LMFDB - Embedded newform 4025.2.a.v.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4025,2,Mod(1,4025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4025, 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("4025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4025 = 5^{2} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4025.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:	$32.1397868136$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 10x^{6} + 9x^{5} + 28x^{4} - 22x^{3} - 16x^{2} + 7x + 3 $ x^8 - x^7 - 10x^6 + 9x^5 + 28x^4 - 22x^3 - 16x^2 + 7x + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$2.39863$ of defining polynomial
Character	$\chi$	$=$	4025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.39863 q^{2} +2.12900 q^{3} +3.75344 q^{4} +5.10670 q^{6} -1.00000 q^{7} +4.20585 q^{8} +1.53265 q^{9} +O(q^{10})$	\(q+2.39863 q^{2} +2.12900 q^{3} +3.75344 q^{4} +5.10670 q^{6} -1.00000 q^{7} +4.20585 q^{8} +1.53265 q^{9} +5.51035 q^{11} +7.99108 q^{12} +3.23035 q^{13} -2.39863 q^{14} +2.58142 q^{16} -2.52456 q^{17} +3.67627 q^{18} +3.05486 q^{19} -2.12900 q^{21} +13.2173 q^{22} +1.00000 q^{23} +8.95427 q^{24} +7.74842 q^{26} -3.12398 q^{27} -3.75344 q^{28} -7.64429 q^{29} +3.95570 q^{31} -2.21983 q^{32} +11.7315 q^{33} -6.05549 q^{34} +5.75272 q^{36} +1.87459 q^{37} +7.32748 q^{38} +6.87742 q^{39} -7.43037 q^{41} -5.10670 q^{42} -1.70157 q^{43} +20.6827 q^{44} +2.39863 q^{46} +5.19873 q^{47} +5.49585 q^{48} +1.00000 q^{49} -5.37480 q^{51} +12.1249 q^{52} +5.57648 q^{53} -7.49329 q^{54} -4.20585 q^{56} +6.50380 q^{57} -18.3358 q^{58} +1.05654 q^{59} -5.87905 q^{61} +9.48827 q^{62} -1.53265 q^{63} -10.4874 q^{64} +28.1397 q^{66} +5.43680 q^{67} -9.47578 q^{68} +2.12900 q^{69} -4.81544 q^{71} +6.44611 q^{72} +9.52328 q^{73} +4.49646 q^{74} +11.4662 q^{76} -5.51035 q^{77} +16.4964 q^{78} -5.88424 q^{79} -11.2489 q^{81} -17.8227 q^{82} -10.1058 q^{83} -7.99108 q^{84} -4.08145 q^{86} -16.2747 q^{87} +23.1757 q^{88} +17.8642 q^{89} -3.23035 q^{91} +3.75344 q^{92} +8.42170 q^{93} +12.4698 q^{94} -4.72603 q^{96} -3.99122 q^{97} +2.39863 q^{98} +8.44545 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + q^{2} + 4 q^{3} + 5 q^{4} - q^{6} - 8 q^{7}+O(q^{10}) $ 8 * q + q^2 + 4 * q^3 + 5 * q^4 - q^6 - 8 * q^7	$ 8 q + q^{2} + 4 q^{3} + 5 q^{4} - q^{6} - 8 q^{7} + 3 q^{11} + 9 q^{12} + 5 q^{13} - q^{14} - q^{16} + 5 q^{17} + 2 q^{18} - 2 q^{19} - 4 q^{21} + 21 q^{22} + 8 q^{23} - 6 q^{24} + 18 q^{26} + 7 q^{27} - 5 q^{28} - 9 q^{29} - 3 q^{31} + 6 q^{32} + 4 q^{33} - 10 q^{34} + 16 q^{36} + 6 q^{37} + 4 q^{38} - 2 q^{39} - 7 q^{41} + q^{42} + 8 q^{43} + 4 q^{44} + q^{46} + 22 q^{47} + 9 q^{48} + 8 q^{49} - 12 q^{51} + 11 q^{52} + 21 q^{53} - 15 q^{54} + 8 q^{57} + 16 q^{58} + 14 q^{59} + 8 q^{61} + 12 q^{62} - 40 q^{64} + 55 q^{66} + 21 q^{67} + 3 q^{68} + 4 q^{69} + 11 q^{71} - q^{72} + 26 q^{73} - 41 q^{74} + 21 q^{76} - 3 q^{77} + 17 q^{78} - 16 q^{79} - 20 q^{81} - q^{82} + 20 q^{83} - 9 q^{84} + 14 q^{86} - 29 q^{87} + 32 q^{88} + 15 q^{89} - 5 q^{91} + 5 q^{92} + 19 q^{93} + 21 q^{94} + 52 q^{96} + q^{97} + q^{98} + 15 q^{99}+O(q^{100}) $ 8 * q + q^2 + 4 * q^3 + 5 * q^4 - q^6 - 8 * q^7 + 3 * q^11 + 9 * q^12 + 5 * q^13 - q^14 - q^16 + 5 * q^17 + 2 * q^18 - 2 * q^19 - 4 * q^21 + 21 * q^22 + 8 * q^23 - 6 * q^24 + 18 * q^26 + 7 * q^27 - 5 * q^28 - 9 * q^29 - 3 * q^31 + 6 * q^32 + 4 * q^33 - 10 * q^34 + 16 * q^36 + 6 * q^37 + 4 * q^38 - 2 * q^39 - 7 * q^41 + q^42 + 8 * q^43 + 4 * q^44 + q^46 + 22 * q^47 + 9 * q^48 + 8 * q^49 - 12 * q^51 + 11 * q^52 + 21 * q^53 - 15 * q^54 + 8 * q^57 + 16 * q^58 + 14 * q^59 + 8 * q^61 + 12 * q^62 - 40 * q^64 + 55 * q^66 + 21 * q^67 + 3 * q^68 + 4 * q^69 + 11 * q^71 - q^72 + 26 * q^73 - 41 * q^74 + 21 * q^76 - 3 * q^77 + 17 * q^78 - 16 * q^79 - 20 * q^81 - q^82 + 20 * q^83 - 9 * q^84 + 14 * q^86 - 29 * q^87 + 32 * q^88 + 15 * q^89 - 5 * q^91 + 5 * q^92 + 19 * q^93 + 21 * q^94 + 52 * q^96 + q^97 + q^98 + 15 * 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 $

$2$	2.39863	1.69609	0.848045	−	0.529925i	$-0.177780\pi$
$2$	2.39863	1.69609	0.848045	+	0.529925i	$0.177780\pi$
$3$	2.12900	1.22918	0.614590	−	0.788847i	$-0.289321\pi$
$3$	2.12900	1.22918	0.614590	+	0.788847i	$0.289321\pi$
$4$	3.75344	1.87672
$5$	0	0
$5$	0	0
$6$	5.10670	2.08480
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	4.20585	1.48699
$9$	1.53265	0.510885
$10$	0	0
$11$	5.51035	1.66143	0.830716	−	0.556697i	$-0.187931\pi$
$11$	5.51035	1.66143	0.830716	+	0.556697i	$0.187931\pi$
$12$	7.99108	2.30683
$13$	3.23035	0.895938	0.447969	−	0.894049i	$-0.352148\pi$
$13$	3.23035	0.895938	0.447969	+	0.894049i	$0.352148\pi$
$14$	−2.39863	−0.641061
$15$	0	0
$16$	2.58142	0.645354
$17$	−2.52456	−0.612296	−0.306148	−	0.951984i	$-0.599040\pi$
$17$	−2.52456	−0.612296	−0.306148	+	0.951984i	$0.599040\pi$
$18$	3.67627	0.866506
$19$	3.05486	0.700832	0.350416	−	0.936594i	$-0.386040\pi$
$19$	3.05486	0.700832	0.350416	+	0.936594i	$0.386040\pi$
$20$	0	0
$21$	−2.12900	−0.464587
$22$	13.2173	2.81794
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	8.95427	1.82778
$25$	0	0
$26$	7.74842	1.51959
$27$	−3.12398	−0.601211
$28$	−3.75344	−0.709333
$29$	−7.64429	−1.41951	−0.709754	−	0.704450i	$-0.751194\pi$
$29$	−7.64429	−1.41951	−0.709754	+	0.704450i	$0.751194\pi$
$30$	0	0
$31$	3.95570	0.710465	0.355232	−	0.934778i	$-0.384402\pi$
$31$	3.95570	0.710465	0.355232	+	0.934778i	$0.384402\pi$
$32$	−2.21983	−0.392415
$33$	11.7315	2.04220
$34$	−6.05549	−1.03851
$35$	0	0
$36$	5.75272	0.958787
$37$	1.87459	0.308181	0.154091	−	0.988057i	$-0.450755\pi$
$37$	1.87459	0.308181	0.154091	+	0.988057i	$0.450755\pi$
$38$	7.32748	1.18867
$39$	6.87742	1.10127
$40$	0	0
$41$	−7.43037	−1.16043	−0.580215	−	0.814464i	$-0.697031\pi$
$41$	−7.43037	−1.16043	−0.580215	+	0.814464i	$0.697031\pi$
$42$	−5.10670	−0.787980
$43$	−1.70157	−0.259487	−0.129744	−	0.991548i	$-0.541415\pi$
$43$	−1.70157	−0.259487	−0.129744	+	0.991548i	$0.541415\pi$
$44$	20.6827	3.11804
$45$	0	0
$46$	2.39863	0.353659
$47$	5.19873	0.758312	0.379156	−	0.925333i	$-0.376214\pi$
$47$	5.19873	0.758312	0.379156	+	0.925333i	$0.376214\pi$
$48$	5.49585	0.793257
$49$	1.00000	0.142857
$50$	0	0
$51$	−5.37480	−0.752622
$52$	12.1249	1.68142
$53$	5.57648	0.765989	0.382994	−	0.923751i	$-0.374893\pi$
$53$	5.57648	0.765989	0.382994	+	0.923751i	$0.374893\pi$
$54$	−7.49329	−1.01971
$55$	0	0
$56$	−4.20585	−0.562031
$57$	6.50380	0.861449
$58$	−18.3358	−2.40761
$59$	1.05654	0.137550	0.0687750	−	0.997632i	$-0.478091\pi$
$59$	1.05654	0.137550	0.0687750	+	0.997632i	$0.478091\pi$
$60$	0	0
$61$	−5.87905	−0.752735	−0.376368	−	0.926470i	$-0.622827\pi$
$61$	−5.87905	−0.752735	−0.376368	+	0.926470i	$0.622827\pi$
$62$	9.48827	1.20501
$63$	−1.53265	−0.193096
$64$	−10.4874	−1.31092
$65$	0	0
$66$	28.1397	3.46375
$67$	5.43680	0.664211	0.332105	−	0.943242i	$-0.392241\pi$
$67$	5.43680	0.664211	0.332105	+	0.943242i	$0.392241\pi$
$68$	−9.47578	−1.14911
$69$	2.12900	0.256302
$70$	0	0
$71$	−4.81544	−0.571487	−0.285744	−	0.958306i	$-0.592241\pi$
$71$	−4.81544	−0.571487	−0.285744	+	0.958306i	$0.592241\pi$
$72$	6.44611	0.759682
$73$	9.52328	1.11462	0.557308	−	0.830306i	$-0.311834\pi$
$73$	9.52328	1.11462	0.557308	+	0.830306i	$0.311834\pi$
$74$	4.49646	0.522703
$75$	0	0
$76$	11.4662	1.31527
$77$	−5.51035	−0.627962
$78$	16.4964	1.86785
$79$	−5.88424	−0.662028	−0.331014	−	0.943626i	$-0.607391\pi$
$79$	−5.88424	−0.662028	−0.331014	+	0.943626i	$0.607391\pi$
$80$	0	0
$81$	−11.2489	−1.24988
$82$	−17.8227	−1.96819
$83$	−10.1058	−1.10925	−0.554627	−	0.832099i	$-0.687139\pi$
$83$	−10.1058	−1.10925	−0.554627	+	0.832099i	$0.687139\pi$
$84$	−7.99108	−0.871898
$85$	0	0
$86$	−4.08145	−0.440114
$87$	−16.2747	−1.74483
$88$	23.1757	2.47054
$89$	17.8642	1.89360	0.946800	−	0.321822i	$-0.104295\pi$
$89$	17.8642	1.89360	0.946800	+	0.321822i	$0.104295\pi$
$90$	0	0
$91$	−3.23035	−0.338633
$92$	3.75344	0.391323
$93$	8.42170	0.873290
$94$	12.4698	1.28617
$95$	0	0
$96$	−4.72603	−0.482348
$97$	−3.99122	−0.405247	−0.202623	−	0.979257i	$-0.564947\pi$
$97$	−3.99122	−0.405247	−0.202623	+	0.979257i	$0.564947\pi$
$98$	2.39863	0.242298
$99$	8.44545	0.848800
$100$	0	0
$101$	−0.814694	−0.0810650	−0.0405325	−	0.999178i	$-0.512905\pi$
$101$	−0.814694	−0.0810650	−0.0405325	+	0.999178i	$0.512905\pi$
$102$	−12.8922	−1.27651
$103$	10.0727	0.992494	0.496247	−	0.868181i	$-0.334711\pi$
$103$	10.0727	0.992494	0.496247	+	0.868181i	$0.334711\pi$
$104$	13.5864	1.33225
$105$	0	0
$106$	13.3759	1.29919
$107$	−14.4811	−1.39994	−0.699971	−	0.714172i	$-0.746804\pi$
$107$	−14.4811	−1.39994	−0.699971	+	0.714172i	$0.746804\pi$
$108$	−11.7257	−1.12830
$109$	−19.7856	−1.89511	−0.947557	−	0.319588i	$-0.896455\pi$
$109$	−19.7856	−1.89511	−0.947557	+	0.319588i	$0.896455\pi$
$110$	0	0
$111$	3.99101	0.378810
$112$	−2.58142	−0.243921
$113$	12.6513	1.19013	0.595067	−	0.803676i	$-0.297126\pi$
$113$	12.6513	1.19013	0.595067	+	0.803676i	$0.297126\pi$
$114$	15.6002	1.46109
$115$	0	0
$116$	−28.6923	−2.66402
$117$	4.95101	0.457721
$118$	2.53426	0.233297
$119$	2.52456	0.231426
$120$	0	0
$121$	19.3639	1.76036
$122$	−14.1017	−1.27671
$123$	−15.8193	−1.42638
$124$	14.8475	1.33334
$125$	0	0
$126$	−3.67627	−0.327508
$127$	1.10675	0.0982085	0.0491042	−	0.998794i	$-0.484363\pi$
$127$	1.10675	0.0982085	0.0491042	+	0.998794i	$0.484363\pi$
$128$	−20.7157	−1.83103
$129$	−3.62265	−0.318957
$130$	0	0
$131$	−14.3514	−1.25388	−0.626942	−	0.779066i	$-0.715694\pi$
$131$	−14.3514	−1.25388	−0.626942	+	0.779066i	$0.715694\pi$
$132$	44.0336	3.83263
$133$	−3.05486	−0.264890
$134$	13.0409	1.12656
$135$	0	0
$136$	−10.6179	−0.910480
$137$	−12.4418	−1.06298	−0.531489	−	0.847065i	$-0.678367\pi$
$137$	−12.4418	−1.06298	−0.531489	+	0.847065i	$0.678367\pi$
$138$	5.10670	0.434711
$139$	1.03498	0.0877862	0.0438931	−	0.999036i	$-0.486024\pi$
$139$	1.03498	0.0877862	0.0438931	+	0.999036i	$0.486024\pi$
$140$	0	0
$141$	11.0681	0.932102
$142$	−11.5505	−0.969293
$143$	17.8003	1.48854
$144$	3.95642	0.329702
$145$	0	0
$146$	22.8428	1.89049
$147$	2.12900	0.175597
$148$	7.03617	0.578369
$149$	13.6330	1.11686	0.558431	−	0.829551i	$-0.311403\pi$
$149$	13.6330	1.11686	0.558431	+	0.829551i	$0.311403\pi$
$150$	0	0
$151$	15.6589	1.27431	0.637153	−	0.770737i	$-0.280112\pi$
$151$	15.6589	1.27431	0.637153	+	0.770737i	$0.280112\pi$
$152$	12.8483	1.04213
$153$	−3.86928	−0.312813
$154$	−13.2173	−1.06508
$155$	0	0
$156$	25.8140	2.06677
$157$	6.28739	0.501788	0.250894	−	0.968015i	$-0.419275\pi$
$157$	6.28739	0.501788	0.250894	+	0.968015i	$0.419275\pi$
$158$	−14.1141	−1.12286
$159$	11.8723	0.941538
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	−26.9821	−2.11991
$163$	−7.85687	−0.615397	−0.307699	−	0.951484i	$-0.599559\pi$
$163$	−7.85687	−0.615397	−0.307699	+	0.951484i	$0.599559\pi$
$164$	−27.8894	−2.17780
$165$	0	0
$166$	−24.2401	−1.88139
$167$	−20.6355	−1.59682	−0.798411	−	0.602113i	$-0.794326\pi$
$167$	−20.6355	−1.59682	−0.798411	+	0.602113i	$0.794326\pi$
$168$	−8.95427	−0.690837
$169$	−2.56485	−0.197296
$170$	0	0
$171$	4.68204	0.358044
$172$	−6.38675	−0.486985
$173$	−6.06862	−0.461389	−0.230694	−	0.973026i	$-0.574100\pi$
$173$	−6.06862	−0.461389	−0.230694	+	0.973026i	$0.574100\pi$
$174$	−39.0370	−2.95939
$175$	0	0
$176$	14.2245	1.07221
$177$	2.24938	0.169074
$178$	42.8496	3.21172
$179$	−20.2907	−1.51660	−0.758299	−	0.651907i	$-0.773969\pi$
$179$	−20.2907	−1.51660	−0.758299	+	0.651907i	$0.773969\pi$
$180$	0	0
$181$	6.10372	0.453686	0.226843	−	0.973931i	$-0.427160\pi$
$181$	6.10372	0.453686	0.226843	+	0.973931i	$0.427160\pi$
$182$	−7.74842	−0.574351
$183$	−12.5165	−0.925247
$184$	4.20585	0.310059
$185$	0	0
$186$	20.2006	1.48118
$187$	−13.9112	−1.01729
$188$	19.5131	1.42314
$189$	3.12398	0.227236
$190$	0	0
$191$	9.94685	0.719729	0.359865	−	0.933005i	$-0.382823\pi$
$191$	9.94685	0.719729	0.359865	+	0.933005i	$0.382823\pi$
$192$	−22.3277	−1.61136
$193$	15.4732	1.11379	0.556893	−	0.830584i	$-0.311993\pi$
$193$	15.4732	1.11379	0.556893	+	0.830584i	$0.311993\pi$
$194$	−9.57346	−0.687335
$195$	0	0
$196$	3.75344	0.268103
$197$	−9.54246	−0.679872	−0.339936	−	0.940449i	$-0.610405\pi$
$197$	−9.54246	−0.679872	−0.339936	+	0.940449i	$0.610405\pi$
$198$	20.2575	1.43964
$199$	5.80918	0.411802	0.205901	−	0.978573i	$-0.433988\pi$
$199$	5.80918	0.411802	0.205901	+	0.978573i	$0.433988\pi$
$200$	0	0
$201$	11.5750	0.816435
$202$	−1.95415	−0.137494
$203$	7.64429	0.536524
$204$	−20.1740	−1.41246
$205$	0	0
$206$	24.1607	1.68336
$207$	1.53265	0.106527
$208$	8.33888	0.578197
$209$	16.8333	1.16439
$210$	0	0
$211$	−24.9674	−1.71882	−0.859412	−	0.511283i	$-0.829170\pi$
$211$	−24.9674	−1.71882	−0.859412	+	0.511283i	$0.829170\pi$
$212$	20.9310	1.43755
$213$	−10.2521	−0.702461
$214$	−34.7349	−2.37443
$215$	0	0
$216$	−13.1390	−0.893997
$217$	−3.95570	−0.268530
$218$	−47.4583	−3.21428
$219$	20.2751	1.37006
$220$	0	0
$221$	−8.15521	−0.548579
$222$	9.57297	0.642496
$223$	22.0644	1.47754	0.738771	−	0.673957i	$-0.235407\pi$
$223$	22.0644	1.47754	0.738771	+	0.673957i	$0.235407\pi$
$224$	2.21983	0.148319
$225$	0	0
$226$	30.3458	2.01857
$227$	−27.4282	−1.82047	−0.910237	−	0.414087i	$-0.864101\pi$
$227$	−27.4282	−1.82047	−0.910237	+	0.414087i	$0.864101\pi$
$228$	24.4116	1.61670
$229$	18.7040	1.23599	0.617996	−	0.786181i	$-0.287945\pi$
$229$	18.7040	1.23599	0.617996	+	0.786181i	$0.287945\pi$
$230$	0	0
$231$	−11.7315	−0.771879
$232$	−32.1507	−2.11080
$233$	−21.7813	−1.42694	−0.713472	−	0.700684i	$-0.752878\pi$
$233$	−21.7813	−1.42694	−0.713472	+	0.700684i	$0.752878\pi$
$234$	11.8756	0.776335
$235$	0	0
$236$	3.96567	0.258143
$237$	−12.5276	−0.813752
$238$	6.05549	0.392519
$239$	−14.2026	−0.918688	−0.459344	−	0.888258i	$-0.651916\pi$
$239$	−14.2026	−0.918688	−0.459344	+	0.888258i	$0.651916\pi$
$240$	0	0
$241$	0.677271	0.0436269	0.0218134	−	0.999762i	$-0.493056\pi$
$241$	0.677271	0.0436269	0.0218134	+	0.999762i	$0.493056\pi$
$242$	46.4469	2.98572
$243$	−14.5771	−0.935119
$244$	−22.0666	−1.41267
$245$	0	0
$246$	−37.9446	−2.41926
$247$	9.86825	0.627902
$248$	16.6371	1.05646
$249$	−21.5152	−1.36347
$250$	0	0
$251$	24.2726	1.53207	0.766036	−	0.642798i	$-0.222227\pi$
$251$	24.2726	1.53207	0.766036	+	0.642798i	$0.222227\pi$
$252$	−5.75272	−0.362387
$253$	5.51035	0.346432
$254$	2.65469	0.166570
$255$	0	0
$256$	−28.7147	−1.79467
$257$	21.5650	1.34519	0.672594	−	0.740012i	$-0.265180\pi$
$257$	21.5650	1.34519	0.672594	+	0.740012i	$0.265180\pi$
$258$	−8.68941	−0.540979
$259$	−1.87459	−0.116482
$260$	0	0
$261$	−11.7160	−0.725205
$262$	−34.4236	−2.12670
$263$	12.3482	0.761420	0.380710	−	0.924694i	$-0.375680\pi$
$263$	12.3482	0.761420	0.380710	+	0.924694i	$0.375680\pi$
$264$	49.3411	3.03674
$265$	0	0
$266$	−7.32748	−0.449277
$267$	38.0329	2.32758
$268$	20.4067	1.24654
$269$	−22.2315	−1.35548	−0.677738	−	0.735303i	$-0.737040\pi$
$269$	−22.2315	−1.35548	−0.677738	+	0.735303i	$0.737040\pi$
$270$	0	0
$271$	−24.1843	−1.46909	−0.734547	−	0.678558i	$-0.762605\pi$
$271$	−24.1843	−1.46909	−0.734547	+	0.678558i	$0.762605\pi$
$272$	−6.51694	−0.395148
$273$	−6.87742	−0.416241
$274$	−29.8434	−1.80290
$275$	0	0
$276$	7.99108	0.481006
$277$	13.7035	0.823365	0.411682	−	0.911327i	$-0.364941\pi$
$277$	13.7035	0.823365	0.411682	+	0.911327i	$0.364941\pi$
$278$	2.48255	0.148893
$279$	6.06272	0.362966
$280$	0	0
$281$	9.10918	0.543408	0.271704	−	0.962381i	$-0.412413\pi$
$281$	9.10918	0.543408	0.271704	+	0.962381i	$0.412413\pi$
$282$	26.5483	1.58093
$283$	1.16628	0.0693279	0.0346639	−	0.999399i	$-0.488964\pi$
$283$	1.16628	0.0693279	0.0346639	+	0.999399i	$0.488964\pi$
$284$	−18.0744	−1.07252
$285$	0	0
$286$	42.6965	2.52470
$287$	7.43037	0.438601
$288$	−3.40223	−0.200479
$289$	−10.6266	−0.625094
$290$	0	0
$291$	−8.49731	−0.498121
$292$	35.7450	2.09182
$293$	10.9879	0.641918	0.320959	−	0.947093i	$-0.395995\pi$
$293$	10.9879	0.641918	0.320959	+	0.947093i	$0.395995\pi$
$294$	5.10670	0.297829
$295$	0	0
$296$	7.88426	0.458263
$297$	−17.2142	−0.998871
$298$	32.7007	1.89430
$299$	3.23035	0.186816
$300$	0	0
$301$	1.70157	0.0980770
$302$	37.5600	2.16134
$303$	−1.73449	−0.0996436
$304$	7.88586	0.452285
$305$	0	0
$306$	−9.28097	−0.530558
$307$	−8.24449	−0.470538	−0.235269	−	0.971930i	$-0.575597\pi$
$307$	−8.24449	−0.470538	−0.235269	+	0.971930i	$0.575597\pi$
$308$	−20.6827	−1.17851
$309$	21.4448	1.21995
$310$	0	0
$311$	1.68609	0.0956092	0.0478046	−	0.998857i	$-0.484778\pi$
$311$	1.68609	0.0956092	0.0478046	+	0.998857i	$0.484778\pi$
$312$	28.9254	1.63758
$313$	−7.55110	−0.426813	−0.213407	−	0.976963i	$-0.568456\pi$
$313$	−7.55110	−0.426813	−0.213407	+	0.976963i	$0.568456\pi$
$314$	15.0811	0.851078
$315$	0	0
$316$	−22.0861	−1.24244
$317$	−3.54214	−0.198946	−0.0994731	−	0.995040i	$-0.531716\pi$
$317$	−3.54214	−0.198946	−0.0994731	+	0.995040i	$0.531716\pi$
$318$	28.4774	1.59693
$319$	−42.1227	−2.35842
$320$	0	0
$321$	−30.8303	−1.72078
$322$	−2.39863	−0.133671
$323$	−7.71217	−0.429117
$324$	−42.2222	−2.34568
$325$	0	0
$326$	−18.8457	−1.04377
$327$	−42.1235	−2.32944
$328$	−31.2510	−1.72555
$329$	−5.19873	−0.286615
$330$	0	0
$331$	−7.20201	−0.395858	−0.197929	−	0.980216i	$-0.563422\pi$
$331$	−7.20201	−0.395858	−0.197929	+	0.980216i	$0.563422\pi$
$332$	−37.9314	−2.08176
$333$	2.87310	0.157445
$334$	−49.4970	−2.70835
$335$	0	0
$336$	−5.49585	−0.299823
$337$	8.06778	0.439480	0.219740	−	0.975558i	$-0.429479\pi$
$337$	8.06778	0.439480	0.219740	+	0.975558i	$0.429479\pi$
$338$	−6.15213	−0.334632
$339$	26.9346	1.46289
$340$	0	0
$341$	21.7973	1.18039
$342$	11.2305	0.607275
$343$	−1.00000	−0.0539949
$344$	−7.15656	−0.385856
$345$	0	0
$346$	−14.5564	−0.782557
$347$	−14.4666	−0.776606	−0.388303	−	0.921532i	$-0.626939\pi$
$347$	−14.4666	−0.776606	−0.388303	+	0.921532i	$0.626939\pi$
$348$	−61.0861	−3.27456
$349$	−21.0982	−1.12936	−0.564681	−	0.825309i	$-0.691001\pi$
$349$	−21.0982	−1.12936	−0.564681	+	0.825309i	$0.691001\pi$
$350$	0	0
$351$	−10.0916	−0.538648
$352$	−12.2320	−0.651970
$353$	−4.40360	−0.234380	−0.117190	−	0.993110i	$-0.537389\pi$
$353$	−4.40360	−0.234380	−0.117190	+	0.993110i	$0.537389\pi$
$354$	5.39544	0.286764
$355$	0	0
$356$	67.0521	3.55376
$357$	5.37480	0.284464
$358$	−48.6699	−2.57229
$359$	−6.30099	−0.332553	−0.166277	−	0.986079i	$-0.553174\pi$
$359$	−6.30099	−0.332553	−0.166277	+	0.986079i	$0.553174\pi$
$360$	0	0
$361$	−9.66785	−0.508834
$362$	14.6406	0.769492
$363$	41.2258	2.16380
$364$	−12.1249	−0.635518
$365$	0	0
$366$	−30.0225	−1.56930
$367$	15.6835	0.818671	0.409336	−	0.912384i	$-0.365760\pi$
$367$	15.6835	0.818671	0.409336	+	0.912384i	$0.365760\pi$
$368$	2.58142	0.134566
$369$	−11.3882	−0.592845
$370$	0	0
$371$	−5.57648	−0.289516
$372$	31.6103	1.63892
$373$	15.6547	0.810570	0.405285	−	0.914190i	$-0.367172\pi$
$373$	15.6547	0.810570	0.405285	+	0.914190i	$0.367172\pi$
$374$	−33.3679	−1.72541
$375$	0	0
$376$	21.8651	1.12760
$377$	−24.6937	−1.27179
$378$	7.49329	0.385413
$379$	−26.5392	−1.36323	−0.681615	−	0.731711i	$-0.738722\pi$
$379$	−26.5392	−1.36323	−0.681615	+	0.731711i	$0.738722\pi$
$380$	0	0
$381$	2.35628	0.120716
$382$	23.8588	1.22072
$383$	−6.58722	−0.336591	−0.168296	−	0.985737i	$-0.553826\pi$
$383$	−6.58722	−0.336591	−0.168296	+	0.985737i	$0.553826\pi$
$384$	−44.1039	−2.25067
$385$	0	0
$386$	37.1146	1.88908
$387$	−2.60792	−0.132568
$388$	−14.9808	−0.760534
$389$	−6.22390	−0.315564	−0.157782	−	0.987474i	$-0.550434\pi$
$389$	−6.22390	−0.315564	−0.157782	+	0.987474i	$0.550434\pi$
$390$	0	0
$391$	−2.52456	−0.127673
$392$	4.20585	0.212428
$393$	−30.5541	−1.54125
$394$	−22.8888	−1.15312
$395$	0	0
$396$	31.6995	1.59296
$397$	−29.5951	−1.48533	−0.742667	−	0.669660i	$-0.766440\pi$
$397$	−29.5951	−1.48533	−0.742667	+	0.669660i	$0.766440\pi$
$398$	13.9341	0.698453
$399$	−6.50380	−0.325597
$400$	0	0
$401$	3.64577	0.182061	0.0910304	−	0.995848i	$-0.470984\pi$
$401$	3.64577	0.182061	0.0910304	+	0.995848i	$0.470984\pi$
$402$	27.7641	1.38475
$403$	12.7783	0.636532
$404$	−3.05790	−0.152136
$405$	0	0
$406$	18.3358	0.909992
$407$	10.3297	0.512022
$408$	−22.6056	−1.11914
$409$	−20.7501	−1.02603	−0.513013	−	0.858381i	$-0.671471\pi$
$409$	−20.7501	−1.02603	−0.513013	+	0.858381i	$0.671471\pi$
$410$	0	0
$411$	−26.4887	−1.30659
$412$	37.8073	1.86263
$413$	−1.05654	−0.0519890
$414$	3.67627	0.180679
$415$	0	0
$416$	−7.17083	−0.351579
$417$	2.20349	0.107905
$418$	40.3769	1.97490
$419$	3.45542	0.168808	0.0844042	−	0.996432i	$-0.473101\pi$
$419$	3.45542	0.168808	0.0844042	+	0.996432i	$0.473101\pi$
$420$	0	0
$421$	−22.3709	−1.09029	−0.545146	−	0.838341i	$-0.683526\pi$
$421$	−22.3709	−1.09029	−0.545146	+	0.838341i	$0.683526\pi$
$422$	−59.8876	−2.91528
$423$	7.96785	0.387410
$424$	23.4539	1.13902
$425$	0	0
$426$	−24.5910	−1.19144
$427$	5.87905	0.284507
$428$	−54.3539	−2.62730
$429$	37.8970	1.82968
$430$	0	0
$431$	−31.0609	−1.49615	−0.748075	−	0.663614i	$-0.769022\pi$
$431$	−31.0609	−1.49615	−0.748075	+	0.663614i	$0.769022\pi$
$432$	−8.06431	−0.387994
$433$	9.38424	0.450978	0.225489	−	0.974246i	$-0.427602\pi$
$433$	9.38424	0.450978	0.225489	+	0.974246i	$0.427602\pi$
$434$	−9.48827	−0.455452
$435$	0	0
$436$	−74.2639	−3.55659
$437$	3.05486	0.146134
$438$	48.6325	2.32375
$439$	23.6581	1.12914	0.564570	−	0.825386i	$-0.309042\pi$
$439$	23.6581	1.12914	0.564570	+	0.825386i	$0.309042\pi$
$440$	0	0
$441$	1.53265	0.0729835
$442$	−19.5614	−0.930439
$443$	24.0535	1.14282	0.571409	−	0.820665i	$-0.306397\pi$
$443$	24.0535	1.14282	0.571409	+	0.820665i	$0.306397\pi$
$444$	14.9800	0.710920
$445$	0	0
$446$	52.9244	2.50604
$447$	29.0248	1.37283
$448$	10.4874	0.495483
$449$	9.77210	0.461174	0.230587	−	0.973052i	$-0.425935\pi$
$449$	9.77210	0.461174	0.230587	+	0.973052i	$0.425935\pi$
$450$	0	0
$451$	−40.9439	−1.92797
$452$	47.4858	2.23355
$453$	33.3379	1.56635
$454$	−65.7902	−3.08769
$455$	0	0
$456$	27.3540	1.28097
$457$	15.0327	0.703199	0.351600	−	0.936150i	$-0.385638\pi$
$457$	15.0327	0.703199	0.351600	+	0.936150i	$0.385638\pi$
$458$	44.8639	2.09635
$459$	7.88669	0.368119
$460$	0	0
$461$	12.4330	0.579062	0.289531	−	0.957169i	$-0.406501\pi$
$461$	12.4330	0.579062	0.289531	+	0.957169i	$0.406501\pi$
$462$	−28.1397	−1.30918
$463$	13.4825	0.626585	0.313292	−	0.949657i	$-0.398568\pi$
$463$	13.4825	0.626585	0.313292	+	0.949657i	$0.398568\pi$
$464$	−19.7331	−0.916086
$465$	0	0
$466$	−52.2454	−2.42022
$467$	37.7706	1.74781	0.873907	−	0.486092i	$-0.161578\pi$
$467$	37.7706	1.74781	0.873907	+	0.486092i	$0.161578\pi$
$468$	18.5833	0.859013
$469$	−5.43680	−0.251048
$470$	0	0
$471$	13.3859	0.616788
$472$	4.44366	0.204536
$473$	−9.37626	−0.431121
$474$	−30.0490	−1.38020
$475$	0	0
$476$	9.47578	0.434322
$477$	8.54681	0.391332
$478$	−34.0668	−1.55818
$479$	27.9239	1.27587	0.637937	−	0.770089i	$-0.279788\pi$
$479$	27.9239	1.27587	0.637937	+	0.770089i	$0.279788\pi$
$480$	0	0
$481$	6.05559	0.276111
$482$	1.62453	0.0739951
$483$	−2.12900	−0.0968730
$484$	72.6812	3.30369
$485$	0	0
$486$	−34.9650	−1.58604
$487$	37.9310	1.71882	0.859409	−	0.511289i	$-0.170832\pi$
$487$	37.9310	1.71882	0.859409	+	0.511289i	$0.170832\pi$
$488$	−24.7264	−1.11931
$489$	−16.7273	−0.756434
$490$	0	0
$491$	−41.7563	−1.88443	−0.942217	−	0.335004i	$-0.891262\pi$
$491$	−41.7563	−1.88443	−0.942217	+	0.335004i	$0.891262\pi$
$492$	−59.3767	−2.67691
$493$	19.2985	0.869159
$494$	23.6703	1.06498
$495$	0	0
$496$	10.2113	0.458502
$497$	4.81544	0.216002
$498$	−51.6072	−2.31257
$499$	−0.962564	−0.0430903	−0.0215451	−	0.999768i	$-0.506859\pi$
$499$	−0.962564	−0.0430903	−0.0215451	+	0.999768i	$0.506859\pi$
$500$	0	0
$501$	−43.9330	−1.96278
$502$	58.2210	2.59853
$503$	26.2435	1.17014	0.585070	−	0.810983i	$-0.301067\pi$
$503$	26.2435	1.17014	0.585070	+	0.810983i	$0.301067\pi$
$504$	−6.44611	−0.287133
$505$	0	0
$506$	13.2173	0.587580
$507$	−5.46057	−0.242512
$508$	4.15413	0.184310
$509$	35.2003	1.56023	0.780113	−	0.625639i	$-0.215162\pi$
$509$	35.2003	1.56023	0.780113	+	0.625639i	$0.215162\pi$
$510$	0	0
$511$	−9.52328	−0.421285
$512$	−27.4444	−1.21288
$513$	−9.54333	−0.421348
$514$	51.7265	2.28156
$515$	0	0
$516$	−13.5974	−0.598592
$517$	28.6468	1.25988
$518$	−4.49646	−0.197563
$519$	−12.9201	−0.567130
$520$	0	0
$521$	23.8932	1.04678	0.523390	−	0.852093i	$-0.324667\pi$
$521$	23.8932	1.04678	0.523390	+	0.852093i	$0.324667\pi$
$522$	−28.1025	−1.23001
$523$	21.8014	0.953309	0.476655	−	0.879091i	$-0.341849\pi$
$523$	21.8014	0.953309	0.476655	+	0.879091i	$0.341849\pi$
$524$	−53.8669	−2.35319
$525$	0	0
$526$	29.6187	1.29144
$527$	−9.98641	−0.435015
$528$	30.2840	1.31794
$529$	1.00000	0.0434783
$530$	0	0
$531$	1.61931	0.0702722
$532$	−11.4662	−0.497123
$533$	−24.0027	−1.03967
$534$	91.2270	3.94778
$535$	0	0
$536$	22.8664	0.987677
$537$	−43.1990	−1.86417
$538$	−53.3251	−2.29901
$539$	5.51035	0.237347
$540$	0	0
$541$	−2.93305	−0.126101	−0.0630507	−	0.998010i	$-0.520083\pi$
$541$	−2.93305	−0.126101	−0.0630507	+	0.998010i	$0.520083\pi$
$542$	−58.0093	−2.49171
$543$	12.9948	0.557662
$544$	5.60410	0.240274
$545$	0	0
$546$	−16.4964	−0.705981
$547$	−14.9801	−0.640501	−0.320251	−	0.947333i	$-0.603767\pi$
$547$	−14.9801	−0.640501	−0.320251	+	0.947333i	$0.603767\pi$
$548$	−46.6996	−1.99491
$549$	−9.01055	−0.384561
$550$	0	0
$551$	−23.3522	−0.994837
$552$	8.95427	0.381119
$553$	5.88424	0.250223
$554$	32.8697	1.39650
$555$	0	0
$556$	3.88475	0.164750
$557$	36.0817	1.52883	0.764415	−	0.644725i	$-0.223028\pi$
$557$	36.0817	1.52883	0.764415	+	0.644725i	$0.223028\pi$
$558$	14.5422	0.615622
$559$	−5.49667	−0.232485
$560$	0	0
$561$	−29.6170	−1.25043
$562$	21.8496	0.921669
$563$	−10.1061	−0.425921	−0.212961	−	0.977061i	$-0.568311\pi$
$563$	−10.1061	−0.425921	−0.212961	+	0.977061i	$0.568311\pi$
$564$	41.5434	1.74929
$565$	0	0
$566$	2.79747	0.117586
$567$	11.2489	0.472411
$568$	−20.2530	−0.849797
$569$	−25.4484	−1.06685	−0.533426	−	0.845847i	$-0.679096\pi$
$569$	−25.4484	−1.06685	−0.533426	+	0.845847i	$0.679096\pi$
$570$	0	0
$571$	22.6764	0.948978	0.474489	−	0.880262i	$-0.342633\pi$
$571$	22.6764	0.948978	0.474489	+	0.880262i	$0.342633\pi$
$572$	66.8125	2.79357
$573$	21.1769	0.884677
$574$	17.8227	0.743906
$575$	0	0
$576$	−16.0735	−0.669731
$577$	−5.59657	−0.232988	−0.116494	−	0.993191i	$-0.537166\pi$
$577$	−5.59657	−0.232988	−0.116494	+	0.993191i	$0.537166\pi$
$578$	−25.4893	−1.06021
$579$	32.9425	1.36904
$580$	0	0
$581$	10.1058	0.419259
$582$	−20.3819	−0.844858
$583$	30.7283	1.27264
$584$	40.0535	1.65743
$585$	0	0
$586$	26.3559	1.08875
$587$	32.9026	1.35804	0.679018	−	0.734122i	$-0.262406\pi$
$587$	32.9026	1.35804	0.679018	+	0.734122i	$0.262406\pi$
$588$	7.99108	0.329547
$589$	12.0841	0.497917
$590$	0	0
$591$	−20.3159	−0.835685
$592$	4.83911	0.198886
$593$	−16.0208	−0.657896	−0.328948	−	0.944348i	$-0.606694\pi$
$593$	−16.0208	−0.657896	−0.328948	+	0.944348i	$0.606694\pi$
$594$	−41.2906	−1.69417
$595$	0	0
$596$	51.1708	2.09604
$597$	12.3678	0.506179
$598$	7.74842	0.316856
$599$	47.3607	1.93511	0.967553	−	0.252667i	$-0.0813077\pi$
$599$	47.3607	1.93511	0.967553	+	0.252667i	$0.0813077\pi$
$600$	0	0
$601$	33.7156	1.37529	0.687644	−	0.726048i	$-0.258645\pi$
$601$	33.7156	1.37529	0.687644	+	0.726048i	$0.258645\pi$
$602$	4.08145	0.166347
$603$	8.33273	0.339335
$604$	58.7749	2.39151
$605$	0	0
$606$	−4.16039	−0.169004
$607$	−10.2526	−0.416142	−0.208071	−	0.978114i	$-0.566718\pi$
$607$	−10.2526	−0.416142	−0.208071	+	0.978114i	$0.566718\pi$
$608$	−6.78127	−0.275017
$609$	16.2747	0.659484
$610$	0	0
$611$	16.7937	0.679400
$612$	−14.5231	−0.587061
$613$	−45.2069	−1.82589	−0.912945	−	0.408082i	$-0.866198\pi$
$613$	−45.2069	−1.82589	−0.912945	+	0.408082i	$0.866198\pi$
$614$	−19.7755	−0.798074
$615$	0	0
$616$	−23.1757	−0.933776
$617$	45.1951	1.81948	0.909742	−	0.415173i	$-0.136279\pi$
$617$	45.1951	1.81948	0.909742	+	0.415173i	$0.136279\pi$
$618$	51.4383	2.06915
$619$	−21.4869	−0.863630	−0.431815	−	0.901962i	$-0.642127\pi$
$619$	−21.4869	−0.863630	−0.431815	+	0.901962i	$0.642127\pi$
$620$	0	0
$621$	−3.12398	−0.125361
$622$	4.04430	0.162162
$623$	−17.8642	−0.715714
$624$	17.7535	0.710709
$625$	0	0
$626$	−18.1123	−0.723914
$627$	35.8382	1.43124
$628$	23.5993	0.941716
$629$	−4.73252	−0.188698
$630$	0	0
$631$	−8.53273	−0.339683	−0.169841	−	0.985471i	$-0.554326\pi$
$631$	−8.53273	−0.339683	−0.169841	+	0.985471i	$0.554326\pi$
$632$	−24.7482	−0.984431
$633$	−53.1556	−2.11275
$634$	−8.49628	−0.337430
$635$	0	0
$636$	44.5621	1.76700
$637$	3.23035	0.127991
$638$	−101.037	−4.00008
$639$	−7.38040	−0.291964
$640$	0	0
$641$	29.2446	1.15509	0.577545	−	0.816359i	$-0.304011\pi$
$641$	29.2446	1.15509	0.577545	+	0.816359i	$0.304011\pi$
$642$	−73.9506	−2.91860
$643$	−14.0744	−0.555039	−0.277519	−	0.960720i	$-0.589512\pi$
$643$	−14.0744	−0.555039	−0.277519	+	0.960720i	$0.589512\pi$
$644$	−3.75344	−0.147906
$645$	0	0
$646$	−18.4987	−0.727820
$647$	36.4310	1.43225	0.716125	−	0.697972i	$-0.245914\pi$
$647$	36.4310	1.43225	0.716125	+	0.697972i	$0.245914\pi$
$648$	−47.3113	−1.85857
$649$	5.82191	0.228530
$650$	0	0
$651$	−8.42170	−0.330072
$652$	−29.4903	−1.15493
$653$	15.8108	0.618726	0.309363	−	0.950944i	$-0.399884\pi$
$653$	15.8108	0.618726	0.309363	+	0.950944i	$0.399884\pi$
$654$	−101.039	−3.95093
$655$	0	0
$656$	−19.1809	−0.748888
$657$	14.5959	0.569440
$658$	−12.4698	−0.486125
$659$	−10.0233	−0.390452	−0.195226	−	0.980758i	$-0.562544\pi$
$659$	−10.0233	−0.390452	−0.195226	+	0.980758i	$0.562544\pi$
$660$	0	0
$661$	−0.748782	−0.0291243	−0.0145621	−	0.999894i	$-0.504635\pi$
$661$	−0.748782	−0.0291243	−0.0145621	+	0.999894i	$0.504635\pi$
$662$	−17.2750	−0.671411
$663$	−17.3625	−0.674302
$664$	−42.5034	−1.64945
$665$	0	0
$666$	6.89151	0.267041
$667$	−7.64429	−0.295988
$668$	−77.4540	−2.99679
$669$	46.9752	1.81617
$670$	0	0
$671$	−32.3956	−1.25062
$672$	4.72603	0.182311
$673$	14.6529	0.564826	0.282413	−	0.959293i	$-0.408865\pi$
$673$	14.6529	0.564826	0.282413	+	0.959293i	$0.408865\pi$
$674$	19.3516	0.745397
$675$	0	0
$676$	−9.62699	−0.370269
$677$	34.8249	1.33843	0.669215	−	0.743069i	$-0.266630\pi$
$677$	34.8249	1.33843	0.669215	+	0.743069i	$0.266630\pi$
$678$	64.6063	2.48119
$679$	3.99122	0.153169
$680$	0	0
$681$	−58.3948	−2.23769
$682$	52.2837	2.00205
$683$	−31.4489	−1.20336	−0.601679	−	0.798738i	$-0.705501\pi$
$683$	−31.4489	−1.20336	−0.601679	+	0.798738i	$0.705501\pi$
$684$	17.5737	0.671949
$685$	0	0
$686$	−2.39863	−0.0915802
$687$	39.8208	1.51926
$688$	−4.39247	−0.167461
$689$	18.0140	0.686278
$690$	0	0
$691$	6.62630	0.252076	0.126038	−	0.992025i	$-0.459774\pi$
$691$	6.62630	0.252076	0.126038	+	0.992025i	$0.459774\pi$
$692$	−22.7782	−0.865897
$693$	−8.44545	−0.320816
$694$	−34.7000	−1.31719
$695$	0	0
$696$	−68.4490	−2.59455
$697$	18.7584	0.710526
$698$	−50.6069	−1.91550
$699$	−46.3726	−1.75397
$700$	0	0
$701$	−40.1035	−1.51469	−0.757343	−	0.653017i	$-0.773503\pi$
$701$	−40.1035	−1.51469	−0.757343	+	0.653017i	$0.773503\pi$
$702$	−24.2059	−0.913594
$703$	5.72661	0.215983
$704$	−57.7892	−2.17801
$705$	0	0
$706$	−10.5626	−0.397529
$707$	0.814694	0.0306397
$708$	8.44291	0.317304
$709$	−0.0938634	−0.00352511	−0.00176256	−	0.999998i	$-0.500561\pi$
$709$	−0.0938634	−0.00352511	−0.00176256	+	0.999998i	$0.500561\pi$
$710$	0	0
$711$	−9.01850	−0.338220
$712$	75.1341	2.81577
$713$	3.95570	0.148142
$714$	12.8922	0.482477
$715$	0	0
$716$	−76.1599	−2.84623
$717$	−30.2373	−1.12923
$718$	−15.1137	−0.564040
$719$	−43.2177	−1.61175	−0.805875	−	0.592086i	$-0.798305\pi$
$719$	−43.2177	−1.61175	−0.805875	+	0.592086i	$0.798305\pi$
$720$	0	0
$721$	−10.0727	−0.375128
$722$	−23.1896	−0.863028
$723$	1.44191	0.0536253
$724$	22.9099	0.851441
$725$	0	0
$726$	98.8856	3.66999
$727$	−10.5889	−0.392720	−0.196360	−	0.980532i	$-0.562912\pi$
$727$	−10.5889	−0.392720	−0.196360	+	0.980532i	$0.562912\pi$
$728$	−13.5864	−0.503544
$729$	2.71220	0.100452
$730$	0	0
$731$	4.29572	0.158883
$732$	−46.9800	−1.73643
$733$	−28.1568	−1.03999	−0.519997	−	0.854168i	$-0.674067\pi$
$733$	−28.1568	−1.03999	−0.519997	+	0.854168i	$0.674067\pi$
$734$	37.6189	1.38854
$735$	0	0
$736$	−2.21983	−0.0818241
$737$	29.9586	1.10354
$738$	−27.3161	−1.00552
$739$	30.6974	1.12922	0.564612	−	0.825357i	$-0.309026\pi$
$739$	30.6974	1.12922	0.564612	+	0.825357i	$0.309026\pi$
$740$	0	0
$741$	21.0095	0.771805
$742$	−13.3759	−0.491046
$743$	37.3623	1.37069	0.685346	−	0.728218i	$-0.259651\pi$
$743$	37.3623	1.37069	0.685346	+	0.728218i	$0.259651\pi$
$744$	35.4204	1.29858
$745$	0	0
$746$	37.5499	1.37480
$747$	−15.4887	−0.566701
$748$	−52.2148	−1.90916
$749$	14.4811	0.529128
$750$	0	0
$751$	5.48551	0.200169	0.100084	−	0.994979i	$-0.468089\pi$
$751$	5.48551	0.200169	0.100084	+	0.994979i	$0.468089\pi$
$752$	13.4201	0.489380
$753$	51.6764	1.88319
$754$	−59.2311	−2.15707
$755$	0	0
$756$	11.7257	0.426459
$757$	−44.8116	−1.62870	−0.814352	−	0.580371i	$-0.802908\pi$
$757$	−44.8116	−1.62870	−0.814352	+	0.580371i	$0.802908\pi$
$758$	−63.6579	−2.31216
$759$	11.7315	0.425828
$760$	0	0
$761$	31.9561	1.15841	0.579204	−	0.815182i	$-0.303363\pi$
$761$	31.9561	1.15841	0.579204	+	0.815182i	$0.303363\pi$
$762$	5.65185	0.204745
$763$	19.7856	0.716285
$764$	37.3349	1.35073
$765$	0	0
$766$	−15.8003	−0.570888
$767$	3.41300	0.123236
$768$	−61.1336	−2.20597
$769$	44.8336	1.61674	0.808371	−	0.588674i	$-0.200350\pi$
$769$	44.8336	1.61674	0.808371	+	0.588674i	$0.200350\pi$
$770$	0	0
$771$	45.9119	1.65348
$772$	58.0777	2.09026
$773$	−9.85872	−0.354594	−0.177297	−	0.984157i	$-0.556735\pi$
$773$	−9.85872	−0.354594	−0.177297	+	0.984157i	$0.556735\pi$
$774$	−6.25545	−0.224847
$775$	0	0
$776$	−16.7865	−0.602599
$777$	−3.99101	−0.143177
$778$	−14.9289	−0.535225
$779$	−22.6987	−0.813266
$780$	0	0
$781$	−26.5347	−0.949487
$782$	−6.05549	−0.216544
$783$	23.8806	0.853424
$784$	2.58142	0.0921935
$785$	0	0
$786$	−73.2880	−2.61410
$787$	15.3257	0.546301	0.273151	−	0.961971i	$-0.411934\pi$
$787$	15.3257	0.546301	0.273151	+	0.961971i	$0.411934\pi$
$788$	−35.8170	−1.27593
$789$	26.2893	0.935923
$790$	0	0
$791$	−12.6513	−0.449828
$792$	35.5203	1.26216
$793$	−18.9914	−0.674404
$794$	−70.9877	−2.51926
$795$	0	0
$796$	21.8044	0.772836
$797$	51.2008	1.81363	0.906813	−	0.421533i	$-0.138508\pi$
$797$	51.2008	1.81363	0.906813	+	0.421533i	$0.138508\pi$
$798$	−15.6002	−0.552242
$799$	−13.1245	−0.464311
$800$	0	0
$801$	27.3796	0.967411
$802$	8.74485	0.308792
$803$	52.4766	1.85186
$804$	43.4459	1.53222
$805$	0	0
$806$	30.6504	1.07962
$807$	−47.3309	−1.66613
$808$	−3.42648	−0.120543
$809$	24.3722	0.856882	0.428441	−	0.903570i	$-0.359063\pi$
$809$	24.3722	0.856882	0.428441	+	0.903570i	$0.359063\pi$
$810$	0	0
$811$	48.5682	1.70546	0.852730	−	0.522353i	$-0.174946\pi$
$811$	48.5682	1.70546	0.852730	+	0.522353i	$0.174946\pi$
$812$	28.6923	1.00690
$813$	−51.4885	−1.80578
$814$	24.7770	0.868435
$815$	0	0
$816$	−13.8746	−0.485708
$817$	−5.19806	−0.181857
$818$	−49.7719	−1.74023
$819$	−4.95101	−0.173002
$820$	0	0
$821$	24.4367	0.852845	0.426423	−	0.904524i	$-0.359774\pi$
$821$	24.4367	0.852845	0.426423	+	0.904524i	$0.359774\pi$
$822$	−63.5366	−2.21609
$823$	13.3332	0.464767	0.232383	−	0.972624i	$-0.425348\pi$
$823$	13.3332	0.464767	0.232383	+	0.972624i	$0.425348\pi$
$824$	42.3643	1.47583
$825$	0	0
$826$	−2.53426	−0.0881781
$827$	−37.0517	−1.28841	−0.644206	−	0.764852i	$-0.722812\pi$
$827$	−37.0517	−1.28841	−0.644206	+	0.764852i	$0.722812\pi$
$828$	5.75272	0.199921
$829$	46.5009	1.61504	0.807522	−	0.589838i	$-0.200808\pi$
$829$	46.5009	1.61504	0.807522	+	0.589838i	$0.200808\pi$
$830$	0	0
$831$	29.1748	1.01206
$832$	−33.8779	−1.17451
$833$	−2.52456	−0.0874708
$834$	5.28535	0.183017
$835$	0	0
$836$	63.1828	2.18522
$837$	−12.3575	−0.427139
$838$	8.28829	0.286314
$839$	10.1973	0.352050	0.176025	−	0.984386i	$-0.443676\pi$
$839$	10.1973	0.352050	0.176025	+	0.984386i	$0.443676\pi$
$840$	0	0
$841$	29.4351	1.01500
$842$	−53.6596	−1.84923
$843$	19.3935	0.667947
$844$	−93.7135	−3.22575
$845$	0	0
$846$	19.1119	0.657082
$847$	−19.3639	−0.665352
$848$	14.3952	0.494334
$849$	2.48300	0.0852164
$850$	0	0
$851$	1.87459	0.0642602
$852$	−38.4805	−1.31832
$853$	30.0361	1.02842	0.514209	−	0.857665i	$-0.328086\pi$
$853$	30.0361	1.02842	0.514209	+	0.857665i	$0.328086\pi$
$854$	14.1017	0.482550
$855$	0	0
$856$	−60.9054	−2.08170
$857$	48.1216	1.64380	0.821902	−	0.569629i	$-0.192913\pi$
$857$	48.1216	1.64380	0.821902	+	0.569629i	$0.192913\pi$
$858$	90.9009	3.10331
$859$	34.6956	1.18380	0.591900	−	0.806012i	$-0.298378\pi$
$859$	34.6956	1.18380	0.591900	+	0.806012i	$0.298378\pi$
$860$	0	0
$861$	15.8193	0.539120
$862$	−74.5036	−2.53760
$863$	−11.8351	−0.402870	−0.201435	−	0.979502i	$-0.564560\pi$
$863$	−11.8351	−0.402870	−0.201435	+	0.979502i	$0.564560\pi$
$864$	6.93472	0.235924
$865$	0	0
$866$	22.5093	0.764898
$867$	−22.6241	−0.768353
$868$	−14.8475	−0.503956
$869$	−32.4242	−1.09991
$870$	0	0
$871$	17.5628	0.595091
$872$	−83.2151	−2.81802
$873$	−6.11715	−0.207034
$874$	7.32748	0.247856
$875$	0	0
$876$	76.1013	2.57122
$877$	28.6758	0.968313	0.484156	−	0.874982i	$-0.339127\pi$
$877$	28.6758	0.968313	0.484156	+	0.874982i	$0.339127\pi$
$878$	56.7471	1.91512
$879$	23.3932	0.789034
$880$	0	0
$881$	−39.1021	−1.31738	−0.658692	−	0.752413i	$-0.728890\pi$
$881$	−39.1021	−1.31738	−0.658692	+	0.752413i	$0.728890\pi$
$882$	3.67627	0.123787
$883$	10.8295	0.364443	0.182222	−	0.983257i	$-0.441671\pi$
$883$	10.8295	0.364443	0.182222	+	0.983257i	$0.441671\pi$
$884$	−30.6101	−1.02953
$885$	0	0
$886$	57.6956	1.93832
$887$	−4.14636	−0.139221	−0.0696105	−	0.997574i	$-0.522176\pi$
$887$	−4.14636	−0.139221	−0.0696105	+	0.997574i	$0.522176\pi$
$888$	16.7856	0.563288
$889$	−1.10675	−0.0371193
$890$	0	0
$891$	−61.9855	−2.07659
$892$	82.8173	2.77293
$893$	15.8814	0.531450
$894$	69.6198	2.32844
$895$	0	0
$896$	20.7157	0.692064
$897$	6.87742	0.229630
$898$	23.4397	0.782192
$899$	−30.2385	−1.00851
$900$	0	0
$901$	−14.0782	−0.469012
$902$	−98.2094	−3.27002
$903$	3.62265	0.120554
$904$	53.2094	1.76972
$905$	0	0
$906$	79.9654	2.65667
$907$	6.01951	0.199874	0.0999372	−	0.994994i	$-0.468136\pi$
$907$	6.01951	0.199874	0.0999372	+	0.994994i	$0.468136\pi$
$908$	−102.950	−3.41652
$909$	−1.24864	−0.0414149
$910$	0	0
$911$	−48.7555	−1.61534	−0.807671	−	0.589633i	$-0.799272\pi$
$911$	−48.7555	−1.61534	−0.807671	+	0.589633i	$0.799272\pi$
$912$	16.7890	0.555940
$913$	−55.6864	−1.84295
$914$	36.0579	1.19269
$915$	0	0
$916$	70.2042	2.31961
$917$	14.3514	0.473924
$918$	18.9173	0.624363
$919$	−17.9232	−0.591231	−0.295615	−	0.955307i	$-0.595525\pi$
$919$	−17.9232	−0.591231	−0.295615	+	0.955307i	$0.595525\pi$
$920$	0	0
$921$	−17.5525	−0.578376
$922$	29.8222	0.982141
$923$	−15.5555	−0.512017
$924$	−44.0336	−1.44860
$925$	0	0
$926$	32.3395	1.06274
$927$	15.4380	0.507050
$928$	16.9690	0.557036
$929$	−26.7816	−0.878676	−0.439338	−	0.898322i	$-0.644787\pi$
$929$	−26.7816	−0.878676	−0.439338	+	0.898322i	$0.644787\pi$
$930$	0	0
$931$	3.05486	0.100119
$932$	−81.7549	−2.67797
$933$	3.58968	0.117521
$934$	90.5978	2.96445
$935$	0	0
$936$	20.8232	0.680627
$937$	21.7572	0.710775	0.355388	−	0.934719i	$-0.384349\pi$
$937$	21.7572	0.710775	0.355388	+	0.934719i	$0.384349\pi$
$938$	−13.0409	−0.425800
$939$	−16.0763	−0.524631
$940$	0	0
$941$	−29.1789	−0.951205	−0.475603	−	0.879660i	$-0.657770\pi$
$941$	−29.1789	−0.951205	−0.475603	+	0.879660i	$0.657770\pi$
$942$	32.1078	1.04613
$943$	−7.43037	−0.241966
$944$	2.72738	0.0887685
$945$	0	0
$946$	−22.4902	−0.731219
$947$	−22.8727	−0.743263	−0.371631	−	0.928380i	$-0.621201\pi$
$947$	−22.8727	−0.743263	−0.371631	+	0.928380i	$0.621201\pi$
$948$	−47.0214	−1.52718
$949$	30.7635	0.998626
$950$	0	0
$951$	−7.54122	−0.244541
$952$	10.6179	0.344129
$953$	7.86688	0.254833	0.127417	−	0.991849i	$-0.459331\pi$
$953$	7.86688	0.254833	0.127417	+	0.991849i	$0.459331\pi$
$954$	20.5007	0.663734
$955$	0	0
$956$	−53.3085	−1.72412
$957$	−89.6793	−2.89892
$958$	66.9791	2.16400
$959$	12.4418	0.401768
$960$	0	0
$961$	−15.3524	−0.495240
$962$	14.5251	0.468309
$963$	−22.1945	−0.715208
$964$	2.54210	0.0818754
$965$	0	0
$966$	−5.10670	−0.164305
$967$	29.4077	0.945688	0.472844	−	0.881146i	$-0.343227\pi$
$967$	29.4077	0.945688	0.472844	+	0.881146i	$0.343227\pi$
$968$	81.4418	2.61764
$969$	−16.4192	−0.527462
$970$	0	0
$971$	43.3889	1.39242	0.696208	−	0.717840i	$-0.254869\pi$
$971$	43.3889	1.39242	0.696208	+	0.717840i	$0.254869\pi$
$972$	−54.7141	−1.75495
$973$	−1.03498	−0.0331801
$974$	90.9825	2.91527
$975$	0	0
$976$	−15.1763	−0.485781
$977$	−47.4500	−1.51806	−0.759029	−	0.651057i	$-0.774326\pi$
$977$	−47.4500	−1.51806	−0.759029	+	0.651057i	$0.774326\pi$
$978$	−40.1226	−1.28298
$979$	98.4379	3.14609
$980$	0	0
$981$	−30.3244	−0.968184
$982$	−100.158	−3.19617
$983$	33.3302	1.06307	0.531535	−	0.847037i	$-0.321616\pi$
$983$	33.3302	1.06307	0.531535	+	0.847037i	$0.321616\pi$
$984$	−66.5336	−2.12101
$985$	0	0
$986$	46.2899	1.47417
$987$	−11.0681	−0.352302
$988$	37.0399	1.17840
$989$	−1.70157	−0.0541069
$990$	0	0
$991$	−3.99598	−0.126937	−0.0634683	−	0.997984i	$-0.520216\pi$
$991$	−3.99598	−0.126937	−0.0634683	+	0.997984i	$0.520216\pi$
$992$	−8.78099	−0.278797
$993$	−15.3331	−0.486581
$994$	11.5505	0.366358
$995$	0	0
$996$	−80.7561	−2.55886
$997$	−14.3336	−0.453949	−0.226974	−	0.973901i	$-0.572883\pi$
$997$	−14.3336	−0.453949	−0.226974	+	0.973901i	$0.572883\pi$
$998$	−2.30884	−0.0730850
$999$	−5.85620	−0.185282

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4025.2.a.v.1.8	yes	8
5.4	even	2		4025.2.a.u.1.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4025.2.a.u.1.1	✓	8	5.4	even	2
4025.2.a.v.1.8	yes	8	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:	\( N \)	\(=\)	\( 4025 = 5^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4025.a (trivial)

\(f(q)\)	\(=\)	\(q+2.39863 q^{2} +2.12900 q^{3} +3.75344 q^{4} +5.10670 q^{6} -1.00000 q^{7} +4.20585 q^{8} +1.53265 q^{9} +O(q^{10})\)	\(q+2.39863 q^{2} +2.12900 q^{3} +3.75344 q^{4} +5.10670 q^{6} -1.00000 q^{7} +4.20585 q^{8} +1.53265 q^{9} +5.51035 q^{11} +7.99108 q^{12} +3.23035 q^{13} -2.39863 q^{14} +2.58142 q^{16} -2.52456 q^{17} +3.67627 q^{18} +3.05486 q^{19} -2.12900 q^{21} +13.2173 q^{22} +1.00000 q^{23} +8.95427 q^{24} +7.74842 q^{26} -3.12398 q^{27} -3.75344 q^{28} -7.64429 q^{29} +3.95570 q^{31} -2.21983 q^{32} +11.7315 q^{33} -6.05549 q^{34} +5.75272 q^{36} +1.87459 q^{37} +7.32748 q^{38} +6.87742 q^{39} -7.43037 q^{41} -5.10670 q^{42} -1.70157 q^{43} +20.6827 q^{44} +2.39863 q^{46} +5.19873 q^{47} +5.49585 q^{48} +1.00000 q^{49} -5.37480 q^{51} +12.1249 q^{52} +5.57648 q^{53} -7.49329 q^{54} -4.20585 q^{56} +6.50380 q^{57} -18.3358 q^{58} +1.05654 q^{59} -5.87905 q^{61} +9.48827 q^{62} -1.53265 q^{63} -10.4874 q^{64} +28.1397 q^{66} +5.43680 q^{67} -9.47578 q^{68} +2.12900 q^{69} -4.81544 q^{71} +6.44611 q^{72} +9.52328 q^{73} +4.49646 q^{74} +11.4662 q^{76} -5.51035 q^{77} +16.4964 q^{78} -5.88424 q^{79} -11.2489 q^{81} -17.8227 q^{82} -10.1058 q^{83} -7.99108 q^{84} -4.08145 q^{86} -16.2747 q^{87} +23.1757 q^{88} +17.8642 q^{89} -3.23035 q^{91} +3.75344 q^{92} +8.42170 q^{93} +12.4698 q^{94} -4.72603 q^{96} -3.99122 q^{97} +2.39863 q^{98} +8.44545 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + q^{2} + 4 q^{3} + 5 q^{4} - q^{6} - 8 q^{7}+O(q^{10}) \) 8 * q + q^2 + 4 * q^3 + 5 * q^4 - q^6 - 8 * q^7	\( 8 q + q^{2} + 4 q^{3} + 5 q^{4} - q^{6} - 8 q^{7} + 3 q^{11} + 9 q^{12} + 5 q^{13} - q^{14} - q^{16} + 5 q^{17} + 2 q^{18} - 2 q^{19} - 4 q^{21} + 21 q^{22} + 8 q^{23} - 6 q^{24} + 18 q^{26} + 7 q^{27} - 5 q^{28} - 9 q^{29} - 3 q^{31} + 6 q^{32} + 4 q^{33} - 10 q^{34} + 16 q^{36} + 6 q^{37} + 4 q^{38} - 2 q^{39} - 7 q^{41} + q^{42} + 8 q^{43} + 4 q^{44} + q^{46} + 22 q^{47} + 9 q^{48} + 8 q^{49} - 12 q^{51} + 11 q^{52} + 21 q^{53} - 15 q^{54} + 8 q^{57} + 16 q^{58} + 14 q^{59} + 8 q^{61} + 12 q^{62} - 40 q^{64} + 55 q^{66} + 21 q^{67} + 3 q^{68} + 4 q^{69} + 11 q^{71} - q^{72} + 26 q^{73} - 41 q^{74} + 21 q^{76} - 3 q^{77} + 17 q^{78} - 16 q^{79} - 20 q^{81} - q^{82} + 20 q^{83} - 9 q^{84} + 14 q^{86} - 29 q^{87} + 32 q^{88} + 15 q^{89} - 5 q^{91} + 5 q^{92} + 19 q^{93} + 21 q^{94} + 52 q^{96} + q^{97} + q^{98} + 15 q^{99}+O(q^{100}) \) 8 * q + q^2 + 4 * q^3 + 5 * q^4 - q^6 - 8 * q^7 + 3 * q^11 + 9 * q^12 + 5 * q^13 - q^14 - q^16 + 5 * q^17 + 2 * q^18 - 2 * q^19 - 4 * q^21 + 21 * q^22 + 8 * q^23 - 6 * q^24 + 18 * q^26 + 7 * q^27 - 5 * q^28 - 9 * q^29 - 3 * q^31 + 6 * q^32 + 4 * q^33 - 10 * q^34 + 16 * q^36 + 6 * q^37 + 4 * q^38 - 2 * q^39 - 7 * q^41 + q^42 + 8 * q^43 + 4 * q^44 + q^46 + 22 * q^47 + 9 * q^48 + 8 * q^49 - 12 * q^51 + 11 * q^52 + 21 * q^53 - 15 * q^54 + 8 * q^57 + 16 * q^58 + 14 * q^59 + 8 * q^61 + 12 * q^62 - 40 * q^64 + 55 * q^66 + 21 * q^67 + 3 * q^68 + 4 * q^69 + 11 * q^71 - q^72 + 26 * q^73 - 41 * q^74 + 21 * q^76 - 3 * q^77 + 17 * q^78 - 16 * q^79 - 20 * q^81 - q^82 + 20 * q^83 - 9 * q^84 + 14 * q^86 - 29 * q^87 + 32 * q^88 + 15 * q^89 - 5 * q^91 + 5 * q^92 + 19 * q^93 + 21 * q^94 + 52 * q^96 + q^97 + q^98 + 15 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more