LMFDB - Embedded newform 6025.2.a.j.1.5

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:	$1$
Dimension:	$25$
Twist minimal:	no (minimal twist has level 1205)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
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)

$f(q)$	$=$	$q-2.31949 q^{2} -0.405385 q^{3} +3.38004 q^{4} +0.940288 q^{6} -1.18682 q^{7} -3.20098 q^{8} -2.83566 q^{9} +O(q^{10})$	\(q-2.31949 q^{2} -0.405385 q^{3} +3.38004 q^{4} +0.940288 q^{6} -1.18682 q^{7} -3.20098 q^{8} -2.83566 q^{9} +2.77887 q^{11} -1.37022 q^{12} -3.13031 q^{13} +2.75282 q^{14} +0.664580 q^{16} +2.57029 q^{17} +6.57729 q^{18} +3.02001 q^{19} +0.481120 q^{21} -6.44556 q^{22} -2.95317 q^{23} +1.29763 q^{24} +7.26071 q^{26} +2.36569 q^{27} -4.01150 q^{28} -1.77022 q^{29} +1.88113 q^{31} +4.86048 q^{32} -1.12651 q^{33} -5.96176 q^{34} -9.58465 q^{36} -9.47803 q^{37} -7.00488 q^{38} +1.26898 q^{39} -4.74304 q^{41} -1.11595 q^{42} +7.70167 q^{43} +9.39267 q^{44} +6.84985 q^{46} +13.1180 q^{47} -0.269411 q^{48} -5.59145 q^{49} -1.04196 q^{51} -10.5805 q^{52} -2.31668 q^{53} -5.48720 q^{54} +3.79900 q^{56} -1.22427 q^{57} +4.10601 q^{58} -1.39815 q^{59} +4.77844 q^{61} -4.36325 q^{62} +3.36543 q^{63} -12.6030 q^{64} +2.61293 q^{66} +14.0509 q^{67} +8.68768 q^{68} +1.19717 q^{69} +0.665489 q^{71} +9.07691 q^{72} -14.8463 q^{73} +21.9842 q^{74} +10.2077 q^{76} -3.29802 q^{77} -2.94339 q^{78} -0.927804 q^{79} +7.54797 q^{81} +11.0014 q^{82} +6.69746 q^{83} +1.62621 q^{84} -17.8640 q^{86} +0.717621 q^{87} -8.89511 q^{88} +2.64088 q^{89} +3.71512 q^{91} -9.98182 q^{92} -0.762581 q^{93} -30.4270 q^{94} -1.97037 q^{96} +15.2175 q^{97} +12.9693 q^{98} -7.87993 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q - 6 q^{2} - 15 q^{3} + 32 q^{4} - q^{6} - 19 q^{7} - 15 q^{8} + 32 q^{9}+O(q^{10}) $ 25 * q - 6 * q^2 - 15 * q^3 + 32 * q^4 - q^6 - 19 * q^7 - 15 * q^8 + 32 * q^9	$ 25 q - 6 q^{2} - 15 q^{3} + 32 q^{4} - q^{6} - 19 q^{7} - 15 q^{8} + 32 q^{9} + 2 q^{11} - 20 q^{12} - 14 q^{13} - 5 q^{14} + 38 q^{16} - 7 q^{17} - 9 q^{18} + 30 q^{19} + q^{21} - q^{22} - 43 q^{23} - 6 q^{24} - 22 q^{26} - 42 q^{27} - 32 q^{28} - 4 q^{29} + 14 q^{31} - 26 q^{32} - 4 q^{33} + 7 q^{34} + 15 q^{36} - 16 q^{37} - 14 q^{38} - 21 q^{39} - q^{41} + 25 q^{42} - 35 q^{43} - 52 q^{44} - 27 q^{46} - 50 q^{47} - 26 q^{48} + 46 q^{49} - 7 q^{51} - 3 q^{52} - 4 q^{53} - 31 q^{54} - 51 q^{56} - 2 q^{58} + 6 q^{59} + 19 q^{61} - 28 q^{63} + 49 q^{64} - 27 q^{66} - 65 q^{67} + 25 q^{68} + 2 q^{69} - 34 q^{71} + 10 q^{72} - 8 q^{73} - 42 q^{74} + 71 q^{76} - q^{77} + 59 q^{78} - 12 q^{79} + 29 q^{81} - 11 q^{82} - 41 q^{83} - 10 q^{84} - 13 q^{86} - 40 q^{87} + 52 q^{88} - 24 q^{89} + 46 q^{91} - 85 q^{92} + 30 q^{93} + 14 q^{94} - 30 q^{96} - 9 q^{97} + 64 q^{98} + 2 q^{99}+O(q^{100}) $ 25 * q - 6 * q^2 - 15 * q^3 + 32 * q^4 - q^6 - 19 * q^7 - 15 * q^8 + 32 * q^9 + 2 * q^11 - 20 * q^12 - 14 * q^13 - 5 * q^14 + 38 * q^16 - 7 * q^17 - 9 * q^18 + 30 * q^19 + q^21 - q^22 - 43 * q^23 - 6 * q^24 - 22 * q^26 - 42 * q^27 - 32 * q^28 - 4 * q^29 + 14 * q^31 - 26 * q^32 - 4 * q^33 + 7 * q^34 + 15 * q^36 - 16 * q^37 - 14 * q^38 - 21 * q^39 - q^41 + 25 * q^42 - 35 * q^43 - 52 * q^44 - 27 * q^46 - 50 * q^47 - 26 * q^48 + 46 * q^49 - 7 * q^51 - 3 * q^52 - 4 * q^53 - 31 * q^54 - 51 * q^56 - 2 * q^58 + 6 * q^59 + 19 * q^61 - 28 * q^63 + 49 * q^64 - 27 * q^66 - 65 * q^67 + 25 * q^68 + 2 * q^69 - 34 * q^71 + 10 * q^72 - 8 * q^73 - 42 * q^74 + 71 * q^76 - q^77 + 59 * q^78 - 12 * q^79 + 29 * q^81 - 11 * q^82 - 41 * q^83 - 10 * q^84 - 13 * q^86 - 40 * q^87 + 52 * q^88 - 24 * q^89 + 46 * q^91 - 85 * q^92 + 30 * q^93 + 14 * q^94 - 30 * q^96 - 9 * q^97 + 64 * q^98 + 2 * 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 $

$2$	−2.31949	−1.64013	−0.820064	−	0.572272i	$-0.806062\pi$
$2$	−2.31949	−1.64013	−0.820064	+	0.572272i	$0.806062\pi$
$3$	−0.405385	−0.234049	−0.117025	−	0.993129i	$-0.537336\pi$
$3$	−0.405385	−0.234049	−0.117025	+	0.993129i	$0.537336\pi$
$4$	3.38004	1.69002
$5$	0	0
$5$	0	0
$6$	0.940288	0.383871
$7$	−1.18682	−0.448577	−0.224288	−	0.974523i	$-0.572006\pi$
$7$	−1.18682	−0.448577	−0.224288	+	0.974523i	$0.572006\pi$
$8$	−3.20098	−1.13172
$9$	−2.83566	−0.945221
$10$	0	0
$11$	2.77887	0.837860	0.418930	−	0.908019i	$-0.362405\pi$
$11$	2.77887	0.837860	0.418930	+	0.908019i	$0.362405\pi$
$12$	−1.37022	−0.395548
$13$	−3.13031	−0.868190	−0.434095	−	0.900867i	$-0.642932\pi$
$13$	−3.13031	−0.868190	−0.434095	+	0.900867i	$0.642932\pi$
$14$	2.75282	0.735723
$15$	0	0
$16$	0.664580	0.166145
$17$	2.57029	0.623387	0.311693	−	0.950183i	$-0.399104\pi$
$17$	2.57029	0.623387	0.311693	+	0.950183i	$0.399104\pi$
$18$	6.57729	1.55028
$19$	3.02001	0.692837	0.346419	−	0.938080i	$-0.387398\pi$
$19$	3.02001	0.692837	0.346419	+	0.938080i	$0.387398\pi$
$20$	0	0
$21$	0.481120	0.104989
$22$	−6.44556	−1.37420
$23$	−2.95317	−0.615778	−0.307889	−	0.951422i	$-0.599623\pi$
$23$	−2.95317	−0.615778	−0.307889	+	0.951422i	$0.599623\pi$
$24$	1.29763	0.264878
$25$	0	0
$26$	7.26071	1.42394
$27$	2.36569	0.455278
$28$	−4.01150	−0.758103
$29$	−1.77022	−0.328721	−0.164361	−	0.986400i	$-0.552556\pi$
$29$	−1.77022	−0.328721	−0.164361	+	0.986400i	$0.552556\pi$
$30$	0	0
$31$	1.88113	0.337860	0.168930	−	0.985628i	$-0.445969\pi$
$31$	1.88113	0.337860	0.168930	+	0.985628i	$0.445969\pi$
$32$	4.86048	0.859220
$33$	−1.12651	−0.196101
$34$	−5.96176	−1.02243
$35$	0	0
$36$	−9.58465	−1.59744
$37$	−9.47803	−1.55818	−0.779090	−	0.626913i	$-0.784318\pi$
$37$	−9.47803	−1.55818	−0.779090	+	0.626913i	$0.784318\pi$
$38$	−7.00488	−1.13634
$39$	1.26898	0.203199
$40$	0	0
$41$	−4.74304	−0.740738	−0.370369	−	0.928885i	$-0.620769\pi$
$41$	−4.74304	−0.740738	−0.370369	+	0.928885i	$0.620769\pi$
$42$	−1.11595	−0.172195
$43$	7.70167	1.17449	0.587247	−	0.809408i	$-0.300212\pi$
$43$	7.70167	1.17449	0.587247	+	0.809408i	$0.300212\pi$
$44$	9.39267	1.41600
$45$	0	0
$46$	6.84985	1.00995
$47$	13.1180	1.91345	0.956725	−	0.290992i	$-0.0939855\pi$
$47$	13.1180	1.91345	0.956725	+	0.290992i	$0.0939855\pi$
$48$	−0.269411	−0.0388861
$49$	−5.59145	−0.798779
$50$	0	0
$51$	−1.04196	−0.145903
$52$	−10.5805	−1.46726
$53$	−2.31668	−0.318221	−0.159110	−	0.987261i	$-0.550863\pi$
$53$	−2.31668	−0.318221	−0.159110	+	0.987261i	$0.550863\pi$
$54$	−5.48720	−0.746714
$55$	0	0
$56$	3.79900	0.507663
$57$	−1.22427	−0.162158
$58$	4.10601	0.539145
$59$	−1.39815	−0.182024	−0.0910119	−	0.995850i	$-0.529010\pi$
$59$	−1.39815	−0.182024	−0.0910119	+	0.995850i	$0.529010\pi$
$60$	0	0
$61$	4.77844	0.611816	0.305908	−	0.952061i	$-0.401040\pi$
$61$	4.77844	0.611816	0.305908	+	0.952061i	$0.401040\pi$
$62$	−4.36325	−0.554134
$63$	3.36543	0.424004
$64$	−12.6030	−1.57538
$65$	0	0
$66$	2.61293	0.321630
$67$	14.0509	1.71659	0.858295	−	0.513157i	$-0.171524\pi$
$67$	14.0509	1.71659	0.858295	+	0.513157i	$0.171524\pi$
$68$	8.68768	1.05354
$69$	1.19717	0.144122
$70$	0	0
$71$	0.665489	0.0789790	0.0394895	−	0.999220i	$-0.487427\pi$
$71$	0.665489	0.0789790	0.0394895	+	0.999220i	$0.487427\pi$
$72$	9.07691	1.06972
$73$	−14.8463	−1.73763	−0.868814	−	0.495139i	$-0.835117\pi$
$73$	−14.8463	−1.73763	−0.868814	+	0.495139i	$0.835117\pi$
$74$	21.9842	2.55561
$75$	0	0
$76$	10.2077	1.17091
$77$	−3.29802	−0.375844
$78$	−2.94339	−0.333273
$79$	−0.927804	−0.104386	−0.0521931	−	0.998637i	$-0.516621\pi$
$79$	−0.927804	−0.104386	−0.0521931	+	0.998637i	$0.516621\pi$
$80$	0	0
$81$	7.54797	0.838663
$82$	11.0014	1.21490
$83$	6.69746	0.735142	0.367571	−	0.929996i	$-0.380190\pi$
$83$	6.69746	0.735142	0.367571	+	0.929996i	$0.380190\pi$
$84$	1.62621	0.177434
$85$	0	0
$86$	−17.8640	−1.92632
$87$	0.717621	0.0769371
$88$	−8.89511	−0.948222
$89$	2.64088	0.279933	0.139967	−	0.990156i	$-0.455301\pi$
$89$	2.64088	0.279933	0.139967	+	0.990156i	$0.455301\pi$
$90$	0	0
$91$	3.71512	0.389450
$92$	−9.98182	−1.04068
$93$	−0.762581	−0.0790759
$94$	−30.4270	−3.13830
$95$	0	0
$96$	−1.97037	−0.201100
$97$	15.2175	1.54510	0.772552	−	0.634952i	$-0.218980\pi$
$97$	15.2175	1.54510	0.772552	+	0.634952i	$0.218980\pi$
$98$	12.9693	1.31010
$99$	−7.87993	−0.791963
$100$	0	0
$101$	−10.8626	−1.08087	−0.540434	−	0.841386i	$-0.681740\pi$
$101$	−10.8626	−1.08087	−0.540434	+	0.841386i	$0.681740\pi$
$102$	2.41681	0.239300
$103$	−5.17014	−0.509429	−0.254714	−	0.967016i	$-0.581981\pi$
$103$	−5.17014	−0.509429	−0.254714	+	0.967016i	$0.581981\pi$
$104$	10.0201	0.982548
$105$	0	0
$106$	5.37352	0.521922
$107$	5.21791	0.504434	0.252217	−	0.967671i	$-0.418840\pi$
$107$	5.21791	0.504434	0.252217	+	0.967671i	$0.418840\pi$
$108$	7.99613	0.769428
$109$	2.89471	0.277263	0.138632	−	0.990344i	$-0.455730\pi$
$109$	2.89471	0.277263	0.138632	+	0.990344i	$0.455730\pi$
$110$	0	0
$111$	3.84226	0.364691
$112$	−0.788738	−0.0745287
$113$	−6.82583	−0.642120	−0.321060	−	0.947059i	$-0.604039\pi$
$113$	−6.82583	−0.642120	−0.321060	+	0.947059i	$0.604039\pi$
$114$	2.83968	0.265960
$115$	0	0
$116$	−5.98341	−0.555545
$117$	8.87649	0.820632
$118$	3.24300	0.298542
$119$	−3.05048	−0.279637
$120$	0	0
$121$	−3.27790	−0.297991
$122$	−11.0835	−1.00346
$123$	1.92276	0.173369
$124$	6.35827	0.570990
$125$	0	0
$126$	−7.80608	−0.695421
$127$	−7.99442	−0.709390	−0.354695	−	0.934982i	$-0.615415\pi$
$127$	−7.99442	−0.709390	−0.354695	+	0.934982i	$0.615415\pi$
$128$	19.5116	1.72460
$129$	−3.12214	−0.274889
$130$	0	0
$131$	5.43217	0.474611	0.237305	−	0.971435i	$-0.423736\pi$
$131$	5.43217	0.474611	0.237305	+	0.971435i	$0.423736\pi$
$132$	−3.80765	−0.331414
$133$	−3.58421	−0.310791
$134$	−32.5909	−2.81543
$135$	0	0
$136$	−8.22746	−0.705499
$137$	−4.29447	−0.366902	−0.183451	−	0.983029i	$-0.558727\pi$
$137$	−4.29447	−0.366902	−0.183451	+	0.983029i	$0.558727\pi$
$138$	−2.77683	−0.236379
$139$	−4.08705	−0.346659	−0.173329	−	0.984864i	$-0.555453\pi$
$139$	−4.08705	−0.346659	−0.173329	+	0.984864i	$0.555453\pi$
$140$	0	0
$141$	−5.31783	−0.447842
$142$	−1.54359	−0.129536
$143$	−8.69870	−0.727422
$144$	−1.88452	−0.157044
$145$	0	0
$146$	34.4359	2.84993
$147$	2.26669	0.186954
$148$	−32.0361	−2.63335
$149$	−22.1808	−1.81712	−0.908561	−	0.417753i	$-0.862818\pi$
$149$	−22.1808	−1.81712	−0.908561	+	0.417753i	$0.862818\pi$
$150$	0	0
$151$	22.5334	1.83374	0.916870	−	0.399185i	$-0.130707\pi$
$151$	22.5334	1.83374	0.916870	+	0.399185i	$0.130707\pi$
$152$	−9.66700	−0.784097
$153$	−7.28848	−0.589238
$154$	7.64973	0.616433
$155$	0	0
$156$	4.28920	0.343411
$157$	16.0364	1.27984	0.639922	−	0.768440i	$-0.278967\pi$
$157$	16.0364	1.27984	0.639922	+	0.768440i	$0.278967\pi$
$158$	2.15203	0.171207
$159$	0.939148	0.0744793
$160$	0	0
$161$	3.50489	0.276224
$162$	−17.5074	−1.37552
$163$	−1.38925	−0.108814	−0.0544071	−	0.998519i	$-0.517327\pi$
$163$	−1.38925	−0.108814	−0.0544071	+	0.998519i	$0.517327\pi$
$164$	−16.0316	−1.25186
$165$	0	0
$166$	−15.5347	−1.20573
$167$	−7.80002	−0.603583	−0.301792	−	0.953374i	$-0.597585\pi$
$167$	−7.80002	−0.603583	−0.301792	+	0.953374i	$0.597585\pi$
$168$	−1.54006	−0.118818
$169$	−3.20119	−0.246245
$170$	0	0
$171$	−8.56372	−0.654884
$172$	26.0319	1.98492
$173$	−25.7027	−1.95414	−0.977070	−	0.212920i	$-0.931703\pi$
$173$	−25.7027	−1.95414	−0.977070	+	0.212920i	$0.931703\pi$
$174$	−1.66452	−0.126187
$175$	0	0
$176$	1.84678	0.139206
$177$	0.566790	0.0426025
$178$	−6.12550	−0.459126
$179$	−8.41738	−0.629144	−0.314572	−	0.949234i	$-0.601861\pi$
$179$	−8.41738	−0.629144	−0.314572	+	0.949234i	$0.601861\pi$
$180$	0	0
$181$	−3.90367	−0.290157	−0.145079	−	0.989420i	$-0.546343\pi$
$181$	−3.90367	−0.290157	−0.145079	+	0.989420i	$0.546343\pi$
$182$	−8.61718	−0.638748
$183$	−1.93711	−0.143195
$184$	9.45305	0.696888
$185$	0	0
$186$	1.76880	0.129695
$187$	7.14249	0.522311
$188$	44.3392	3.23377
$189$	−2.80766	−0.204227
$190$	0	0
$191$	3.68698	0.266780	0.133390	−	0.991064i	$-0.457414\pi$
$191$	3.68698	0.266780	0.133390	+	0.991064i	$0.457414\pi$
$192$	5.10907	0.368716
$193$	24.8859	1.79132	0.895662	−	0.444736i	$-0.146702\pi$
$193$	24.8859	1.79132	0.895662	+	0.444736i	$0.146702\pi$
$194$	−35.2969	−2.53417
$195$	0	0
$196$	−18.8993	−1.34995
$197$	−3.61827	−0.257791	−0.128895	−	0.991658i	$-0.541143\pi$
$197$	−3.61827	−0.257791	−0.128895	+	0.991658i	$0.541143\pi$
$198$	18.2774	1.29892
$199$	13.0965	0.928384	0.464192	−	0.885735i	$-0.346345\pi$
$199$	13.0965	0.928384	0.464192	+	0.885735i	$0.346345\pi$
$200$	0	0
$201$	−5.69602	−0.401767
$202$	25.1957	1.77276
$203$	2.10094	0.147457
$204$	−3.52186	−0.246579
$205$	0	0
$206$	11.9921	0.835528
$207$	8.37419	0.582046
$208$	−2.08034	−0.144245
$209$	8.39220	0.580500
$210$	0	0
$211$	17.3322	1.19319	0.596597	−	0.802541i	$-0.296519\pi$
$211$	17.3322	1.19319	0.596597	+	0.802541i	$0.296519\pi$
$212$	−7.83047	−0.537799
$213$	−0.269779	−0.0184850
$214$	−12.1029	−0.827336
$215$	0	0
$216$	−7.57255	−0.515247
$217$	−2.23256	−0.151556
$218$	−6.71426	−0.454747
$219$	6.01847	0.406691
$220$	0	0
$221$	−8.04579	−0.541219
$222$	−8.91208	−0.598140
$223$	−27.1442	−1.81771	−0.908854	−	0.417115i	$-0.863041\pi$
$223$	−27.1442	−1.81771	−0.908854	+	0.417115i	$0.863041\pi$
$224$	−5.76853	−0.385426
$225$	0	0
$226$	15.8324	1.05316
$227$	−5.83871	−0.387529	−0.193764	−	0.981048i	$-0.562070\pi$
$227$	−5.83871	−0.387529	−0.193764	+	0.981048i	$0.562070\pi$
$228$	−4.13807	−0.274050
$229$	4.19352	0.277116	0.138558	−	0.990354i	$-0.455753\pi$
$229$	4.19352	0.277116	0.138558	+	0.990354i	$0.455753\pi$
$230$	0	0
$231$	1.33697	0.0879661
$232$	5.66645	0.372020
$233$	−3.08769	−0.202282	−0.101141	−	0.994872i	$-0.532249\pi$
$233$	−3.08769	−0.202282	−0.101141	+	0.994872i	$0.532249\pi$
$234$	−20.5889	−1.34594
$235$	0	0
$236$	−4.72580	−0.307624
$237$	0.376118	0.0244315
$238$	7.07555	0.458640
$239$	−11.0889	−0.717278	−0.358639	−	0.933476i	$-0.616759\pi$
$239$	−11.0889	−0.717278	−0.358639	+	0.933476i	$0.616759\pi$
$240$	0	0
$241$	−1.00000	−0.0644157
$241$	−1.00000	−0.0644157
$242$	7.60306	0.488743
$243$	−10.1569	−0.651566
$244$	16.1513	1.03398
$245$	0	0
$246$	−4.45982	−0.284348
$247$	−9.45354	−0.601515
$248$	−6.02145	−0.382363
$249$	−2.71505	−0.172059
$250$	0	0
$251$	−13.4460	−0.848702	−0.424351	−	0.905498i	$-0.639498\pi$
$251$	−13.4460	−0.848702	−0.424351	+	0.905498i	$0.639498\pi$
$252$	11.3753	0.716575
$253$	−8.20646	−0.515936
$254$	18.5430	1.16349
$255$	0	0
$256$	−20.0509	−1.25318
$257$	−1.83610	−0.114533	−0.0572665	−	0.998359i	$-0.518238\pi$
$257$	−1.83610	−0.114533	−0.0572665	+	0.998359i	$0.518238\pi$
$258$	7.24179	0.450854
$259$	11.2487	0.698963
$260$	0	0
$261$	5.01974	0.310714
$262$	−12.5999	−0.778422
$263$	20.6732	1.27477	0.637383	−	0.770547i	$-0.280017\pi$
$263$	20.6732	1.27477	0.637383	+	0.770547i	$0.280017\pi$
$264$	3.60595	0.221931
$265$	0	0
$266$	8.31355	0.509736
$267$	−1.07058	−0.0655182
$268$	47.4925	2.90107
$269$	10.4062	0.634480	0.317240	−	0.948345i	$-0.397244\pi$
$269$	10.4062	0.634480	0.317240	+	0.948345i	$0.397244\pi$
$270$	0	0
$271$	11.9422	0.725440	0.362720	−	0.931898i	$-0.381848\pi$
$271$	11.9422	0.725440	0.362720	+	0.931898i	$0.381848\pi$
$272$	1.70816	0.103573
$273$	−1.50605	−0.0911505
$274$	9.96099	0.601765
$275$	0	0
$276$	4.04648	0.243570
$277$	−13.9911	−0.840643	−0.420321	−	0.907375i	$-0.638083\pi$
$277$	−13.9911	−0.840643	−0.420321	+	0.907375i	$0.638083\pi$
$278$	9.47987	0.568565
$279$	−5.33424	−0.319352
$280$	0	0
$281$	7.63978	0.455751	0.227876	−	0.973690i	$-0.426822\pi$
$281$	7.63978	0.455751	0.227876	+	0.973690i	$0.426822\pi$
$282$	12.3347	0.734518
$283$	18.4956	1.09945	0.549724	−	0.835346i	$-0.314733\pi$
$283$	18.4956	1.09945	0.549724	+	0.835346i	$0.314733\pi$
$284$	2.24938	0.133476
$285$	0	0
$286$	20.1766	1.19306
$287$	5.62914	0.332278
$288$	−13.7827	−0.812153
$289$	−10.3936	−0.611389
$290$	0	0
$291$	−6.16896	−0.361631
$292$	−50.1811	−2.93662
$293$	16.0471	0.937481	0.468741	−	0.883336i	$-0.344708\pi$
$293$	16.0471	0.937481	0.468741	+	0.883336i	$0.344708\pi$
$294$	−5.25757	−0.306628
$295$	0	0
$296$	30.3390	1.76342
$297$	6.57394	0.381459
$298$	51.4481	2.98031
$299$	9.24432	0.534613
$300$	0	0
$301$	−9.14051	−0.526850
$302$	−52.2660	−3.00757
$303$	4.40354	0.252977
$304$	2.00704	0.115111
$305$	0	0
$306$	16.9056	0.966426
$307$	−28.4424	−1.62329	−0.811647	−	0.584148i	$-0.801429\pi$
$307$	−28.4424	−1.62329	−0.811647	+	0.584148i	$0.801429\pi$
$308$	−11.1474	−0.635184
$309$	2.09590	0.119232
$310$	0	0
$311$	11.8065	0.669484	0.334742	−	0.942310i	$-0.391351\pi$
$311$	11.8065	0.669484	0.334742	+	0.942310i	$0.391351\pi$
$312$	−4.06199	−0.229965
$313$	28.0313	1.58442	0.792211	−	0.610247i	$-0.208930\pi$
$313$	28.0313	1.58442	0.792211	+	0.610247i	$0.208930\pi$
$314$	−37.1963	−2.09911
$315$	0	0
$316$	−3.13601	−0.176415
$317$	15.8810	0.891966	0.445983	−	0.895041i	$-0.352854\pi$
$317$	15.8810	0.891966	0.445983	+	0.895041i	$0.352854\pi$
$318$	−2.17835	−0.122156
$319$	−4.91920	−0.275422
$320$	0	0
$321$	−2.11526	−0.118062
$322$	−8.12955	−0.453042
$323$	7.76229	0.431906
$324$	25.5124	1.41736
$325$	0	0
$326$	3.22234	0.178469
$327$	−1.17347	−0.0648932
$328$	15.1824	0.838307
$329$	−15.5687	−0.858329
$330$	0	0
$331$	−9.28769	−0.510497	−0.255249	−	0.966875i	$-0.582157\pi$
$331$	−9.28769	−0.510497	−0.255249	+	0.966875i	$0.582157\pi$
$332$	22.6377	1.24240
$333$	26.8765	1.47282
$334$	18.0921	0.989954
$335$	0	0
$336$	0.319743	0.0174434
$337$	−5.42203	−0.295357	−0.147679	−	0.989035i	$-0.547180\pi$
$337$	−5.42203	−0.295357	−0.147679	+	0.989035i	$0.547180\pi$
$338$	7.42513	0.403874
$339$	2.76709	0.150288
$340$	0	0
$341$	5.22740	0.283079
$342$	19.8635	1.07409
$343$	14.9438	0.806890
$344$	−24.6529	−1.32920
$345$	0	0
$346$	59.6172	3.20504
$347$	−26.1495	−1.40378	−0.701890	−	0.712285i	$-0.747660\pi$
$347$	−26.1495	−1.40378	−0.701890	+	0.712285i	$0.747660\pi$
$348$	2.42559	0.130025
$349$	−13.1428	−0.703520	−0.351760	−	0.936090i	$-0.614417\pi$
$349$	−13.1428	−0.703520	−0.351760	+	0.936090i	$0.614417\pi$
$350$	0	0
$351$	−7.40534	−0.395268
$352$	13.5066	0.719906
$353$	−13.0108	−0.692493	−0.346247	−	0.938144i	$-0.612544\pi$
$353$	−13.0108	−0.692493	−0.346247	+	0.938144i	$0.612544\pi$
$354$	−1.31466	−0.0698736
$355$	0	0
$356$	8.92628	0.473092
$357$	1.23662	0.0654488
$358$	19.5240	1.03188
$359$	16.5392	0.872903	0.436452	−	0.899728i	$-0.356235\pi$
$359$	16.5392	0.872903	0.436452	+	0.899728i	$0.356235\pi$
$360$	0	0
$361$	−9.87956	−0.519977
$362$	9.05452	0.475895
$363$	1.32881	0.0697446
$364$	12.5572	0.658178
$365$	0	0
$366$	4.49311	0.234858
$367$	−28.6948	−1.49786	−0.748929	−	0.662651i	$-0.769431\pi$
$367$	−28.6948	−1.49786	−0.748929	+	0.662651i	$0.769431\pi$
$368$	−1.96262	−0.102308
$369$	13.4497	0.700161
$370$	0	0
$371$	2.74949	0.142746
$372$	−2.57755	−0.133640
$373$	−11.0673	−0.573042	−0.286521	−	0.958074i	$-0.592499\pi$
$373$	−11.0673	−0.573042	−0.286521	+	0.958074i	$0.592499\pi$
$374$	−16.5669	−0.856656
$375$	0	0
$376$	−41.9904	−2.16549
$377$	5.54133	0.285393
$378$	6.51233	0.334958
$379$	13.0039	0.667968	0.333984	−	0.942579i	$-0.391607\pi$
$379$	13.0039	0.667968	0.333984	+	0.942579i	$0.391607\pi$
$380$	0	0
$381$	3.24082	0.166032
$382$	−8.55192	−0.437554
$383$	−34.8805	−1.78231	−0.891155	−	0.453698i	$-0.850104\pi$
$383$	−34.8805	−1.78231	−0.891155	+	0.453698i	$0.850104\pi$
$384$	−7.90971	−0.403641
$385$	0	0
$386$	−57.7225	−2.93800
$387$	−21.8393	−1.11016
$388$	51.4357	2.61125
$389$	−20.3114	−1.02983	−0.514915	−	0.857242i	$-0.672176\pi$
$389$	−20.3114	−1.02983	−0.514915	+	0.857242i	$0.672176\pi$
$390$	0	0
$391$	−7.59050	−0.383868
$392$	17.8982	0.903993
$393$	−2.20212	−0.111082
$394$	8.39254	0.422810
$395$	0	0
$396$	−26.6345	−1.33843
$397$	−13.8380	−0.694511	−0.347256	−	0.937770i	$-0.612886\pi$
$397$	−13.8380	−0.694511	−0.347256	+	0.937770i	$0.612886\pi$
$398$	−30.3771	−1.52267
$399$	1.45299	0.0727403
$400$	0	0
$401$	26.2220	1.30946	0.654732	−	0.755861i	$-0.272781\pi$
$401$	26.2220	1.30946	0.654732	+	0.755861i	$0.272781\pi$
$402$	13.2119	0.658949
$403$	−5.88850	−0.293327
$404$	−36.7160	−1.82669
$405$	0	0
$406$	−4.87310	−0.241848
$407$	−26.3382	−1.30554
$408$	3.33529	0.165122
$409$	15.9036	0.786385	0.393192	−	0.919456i	$-0.371371\pi$
$409$	15.9036	0.786385	0.393192	+	0.919456i	$0.371371\pi$
$410$	0	0
$411$	1.74092	0.0858731
$412$	−17.4753	−0.860944
$413$	1.65936	0.0816516
$414$	−19.4239	−0.954630
$415$	0	0
$416$	−15.2148	−0.745967
$417$	1.65683	0.0811353
$418$	−19.4656	−0.952095
$419$	10.7679	0.526049	0.263024	−	0.964789i	$-0.415280\pi$
$419$	10.7679	0.526049	0.263024	+	0.964789i	$0.415280\pi$
$420$	0	0
$421$	10.8916	0.530826	0.265413	−	0.964135i	$-0.414492\pi$
$421$	10.8916	0.530826	0.265413	+	0.964135i	$0.414492\pi$
$422$	−40.2018	−1.95699
$423$	−37.1981	−1.80863
$424$	7.41566	0.360136
$425$	0	0
$426$	0.625751	0.0303177
$427$	−5.67116	−0.274447
$428$	17.6367	0.852503
$429$	3.52633	0.170253
$430$	0	0
$431$	−1.99774	−0.0962277	−0.0481139	−	0.998842i	$-0.515321\pi$
$431$	−1.99774	−0.0962277	−0.0481139	+	0.998842i	$0.515321\pi$
$432$	1.57219	0.0756421
$433$	−37.3315	−1.79404	−0.897019	−	0.441992i	$-0.854272\pi$
$433$	−37.3315	−1.79404	−0.897019	+	0.441992i	$0.854272\pi$
$434$	5.17840	0.248571
$435$	0	0
$436$	9.78423	0.468580
$437$	−8.91859	−0.426634
$438$	−13.9598	−0.667025
$439$	26.2028	1.25059	0.625296	−	0.780388i	$-0.284978\pi$
$439$	26.2028	1.25059	0.625296	+	0.780388i	$0.284978\pi$
$440$	0	0
$441$	15.8555	0.755023
$442$	18.6621	0.887667
$443$	−9.85912	−0.468421	−0.234210	−	0.972186i	$-0.575250\pi$
$443$	−9.85912	−0.468421	−0.234210	+	0.972186i	$0.575250\pi$
$444$	12.9870	0.616334
$445$	0	0
$446$	62.9606	2.98127
$447$	8.99177	0.425296
$448$	14.9575	0.706677
$449$	−10.6272	−0.501529	−0.250765	−	0.968048i	$-0.580682\pi$
$449$	−10.6272	−0.501529	−0.250765	+	0.968048i	$0.580682\pi$
$450$	0	0
$451$	−13.1803	−0.620634
$452$	−23.0716	−1.08519
$453$	−9.13470	−0.429186
$454$	13.5428	0.635596
$455$	0	0
$456$	3.91886	0.183517
$457$	23.0860	1.07992	0.539958	−	0.841692i	$-0.318440\pi$
$457$	23.0860	1.07992	0.539958	+	0.841692i	$0.318440\pi$
$458$	−9.72684	−0.454505
$459$	6.08052	0.283814
$460$	0	0
$461$	−15.4621	−0.720144	−0.360072	−	0.932924i	$-0.617248\pi$
$461$	−15.4621	−0.720144	−0.360072	+	0.932924i	$0.617248\pi$
$462$	−3.10109	−0.144276
$463$	18.4881	0.859216	0.429608	−	0.903015i	$-0.358652\pi$
$463$	18.4881	0.859216	0.429608	+	0.903015i	$0.358652\pi$
$464$	−1.17645	−0.0546154
$465$	0	0
$466$	7.16188	0.331768
$467$	−8.77991	−0.406286	−0.203143	−	0.979149i	$-0.565116\pi$
$467$	−8.77991	−0.406286	−0.203143	+	0.979149i	$0.565116\pi$
$468$	30.0029	1.38688
$469$	−16.6759	−0.770022
$470$	0	0
$471$	−6.50092	−0.299547
$472$	4.47546	0.206000
$473$	21.4019	0.984061
$474$	−0.872403	−0.0400708
$475$	0	0
$476$	−10.3107	−0.472591
$477$	6.56932	0.300789
$478$	25.7205	1.17643
$479$	−31.7787	−1.45200	−0.726002	−	0.687692i	$-0.758624\pi$
$479$	−31.7787	−1.45200	−0.726002	+	0.687692i	$0.758624\pi$
$480$	0	0
$481$	29.6691	1.35280
$482$	2.31949	0.105650
$483$	−1.42083	−0.0646500
$484$	−11.0794	−0.503610
$485$	0	0
$486$	23.5589	1.06865
$487$	3.79157	0.171813	0.0859063	−	0.996303i	$-0.472621\pi$
$487$	3.79157	0.171813	0.0859063	+	0.996303i	$0.472621\pi$
$488$	−15.2957	−0.692404
$489$	0.563180	0.0254679
$490$	0	0
$491$	−11.8805	−0.536159	−0.268079	−	0.963397i	$-0.586389\pi$
$491$	−11.8805	−0.536159	−0.268079	+	0.963397i	$0.586389\pi$
$492$	6.49899	0.292997
$493$	−4.54998	−0.204921
$494$	21.9274	0.986561
$495$	0	0
$496$	1.25016	0.0561337
$497$	−0.789817	−0.0354281
$498$	6.29754	0.282199
$499$	12.0966	0.541517	0.270758	−	0.962647i	$-0.412726\pi$
$499$	12.0966	0.541517	0.270758	+	0.962647i	$0.412726\pi$
$500$	0	0
$501$	3.16201	0.141268
$502$	31.1878	1.39198
$503$	−28.8004	−1.28415	−0.642073	−	0.766643i	$-0.721926\pi$
$503$	−28.8004	−1.28415	−0.642073	+	0.766643i	$0.721926\pi$
$504$	−10.7727	−0.479853
$505$	0	0
$506$	19.0348	0.846200
$507$	1.29772	0.0576336
$508$	−27.0214	−1.19888
$509$	10.9405	0.484928	0.242464	−	0.970160i	$-0.422044\pi$
$509$	10.9405	0.484928	0.242464	+	0.970160i	$0.422044\pi$
$510$	0	0
$511$	17.6199	0.779459
$512$	7.48480	0.330784
$513$	7.14441	0.315433
$514$	4.25882	0.187849
$515$	0	0
$516$	−10.5530	−0.464568
$517$	36.4531	1.60320
$518$	−26.0913	−1.14639
$519$	10.4195	0.457365
$520$	0	0
$521$	6.58159	0.288345	0.144172	−	0.989553i	$-0.453948\pi$
$521$	6.58159	0.288345	0.144172	+	0.989553i	$0.453948\pi$
$522$	−11.6433	−0.509611
$523$	−8.21637	−0.359277	−0.179638	−	0.983733i	$-0.557493\pi$
$523$	−8.21637	−0.359277	−0.179638	+	0.983733i	$0.557493\pi$
$524$	18.3609	0.802101
$525$	0	0
$526$	−47.9514	−2.09078
$527$	4.83504	0.210618
$528$	−0.748657	−0.0325811
$529$	−14.2788	−0.620817
$530$	0	0
$531$	3.96468	0.172053
$532$	−12.1148	−0.525242
$533$	14.8472	0.643102
$534$	2.48319	0.107458
$535$	0	0
$536$	−44.9767	−1.94270
$537$	3.41228	0.147251
$538$	−24.1372	−1.04063
$539$	−15.5379	−0.669265
$540$	0	0
$541$	−2.11121	−0.0907679	−0.0453840	−	0.998970i	$-0.514451\pi$
$541$	−2.11121	−0.0907679	−0.0453840	+	0.998970i	$0.514451\pi$
$542$	−27.6999	−1.18981
$543$	1.58249	0.0679111
$544$	12.4929	0.535627
$545$	0	0
$546$	3.49328	0.149498
$547$	6.10652	0.261096	0.130548	−	0.991442i	$-0.458326\pi$
$547$	6.10652	0.261096	0.130548	+	0.991442i	$0.458326\pi$
$548$	−14.5155	−0.620071
$549$	−13.5500	−0.578302
$550$	0	0
$551$	−5.34608	−0.227750
$552$	−3.83213	−0.163106
$553$	1.10114	0.0468252
$554$	32.4522	1.37876
$555$	0	0
$556$	−13.8144	−0.585860
$557$	−35.7486	−1.51472	−0.757359	−	0.652998i	$-0.773511\pi$
$557$	−35.7486	−1.51472	−0.757359	+	0.652998i	$0.773511\pi$
$558$	12.3727	0.523779
$559$	−24.1086	−1.01968
$560$	0	0
$561$	−2.89546	−0.122247
$562$	−17.7204	−0.747490
$563$	16.8644	0.710749	0.355374	−	0.934724i	$-0.384353\pi$
$563$	16.8644	0.710749	0.355374	+	0.934724i	$0.384353\pi$
$564$	−17.9745	−0.756861
$565$	0	0
$566$	−42.9004	−1.80324
$567$	−8.95810	−0.376205
$568$	−2.13022	−0.0893820
$569$	−41.5591	−1.74225	−0.871124	−	0.491064i	$-0.836608\pi$
$569$	−41.5591	−1.74225	−0.871124	+	0.491064i	$0.836608\pi$
$570$	0	0
$571$	−1.75255	−0.0733417	−0.0366709	−	0.999327i	$-0.511675\pi$
$571$	−1.75255	−0.0733417	−0.0366709	+	0.999327i	$0.511675\pi$
$572$	−29.4019	−1.22936
$573$	−1.49465	−0.0624398
$574$	−13.0567	−0.544978
$575$	0	0
$576$	35.7379	1.48908
$577$	8.56040	0.356374	0.178187	−	0.983997i	$-0.442977\pi$
$577$	8.56040	0.356374	0.178187	+	0.983997i	$0.442977\pi$
$578$	24.1079	1.00276
$579$	−10.0884	−0.419258
$580$	0	0
$581$	−7.94869	−0.329767
$582$	14.3088	0.593120
$583$	−6.43775	−0.266624
$584$	47.5228	1.96651
$585$	0	0
$586$	−37.2211	−1.53759
$587$	−27.1714	−1.12148	−0.560742	−	0.827991i	$-0.689484\pi$
$587$	−27.1714	−1.12148	−0.560742	+	0.827991i	$0.689484\pi$
$588$	7.66151	0.315955
$589$	5.68101	0.234082
$590$	0	0
$591$	1.46679	0.0603358
$592$	−6.29891	−0.258884
$593$	−14.7676	−0.606434	−0.303217	−	0.952922i	$-0.598061\pi$
$593$	−14.7676	−0.606434	−0.303217	+	0.952922i	$0.598061\pi$
$594$	−15.2482	−0.625641
$595$	0	0
$596$	−74.9719	−3.07097
$597$	−5.30912	−0.217288
$598$	−21.4421	−0.876833
$599$	−39.2943	−1.60552	−0.802760	−	0.596302i	$-0.796636\pi$
$599$	−39.2943	−1.60552	−0.802760	+	0.596302i	$0.796636\pi$
$600$	0	0
$601$	−30.2159	−1.23253	−0.616267	−	0.787537i	$-0.711356\pi$
$601$	−30.2159	−1.23253	−0.616267	+	0.787537i	$0.711356\pi$
$602$	21.2013	0.864102
$603$	−39.8436	−1.62256
$604$	76.1637	3.09906
$605$	0	0
$606$	−10.2140	−0.414914
$607$	−2.88760	−0.117204	−0.0586020	−	0.998281i	$-0.518664\pi$
$607$	−2.88760	−0.117204	−0.0586020	+	0.998281i	$0.518664\pi$
$608$	14.6787	0.595300
$609$	−0.851689	−0.0345122
$610$	0	0
$611$	−41.0632	−1.66124
$612$	−24.6353	−0.995824
$613$	−42.6240	−1.72157	−0.860784	−	0.508970i	$-0.830026\pi$
$613$	−42.6240	−1.72157	−0.860784	+	0.508970i	$0.830026\pi$
$614$	65.9719	2.66241
$615$	0	0
$616$	10.5569	0.425350
$617$	−42.4532	−1.70910	−0.854550	−	0.519368i	$-0.826167\pi$
$617$	−42.4532	−1.70910	−0.854550	+	0.519368i	$0.826167\pi$
$618$	−4.86142	−0.195555
$619$	1.23242	0.0495353	0.0247676	−	0.999693i	$-0.492115\pi$
$619$	1.23242	0.0495353	0.0247676	+	0.999693i	$0.492115\pi$
$620$	0	0
$621$	−6.98629	−0.280350
$622$	−27.3850	−1.09804
$623$	−3.13426	−0.125571
$624$	0.843338	0.0337606
$625$	0	0
$626$	−65.0183	−2.59865
$627$	−3.40207	−0.135866
$628$	54.2037	2.16296
$629$	−24.3613	−0.971348
$630$	0	0
$631$	2.33330	0.0928871	0.0464435	−	0.998921i	$-0.485211\pi$
$631$	2.33330	0.0928871	0.0464435	+	0.998921i	$0.485211\pi$
$632$	2.96989	0.118136
$633$	−7.02620	−0.279267
$634$	−36.8358	−1.46294
$635$	0	0
$636$	3.17436	0.125871
$637$	17.5030	0.693492
$638$	11.4100	0.451728
$639$	−1.88710	−0.0746526
$640$	0	0
$641$	30.6356	1.21003	0.605016	−	0.796213i	$-0.293167\pi$
$641$	30.6356	1.21003	0.605016	+	0.796213i	$0.293167\pi$
$642$	4.90633	0.193638
$643$	−10.7810	−0.425161	−0.212581	−	0.977144i	$-0.568187\pi$
$643$	−10.7810	−0.425161	−0.212581	+	0.977144i	$0.568187\pi$
$644$	11.8466	0.466823
$645$	0	0
$646$	−18.0046	−0.708380
$647$	−9.50868	−0.373825	−0.186912	−	0.982377i	$-0.559848\pi$
$647$	−9.50868	−0.373825	−0.186912	+	0.982377i	$0.559848\pi$
$648$	−24.1609	−0.949131
$649$	−3.88527	−0.152510
$650$	0	0
$651$	0.905048	0.0354716
$652$	−4.69570	−0.183898
$653$	−24.6438	−0.964386	−0.482193	−	0.876065i	$-0.660160\pi$
$653$	−24.6438	−0.964386	−0.482193	+	0.876065i	$0.660160\pi$
$654$	2.72186	0.106433
$655$	0	0
$656$	−3.15213	−0.123070
$657$	42.0991	1.64244
$658$	36.1114	1.40777
$659$	−43.4441	−1.69234	−0.846172	−	0.532910i	$-0.821098\pi$
$659$	−43.4441	−1.69234	−0.846172	+	0.532910i	$0.821098\pi$
$660$	0	0
$661$	7.51370	0.292249	0.146124	−	0.989266i	$-0.453320\pi$
$661$	7.51370	0.292249	0.146124	+	0.989266i	$0.453320\pi$
$662$	21.5427	0.837281
$663$	3.26165	0.126672
$664$	−21.4385	−0.831974
$665$	0	0
$666$	−62.3398	−2.41562
$667$	5.22776	0.202419
$668$	−26.3643	−1.02007
$669$	11.0038	0.425433
$670$	0	0
$671$	13.2786	0.512616
$672$	2.33848	0.0902087
$673$	−30.6706	−1.18227	−0.591133	−	0.806574i	$-0.701319\pi$
$673$	−30.6706	−1.18227	−0.591133	+	0.806574i	$0.701319\pi$
$674$	12.5764	0.484423
$675$	0	0
$676$	−10.8201	−0.416159
$677$	1.58413	0.0608829	0.0304415	−	0.999537i	$-0.490309\pi$
$677$	1.58413	0.0608829	0.0304415	+	0.999537i	$0.490309\pi$
$678$	−6.41824	−0.246491
$679$	−18.0605	−0.693097
$680$	0	0
$681$	2.36693	0.0907008
$682$	−12.1249	−0.464286
$683$	−3.32110	−0.127078	−0.0635392	−	0.997979i	$-0.520239\pi$
$683$	−3.32110	−0.127078	−0.0635392	+	0.997979i	$0.520239\pi$
$684$	−28.9457	−1.10677
$685$	0	0
$686$	−34.6620	−1.32340
$687$	−1.69999	−0.0648588
$688$	5.11837	0.195136
$689$	7.25192	0.276276
$690$	0	0
$691$	4.24642	0.161541	0.0807707	−	0.996733i	$-0.474262\pi$
$691$	4.24642	0.161541	0.0807707	+	0.996733i	$0.474262\pi$
$692$	−86.8761	−3.30253
$693$	9.35207	0.355256
$694$	60.6536	2.30238
$695$	0	0
$696$	−2.29709	−0.0870711
$697$	−12.1910	−0.461766
$698$	30.4847	1.15386
$699$	1.25171	0.0473439
$700$	0	0
$701$	−29.8099	−1.12590	−0.562952	−	0.826489i	$-0.690335\pi$
$701$	−29.8099	−1.12590	−0.562952	+	0.826489i	$0.690335\pi$
$702$	17.1766	0.648290
$703$	−28.6237	−1.07956
$704$	−35.0221	−1.31994
$705$	0	0
$706$	30.1784	1.13578
$707$	12.8920	0.484852
$708$	1.91577	0.0719991
$709$	−10.8815	−0.408662	−0.204331	−	0.978902i	$-0.565502\pi$
$709$	−10.8815	−0.408662	−0.204331	+	0.978902i	$0.565502\pi$
$710$	0	0
$711$	2.63094	0.0986680
$712$	−8.45343	−0.316806
$713$	−5.55528	−0.208047
$714$	−2.86833	−0.107344
$715$	0	0
$716$	−28.4511	−1.06327
$717$	4.49526	0.167879
$718$	−38.3624	−1.43167
$719$	−3.47427	−0.129568	−0.0647842	−	0.997899i	$-0.520636\pi$
$719$	−3.47427	−0.129568	−0.0647842	+	0.997899i	$0.520636\pi$
$720$	0	0
$721$	6.13603	0.228518
$722$	22.9155	0.852828
$723$	0.405385	0.0150764
$724$	−13.1945	−0.490371
$725$	0	0
$726$	−3.08217	−0.114390
$727$	−9.75102	−0.361645	−0.180823	−	0.983516i	$-0.557876\pi$
$727$	−9.75102	−0.361645	−0.180823	+	0.983516i	$0.557876\pi$
$728$	−11.8920	−0.440748
$729$	−18.5264	−0.686165
$730$	0	0
$731$	19.7955	0.732164
$732$	−6.54750	−0.242003
$733$	35.7204	1.31936	0.659682	−	0.751545i	$-0.270691\pi$
$733$	35.7204	1.31936	0.659682	+	0.751545i	$0.270691\pi$
$734$	66.5574	2.45668
$735$	0	0
$736$	−14.3538	−0.529089
$737$	39.0455	1.43826
$738$	−31.1963	−1.14835
$739$	17.8243	0.655677	0.327839	−	0.944734i	$-0.393680\pi$
$739$	17.8243	0.655677	0.327839	+	0.944734i	$0.393680\pi$
$740$	0	0
$741$	3.83233	0.140784
$742$	−6.37741	−0.234122
$743$	−21.3416	−0.782947	−0.391473	−	0.920189i	$-0.628034\pi$
$743$	−21.3416	−0.782947	−0.391473	+	0.920189i	$0.628034\pi$
$744$	2.44101	0.0894917
$745$	0	0
$746$	25.6705	0.939863
$747$	−18.9917	−0.694871
$748$	24.1419	0.882715
$749$	−6.19273	−0.226277
$750$	0	0
$751$	−17.7669	−0.648322	−0.324161	−	0.946002i	$-0.605082\pi$
$751$	−17.7669	−0.648322	−0.324161	+	0.946002i	$0.605082\pi$
$752$	8.71793	0.317910
$753$	5.45080	0.198638
$754$	−12.8531	−0.468081
$755$	0	0
$756$	−9.48998	−0.345147
$757$	30.7633	1.11811	0.559056	−	0.829130i	$-0.311164\pi$
$757$	30.7633	1.11811	0.559056	+	0.829130i	$0.311164\pi$
$758$	−30.1625	−1.09555
$759$	3.32678	0.120754
$760$	0	0
$761$	−29.7689	−1.07912	−0.539561	−	0.841946i	$-0.681410\pi$
$761$	−29.7689	−1.07912	−0.539561	+	0.841946i	$0.681410\pi$
$762$	−7.51706	−0.272314
$763$	−3.43551	−0.124374
$764$	12.4621	0.450864
$765$	0	0
$766$	80.9050	2.92322
$767$	4.37664	0.158031
$768$	8.12836	0.293307
$769$	30.8952	1.11411	0.557056	−	0.830475i	$-0.311931\pi$
$769$	30.8952	1.11411	0.557056	+	0.830475i	$0.311931\pi$
$770$	0	0
$771$	0.744329	0.0268064
$772$	84.1152	3.02737
$773$	−29.0199	−1.04377	−0.521887	−	0.853015i	$-0.674772\pi$
$773$	−29.0199	−1.04377	−0.521887	+	0.853015i	$0.674772\pi$
$774$	50.6561	1.82080
$775$	0	0
$776$	−48.7110	−1.74862
$777$	−4.56007	−0.163592
$778$	47.1121	1.68905
$779$	−14.3240	−0.513211
$780$	0	0
$781$	1.84930	0.0661733
$782$	17.6061	0.629593
$783$	−4.18779	−0.149660
$784$	−3.71597	−0.132713
$785$	0	0
$786$	5.10780	0.182189
$787$	−31.1296	−1.10965	−0.554826	−	0.831967i	$-0.687215\pi$
$787$	−31.1296	−1.10965	−0.554826	+	0.831967i	$0.687215\pi$
$788$	−12.2299	−0.435671
$789$	−8.38063	−0.298358
$790$	0	0
$791$	8.10105	0.288040
$792$	25.2235	0.896279
$793$	−14.9580	−0.531173
$794$	32.0972	1.13909
$795$	0	0
$796$	44.2666	1.56899
$797$	−12.5369	−0.444079	−0.222039	−	0.975038i	$-0.571271\pi$
$797$	−12.5369	−0.444079	−0.222039	+	0.975038i	$0.571271\pi$
$798$	−3.37019	−0.119303
$799$	33.7170	1.19282
$800$	0	0
$801$	−7.48865	−0.264599
$802$	−60.8217	−2.14769
$803$	−41.2559	−1.45589
$804$	−19.2528	−0.678993
$805$	0	0
$806$	13.6583	0.481093
$807$	−4.21854	−0.148500
$808$	34.7710	1.22324
$809$	−32.2296	−1.13313	−0.566566	−	0.824016i	$-0.691728\pi$
$809$	−32.2296	−1.13313	−0.566566	+	0.824016i	$0.691728\pi$
$810$	0	0
$811$	2.37365	0.0833500	0.0416750	−	0.999131i	$-0.486731\pi$
$811$	2.37365	0.0833500	0.0416750	+	0.999131i	$0.486731\pi$
$812$	7.10124	0.249205
$813$	−4.84121	−0.169789
$814$	61.0912	2.14125
$815$	0	0
$816$	−0.692464	−0.0242411
$817$	23.2591	0.813733
$818$	−36.8884	−1.28977
$819$	−10.5348	−0.368116
$820$	0	0
$821$	−30.0708	−1.04948	−0.524739	−	0.851263i	$-0.675837\pi$
$821$	−30.0708	−1.04948	−0.524739	+	0.851263i	$0.675837\pi$
$822$	−4.03804	−0.140843
$823$	−25.0100	−0.871794	−0.435897	−	0.899997i	$-0.643569\pi$
$823$	−25.0100	−0.871794	−0.435897	+	0.899997i	$0.643569\pi$
$824$	16.5495	0.576530
$825$	0	0
$826$	−3.84886	−0.133919
$827$	40.7256	1.41617	0.708084	−	0.706128i	$-0.249560\pi$
$827$	40.7256	1.41617	0.708084	+	0.706128i	$0.249560\pi$
$828$	28.3051	0.983669
$829$	−36.0803	−1.25312	−0.626560	−	0.779373i	$-0.715538\pi$
$829$	−36.0803	−1.25312	−0.626560	+	0.779373i	$0.715538\pi$
$830$	0	0
$831$	5.67178	0.196752
$832$	39.4513	1.36773
$833$	−14.3717	−0.497948
$834$	−3.84300	−0.133072
$835$	0	0
$836$	28.3659	0.981057
$837$	4.45016	0.153820
$838$	−24.9762	−0.862787
$839$	20.2871	0.700390	0.350195	−	0.936677i	$-0.386115\pi$
$839$	20.2871	0.700390	0.350195	+	0.936677i	$0.386115\pi$
$840$	0	0
$841$	−25.8663	−0.891942
$842$	−25.2631	−0.870623
$843$	−3.09706	−0.106668
$844$	58.5833	2.01652
$845$	0	0
$846$	86.2807	2.96639
$847$	3.89029	0.133672
$848$	−1.53962	−0.0528707
$849$	−7.49784	−0.257325
$850$	0	0
$851$	27.9902	0.959493
$852$	−0.911865	−0.0312400
$853$	−47.0287	−1.61023	−0.805116	−	0.593117i	$-0.797897\pi$
$853$	−47.0287	−1.61023	−0.805116	+	0.593117i	$0.797897\pi$
$854$	13.1542	0.450127
$855$	0	0
$856$	−16.7024	−0.570878
$857$	−20.9248	−0.714779	−0.357389	−	0.933955i	$-0.616333\pi$
$857$	−20.9248	−0.714779	−0.357389	+	0.933955i	$0.616333\pi$
$858$	−8.17928	−0.279236
$859$	−1.85736	−0.0633724	−0.0316862	−	0.999498i	$-0.510088\pi$
$859$	−1.85736	−0.0633724	−0.0316862	+	0.999498i	$0.510088\pi$
$860$	0	0
$861$	−2.28197	−0.0777694
$862$	4.63374	0.157826
$863$	56.2937	1.91626	0.958129	−	0.286335i	$-0.0924372\pi$
$863$	56.2937	1.91626	0.958129	+	0.286335i	$0.0924372\pi$
$864$	11.4984	0.391184
$865$	0	0
$866$	86.5901	2.94245
$867$	4.21342	0.143095
$868$	−7.54614	−0.256133
$869$	−2.57824	−0.0874610
$870$	0	0
$871$	−43.9836	−1.49033
$872$	−9.26593	−0.313784
$873$	−43.1517	−1.46046
$874$	20.6866	0.699734
$875$	0	0
$876$	20.3427	0.687315
$877$	22.0063	0.743100	0.371550	−	0.928413i	$-0.378826\pi$
$877$	22.0063	0.743100	0.371550	+	0.928413i	$0.378826\pi$
$878$	−60.7772	−2.05113
$879$	−6.50526	−0.219417
$880$	0	0
$881$	−39.7181	−1.33814	−0.669068	−	0.743201i	$-0.733307\pi$
$881$	−39.7181	−1.33814	−0.669068	+	0.743201i	$0.733307\pi$
$882$	−36.7766	−1.23833
$883$	42.4186	1.42750	0.713750	−	0.700400i	$-0.246995\pi$
$883$	42.4186	1.42750	0.713750	+	0.700400i	$0.246995\pi$
$884$	−27.1951	−0.914669
$885$	0	0
$886$	22.8681	0.768270
$887$	1.87428	0.0629322	0.0314661	−	0.999505i	$-0.489982\pi$
$887$	1.87428	0.0629322	0.0314661	+	0.999505i	$0.489982\pi$
$888$	−12.2990	−0.412728
$889$	9.48796	0.318216
$890$	0	0
$891$	20.9748	0.702682
$892$	−91.7483	−3.07196
$893$	39.6163	1.32571
$894$	−20.8563	−0.697540
$895$	0	0
$896$	−23.1568	−0.773614
$897$	−3.74751	−0.125126
$898$	24.6497	0.822572
$899$	−3.33000	−0.111062
$900$	0	0
$901$	−5.95454	−0.198375
$902$	30.5715	1.01792
$903$	3.70543	0.123309
$904$	21.8494	0.726700
$905$	0	0
$906$	21.1879	0.703920
$907$	−34.4174	−1.14281	−0.571406	−	0.820668i	$-0.693602\pi$
$907$	−34.4174	−1.14281	−0.571406	+	0.820668i	$0.693602\pi$
$908$	−19.7351	−0.654931
$909$	30.8027	1.02166
$910$	0	0
$911$	−9.44483	−0.312921	−0.156461	−	0.987684i	$-0.550008\pi$
$911$	−9.44483	−0.312921	−0.156461	+	0.987684i	$0.550008\pi$
$912$	−0.813623	−0.0269418
$913$	18.6113	0.615946
$914$	−53.5477	−1.77120
$915$	0	0
$916$	14.1743	0.468331
$917$	−6.44702	−0.212899
$918$	−14.1037	−0.465491
$919$	38.3104	1.26374	0.631872	−	0.775073i	$-0.282287\pi$
$919$	38.3104	1.26374	0.631872	+	0.775073i	$0.282287\pi$
$920$	0	0
$921$	11.5301	0.379931
$922$	35.8643	1.18113
$923$	−2.08318	−0.0685688
$924$	4.51901	0.148664
$925$	0	0
$926$	−42.8831	−1.40922
$927$	14.6608	0.481523
$928$	−8.60412	−0.282444
$929$	36.2130	1.18811	0.594055	−	0.804425i	$-0.297526\pi$
$929$	36.2130	1.18811	0.594055	+	0.804425i	$0.297526\pi$
$930$	0	0
$931$	−16.8862	−0.553424
$932$	−10.4365	−0.341860
$933$	−4.78618	−0.156692
$934$	20.3649	0.666361
$935$	0	0
$936$	−28.4135	−0.928725
$937$	4.62244	0.151009	0.0755043	−	0.997145i	$-0.475943\pi$
$937$	4.62244	0.151009	0.0755043	+	0.997145i	$0.475943\pi$
$938$	38.6796	1.26293
$939$	−11.3635	−0.370833
$940$	0	0
$941$	−12.4396	−0.405520	−0.202760	−	0.979228i	$-0.564991\pi$
$941$	−12.4396	−0.405520	−0.202760	+	0.979228i	$0.564991\pi$
$942$	15.0788	0.491295
$943$	14.0070	0.456130
$944$	−0.929183	−0.0302423
$945$	0	0
$946$	−49.6415	−1.61399
$947$	59.1336	1.92158	0.960791	−	0.277272i	$-0.0894303\pi$
$947$	59.1336	1.92158	0.960791	+	0.277272i	$0.0894303\pi$
$948$	1.27129	0.0412897
$949$	46.4734	1.50859
$950$	0	0
$951$	−6.43793	−0.208764
$952$	9.76453	0.316470
$953$	−21.6141	−0.700150	−0.350075	−	0.936722i	$-0.613844\pi$
$953$	−21.6141	−0.700150	−0.350075	+	0.936722i	$0.613844\pi$
$954$	−15.2375	−0.493332
$955$	0	0
$956$	−37.4807	−1.21221
$957$	1.99417	0.0644625
$958$	73.7103	2.38147
$959$	5.09678	0.164583
$960$	0	0
$961$	−27.4614	−0.885851
$962$	−68.8173	−2.21876
$963$	−14.7962	−0.476802
$964$	−3.38004	−0.108864
$965$	0	0
$966$	3.29560	0.106034
$967$	−27.7489	−0.892344	−0.446172	−	0.894947i	$-0.647213\pi$
$967$	−27.7489	−0.892344	−0.446172	+	0.894947i	$0.647213\pi$
$968$	10.4925	0.337242
$969$	−3.14672	−0.101087
$970$	0	0
$971$	−3.57855	−0.114841	−0.0574206	−	0.998350i	$-0.518288\pi$
$971$	−3.57855	−0.114841	−0.0574206	+	0.998350i	$0.518288\pi$
$972$	−34.3308	−1.10116
$973$	4.85060	0.155503
$974$	−8.79452	−0.281795
$975$	0	0
$976$	3.17565	0.101650
$977$	45.2401	1.44736	0.723680	−	0.690136i	$-0.242449\pi$
$977$	45.2401	1.44736	0.723680	+	0.690136i	$0.242449\pi$
$978$	−1.30629	−0.0417706
$979$	7.33866	0.234545
$980$	0	0
$981$	−8.20843	−0.262075
$982$	27.5567	0.879369
$983$	−28.2238	−0.900200	−0.450100	−	0.892978i	$-0.648612\pi$
$983$	−28.2238	−0.900200	−0.450100	+	0.892978i	$0.648612\pi$
$984$	−6.15472	−0.196205
$985$	0	0
$986$	10.5536	0.336096
$987$	6.31132	0.200891
$988$	−31.9533	−1.01657
$989$	−22.7443	−0.723227
$990$	0	0
$991$	−30.1101	−0.956478	−0.478239	−	0.878230i	$-0.658725\pi$
$991$	−30.1101	−0.956478	−0.478239	+	0.878230i	$0.658725\pi$
$992$	9.14318	0.290296
$993$	3.76509	0.119482
$994$	1.83197	0.0581066
$995$	0	0
$996$	−9.17698	−0.290784
$997$	−39.6372	−1.25532	−0.627662	−	0.778486i	$-0.715988\pi$
$997$	−39.6372	−1.25532	−0.627662	+	0.778486i	$0.715988\pi$
$998$	−28.0579	−0.888157
$999$	−22.4221	−0.709404

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.j.1.5		25
5.4	even	2		1205.2.a.e.1.21	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1205.2.a.e.1.21	✓	25	5.4	even	2
6025.2.a.j.1.5		25	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)

\(f(q)\)	\(=\)	\(q-2.31949 q^{2} -0.405385 q^{3} +3.38004 q^{4} +0.940288 q^{6} -1.18682 q^{7} -3.20098 q^{8} -2.83566 q^{9} +O(q^{10})\)	\(q-2.31949 q^{2} -0.405385 q^{3} +3.38004 q^{4} +0.940288 q^{6} -1.18682 q^{7} -3.20098 q^{8} -2.83566 q^{9} +2.77887 q^{11} -1.37022 q^{12} -3.13031 q^{13} +2.75282 q^{14} +0.664580 q^{16} +2.57029 q^{17} +6.57729 q^{18} +3.02001 q^{19} +0.481120 q^{21} -6.44556 q^{22} -2.95317 q^{23} +1.29763 q^{24} +7.26071 q^{26} +2.36569 q^{27} -4.01150 q^{28} -1.77022 q^{29} +1.88113 q^{31} +4.86048 q^{32} -1.12651 q^{33} -5.96176 q^{34} -9.58465 q^{36} -9.47803 q^{37} -7.00488 q^{38} +1.26898 q^{39} -4.74304 q^{41} -1.11595 q^{42} +7.70167 q^{43} +9.39267 q^{44} +6.84985 q^{46} +13.1180 q^{47} -0.269411 q^{48} -5.59145 q^{49} -1.04196 q^{51} -10.5805 q^{52} -2.31668 q^{53} -5.48720 q^{54} +3.79900 q^{56} -1.22427 q^{57} +4.10601 q^{58} -1.39815 q^{59} +4.77844 q^{61} -4.36325 q^{62} +3.36543 q^{63} -12.6030 q^{64} +2.61293 q^{66} +14.0509 q^{67} +8.68768 q^{68} +1.19717 q^{69} +0.665489 q^{71} +9.07691 q^{72} -14.8463 q^{73} +21.9842 q^{74} +10.2077 q^{76} -3.29802 q^{77} -2.94339 q^{78} -0.927804 q^{79} +7.54797 q^{81} +11.0014 q^{82} +6.69746 q^{83} +1.62621 q^{84} -17.8640 q^{86} +0.717621 q^{87} -8.89511 q^{88} +2.64088 q^{89} +3.71512 q^{91} -9.98182 q^{92} -0.762581 q^{93} -30.4270 q^{94} -1.97037 q^{96} +15.2175 q^{97} +12.9693 q^{98} -7.87993 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 6 q^{2} - 15 q^{3} + 32 q^{4} - q^{6} - 19 q^{7} - 15 q^{8} + 32 q^{9}+O(q^{10}) \) 25 * q - 6 * q^2 - 15 * q^3 + 32 * q^4 - q^6 - 19 * q^7 - 15 * q^8 + 32 * q^9	\( 25 q - 6 q^{2} - 15 q^{3} + 32 q^{4} - q^{6} - 19 q^{7} - 15 q^{8} + 32 q^{9} + 2 q^{11} - 20 q^{12} - 14 q^{13} - 5 q^{14} + 38 q^{16} - 7 q^{17} - 9 q^{18} + 30 q^{19} + q^{21} - q^{22} - 43 q^{23} - 6 q^{24} - 22 q^{26} - 42 q^{27} - 32 q^{28} - 4 q^{29} + 14 q^{31} - 26 q^{32} - 4 q^{33} + 7 q^{34} + 15 q^{36} - 16 q^{37} - 14 q^{38} - 21 q^{39} - q^{41} + 25 q^{42} - 35 q^{43} - 52 q^{44} - 27 q^{46} - 50 q^{47} - 26 q^{48} + 46 q^{49} - 7 q^{51} - 3 q^{52} - 4 q^{53} - 31 q^{54} - 51 q^{56} - 2 q^{58} + 6 q^{59} + 19 q^{61} - 28 q^{63} + 49 q^{64} - 27 q^{66} - 65 q^{67} + 25 q^{68} + 2 q^{69} - 34 q^{71} + 10 q^{72} - 8 q^{73} - 42 q^{74} + 71 q^{76} - q^{77} + 59 q^{78} - 12 q^{79} + 29 q^{81} - 11 q^{82} - 41 q^{83} - 10 q^{84} - 13 q^{86} - 40 q^{87} + 52 q^{88} - 24 q^{89} + 46 q^{91} - 85 q^{92} + 30 q^{93} + 14 q^{94} - 30 q^{96} - 9 q^{97} + 64 q^{98} + 2 q^{99}+O(q^{100}) \) 25 * q - 6 * q^2 - 15 * q^3 + 32 * q^4 - q^6 - 19 * q^7 - 15 * q^8 + 32 * q^9 + 2 * q^11 - 20 * q^12 - 14 * q^13 - 5 * q^14 + 38 * q^16 - 7 * q^17 - 9 * q^18 + 30 * q^19 + q^21 - q^22 - 43 * q^23 - 6 * q^24 - 22 * q^26 - 42 * q^27 - 32 * q^28 - 4 * q^29 + 14 * q^31 - 26 * q^32 - 4 * q^33 + 7 * q^34 + 15 * q^36 - 16 * q^37 - 14 * q^38 - 21 * q^39 - q^41 + 25 * q^42 - 35 * q^43 - 52 * q^44 - 27 * q^46 - 50 * q^47 - 26 * q^48 + 46 * q^49 - 7 * q^51 - 3 * q^52 - 4 * q^53 - 31 * q^54 - 51 * q^56 - 2 * q^58 + 6 * q^59 + 19 * q^61 - 28 * q^63 + 49 * q^64 - 27 * q^66 - 65 * q^67 + 25 * q^68 + 2 * q^69 - 34 * q^71 + 10 * q^72 - 8 * q^73 - 42 * q^74 + 71 * q^76 - q^77 + 59 * q^78 - 12 * q^79 + 29 * q^81 - 11 * q^82 - 41 * q^83 - 10 * q^84 - 13 * q^86 - 40 * q^87 + 52 * q^88 - 24 * q^89 + 46 * q^91 - 85 * q^92 + 30 * q^93 + 14 * q^94 - 30 * q^96 - 9 * q^97 + 64 * q^98 + 2 * q^99

Introduction

L-functions

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Database

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