LMFDB - Embedded newform 4025.2.a.v.1.7

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			1.7
Root			$2.02014$ 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)

$f(q)$	$=$	$q+2.02014 q^{2} -1.91409 q^{3} +2.08098 q^{4} -3.86674 q^{6} -1.00000 q^{7} +0.163592 q^{8} +0.663753 q^{9} +O(q^{10})$	\(q+2.02014 q^{2} -1.91409 q^{3} +2.08098 q^{4} -3.86674 q^{6} -1.00000 q^{7} +0.163592 q^{8} +0.663753 q^{9} -0.288897 q^{11} -3.98319 q^{12} +1.24906 q^{13} -2.02014 q^{14} -3.83148 q^{16} +2.73499 q^{17} +1.34088 q^{18} -1.21255 q^{19} +1.91409 q^{21} -0.583614 q^{22} +1.00000 q^{23} -0.313130 q^{24} +2.52328 q^{26} +4.47180 q^{27} -2.08098 q^{28} +8.65753 q^{29} -5.24288 q^{31} -8.06733 q^{32} +0.552976 q^{33} +5.52506 q^{34} +1.38126 q^{36} -10.7713 q^{37} -2.44952 q^{38} -2.39081 q^{39} -5.72220 q^{41} +3.86674 q^{42} +4.87768 q^{43} -0.601190 q^{44} +2.02014 q^{46} +10.4853 q^{47} +7.33381 q^{48} +1.00000 q^{49} -5.23502 q^{51} +2.59927 q^{52} +5.62306 q^{53} +9.03367 q^{54} -0.163592 q^{56} +2.32092 q^{57} +17.4895 q^{58} +13.2076 q^{59} +1.33657 q^{61} -10.5914 q^{62} -0.663753 q^{63} -8.63420 q^{64} +1.11709 q^{66} +11.2975 q^{67} +5.69145 q^{68} -1.91409 q^{69} -9.91727 q^{71} +0.108584 q^{72} +3.84869 q^{73} -21.7596 q^{74} -2.52328 q^{76} +0.288897 q^{77} -4.82979 q^{78} -3.22963 q^{79} -10.5507 q^{81} -11.5597 q^{82} +16.4634 q^{83} +3.98319 q^{84} +9.85362 q^{86} -16.5713 q^{87} -0.0472612 q^{88} +15.9409 q^{89} -1.24906 q^{91} +2.08098 q^{92} +10.0354 q^{93} +21.1818 q^{94} +15.4416 q^{96} -2.42844 q^{97} +2.02014 q^{98} -0.191756 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.02014	1.42846	0.714229	−	0.699912i	$-0.246778\pi$
$2$	2.02014	1.42846	0.714229	+	0.699912i	$0.246778\pi$
$3$	−1.91409	−1.10510	−0.552551	−	0.833479i	$-0.686346\pi$
$3$	−1.91409	−1.10510	−0.552551	+	0.833479i	$0.686346\pi$
$4$	2.08098	1.04049
$5$	0	0
$5$	0	0
$6$	−3.86674	−1.57859
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0.163592	0.0578384
$9$	0.663753	0.221251
$10$	0	0
$11$	−0.288897	−0.0871058	−0.0435529	−	0.999051i	$-0.513868\pi$
$11$	−0.288897	−0.0871058	−0.0435529	+	0.999051i	$0.513868\pi$
$12$	−3.98319	−1.14985
$13$	1.24906	0.346427	0.173213	−	0.984884i	$-0.444585\pi$
$13$	1.24906	0.346427	0.173213	+	0.984884i	$0.444585\pi$
$14$	−2.02014	−0.539906
$15$	0	0
$16$	−3.83148	−0.957870
$17$	2.73499	0.663331	0.331666	−	0.943397i	$-0.392389\pi$
$17$	2.73499	0.663331	0.331666	+	0.943397i	$0.392389\pi$
$18$	1.34088	0.316047
$19$	−1.21255	−0.278177	−0.139088	−	0.990280i	$-0.544417\pi$
$19$	−1.21255	−0.278177	−0.139088	+	0.990280i	$0.544417\pi$
$20$	0	0
$21$	1.91409	0.417689
$22$	−0.583614	−0.124427
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	−0.313130	−0.0639174
$25$	0	0
$26$	2.52328	0.494856
$27$	4.47180	0.860597
$28$	−2.08098	−0.393268
$29$	8.65753	1.60766	0.803832	−	0.594857i	$-0.202791\pi$
$29$	8.65753	1.60766	0.803832	+	0.594857i	$0.202791\pi$
$30$	0	0
$31$	−5.24288	−0.941649	−0.470825	−	0.882227i	$-0.656044\pi$
$31$	−5.24288	−0.941649	−0.470825	+	0.882227i	$0.656044\pi$
$32$	−8.06733	−1.42612
$33$	0.552976	0.0962608
$34$	5.52506	0.947541
$35$	0	0
$36$	1.38126	0.230209
$37$	−10.7713	−1.77079	−0.885397	−	0.464836i	$-0.846113\pi$
$37$	−10.7713	−1.77079	−0.885397	+	0.464836i	$0.846113\pi$
$38$	−2.44952	−0.397364
$39$	−2.39081	−0.382837
$40$	0	0
$41$	−5.72220	−0.893657	−0.446828	−	0.894620i	$-0.647447\pi$
$41$	−5.72220	−0.893657	−0.446828	+	0.894620i	$0.647447\pi$
$42$	3.86674	0.596651
$43$	4.87768	0.743839	0.371920	−	0.928265i	$-0.378700\pi$
$43$	4.87768	0.743839	0.371920	+	0.928265i	$0.378700\pi$
$44$	−0.601190	−0.0906328
$45$	0	0
$46$	2.02014	0.297854
$47$	10.4853	1.52944	0.764720	−	0.644362i	$-0.222877\pi$
$47$	10.4853	1.52944	0.764720	+	0.644362i	$0.222877\pi$
$48$	7.33381	1.05854
$49$	1.00000	0.142857
$50$	0	0
$51$	−5.23502	−0.733049
$52$	2.59927	0.360453
$53$	5.62306	0.772387	0.386193	−	0.922418i	$-0.373790\pi$
$53$	5.62306	0.772387	0.386193	+	0.922418i	$0.373790\pi$
$54$	9.03367	1.22933
$55$	0	0
$56$	−0.163592	−0.0218609
$57$	2.32092	0.307414
$58$	17.4895	2.29648
$59$	13.2076	1.71949	0.859744	−	0.510725i	$-0.170623\pi$
$59$	13.2076	1.71949	0.859744	+	0.510725i	$0.170623\pi$
$60$	0	0
$61$	1.33657	0.171130	0.0855651	−	0.996333i	$-0.472730\pi$
$61$	1.33657	0.171130	0.0855651	+	0.996333i	$0.472730\pi$
$62$	−10.5914	−1.34511
$63$	−0.663753	−0.0836250
$64$	−8.63420	−1.07927
$65$	0	0
$66$	1.11709	0.137505
$67$	11.2975	1.38021	0.690104	−	0.723710i	$-0.257565\pi$
$67$	11.2975	1.38021	0.690104	+	0.723710i	$0.257565\pi$
$68$	5.69145	0.690190
$69$	−1.91409	−0.230430
$70$	0	0
$71$	−9.91727	−1.17696	−0.588482	−	0.808510i	$-0.700274\pi$
$71$	−9.91727	−1.17696	−0.588482	+	0.808510i	$0.700274\pi$
$72$	0.108584	0.0127968
$73$	3.84869	0.450455	0.225228	−	0.974306i	$-0.427687\pi$
$73$	3.84869	0.450455	0.225228	+	0.974306i	$0.427687\pi$
$74$	−21.7596	−2.52950
$75$	0	0
$76$	−2.52328	−0.289440
$77$	0.288897	0.0329229
$78$	−4.82979	−0.546866
$79$	−3.22963	−0.363362	−0.181681	−	0.983358i	$-0.558154\pi$
$79$	−3.22963	−0.363362	−0.181681	+	0.983358i	$0.558154\pi$
$80$	0	0
$81$	−10.5507	−1.17230
$82$	−11.5597	−1.27655
$83$	16.4634	1.80710	0.903549	−	0.428486i	$-0.140953\pi$
$83$	16.4634	1.80710	0.903549	+	0.428486i	$0.140953\pi$
$84$	3.98319	0.434602
$85$	0	0
$86$	9.85362	1.06254
$87$	−16.5713	−1.77663
$88$	−0.0472612	−0.00503807
$89$	15.9409	1.68973	0.844865	−	0.534980i	$-0.179681\pi$
$89$	15.9409	1.68973	0.844865	+	0.534980i	$0.179681\pi$
$90$	0	0
$91$	−1.24906	−0.130937
$92$	2.08098	0.216957
$93$	10.0354	1.04062
$94$	21.1818	2.18474
$95$	0	0
$96$	15.4416	1.57600
$97$	−2.42844	−0.246570	−0.123285	−	0.992371i	$-0.539343\pi$
$97$	−2.42844	−0.246570	−0.123285	+	0.992371i	$0.539343\pi$
$98$	2.02014	0.204065
$99$	−0.191756	−0.0192722
$100$	0	0
$101$	11.3142	1.12580	0.562901	−	0.826524i	$-0.309685\pi$
$101$	11.3142	1.12580	0.562901	+	0.826524i	$0.309685\pi$
$102$	−10.5755	−1.04713
$103$	10.2282	1.00781	0.503907	−	0.863758i	$-0.331895\pi$
$103$	10.2282	1.00781	0.503907	+	0.863758i	$0.331895\pi$
$104$	0.204336	0.0200368
$105$	0	0
$106$	11.3594	1.10332
$107$	11.9190	1.15226	0.576129	−	0.817359i	$-0.304563\pi$
$107$	11.9190	1.15226	0.576129	+	0.817359i	$0.304563\pi$
$108$	9.30572	0.895443
$109$	−3.36864	−0.322657	−0.161328	−	0.986901i	$-0.551578\pi$
$109$	−3.36864	−0.322657	−0.161328	+	0.986901i	$0.551578\pi$
$110$	0	0
$111$	20.6173	1.95691
$112$	3.83148	0.362041
$113$	−3.02032	−0.284128	−0.142064	−	0.989857i	$-0.545374\pi$
$113$	−3.02032	−0.284128	−0.142064	+	0.989857i	$0.545374\pi$
$114$	4.68860	0.439128
$115$	0	0
$116$	18.0162	1.67276
$117$	0.829066	0.0766472
$118$	26.6813	2.45622
$119$	−2.73499	−0.250716
$120$	0	0
$121$	−10.9165	−0.992413
$122$	2.70006	0.244452
$123$	10.9528	0.987582
$124$	−10.9103	−0.979777
$125$	0	0
$126$	−1.34088	−0.119455
$127$	4.99218	0.442985	0.221492	−	0.975162i	$-0.428907\pi$
$127$	4.99218	0.442985	0.221492	+	0.975162i	$0.428907\pi$
$128$	−1.30766	−0.115582
$129$	−9.33634	−0.822019
$130$	0	0
$131$	5.57716	0.487279	0.243639	−	0.969866i	$-0.421659\pi$
$131$	5.57716	0.487279	0.243639	+	0.969866i	$0.421659\pi$
$132$	1.15073	0.100158
$133$	1.21255	0.105141
$134$	22.8226	1.97157
$135$	0	0
$136$	0.447421	0.0383661
$137$	−7.37899	−0.630430	−0.315215	−	0.949020i	$-0.602077\pi$
$137$	−7.37899	−0.630430	−0.315215	+	0.949020i	$0.602077\pi$
$138$	−3.86674	−0.329159
$139$	−13.5718	−1.15115	−0.575574	−	0.817749i	$-0.695221\pi$
$139$	−13.5718	−1.15115	−0.575574	+	0.817749i	$0.695221\pi$
$140$	0	0
$141$	−20.0699	−1.69019
$142$	−20.0343	−1.68124
$143$	−0.360850	−0.0301758
$144$	−2.54316	−0.211930
$145$	0	0
$146$	7.77491	0.643456
$147$	−1.91409	−0.157872
$148$	−22.4149	−1.84249
$149$	−0.177018	−0.0145018	−0.00725092	−	0.999974i	$-0.502308\pi$
$149$	−0.177018	−0.0145018	−0.00725092	+	0.999974i	$0.502308\pi$
$150$	0	0
$151$	11.0369	0.898170	0.449085	−	0.893489i	$-0.351750\pi$
$151$	11.0369	0.898170	0.449085	+	0.893489i	$0.351750\pi$
$152$	−0.198362	−0.0160893
$153$	1.81535	0.146763
$154$	0.583614	0.0470290
$155$	0	0
$156$	−4.97524	−0.398338
$157$	1.37058	0.109385	0.0546923	−	0.998503i	$-0.482582\pi$
$157$	1.37058	0.109385	0.0546923	+	0.998503i	$0.482582\pi$
$158$	−6.52431	−0.519046
$159$	−10.7631	−0.853566
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	−21.3139	−1.67458
$163$	−0.0858672	−0.00672564	−0.00336282	−	0.999994i	$-0.501070\pi$
$163$	−0.0858672	−0.00672564	−0.00336282	+	0.999994i	$0.501070\pi$
$164$	−11.9078	−0.929841
$165$	0	0
$166$	33.2585	2.58136
$167$	21.8753	1.69276	0.846380	−	0.532580i	$-0.178777\pi$
$167$	21.8753	1.69276	0.846380	+	0.532580i	$0.178777\pi$
$168$	0.313130	0.0241585
$169$	−11.4399	−0.879989
$170$	0	0
$171$	−0.804830	−0.0615469
$172$	10.1504	0.773958
$173$	−14.8387	−1.12816	−0.564082	−	0.825718i	$-0.690770\pi$
$173$	−14.8387	−1.12816	−0.564082	+	0.825718i	$0.690770\pi$
$174$	−33.4765	−2.53784
$175$	0	0
$176$	1.10690	0.0834361
$177$	−25.2807	−1.90021
$178$	32.2029	2.41371
$179$	12.7872	0.955758	0.477879	−	0.878426i	$-0.341406\pi$
$179$	12.7872	0.955758	0.477879	+	0.878426i	$0.341406\pi$
$180$	0	0
$181$	0.102143	0.00759220	0.00379610	−	0.999993i	$-0.498792\pi$
$181$	0.102143	0.00759220	0.00379610	+	0.999993i	$0.498792\pi$
$182$	−2.52328	−0.187038
$183$	−2.55832	−0.189116
$184$	0.163592	0.0120601
$185$	0	0
$186$	20.2729	1.48648
$187$	−0.790130	−0.0577800
$188$	21.8197	1.59137
$189$	−4.47180	−0.325275
$190$	0	0
$191$	−13.6172	−0.985303	−0.492652	−	0.870227i	$-0.663972\pi$
$191$	−13.6172	−0.985303	−0.492652	+	0.870227i	$0.663972\pi$
$192$	16.5267	1.19271
$193$	3.75151	0.270040	0.135020	−	0.990843i	$-0.456890\pi$
$193$	3.75151	0.270040	0.135020	+	0.990843i	$0.456890\pi$
$194$	−4.90579	−0.352215
$195$	0	0
$196$	2.08098	0.148641
$197$	23.9146	1.70385	0.851923	−	0.523667i	$-0.175436\pi$
$197$	23.9146	1.70385	0.851923	+	0.523667i	$0.175436\pi$
$198$	−0.387375	−0.0275296
$199$	3.57665	0.253542	0.126771	−	0.991932i	$-0.459539\pi$
$199$	3.57665	0.253542	0.126771	+	0.991932i	$0.459539\pi$
$200$	0	0
$201$	−21.6245	−1.52527
$202$	22.8563	1.60816
$203$	−8.65753	−0.607640
$204$	−10.8940	−0.762730
$205$	0	0
$206$	20.6624	1.43962
$207$	0.663753	0.0461340
$208$	−4.78575	−0.331832
$209$	0.350301	0.0242308
$210$	0	0
$211$	−27.0810	−1.86433	−0.932167	−	0.362029i	$-0.882084\pi$
$211$	−27.0810	−1.86433	−0.932167	+	0.362029i	$0.882084\pi$
$212$	11.7015	0.803661
$213$	18.9826	1.30067
$214$	24.0782	1.64595
$215$	0	0
$216$	0.731549	0.0497756
$217$	5.24288	0.355910
$218$	−6.80513	−0.460902
$219$	−7.36675	−0.497799
$220$	0	0
$221$	3.41616	0.229796
$222$	41.6499	2.79536
$223$	−1.08739	−0.0728172	−0.0364086	−	0.999337i	$-0.511592\pi$
$223$	−1.08739	−0.0728172	−0.0364086	+	0.999337i	$0.511592\pi$
$224$	8.06733	0.539021
$225$	0	0
$226$	−6.10148	−0.405865
$227$	−12.0559	−0.800178	−0.400089	−	0.916476i	$-0.631021\pi$
$227$	−12.0559	−0.800178	−0.400089	+	0.916476i	$0.631021\pi$
$228$	4.82980	0.319861
$229$	−13.6197	−0.900014	−0.450007	−	0.893025i	$-0.648579\pi$
$229$	−13.6197	−0.900014	−0.450007	+	0.893025i	$0.648579\pi$
$230$	0	0
$231$	−0.552976	−0.0363832
$232$	1.41630	0.0929847
$233$	−3.35486	−0.219784	−0.109892	−	0.993944i	$-0.535051\pi$
$233$	−3.35486	−0.219784	−0.109892	+	0.993944i	$0.535051\pi$
$234$	1.67483	0.109487
$235$	0	0
$236$	27.4848	1.78911
$237$	6.18181	0.401552
$238$	−5.52506	−0.358137
$239$	13.5416	0.875936	0.437968	−	0.898991i	$-0.355698\pi$
$239$	13.5416	0.875936	0.437968	+	0.898991i	$0.355698\pi$
$240$	0	0
$241$	27.0034	1.73944	0.869721	−	0.493544i	$-0.164299\pi$
$241$	27.0034	1.73944	0.869721	+	0.493544i	$0.164299\pi$
$242$	−22.0530	−1.41762
$243$	6.77962	0.434913
$244$	2.78137	0.178059
$245$	0	0
$246$	22.1263	1.41072
$247$	−1.51454	−0.0963679
$248$	−0.857692	−0.0544635
$249$	−31.5126	−1.99703
$250$	0	0
$251$	−13.9013	−0.877442	−0.438721	−	0.898623i	$-0.644568\pi$
$251$	−13.9013	−0.877442	−0.438721	+	0.898623i	$0.644568\pi$
$252$	−1.38126	−0.0870109
$253$	−0.288897	−0.0181628
$254$	10.0849	0.632785
$255$	0	0
$256$	14.6267	0.914170
$257$	−29.5045	−1.84044	−0.920221	−	0.391399i	$-0.871991\pi$
$257$	−29.5045	−1.84044	−0.920221	+	0.391399i	$0.871991\pi$
$258$	−18.8607	−1.17422
$259$	10.7713	0.669297
$260$	0	0
$261$	5.74646	0.355697
$262$	11.2667	0.696057
$263$	−4.37263	−0.269628	−0.134814	−	0.990871i	$-0.543044\pi$
$263$	−4.37263	−0.269628	−0.134814	+	0.990871i	$0.543044\pi$
$264$	0.0904624	0.00556758
$265$	0	0
$266$	2.44952	0.150189
$267$	−30.5123	−1.86732
$268$	23.5099	1.43609
$269$	17.6575	1.07660	0.538298	−	0.842755i	$-0.319068\pi$
$269$	17.6575	1.07660	0.538298	+	0.842755i	$0.319068\pi$
$270$	0	0
$271$	15.7955	0.959506	0.479753	−	0.877403i	$-0.340726\pi$
$271$	15.7955	0.959506	0.479753	+	0.877403i	$0.340726\pi$
$272$	−10.4790	−0.635386
$273$	2.39081	0.144699
$274$	−14.9066	−0.900542
$275$	0	0
$276$	−3.98319	−0.239760
$277$	6.79460	0.408248	0.204124	−	0.978945i	$-0.434565\pi$
$277$	6.79460	0.408248	0.204124	+	0.978945i	$0.434565\pi$
$278$	−27.4171	−1.64437
$279$	−3.47998	−0.208341
$280$	0	0
$281$	−3.71714	−0.221746	−0.110873	−	0.993835i	$-0.535365\pi$
$281$	−3.71714	−0.221746	−0.110873	+	0.993835i	$0.535365\pi$
$282$	−40.5440	−2.41436
$283$	23.7360	1.41096	0.705481	−	0.708729i	$-0.250731\pi$
$283$	23.7360	1.41096	0.705481	+	0.708729i	$0.250731\pi$
$284$	−20.6376	−1.22462
$285$	0	0
$286$	−0.728968	−0.0431048
$287$	5.72220	0.337771
$288$	−5.35471	−0.315529
$289$	−9.51985	−0.559991
$290$	0	0
$291$	4.64825	0.272485
$292$	8.00905	0.468694
$293$	1.13095	0.0660710	0.0330355	−	0.999454i	$-0.489483\pi$
$293$	1.13095	0.0660710	0.0330355	+	0.999454i	$0.489483\pi$
$294$	−3.86674	−0.225513
$295$	0	0
$296$	−1.76210	−0.102420
$297$	−1.29189	−0.0749631
$298$	−0.357601	−0.0207153
$299$	1.24906	0.0722349
$300$	0	0
$301$	−4.87768	−0.281145
$302$	22.2961	1.28300
$303$	−21.6564	−1.24413
$304$	4.64584	0.266457
$305$	0	0
$306$	3.66728	0.209644
$307$	22.2815	1.27167	0.635835	−	0.771825i	$-0.280656\pi$
$307$	22.2815	1.27167	0.635835	+	0.771825i	$0.280656\pi$
$308$	0.601190	0.0342560
$309$	−19.5777	−1.11374
$310$	0	0
$311$	29.7766	1.68848	0.844239	−	0.535967i	$-0.180053\pi$
$311$	29.7766	1.68848	0.844239	+	0.535967i	$0.180053\pi$
$312$	−0.391118	−0.0221427
$313$	−5.97442	−0.337694	−0.168847	−	0.985642i	$-0.554004\pi$
$313$	−5.97442	−0.337694	−0.168847	+	0.985642i	$0.554004\pi$
$314$	2.76878	0.156251
$315$	0	0
$316$	−6.72079	−0.378074
$317$	−1.30485	−0.0732879	−0.0366440	−	0.999328i	$-0.511667\pi$
$317$	−1.30485	−0.0732879	−0.0366440	+	0.999328i	$0.511667\pi$
$318$	−21.7429	−1.21928
$319$	−2.50114	−0.140037
$320$	0	0
$321$	−22.8142	−1.27336
$322$	−2.02014	−0.112578
$323$	−3.31629	−0.184524
$324$	−21.9558	−1.21977
$325$	0	0
$326$	−0.173464	−0.00960729
$327$	6.44789	0.356569
$328$	−0.936105	−0.0516877
$329$	−10.4853	−0.578074
$330$	0	0
$331$	−26.9399	−1.48075	−0.740374	−	0.672195i	$-0.765352\pi$
$331$	−26.9399	−1.48075	−0.740374	+	0.672195i	$0.765352\pi$
$332$	34.2601	1.88027
$333$	−7.14949	−0.391790
$334$	44.1912	2.41803
$335$	0	0
$336$	−7.33381	−0.400092
$337$	28.3735	1.54560	0.772802	−	0.634647i	$-0.218855\pi$
$337$	28.3735	1.54560	0.772802	+	0.634647i	$0.218855\pi$
$338$	−23.1101	−1.25703
$339$	5.78118	0.313990
$340$	0	0
$341$	1.51465	0.0820231
$342$	−1.62587	−0.0879171
$343$	−1.00000	−0.0539949
$344$	0.797949	0.0430225
$345$	0	0
$346$	−29.9763	−1.61154
$347$	−4.75267	−0.255137	−0.127568	−	0.991830i	$-0.540717\pi$
$347$	−4.75267	−0.255137	−0.127568	+	0.991830i	$0.540717\pi$
$348$	−34.4846	−1.84857
$349$	16.2687	0.870846	0.435423	−	0.900226i	$-0.356599\pi$
$349$	16.2687	0.870846	0.435423	+	0.900226i	$0.356599\pi$
$350$	0	0
$351$	5.58554	0.298134
$352$	2.33063	0.124223
$353$	−11.3235	−0.602689	−0.301345	−	0.953515i	$-0.597435\pi$
$353$	−11.3235	−0.602689	−0.301345	+	0.953515i	$0.597435\pi$
$354$	−51.0706	−2.71437
$355$	0	0
$356$	33.1727	1.75815
$357$	5.23502	0.277067
$358$	25.8319	1.36526
$359$	23.7745	1.25477	0.627384	−	0.778710i	$-0.284126\pi$
$359$	23.7745	1.25477	0.627384	+	0.778710i	$0.284126\pi$
$360$	0	0
$361$	−17.5297	−0.922618
$362$	0.206343	0.0108451
$363$	20.8953	1.09672
$364$	−2.59927	−0.136239
$365$	0	0
$366$	−5.16817	−0.270145
$367$	8.19255	0.427648	0.213824	−	0.976872i	$-0.431408\pi$
$367$	8.19255	0.427648	0.213824	+	0.976872i	$0.431408\pi$
$368$	−3.83148	−0.199730
$369$	−3.79812	−0.197722
$370$	0	0
$371$	−5.62306	−0.291935
$372$	20.8834	1.08275
$373$	−22.9614	−1.18889	−0.594447	−	0.804135i	$-0.702629\pi$
$373$	−22.9614	−1.18889	−0.594447	+	0.804135i	$0.702629\pi$
$374$	−1.59618	−0.0825363
$375$	0	0
$376$	1.71531	0.0884605
$377$	10.8138	0.556937
$378$	−9.03367	−0.464642
$379$	5.12476	0.263241	0.131621	−	0.991300i	$-0.457982\pi$
$379$	5.12476	0.263241	0.131621	+	0.991300i	$0.457982\pi$
$380$	0	0
$381$	−9.55550	−0.489543
$382$	−27.5086	−1.40746
$383$	−29.8007	−1.52274	−0.761372	−	0.648315i	$-0.775474\pi$
$383$	−29.8007	−1.52274	−0.761372	+	0.648315i	$0.775474\pi$
$384$	2.50299	0.127730
$385$	0	0
$386$	7.57860	0.385741
$387$	3.23757	0.164575
$388$	−5.05353	−0.256554
$389$	−33.1973	−1.68317	−0.841585	−	0.540125i	$-0.818377\pi$
$389$	−33.1973	−1.68317	−0.841585	+	0.540125i	$0.818377\pi$
$390$	0	0
$391$	2.73499	0.138314
$392$	0.163592	0.00826263
$393$	−10.6752	−0.538493
$394$	48.3110	2.43387
$395$	0	0
$396$	−0.399041	−0.0200526
$397$	10.6613	0.535076	0.267538	−	0.963547i	$-0.413790\pi$
$397$	10.6613	0.535076	0.267538	+	0.963547i	$0.413790\pi$
$398$	7.22536	0.362174
$399$	−2.32092	−0.116192
$400$	0	0
$401$	−34.2523	−1.71048	−0.855238	−	0.518235i	$-0.826589\pi$
$401$	−34.2523	−1.71048	−0.855238	+	0.518235i	$0.826589\pi$
$402$	−43.6845	−2.17878
$403$	−6.54867	−0.326212
$404$	23.5446	1.17139
$405$	0	0
$406$	−17.4895	−0.867987
$407$	3.11180	0.154246
$408$	−0.856406	−0.0423984
$409$	0.321476	0.0158960	0.00794798	−	0.999968i	$-0.497470\pi$
$409$	0.321476	0.0158960	0.00794798	+	0.999968i	$0.497470\pi$
$410$	0	0
$411$	14.1241	0.696690
$412$	21.2847	1.04862
$413$	−13.2076	−0.649906
$414$	1.34088	0.0659004
$415$	0	0
$416$	−10.0766	−0.494044
$417$	25.9778	1.27214
$418$	0.707659	0.0346127
$419$	−35.6073	−1.73953	−0.869765	−	0.493465i	$-0.835730\pi$
$419$	−35.6073	−1.73953	−0.869765	+	0.493465i	$0.835730\pi$
$420$	0	0
$421$	−13.7573	−0.670490	−0.335245	−	0.942131i	$-0.608819\pi$
$421$	−13.7573	−0.670490	−0.335245	+	0.942131i	$0.608819\pi$
$422$	−54.7075	−2.66312
$423$	6.95966	0.338390
$424$	0.919887	0.0446736
$425$	0	0
$426$	38.3475	1.85794
$427$	−1.33657	−0.0646811
$428$	24.8033	1.19891
$429$	0.690700	0.0333473
$430$	0	0
$431$	13.5482	0.652594	0.326297	−	0.945267i	$-0.394199\pi$
$431$	13.5482	0.652594	0.326297	+	0.945267i	$0.394199\pi$
$432$	−17.1336	−0.824341
$433$	−1.21089	−0.0581916	−0.0290958	−	0.999577i	$-0.509263\pi$
$433$	−1.21089	−0.0581916	−0.0290958	+	0.999577i	$0.509263\pi$
$434$	10.5914	0.508402
$435$	0	0
$436$	−7.01007	−0.335721
$437$	−1.21255	−0.0580039
$438$	−14.8819	−0.711084
$439$	0.789991	0.0377042	0.0188521	−	0.999822i	$-0.493999\pi$
$439$	0.789991	0.0377042	0.0188521	+	0.999822i	$0.493999\pi$
$440$	0	0
$441$	0.663753	0.0316073
$442$	6.90113	0.328253
$443$	29.0674	1.38103	0.690516	−	0.723317i	$-0.257383\pi$
$443$	29.0674	1.38103	0.690516	+	0.723317i	$0.257383\pi$
$444$	42.9042	2.03614
$445$	0	0
$446$	−2.19669	−0.104016
$447$	0.338828	0.0160260
$448$	8.63420	0.407927
$449$	10.3037	0.486260	0.243130	−	0.969994i	$-0.421826\pi$
$449$	10.3037	0.486260	0.243130	+	0.969994i	$0.421826\pi$
$450$	0	0
$451$	1.65313	0.0778427
$452$	−6.28523	−0.295632
$453$	−21.1257	−0.992570
$454$	−24.3547	−1.14302
$455$	0	0
$456$	0.379684	0.0177803
$457$	5.57079	0.260591	0.130295	−	0.991475i	$-0.458407\pi$
$457$	5.57079	0.260591	0.130295	+	0.991475i	$0.458407\pi$
$458$	−27.5137	−1.28563
$459$	12.2303	0.570861
$460$	0	0
$461$	−25.0129	−1.16497	−0.582483	−	0.812843i	$-0.697919\pi$
$461$	−25.0129	−1.16497	−0.582483	+	0.812843i	$0.697919\pi$
$462$	−1.11709	−0.0519718
$463$	14.0148	0.651323	0.325662	−	0.945486i	$-0.394413\pi$
$463$	14.0148	0.651323	0.325662	+	0.945486i	$0.394413\pi$
$464$	−33.1712	−1.53993
$465$	0	0
$466$	−6.77731	−0.313953
$467$	30.4498	1.40905	0.704524	−	0.709680i	$-0.251161\pi$
$467$	30.4498	1.40905	0.704524	+	0.709680i	$0.251161\pi$
$468$	1.72527	0.0797506
$469$	−11.2975	−0.521670
$470$	0	0
$471$	−2.62343	−0.120881
$472$	2.16066	0.0994525
$473$	−1.40915	−0.0647928
$474$	12.4881	0.573599
$475$	0	0
$476$	−5.69145	−0.260867
$477$	3.73232	0.170891
$478$	27.3561	1.25124
$479$	−10.4389	−0.476966	−0.238483	−	0.971147i	$-0.576650\pi$
$479$	−10.4389	−0.476966	−0.238483	+	0.971147i	$0.576650\pi$
$480$	0	0
$481$	−13.4540	−0.613450
$482$	54.5507	2.48472
$483$	1.91409	0.0870943
$484$	−22.7171	−1.03260
$485$	0	0
$486$	13.6958	0.621254
$487$	−6.31505	−0.286162	−0.143081	−	0.989711i	$-0.545701\pi$
$487$	−6.31505	−0.286162	−0.143081	+	0.989711i	$0.545701\pi$
$488$	0.218652	0.00989790
$489$	0.164358	0.00743252
$490$	0	0
$491$	20.2480	0.913779	0.456889	−	0.889523i	$-0.348963\pi$
$491$	20.2480	0.913779	0.456889	+	0.889523i	$0.348963\pi$
$492$	22.7926	1.02757
$493$	23.6782	1.06641
$494$	−3.05959	−0.137657
$495$	0	0
$496$	20.0880	0.901978
$497$	9.91727	0.444850
$498$	−63.6599	−2.85267
$499$	−33.7801	−1.51221	−0.756103	−	0.654452i	$-0.772899\pi$
$499$	−33.7801	−1.51221	−0.756103	+	0.654452i	$0.772899\pi$
$500$	0	0
$501$	−41.8713	−1.87067
$502$	−28.0826	−1.25339
$503$	−19.6848	−0.877701	−0.438850	−	0.898560i	$-0.644614\pi$
$503$	−19.6848	−0.877701	−0.438850	+	0.898560i	$0.644614\pi$
$504$	−0.108584	−0.00483674
$505$	0	0
$506$	−0.583614	−0.0259448
$507$	21.8969	0.972477
$508$	10.3886	0.460921
$509$	−3.44241	−0.152582	−0.0762911	−	0.997086i	$-0.524308\pi$
$509$	−3.44241	−0.152582	−0.0762911	+	0.997086i	$0.524308\pi$
$510$	0	0
$511$	−3.84869	−0.170256
$512$	32.1634	1.42144
$513$	−5.42225	−0.239398
$514$	−59.6034	−2.62899
$515$	0	0
$516$	−19.4287	−0.855302
$517$	−3.02918	−0.133223
$518$	21.7596	0.956062
$519$	28.4026	1.24674
$520$	0	0
$521$	2.77397	0.121530	0.0607649	−	0.998152i	$-0.480646\pi$
$521$	2.77397	0.121530	0.0607649	+	0.998152i	$0.480646\pi$
$522$	11.6087	0.508098
$523$	−2.91148	−0.127310	−0.0636550	−	0.997972i	$-0.520276\pi$
$523$	−2.91148	−0.127310	−0.0636550	+	0.997972i	$0.520276\pi$
$524$	11.6060	0.507009
$525$	0	0
$526$	−8.83333	−0.385152
$527$	−14.3392	−0.624625
$528$	−2.11872	−0.0922054
$529$	1.00000	0.0434783
$530$	0	0
$531$	8.76661	0.380438
$532$	2.52328	0.109398
$533$	−7.14736	−0.309587
$534$	−61.6393	−2.66739
$535$	0	0
$536$	1.84818	0.0798291
$537$	−24.4758	−1.05621
$538$	35.6707	1.53787
$539$	−0.288897	−0.0124437
$540$	0	0
$541$	−37.8244	−1.62620	−0.813100	−	0.582125i	$-0.802222\pi$
$541$	−37.8244	−1.62620	−0.813100	+	0.582125i	$0.802222\pi$
$542$	31.9091	1.37061
$543$	−0.195510	−0.00839016
$544$	−22.0640	−0.945987
$545$	0	0
$546$	4.82979	0.206696
$547$	−23.5763	−1.00805	−0.504024	−	0.863690i	$-0.668148\pi$
$547$	−23.5763	−1.00805	−0.504024	+	0.863690i	$0.668148\pi$
$548$	−15.3555	−0.655956
$549$	0.887151	0.0378627
$550$	0	0
$551$	−10.4976	−0.447215
$552$	−0.313130	−0.0133277
$553$	3.22963	0.137338
$554$	13.7261	0.583165
$555$	0	0
$556$	−28.2427	−1.19776
$557$	−3.87212	−0.164067	−0.0820336	−	0.996630i	$-0.526141\pi$
$557$	−3.87212	−0.164067	−0.0820336	+	0.996630i	$0.526141\pi$
$558$	−7.03005	−0.297606
$559$	6.09251	0.257686
$560$	0	0
$561$	1.51238	0.0638529
$562$	−7.50915	−0.316754
$563$	−7.74307	−0.326331	−0.163166	−	0.986599i	$-0.552171\pi$
$563$	−7.74307	−0.326331	−0.163166	+	0.986599i	$0.552171\pi$
$564$	−41.7650	−1.75862
$565$	0	0
$566$	47.9502	2.01550
$567$	10.5507	0.443087
$568$	−1.62238	−0.0680737
$569$	34.6792	1.45383	0.726913	−	0.686729i	$-0.240954\pi$
$569$	34.6792	1.45383	0.726913	+	0.686729i	$0.240954\pi$
$570$	0	0
$571$	34.1099	1.42745	0.713727	−	0.700424i	$-0.247006\pi$
$571$	34.1099	1.42745	0.713727	+	0.700424i	$0.247006\pi$
$572$	−0.750921	−0.0313976
$573$	26.0645	1.08886
$574$	11.5597	0.482491
$575$	0	0
$576$	−5.73097	−0.238790
$577$	9.47800	0.394574	0.197287	−	0.980346i	$-0.436787\pi$
$577$	9.47800	0.394574	0.197287	+	0.980346i	$0.436787\pi$
$578$	−19.2315	−0.799924
$579$	−7.18075	−0.298422
$580$	0	0
$581$	−16.4634	−0.683019
$582$	9.39014	0.389234
$583$	−1.62449	−0.0672794
$584$	0.629614	0.0260536
$585$	0	0
$586$	2.28469	0.0943797
$587$	−23.0062	−0.949566	−0.474783	−	0.880103i	$-0.657474\pi$
$587$	−23.0062	−0.949566	−0.474783	+	0.880103i	$0.657474\pi$
$588$	−3.98319	−0.164264
$589$	6.35723	0.261945
$590$	0	0
$591$	−45.7748	−1.88292
$592$	41.2701	1.69619
$593$	6.59031	0.270632	0.135316	−	0.990803i	$-0.456795\pi$
$593$	6.59031	0.270632	0.135316	+	0.990803i	$0.456795\pi$
$594$	−2.60980	−0.107082
$595$	0	0
$596$	−0.368370	−0.0150890
$597$	−6.84605	−0.280190
$598$	2.52328	0.103185
$599$	−26.9282	−1.10026	−0.550128	−	0.835080i	$-0.685421\pi$
$599$	−26.9282	−1.10026	−0.550128	+	0.835080i	$0.685421\pi$
$600$	0	0
$601$	−3.68774	−0.150426	−0.0752130	−	0.997167i	$-0.523964\pi$
$601$	−3.68774	−0.150426	−0.0752130	+	0.997167i	$0.523964\pi$
$602$	−9.85362	−0.401603
$603$	7.49874	0.305372
$604$	22.9676	0.934537
$605$	0	0
$606$	−43.7490	−1.77718
$607$	6.77554	0.275011	0.137505	−	0.990501i	$-0.456092\pi$
$607$	6.77554	0.275011	0.137505	+	0.990501i	$0.456092\pi$
$608$	9.78200	0.396712
$609$	16.5713	0.671504
$610$	0	0
$611$	13.0968	0.529839
$612$	3.77772	0.152705
$613$	48.9684	1.97781	0.988907	−	0.148535i	$-0.0474557\pi$
$613$	48.9684	1.97781	0.988907	+	0.148535i	$0.0474557\pi$
$614$	45.0118	1.81653
$615$	0	0
$616$	0.0472612	0.00190421
$617$	−21.5060	−0.865797	−0.432899	−	0.901443i	$-0.642509\pi$
$617$	−21.5060	−0.865797	−0.432899	+	0.901443i	$0.642509\pi$
$618$	−39.5498	−1.59093
$619$	−2.33973	−0.0940419	−0.0470209	−	0.998894i	$-0.514973\pi$
$619$	−2.33973	−0.0940419	−0.0470209	+	0.998894i	$0.514973\pi$
$620$	0	0
$621$	4.47180	0.179447
$622$	60.1531	2.41192
$623$	−15.9409	−0.638658
$624$	9.16036	0.366708
$625$	0	0
$626$	−12.0692	−0.482381
$627$	−0.670509	−0.0267775
$628$	2.85216	0.113814
$629$	−29.4594	−1.17462
$630$	0	0
$631$	−10.4866	−0.417464	−0.208732	−	0.977973i	$-0.566934\pi$
$631$	−10.4866	−0.417464	−0.208732	+	0.977973i	$0.566934\pi$
$632$	−0.528341	−0.0210163
$633$	51.8356	2.06028
$634$	−2.63599	−0.104689
$635$	0	0
$636$	−22.3977	−0.888127
$637$	1.24906	0.0494895
$638$	−5.05266	−0.200037
$639$	−6.58261	−0.260404
$640$	0	0
$641$	29.6289	1.17027	0.585136	−	0.810935i	$-0.301041\pi$
$641$	29.6289	1.17027	0.585136	+	0.810935i	$0.301041\pi$
$642$	−46.0879	−1.81894
$643$	6.03764	0.238101	0.119051	−	0.992888i	$-0.462015\pi$
$643$	6.03764	0.238101	0.119051	+	0.992888i	$0.462015\pi$
$644$	−2.08098	−0.0820021
$645$	0	0
$646$	−6.69939	−0.263584
$647$	6.06691	0.238515	0.119257	−	0.992863i	$-0.461949\pi$
$647$	6.06691	0.238515	0.119257	+	0.992863i	$0.461949\pi$
$648$	−1.72601	−0.0678039
$649$	−3.81565	−0.149777
$650$	0	0
$651$	−10.0354	−0.393317
$652$	−0.178688	−0.00699796
$653$	−2.05945	−0.0805923	−0.0402962	−	0.999188i	$-0.512830\pi$
$653$	−2.05945	−0.0805923	−0.0402962	+	0.999188i	$0.512830\pi$
$654$	13.0257	0.509344
$655$	0	0
$656$	21.9245	0.856008
$657$	2.55458	0.0996636
$658$	−21.1818	−0.825754
$659$	−11.7855	−0.459098	−0.229549	−	0.973297i	$-0.573725\pi$
$659$	−11.7855	−0.459098	−0.229549	+	0.973297i	$0.573725\pi$
$660$	0	0
$661$	11.9257	0.463855	0.231928	−	0.972733i	$-0.425497\pi$
$661$	11.9257	0.463855	0.231928	+	0.972733i	$0.425497\pi$
$662$	−54.4224	−2.11519
$663$	−6.53884	−0.253948
$664$	2.69328	0.104520
$665$	0	0
$666$	−14.4430	−0.559655
$667$	8.65753	0.335221
$668$	45.5220	1.76130
$669$	2.08137	0.0804704
$670$	0	0
$671$	−0.386131	−0.0149064
$672$	−15.4416	−0.595673
$673$	14.8572	0.572704	0.286352	−	0.958124i	$-0.407557\pi$
$673$	14.8572	0.572704	0.286352	+	0.958124i	$0.407557\pi$
$674$	57.3186	2.20783
$675$	0	0
$676$	−23.8061	−0.915619
$677$	27.7131	1.06510	0.532551	−	0.846398i	$-0.321234\pi$
$677$	27.7131	1.06510	0.532551	+	0.846398i	$0.321234\pi$
$678$	11.6788	0.448522
$679$	2.42844	0.0931948
$680$	0	0
$681$	23.0761	0.884279
$682$	3.05982	0.117167
$683$	8.03921	0.307612	0.153806	−	0.988101i	$-0.450847\pi$
$683$	8.03921	0.307612	0.153806	+	0.988101i	$0.450847\pi$
$684$	−1.67484	−0.0640389
$685$	0	0
$686$	−2.02014	−0.0771294
$687$	26.0693	0.994607
$688$	−18.6887	−0.712502
$689$	7.02353	0.267575
$690$	0	0
$691$	−17.8600	−0.679428	−0.339714	−	0.940529i	$-0.610330\pi$
$691$	−17.8600	−0.679428	−0.339714	+	0.940529i	$0.610330\pi$
$692$	−30.8790	−1.17384
$693$	0.191756	0.00728422
$694$	−9.60107	−0.364452
$695$	0	0
$696$	−2.71093	−0.102758
$697$	−15.6501	−0.592791
$698$	32.8652	1.24397
$699$	6.42152	0.242884
$700$	0	0
$701$	−33.3687	−1.26032	−0.630158	−	0.776467i	$-0.717010\pi$
$701$	−33.3687	−1.26032	−0.630158	+	0.776467i	$0.717010\pi$
$702$	11.2836	0.425871
$703$	13.0607	0.492594
$704$	2.49440	0.0940111
$705$	0	0
$706$	−22.8751	−0.860916
$707$	−11.3142	−0.425513
$708$	−52.6085	−1.97715
$709$	44.8527	1.68448	0.842239	−	0.539105i	$-0.181237\pi$
$709$	44.8527	1.68448	0.842239	+	0.539105i	$0.181237\pi$
$710$	0	0
$711$	−2.14367	−0.0803941
$712$	2.60780	0.0977313
$713$	−5.24288	−0.196347
$714$	10.5755	0.395778
$715$	0	0
$716$	26.6099	0.994457
$717$	−25.9200	−0.967999
$718$	48.0278	1.79238
$719$	4.02126	0.149968	0.0749838	−	0.997185i	$-0.476109\pi$
$719$	4.02126	0.149968	0.0749838	+	0.997185i	$0.476109\pi$
$720$	0	0
$721$	−10.2282	−0.380918
$722$	−35.4126	−1.31792
$723$	−51.6870	−1.92226
$724$	0.212557	0.00789961
$725$	0	0
$726$	42.2114	1.56661
$727$	6.17662	0.229078	0.114539	−	0.993419i	$-0.463461\pi$
$727$	6.17662	0.229078	0.114539	+	0.993419i	$0.463461\pi$
$728$	−0.204336	−0.00757319
$729$	18.6753	0.691676
$730$	0	0
$731$	13.3404	0.493412
$732$	−5.32381	−0.196774
$733$	−35.6211	−1.31570	−0.657848	−	0.753151i	$-0.728533\pi$
$733$	−35.6211	−1.31570	−0.657848	+	0.753151i	$0.728533\pi$
$734$	16.5501	0.610877
$735$	0	0
$736$	−8.06733	−0.297366
$737$	−3.26382	−0.120224
$738$	−7.67275	−0.282438
$739$	−10.7377	−0.394994	−0.197497	−	0.980304i	$-0.563281\pi$
$739$	−10.7377	−0.394994	−0.197497	+	0.980304i	$0.563281\pi$
$740$	0	0
$741$	2.89897	0.106496
$742$	−11.3594	−0.417016
$743$	−7.99442	−0.293287	−0.146643	−	0.989189i	$-0.546847\pi$
$743$	−7.99442	−0.293287	−0.146643	+	0.989189i	$0.546847\pi$
$744$	1.64170	0.0601877
$745$	0	0
$746$	−46.3852	−1.69828
$747$	10.9276	0.399822
$748$	−1.64425	−0.0601196
$749$	−11.9190	−0.435512
$750$	0	0
$751$	8.01968	0.292642	0.146321	−	0.989237i	$-0.453257\pi$
$751$	8.01968	0.292642	0.146321	+	0.989237i	$0.453257\pi$
$752$	−40.1743	−1.46501
$753$	26.6084	0.969663
$754$	21.8454	0.795561
$755$	0	0
$756$	−9.30572	−0.338446
$757$	−19.5557	−0.710762	−0.355381	−	0.934721i	$-0.615649\pi$
$757$	−19.5557	−0.710762	−0.355381	+	0.934721i	$0.615649\pi$
$758$	10.3528	0.376029
$759$	0.552976	0.0200718
$760$	0	0
$761$	4.11192	0.149057	0.0745285	−	0.997219i	$-0.476255\pi$
$761$	4.11192	0.149057	0.0745285	+	0.997219i	$0.476255\pi$
$762$	−19.3035	−0.699292
$763$	3.36864	0.121953
$764$	−28.3370	−1.02520
$765$	0	0
$766$	−60.2017	−2.17517
$767$	16.4971	0.595677
$768$	−27.9969	−1.01025
$769$	5.29113	0.190803	0.0954014	−	0.995439i	$-0.469587\pi$
$769$	5.29113	0.190803	0.0954014	+	0.995439i	$0.469587\pi$
$770$	0	0
$771$	56.4744	2.03388
$772$	7.80683	0.280974
$773$	37.9458	1.36482	0.682408	−	0.730972i	$-0.260933\pi$
$773$	37.9458	1.36482	0.682408	+	0.730972i	$0.260933\pi$
$774$	6.54036	0.235089
$775$	0	0
$776$	−0.397272	−0.0142612
$777$	−20.6173	−0.739642
$778$	−67.0633	−2.40434
$779$	6.93842	0.248595
$780$	0	0
$781$	2.86507	0.102520
$782$	5.52506	0.197576
$783$	38.7147	1.38355
$784$	−3.83148	−0.136839
$785$	0	0
$786$	−21.5655	−0.769214
$787$	−30.5731	−1.08982	−0.544908	−	0.838496i	$-0.683435\pi$
$787$	−30.5731	−1.08982	−0.544908	+	0.838496i	$0.683435\pi$
$788$	49.7658	1.77283
$789$	8.36961	0.297966
$790$	0	0
$791$	3.02032	0.107390
$792$	−0.0313698	−0.00111468
$793$	1.66945	0.0592840
$794$	21.5374	0.764333
$795$	0	0
$796$	7.44295	0.263808
$797$	49.6975	1.76038	0.880189	−	0.474624i	$-0.157416\pi$
$797$	49.6975	1.76038	0.880189	+	0.474624i	$0.157416\pi$
$798$	−4.68860	−0.165975
$799$	28.6772	1.01453
$800$	0	0
$801$	10.5808	0.373854
$802$	−69.1945	−2.44334
$803$	−1.11188	−0.0392373
$804$	−45.0001	−1.58703
$805$	0	0
$806$	−13.2292	−0.465980
$807$	−33.7981	−1.18975
$808$	1.85091	0.0651146
$809$	45.1229	1.58644	0.793218	−	0.608938i	$-0.208404\pi$
$809$	45.1229	1.58644	0.793218	+	0.608938i	$0.208404\pi$
$810$	0	0
$811$	−8.10266	−0.284523	−0.142261	−	0.989829i	$-0.545437\pi$
$811$	−8.10266	−0.284523	−0.142261	+	0.989829i	$0.545437\pi$
$812$	−18.0162	−0.632243
$813$	−30.2340	−1.06035
$814$	6.28629	0.220334
$815$	0	0
$816$	20.0579	0.702166
$817$	−5.91441	−0.206919
$818$	0.649428	0.0227067
$819$	−0.829066	−0.0289699
$820$	0	0
$821$	−21.5517	−0.752159	−0.376079	−	0.926587i	$-0.622728\pi$
$821$	−21.5517	−0.752159	−0.376079	+	0.926587i	$0.622728\pi$
$822$	28.5327	0.995191
$823$	44.5085	1.55147	0.775735	−	0.631059i	$-0.217380\pi$
$823$	44.5085	1.55147	0.775735	+	0.631059i	$0.217380\pi$
$824$	1.67325	0.0582904
$825$	0	0
$826$	−26.6813	−0.928362
$827$	1.02322	0.0355808	0.0177904	−	0.999842i	$-0.494337\pi$
$827$	1.02322	0.0355808	0.0177904	+	0.999842i	$0.494337\pi$
$828$	1.38126	0.0480020
$829$	5.89560	0.204763	0.102381	−	0.994745i	$-0.467354\pi$
$829$	5.89560	0.204763	0.102381	+	0.994745i	$0.467354\pi$
$830$	0	0
$831$	−13.0055	−0.451156
$832$	−10.7846	−0.373889
$833$	2.73499	0.0947616
$834$	52.4788	1.81719
$835$	0	0
$836$	0.728970	0.0252119
$837$	−23.4451	−0.810381
$838$	−71.9319	−2.48485
$839$	0.805263	0.0278008	0.0139004	−	0.999903i	$-0.495575\pi$
$839$	0.805263	0.0278008	0.0139004	+	0.999903i	$0.495575\pi$
$840$	0	0
$841$	45.9529	1.58458
$842$	−27.7917	−0.957766
$843$	7.11495	0.245052
$844$	−56.3551	−1.93982
$845$	0	0
$846$	14.0595	0.483376
$847$	10.9165	0.375097
$848$	−21.5446	−0.739846
$849$	−45.4330	−1.55926
$850$	0	0
$851$	−10.7713	−0.369236
$852$	39.5024	1.35333
$853$	−28.3539	−0.970818	−0.485409	−	0.874287i	$-0.661329\pi$
$853$	−28.3539	−0.970818	−0.485409	+	0.874287i	$0.661329\pi$
$854$	−2.70006	−0.0923942
$855$	0	0
$856$	1.94986	0.0666448
$857$	−31.3454	−1.07074	−0.535368	−	0.844619i	$-0.679827\pi$
$857$	−31.3454	−1.07074	−0.535368	+	0.844619i	$0.679827\pi$
$858$	1.39531	0.0476352
$859$	−0.926887	−0.0316250	−0.0158125	−	0.999875i	$-0.505033\pi$
$859$	−0.926887	−0.0316250	−0.0158125	+	0.999875i	$0.505033\pi$
$860$	0	0
$861$	−10.9528	−0.373271
$862$	27.3693	0.932203
$863$	−50.8527	−1.73104	−0.865522	−	0.500870i	$-0.833013\pi$
$863$	−50.8527	−1.73104	−0.865522	+	0.500870i	$0.833013\pi$
$864$	−36.0754	−1.22731
$865$	0	0
$866$	−2.44617	−0.0831243
$867$	18.2219	0.618848
$868$	10.9103	0.370321
$869$	0.933031	0.0316509
$870$	0	0
$871$	14.1112	0.478141
$872$	−0.551082	−0.0186620
$873$	−1.61188	−0.0545539
$874$	−2.44952	−0.0828561
$875$	0	0
$876$	−15.3301	−0.517955
$877$	−27.1721	−0.917536	−0.458768	−	0.888556i	$-0.651709\pi$
$877$	−27.1721	−0.917536	−0.458768	+	0.888556i	$0.651709\pi$
$878$	1.59590	0.0538589
$879$	−2.16475	−0.0730152
$880$	0	0
$881$	−14.2898	−0.481437	−0.240718	−	0.970595i	$-0.577383\pi$
$881$	−14.2898	−0.481437	−0.240718	+	0.970595i	$0.577383\pi$
$882$	1.34088	0.0451496
$883$	−14.6190	−0.491967	−0.245984	−	0.969274i	$-0.579111\pi$
$883$	−14.6190	−0.491967	−0.245984	+	0.969274i	$0.579111\pi$
$884$	7.10896	0.239100
$885$	0	0
$886$	58.7202	1.97275
$887$	−29.0006	−0.973746	−0.486873	−	0.873473i	$-0.661863\pi$
$887$	−29.0006	−0.973746	−0.486873	+	0.873473i	$0.661863\pi$
$888$	3.37282	0.113184
$889$	−4.99218	−0.167432
$890$	0	0
$891$	3.04807	0.102114
$892$	−2.26284	−0.0757656
$893$	−12.7139	−0.425455
$894$	0.684481	0.0228925
$895$	0	0
$896$	1.30766	0.0436859
$897$	−2.39081	−0.0798270
$898$	20.8149	0.694602
$899$	−45.3904	−1.51385
$900$	0	0
$901$	15.3790	0.512348
$902$	3.33956	0.111195
$903$	9.33634	0.310694
$904$	−0.494100	−0.0164335
$905$	0	0
$906$	−42.6769	−1.41784
$907$	−19.1097	−0.634527	−0.317263	−	0.948337i	$-0.602764\pi$
$907$	−19.1097	−0.634527	−0.317263	+	0.948337i	$0.602764\pi$
$908$	−25.0881	−0.832578
$909$	7.50981	0.249085
$910$	0	0
$911$	22.1076	0.732457	0.366229	−	0.930525i	$-0.380649\pi$
$911$	22.1076	0.732457	0.366229	+	0.930525i	$0.380649\pi$
$912$	−8.89258	−0.294463
$913$	−4.75624	−0.157409
$914$	11.2538	0.372243
$915$	0	0
$916$	−28.3423	−0.936456
$917$	−5.57716	−0.184174
$918$	24.7070	0.815451
$919$	51.7395	1.70673	0.853365	−	0.521314i	$-0.174558\pi$
$919$	51.7395	1.70673	0.853365	+	0.521314i	$0.174558\pi$
$920$	0	0
$921$	−42.6488	−1.40533
$922$	−50.5296	−1.66410
$923$	−12.3873	−0.407732
$924$	−1.15073	−0.0378563
$925$	0	0
$926$	28.3119	0.930387
$927$	6.78900	0.222980
$928$	−69.8431	−2.29271
$929$	−2.47995	−0.0813647	−0.0406823	−	0.999172i	$-0.512953\pi$
$929$	−2.47995	−0.0813647	−0.0406823	+	0.999172i	$0.512953\pi$
$930$	0	0
$931$	−1.21255	−0.0397396
$932$	−6.98141	−0.228684
$933$	−56.9953	−1.86594
$934$	61.5129	2.01276
$935$	0	0
$936$	0.135628	0.00443315
$937$	−38.1815	−1.24733	−0.623667	−	0.781690i	$-0.714358\pi$
$937$	−38.1815	−1.24733	−0.623667	+	0.781690i	$0.714358\pi$
$938$	−22.8226	−0.745183
$939$	11.4356	0.373186
$940$	0	0
$941$	4.13993	0.134958	0.0674789	−	0.997721i	$-0.478504\pi$
$941$	4.13993	0.134958	0.0674789	+	0.997721i	$0.478504\pi$
$942$	−5.29970	−0.172673
$943$	−5.72220	−0.186340
$944$	−50.6048	−1.64705
$945$	0	0
$946$	−2.84668	−0.0925537
$947$	30.5369	0.992315	0.496157	−	0.868233i	$-0.334744\pi$
$947$	30.5369	0.992315	0.496157	+	0.868233i	$0.334744\pi$
$948$	12.8642	0.417811
$949$	4.80724	0.156050
$950$	0	0
$951$	2.49761	0.0809906
$952$	−0.447421	−0.0145010
$953$	21.4356	0.694366	0.347183	−	0.937797i	$-0.387138\pi$
$953$	21.4356	0.694366	0.347183	+	0.937797i	$0.387138\pi$
$954$	7.53982	0.244111
$955$	0	0
$956$	28.1799	0.911403
$957$	4.78741	0.154755
$958$	−21.0881	−0.681326
$959$	7.37899	0.238280
$960$	0	0
$961$	−3.51221	−0.113297
$962$	−27.1790	−0.876287
$963$	7.91130	0.254938
$964$	56.1935	1.80987
$965$	0	0
$966$	3.86674	0.124410
$967$	47.5812	1.53011	0.765054	−	0.643967i	$-0.222712\pi$
$967$	47.5812	1.53011	0.765054	+	0.643967i	$0.222712\pi$
$968$	−1.78586	−0.0573996
$969$	6.34769	0.203917
$970$	0	0
$971$	34.3631	1.10276	0.551382	−	0.834253i	$-0.314101\pi$
$971$	34.3631	1.10276	0.551382	+	0.834253i	$0.314101\pi$
$972$	14.1083	0.452522
$973$	13.5718	0.435093
$974$	−12.7573	−0.408770
$975$	0	0
$976$	−5.12104	−0.163920
$977$	19.9000	0.636657	0.318329	−	0.947980i	$-0.396878\pi$
$977$	19.9000	0.636657	0.318329	+	0.947980i	$0.396878\pi$
$978$	0.332026	0.0106170
$979$	−4.60528	−0.147185
$980$	0	0
$981$	−2.23594	−0.0713881
$982$	40.9038	1.30529
$983$	−16.7671	−0.534787	−0.267393	−	0.963587i	$-0.586162\pi$
$983$	−16.7671	−0.534787	−0.267393	+	0.963587i	$0.586162\pi$
$984$	1.79179	0.0571202
$985$	0	0
$986$	47.8334	1.52333
$987$	20.0699	0.638831
$988$	−3.15173	−0.100270
$989$	4.87768	0.155101
$990$	0	0
$991$	−28.0399	−0.890718	−0.445359	−	0.895352i	$-0.646924\pi$
$991$	−28.0399	−0.890718	−0.445359	+	0.895352i	$0.646924\pi$
$992$	42.2960	1.34290
$993$	51.5654	1.63638
$994$	20.0343	0.635450
$995$	0	0
$996$	−65.5770	−2.07789
$997$	−39.6366	−1.25530	−0.627652	−	0.778494i	$-0.715984\pi$
$997$	−39.6366	−1.25530	−0.627652	+	0.778494i	$0.715984\pi$
$998$	−68.2407	−2.16012
$999$	−48.1671	−1.52394

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4025.2.a.v.1.7	yes	8
5.4	even	2		4025.2.a.u.1.2	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4025.2.a.u.1.2	✓	8	5.4	even	2
4025.2.a.v.1.7	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)

\(f(q)\)	\(=\)	\(q+2.02014 q^{2} -1.91409 q^{3} +2.08098 q^{4} -3.86674 q^{6} -1.00000 q^{7} +0.163592 q^{8} +0.663753 q^{9} +O(q^{10})\)	\(q+2.02014 q^{2} -1.91409 q^{3} +2.08098 q^{4} -3.86674 q^{6} -1.00000 q^{7} +0.163592 q^{8} +0.663753 q^{9} -0.288897 q^{11} -3.98319 q^{12} +1.24906 q^{13} -2.02014 q^{14} -3.83148 q^{16} +2.73499 q^{17} +1.34088 q^{18} -1.21255 q^{19} +1.91409 q^{21} -0.583614 q^{22} +1.00000 q^{23} -0.313130 q^{24} +2.52328 q^{26} +4.47180 q^{27} -2.08098 q^{28} +8.65753 q^{29} -5.24288 q^{31} -8.06733 q^{32} +0.552976 q^{33} +5.52506 q^{34} +1.38126 q^{36} -10.7713 q^{37} -2.44952 q^{38} -2.39081 q^{39} -5.72220 q^{41} +3.86674 q^{42} +4.87768 q^{43} -0.601190 q^{44} +2.02014 q^{46} +10.4853 q^{47} +7.33381 q^{48} +1.00000 q^{49} -5.23502 q^{51} +2.59927 q^{52} +5.62306 q^{53} +9.03367 q^{54} -0.163592 q^{56} +2.32092 q^{57} +17.4895 q^{58} +13.2076 q^{59} +1.33657 q^{61} -10.5914 q^{62} -0.663753 q^{63} -8.63420 q^{64} +1.11709 q^{66} +11.2975 q^{67} +5.69145 q^{68} -1.91409 q^{69} -9.91727 q^{71} +0.108584 q^{72} +3.84869 q^{73} -21.7596 q^{74} -2.52328 q^{76} +0.288897 q^{77} -4.82979 q^{78} -3.22963 q^{79} -10.5507 q^{81} -11.5597 q^{82} +16.4634 q^{83} +3.98319 q^{84} +9.85362 q^{86} -16.5713 q^{87} -0.0472612 q^{88} +15.9409 q^{89} -1.24906 q^{91} +2.08098 q^{92} +10.0354 q^{93} +21.1818 q^{94} +15.4416 q^{96} -2.42844 q^{97} +2.02014 q^{98} -0.191756 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + q^{2} + 4 q^{3} + 5 q^{4} - q^{6} - 8 q^{7}+O(q^{10}) \) 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

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Groups

Database

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