^{38} - 8 q^{39} + 15 q^{41} - 11 q^{42} + 13 q^{43} - 5 q^{44} + 18 q^{46} + 20 q^{47} + 20 q^{48} + 20 q^{49} + 24 q^{51} - q^{52} + 5 q^{53} + 17 q^{54} + 7 q^{57} + 33 q^{58} - 17 q^{59} + 3 q^{61} - 14 q^{62} + 22 q^{63} - 17 q^{64} - 2 q^{66} + 28 q^{67} + 25 q^{68} - 2 q^{69} - 6 q^{71} + 26 q^{72} + 16 q^{73} - 21 q^{74} + 5 q^{76} - 11 q^{77} - 29 q^{78} + 3 q^{79} + q^{81} - 2 q^{82} + 33 q^{83} + 33 q^{84} + 10 q^{86} + 3 q^{88} - 16 q^{89} - 22 q^{91} + 19 q^{92} + 26 q^{93} - 10 q^{94} + 5 q^{96} + 14 q^{97} - 10 q^{98} - 8 q^{99}+O(q^{100}) \) 5 * q + 3 * q^2 + 7 * q^3 + 5 * q^4 + 2 * q^6 + 11 * q^7 - 3 * q^8 + 8 * q^9 - 5 * q^11 + 7 * q^12 - q^13 - 3 * q^16 + 3 * q^17 + 7 * q^18 + 5 * q^19 + 11 * q^21 - 3 * q^22 + 8 * q^23 + 9 * q^24 - 16 * q^26 + 10 * q^27 + 22 * q^28 + 11 * q^29 - 5 * q^31 + 2 * q^32 - 7 * q^33 + 4 * q^34 - 3 * q^36 + 9 * q^37 + 3 * q^38 - 8 * q^39 + 15 * q^41 - 11 * q^42 + 13 * q^43 - 5 * q^44 + 18 * q^46 + 20 * q^47 + 20 * q^48 + 20 * q^49 + 24 * q^51 - q^52 + 5 * q^53 + 17 * q^54 + 7 * q^57 + 33 * q^58 - 17 * q^59 + 3 * q^61 - 14 * q^62 + 22 * q^63 - 17 * q^64 - 2 * q^66 + 28 * q^67 + 25 * q^68 - 2 * q^69 - 6 * q^71 + 26 * q^72 + 16 * q^73 - 21 * q^74 + 5 * q^76 - 11 * q^77 - 29 * q^78 + 3 * q^79 + q^81 - 2 * q^82 + 33 * q^83 + 33 * q^84 + 10 * q^86 + 3 * q^88 - 16 * q^89 - 22 * q^91 + 19 * q^92 + 26 * q^93 - 10 * q^94 + 5 * q^96 + 14 * q^97 - 10 * q^98 - 8 * q^99

Introduction

L-functions

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