LMFDB - Embedded newform 1925.2.a.bd.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1925,2,Mod(1,1925)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1925, 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("1925.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1925 = 5^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1925.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:	$15.3712023891$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 13x^{5} + 12x^{4} + 47x^{3} - 37x^{2} - 35x + 6 $ x^7 - x^6 - 13x^5 + 12x^4 + 47x^3 - 37x^2 - 35*x + 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.59433$ of defining polynomial
Character	$\chi$	$=$	1925.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.59433 q^{2} -3.08438 q^{3} +0.541890 q^{4} -4.91751 q^{6} +1.00000 q^{7} -2.32471 q^{8} +6.51337 q^{9} +O(q^{10})$	\(q+1.59433 q^{2} -3.08438 q^{3} +0.541890 q^{4} -4.91751 q^{6} +1.00000 q^{7} -2.32471 q^{8} +6.51337 q^{9} +1.00000 q^{11} -1.67139 q^{12} -1.37503 q^{13} +1.59433 q^{14} -4.79013 q^{16} -5.12233 q^{17} +10.3845 q^{18} +1.54189 q^{19} -3.08438 q^{21} +1.59433 q^{22} +1.77568 q^{23} +7.17028 q^{24} -2.19225 q^{26} -10.8366 q^{27} +0.541890 q^{28} +5.66269 q^{29} -4.69319 q^{31} -2.98764 q^{32} -3.08438 q^{33} -8.16669 q^{34} +3.52953 q^{36} -0.0524408 q^{37} +2.45828 q^{38} +4.24110 q^{39} -1.22662 q^{41} -4.91751 q^{42} -3.84404 q^{43} +0.541890 q^{44} +2.83102 q^{46} +12.7556 q^{47} +14.7746 q^{48} +1.00000 q^{49} +15.7992 q^{51} -0.745113 q^{52} +7.74716 q^{53} -17.2770 q^{54} -2.32471 q^{56} -4.75577 q^{57} +9.02821 q^{58} +12.6697 q^{59} +7.02163 q^{61} -7.48250 q^{62} +6.51337 q^{63} +4.81699 q^{64} -4.91751 q^{66} +9.10448 q^{67} -2.77574 q^{68} -5.47685 q^{69} +12.3798 q^{71} -15.1417 q^{72} +8.11874 q^{73} -0.0836080 q^{74} +0.835534 q^{76} +1.00000 q^{77} +6.76172 q^{78} +10.9299 q^{79} +13.8839 q^{81} -1.95563 q^{82} -15.2765 q^{83} -1.67139 q^{84} -6.12867 q^{86} -17.4659 q^{87} -2.32471 q^{88} -10.1105 q^{89} -1.37503 q^{91} +0.962221 q^{92} +14.4756 q^{93} +20.3366 q^{94} +9.21500 q^{96} -10.5846 q^{97} +1.59433 q^{98} +6.51337 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + q^{2} + 13 q^{4} + 3 q^{6} + 7 q^{7} + 9 q^{9}+O(q^{10}) $ 7 * q + q^2 + 13 * q^4 + 3 * q^6 + 7 * q^7 + 9 * q^9	$ 7 q + q^{2} + 13 q^{4} + 3 q^{6} + 7 q^{7} + 9 q^{9} + 7 q^{11} - 9 q^{12} - 3 q^{13} + q^{14} + 21 q^{16} - 2 q^{17} + 8 q^{18} + 20 q^{19} + q^{22} + 11 q^{23} + 18 q^{24} - 13 q^{26} - 12 q^{27} + 13 q^{28} + 4 q^{29} + 6 q^{31} + q^{32} - 7 q^{34} + 12 q^{36} + 19 q^{37} + 3 q^{38} - 10 q^{39} + 24 q^{41} + 3 q^{42} + 13 q^{44} + 33 q^{46} - q^{47} - 15 q^{48} + 7 q^{49} + 19 q^{51} - 29 q^{52} + 7 q^{53} + 9 q^{54} - 9 q^{57} + 37 q^{58} + 9 q^{59} + 18 q^{61} - 40 q^{62} + 9 q^{63} + 8 q^{64} + 3 q^{66} + 18 q^{68} - 15 q^{69} + 18 q^{71} - 64 q^{72} + 5 q^{73} - 24 q^{74} + 88 q^{76} + 7 q^{77} + 79 q^{78} + 25 q^{79} - q^{81} - 60 q^{82} - 17 q^{83} - 9 q^{84} - 41 q^{86} - 24 q^{87} - 16 q^{89} - 3 q^{91} + 28 q^{92} + 26 q^{93} - 31 q^{94} + 17 q^{96} - 4 q^{97} + q^{98} + 9 q^{99}+O(q^{100}) $ 7 * q + q^2 + 13 * q^4 + 3 * q^6 + 7 * q^7 + 9 * q^9 + 7 * q^11 - 9 * q^12 - 3 * q^13 + q^14 + 21 * q^16 - 2 * q^17 + 8 * q^18 + 20 * q^19 + q^22 + 11 * q^23 + 18 * q^24 - 13 * q^26 - 12 * q^27 + 13 * q^28 + 4 * q^29 + 6 * q^31 + q^32 - 7 * q^34 + 12 * q^36 + 19 * q^37 + 3 * q^38 - 10 * q^39 + 24 * q^41 + 3 * q^42 + 13 * q^44 + 33 * q^46 - q^47 - 15 * q^48 + 7 * q^49 + 19 * q^51 - 29 * q^52 + 7 * q^53 + 9 * q^54 - 9 * q^57 + 37 * q^58 + 9 * q^59 + 18 * q^61 - 40 * q^62 + 9 * q^63 + 8 * q^64 + 3 * q^66 + 18 * q^68 - 15 * q^69 + 18 * q^71 - 64 * q^72 + 5 * q^73 - 24 * q^74 + 88 * q^76 + 7 * q^77 + 79 * q^78 + 25 * q^79 - q^81 - 60 * q^82 - 17 * q^83 - 9 * q^84 - 41 * q^86 - 24 * q^87 - 16 * q^89 - 3 * q^91 + 28 * q^92 + 26 * q^93 - 31 * q^94 + 17 * q^96 - 4 * q^97 + q^98 + 9 * 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 $

$2$	1.59433	1.12736	0.563681	−	0.825993i	$-0.309385\pi$
$2$	1.59433	1.12736	0.563681	+	0.825993i	$0.309385\pi$
$3$	−3.08438	−1.78076	−0.890382	−	0.455213i	$-0.849563\pi$
$3$	−3.08438	−1.78076	−0.890382	+	0.455213i	$0.849563\pi$
$4$	0.541890	0.270945
$5$	0	0
$5$	0	0
$6$	−4.91751	−2.00757
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−2.32471	−0.821909
$9$	6.51337	2.17112
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−1.67139	−0.482489
$13$	−1.37503	−0.381364	−0.190682	−	0.981652i	$-0.561070\pi$
$13$	−1.37503	−0.381364	−0.190682	+	0.981652i	$0.561070\pi$
$14$	1.59433	0.426103
$15$	0	0
$16$	−4.79013	−1.19753
$17$	−5.12233	−1.24235	−0.621174	−	0.783673i	$-0.713344\pi$
$17$	−5.12233	−1.24235	−0.621174	+	0.783673i	$0.713344\pi$
$18$	10.3845	2.44764
$19$	1.54189	0.353734	0.176867	−	0.984235i	$-0.443404\pi$
$19$	1.54189	0.353734	0.176867	+	0.984235i	$0.443404\pi$
$20$	0	0
$21$	−3.08438	−0.673066
$22$	1.59433	0.339912
$23$	1.77568	0.370254	0.185127	−	0.982715i	$-0.440730\pi$
$23$	1.77568	0.370254	0.185127	+	0.982715i	$0.440730\pi$
$24$	7.17028	1.46363
$25$	0	0
$26$	−2.19225	−0.429935
$27$	−10.8366	−2.08550
$28$	0.541890	0.102408
$29$	5.66269	1.05154	0.525768	−	0.850628i	$-0.323778\pi$
$29$	5.66269	1.05154	0.525768	+	0.850628i	$0.323778\pi$
$30$	0	0
$31$	−4.69319	−0.842922	−0.421461	−	0.906847i	$-0.638483\pi$
$31$	−4.69319	−0.842922	−0.421461	+	0.906847i	$0.638483\pi$
$32$	−2.98764	−0.528145
$33$	−3.08438	−0.536921
$34$	−8.16669	−1.40058
$35$	0	0
$36$	3.52953	0.588255
$37$	−0.0524408	−0.00862122	−0.00431061	−	0.999991i	$-0.501372\pi$
$37$	−0.0524408	−0.00862122	−0.00431061	+	0.999991i	$0.501372\pi$
$38$	2.45828	0.398786
$39$	4.24110	0.679120
$40$	0	0
$41$	−1.22662	−0.191565	−0.0957827	−	0.995402i	$-0.530535\pi$
$41$	−1.22662	−0.191565	−0.0957827	+	0.995402i	$0.530535\pi$
$42$	−4.91751	−0.758789
$43$	−3.84404	−0.586211	−0.293105	−	0.956080i	$-0.594689\pi$
$43$	−3.84404	−0.586211	−0.293105	+	0.956080i	$0.594689\pi$
$44$	0.541890	0.0816929
$45$	0	0
$46$	2.83102	0.417410
$47$	12.7556	1.86059	0.930297	−	0.366806i	$-0.119549\pi$
$47$	12.7556	1.86059	0.930297	+	0.366806i	$0.119549\pi$
$48$	14.7746	2.13253
$49$	1.00000	0.142857
$50$	0	0
$51$	15.7992	2.21233
$52$	−0.745113	−0.103329
$53$	7.74716	1.06415	0.532077	−	0.846696i	$-0.321412\pi$
$53$	7.74716	1.06415	0.532077	+	0.846696i	$0.321412\pi$
$54$	−17.2770	−2.35111
$55$	0	0
$56$	−2.32471	−0.310652
$57$	−4.75577	−0.629917
$58$	9.02821	1.18546
$59$	12.6697	1.64945	0.824727	−	0.565532i	$-0.191329\pi$
$59$	12.6697	1.64945	0.824727	+	0.565532i	$0.191329\pi$
$60$	0	0
$61$	7.02163	0.899027	0.449514	−	0.893273i	$-0.351597\pi$
$61$	7.02163	0.899027	0.449514	+	0.893273i	$0.351597\pi$
$62$	−7.48250	−0.950278
$63$	6.51337	0.820608
$64$	4.81699	0.602123
$65$	0	0
$66$	−4.91751	−0.605304
$67$	9.10448	1.11229	0.556144	−	0.831086i	$-0.312280\pi$
$67$	9.10448	1.11229	0.556144	+	0.831086i	$0.312280\pi$
$68$	−2.77574	−0.336608
$69$	−5.47685	−0.659336
$70$	0	0
$71$	12.3798	1.46921	0.734606	−	0.678493i	$-0.237367\pi$
$71$	12.3798	1.46921	0.734606	+	0.678493i	$0.237367\pi$
$72$	−15.1417	−1.78447
$73$	8.11874	0.950227	0.475114	−	0.879924i	$-0.342407\pi$
$73$	8.11874	0.950227	0.475114	+	0.879924i	$0.342407\pi$
$74$	−0.0836080	−0.00971923
$75$	0	0
$76$	0.835534	0.0958423
$77$	1.00000	0.113961
$78$	6.76172	0.765614
$79$	10.9299	1.22971	0.614857	−	0.788639i	$-0.289214\pi$
$79$	10.9299	1.22971	0.614857	+	0.788639i	$0.289214\pi$
$80$	0	0
$81$	13.8839	1.54265
$82$	−1.95563	−0.215964
$83$	−15.2765	−1.67681	−0.838404	−	0.545049i	$-0.816511\pi$
$83$	−15.2765	−1.67681	−0.838404	+	0.545049i	$0.816511\pi$
$84$	−1.67139	−0.182364
$85$	0	0
$86$	−6.12867	−0.660872
$87$	−17.4659	−1.87254
$88$	−2.32471	−0.247815
$89$	−10.1105	−1.07171	−0.535857	−	0.844309i	$-0.680011\pi$
$89$	−10.1105	−1.07171	−0.535857	+	0.844309i	$0.680011\pi$
$90$	0	0
$91$	−1.37503	−0.144142
$92$	0.962221	0.100318
$93$	14.4756	1.50105
$94$	20.3366	2.09756
$95$	0	0
$96$	9.21500	0.940502
$97$	−10.5846	−1.07470	−0.537349	−	0.843360i	$-0.680574\pi$
$97$	−10.5846	−1.07470	−0.537349	+	0.843360i	$0.680574\pi$
$98$	1.59433	0.161052
$99$	6.51337	0.654618
$100$	0	0
$101$	8.65019	0.860726	0.430363	−	0.902656i	$-0.358386\pi$
$101$	8.65019	0.860726	0.430363	+	0.902656i	$0.358386\pi$
$102$	25.1891	2.49410
$103$	−5.38243	−0.530347	−0.265173	−	0.964201i	$-0.585429\pi$
$103$	−5.38243	−0.530347	−0.265173	+	0.964201i	$0.585429\pi$
$104$	3.19654	0.313447
$105$	0	0
$106$	12.3515	1.19969
$107$	6.85400	0.662602	0.331301	−	0.943525i	$-0.392513\pi$
$107$	6.85400	0.662602	0.331301	+	0.943525i	$0.392513\pi$
$108$	−5.87222	−0.565054
$109$	1.74671	0.167304	0.0836521	−	0.996495i	$-0.473342\pi$
$109$	1.74671	0.167304	0.0836521	+	0.996495i	$0.473342\pi$
$110$	0	0
$111$	0.161747	0.0153524
$112$	−4.79013	−0.452625
$113$	−16.7795	−1.57848	−0.789239	−	0.614086i	$-0.789525\pi$
$113$	−16.7795	−1.57848	−0.789239	+	0.614086i	$0.789525\pi$
$114$	−7.58226	−0.710144
$115$	0	0
$116$	3.06856	0.284908
$117$	−8.95607	−0.827989
$118$	20.1997	1.85953
$119$	−5.12233	−0.469563
$120$	0	0
$121$	1.00000	0.0909091
$122$	11.1948	1.01353
$123$	3.78335	0.341133
$124$	−2.54319	−0.228385
$125$	0	0
$126$	10.3845	0.925122
$127$	10.7958	0.957976	0.478988	−	0.877821i	$-0.341004\pi$
$127$	10.7958	0.957976	0.478988	+	0.877821i	$0.341004\pi$
$128$	13.6551	1.20696
$129$	11.8565	1.04390
$130$	0	0
$131$	−4.92239	−0.430071	−0.215036	−	0.976606i	$-0.568987\pi$
$131$	−4.92239	−0.430071	−0.215036	+	0.976606i	$0.568987\pi$
$132$	−1.67139	−0.145476
$133$	1.54189	0.133699
$134$	14.5155	1.25395
$135$	0	0
$136$	11.9079	1.02110
$137$	11.7835	1.00673	0.503366	−	0.864073i	$-0.332095\pi$
$137$	11.7835	1.00673	0.503366	+	0.864073i	$0.332095\pi$
$138$	−8.73191	−0.743310
$139$	15.0887	1.27981	0.639904	−	0.768454i	$-0.278974\pi$
$139$	15.0887	1.27981	0.639904	+	0.768454i	$0.278974\pi$
$140$	0	0
$141$	−39.3430	−3.31328
$142$	19.7375	1.65633
$143$	−1.37503	−0.114986
$144$	−31.1999	−2.59999
$145$	0	0
$146$	12.9440	1.07125
$147$	−3.08438	−0.254395
$148$	−0.0284171	−0.00233587
$149$	−18.0250	−1.47667	−0.738335	−	0.674435i	$-0.764387\pi$
$149$	−18.0250	−1.47667	−0.738335	+	0.674435i	$0.764387\pi$
$150$	0	0
$151$	16.3085	1.32717	0.663585	−	0.748101i	$-0.269034\pi$
$151$	16.3085	1.32717	0.663585	+	0.748101i	$0.269034\pi$
$152$	−3.58445	−0.290737
$153$	−33.3636	−2.69729
$154$	1.59433	0.128475
$155$	0	0
$156$	2.29821	0.184004
$157$	9.03944	0.721426	0.360713	−	0.932677i	$-0.382533\pi$
$157$	9.03944	0.721426	0.360713	+	0.932677i	$0.382533\pi$
$158$	17.4259	1.38633
$159$	−23.8951	−1.89501
$160$	0	0
$161$	1.77568	0.139943
$162$	22.1355	1.73913
$163$	−4.45489	−0.348934	−0.174467	−	0.984663i	$-0.555820\pi$
$163$	−4.45489	−0.348934	−0.174467	+	0.984663i	$0.555820\pi$
$164$	−0.664691	−0.0519037
$165$	0	0
$166$	−24.3557	−1.89037
$167$	13.5912	1.05172	0.525859	−	0.850572i	$-0.323744\pi$
$167$	13.5912	1.05172	0.525859	+	0.850572i	$0.323744\pi$
$168$	7.17028	0.553199
$169$	−11.1093	−0.854561
$170$	0	0
$171$	10.0429	0.768000
$172$	−2.08305	−0.158831
$173$	17.6126	1.33906	0.669530	−	0.742785i	$-0.266495\pi$
$173$	17.6126	1.33906	0.669530	+	0.742785i	$0.266495\pi$
$174$	−27.8464	−2.11103
$175$	0	0
$176$	−4.79013	−0.361070
$177$	−39.0781	−2.93729
$178$	−16.1195	−1.20821
$179$	−14.0263	−1.04838	−0.524189	−	0.851602i	$-0.675631\pi$
$179$	−14.0263	−1.04838	−0.524189	+	0.851602i	$0.675631\pi$
$180$	0	0
$181$	−15.7305	−1.16924	−0.584620	−	0.811307i	$-0.698756\pi$
$181$	−15.7305	−1.16924	−0.584620	+	0.811307i	$0.698756\pi$
$182$	−2.19225	−0.162500
$183$	−21.6573	−1.60096
$184$	−4.12793	−0.304315
$185$	0	0
$186$	23.0788	1.69222
$187$	−5.12233	−0.374582
$188$	6.91212	0.504118
$189$	−10.8366	−0.788243
$190$	0	0
$191$	9.60141	0.694734	0.347367	−	0.937729i	$-0.387076\pi$
$191$	9.60141	0.694734	0.347367	+	0.937729i	$0.387076\pi$
$192$	−14.8574	−1.07224
$193$	−5.95647	−0.428756	−0.214378	−	0.976751i	$-0.568772\pi$
$193$	−5.95647	−0.428756	−0.214378	+	0.976751i	$0.568772\pi$
$194$	−16.8753	−1.21157
$195$	0	0
$196$	0.541890	0.0387064
$197$	23.0571	1.64275	0.821376	−	0.570387i	$-0.193207\pi$
$197$	23.0571	1.64275	0.821376	+	0.570387i	$0.193207\pi$
$198$	10.3845	0.737992
$199$	22.6887	1.60836	0.804181	−	0.594384i	$-0.202604\pi$
$199$	22.6887	1.60836	0.804181	+	0.594384i	$0.202604\pi$
$200$	0	0
$201$	−28.0816	−1.98072
$202$	13.7913	0.970349
$203$	5.66269	0.397443
$204$	8.56142	0.599419
$205$	0	0
$206$	−8.58137	−0.597893
$207$	11.5656	0.803868
$208$	6.58657	0.456696
$209$	1.54189	0.106655
$210$	0	0
$211$	17.3081	1.19154	0.595768	−	0.803156i	$-0.296848\pi$
$211$	17.3081	1.19154	0.595768	+	0.803156i	$0.296848\pi$
$212$	4.19810	0.288327
$213$	−38.1840	−2.61632
$214$	10.9275	0.746992
$215$	0	0
$216$	25.1918	1.71409
$217$	−4.69319	−0.318595
$218$	2.78483	0.188612
$219$	−25.0413	−1.69213
$220$	0	0
$221$	7.04335	0.473787
$222$	0.257878	0.0173077
$223$	−22.8306	−1.52885	−0.764426	−	0.644712i	$-0.776977\pi$
$223$	−22.8306	−1.52885	−0.764426	+	0.644712i	$0.776977\pi$
$224$	−2.98764	−0.199620
$225$	0	0
$226$	−26.7520	−1.77952
$227$	−27.5380	−1.82776	−0.913882	−	0.405980i	$-0.866930\pi$
$227$	−27.5380	−1.82776	−0.913882	+	0.405980i	$0.866930\pi$
$228$	−2.57710	−0.170673
$229$	20.1798	1.33352	0.666759	−	0.745273i	$-0.267681\pi$
$229$	20.1798	1.33352	0.666759	+	0.745273i	$0.267681\pi$
$230$	0	0
$231$	−3.08438	−0.202937
$232$	−13.1641	−0.864267
$233$	−19.4685	−1.27543	−0.637713	−	0.770274i	$-0.720119\pi$
$233$	−19.4685	−1.27543	−0.637713	+	0.770274i	$0.720119\pi$
$234$	−14.2789	−0.933443
$235$	0	0
$236$	6.86557	0.446911
$237$	−33.7120	−2.18983
$238$	−8.16669	−0.529368
$239$	−14.5721	−0.942588	−0.471294	−	0.881976i	$-0.656213\pi$
$239$	−14.5721	−0.942588	−0.471294	+	0.881976i	$0.656213\pi$
$240$	0	0
$241$	−12.1372	−0.781823	−0.390911	−	0.920428i	$-0.627840\pi$
$241$	−12.1372	−0.781823	−0.390911	+	0.920428i	$0.627840\pi$
$242$	1.59433	0.102487
$243$	−10.3135	−0.661608
$244$	3.80495	0.243587
$245$	0	0
$246$	6.03191	0.384580
$247$	−2.12014	−0.134901
$248$	10.9103	0.692805
$249$	47.1183	2.98600
$250$	0	0
$251$	−17.5168	−1.10565	−0.552824	−	0.833298i	$-0.686450\pi$
$251$	−17.5168	−1.10565	−0.552824	+	0.833298i	$0.686450\pi$
$252$	3.52953	0.222339
$253$	1.77568	0.111636
$254$	17.2121	1.07999
$255$	0	0
$256$	12.1368	0.758553
$257$	1.18600	0.0739809	0.0369904	−	0.999316i	$-0.488223\pi$
$257$	1.18600	0.0739809	0.0369904	+	0.999316i	$0.488223\pi$
$258$	18.9031	1.17686
$259$	−0.0524408	−0.00325851
$260$	0	0
$261$	36.8832	2.28301
$262$	−7.84791	−0.484846
$263$	1.73827	0.107187	0.0535933	−	0.998563i	$-0.482933\pi$
$263$	1.73827	0.107187	0.0535933	+	0.998563i	$0.482933\pi$
$264$	7.17028	0.441300
$265$	0	0
$266$	2.45828	0.150727
$267$	31.1846	1.90847
$268$	4.93362	0.301369
$269$	−24.2685	−1.47968	−0.739839	−	0.672784i	$-0.765098\pi$
$269$	−24.2685	−1.47968	−0.739839	+	0.672784i	$0.765098\pi$
$270$	0	0
$271$	−7.88490	−0.478974	−0.239487	−	0.970900i	$-0.576979\pi$
$271$	−7.88490	−0.478974	−0.239487	+	0.970900i	$0.576979\pi$
$272$	24.5367	1.48775
$273$	4.24110	0.256683
$274$	18.7868	1.13495
$275$	0	0
$276$	−2.96785	−0.178644
$277$	−25.5811	−1.53702	−0.768510	−	0.639838i	$-0.779002\pi$
$277$	−25.5811	−1.53702	−0.768510	+	0.639838i	$0.779002\pi$
$278$	24.0564	1.44281
$279$	−30.5685	−1.83009
$280$	0	0
$281$	1.32015	0.0787533	0.0393767	−	0.999224i	$-0.487463\pi$
$281$	1.32015	0.0787533	0.0393767	+	0.999224i	$0.487463\pi$
$282$	−62.7258	−3.73527
$283$	20.2165	1.20175	0.600874	−	0.799344i	$-0.294819\pi$
$283$	20.2165	1.20175	0.600874	+	0.799344i	$0.294819\pi$
$284$	6.70849	0.398076
$285$	0	0
$286$	−2.19225	−0.129630
$287$	−1.22662	−0.0724049
$288$	−19.4596	−1.14667
$289$	9.23828	0.543428
$290$	0	0
$291$	32.6467	1.91379
$292$	4.39946	0.257459
$293$	27.3720	1.59909	0.799545	−	0.600607i	$-0.205074\pi$
$293$	27.3720	1.59909	0.799545	+	0.600607i	$0.205074\pi$
$294$	−4.91751	−0.286795
$295$	0	0
$296$	0.121910	0.00708586
$297$	−10.8366	−0.628801
$298$	−28.7379	−1.66474
$299$	−2.44160	−0.141202
$300$	0	0
$301$	−3.84404	−0.221567
$302$	26.0012	1.49620
$303$	−26.6804	−1.53275
$304$	−7.38586	−0.423608
$305$	0	0
$306$	−53.1927	−3.04082
$307$	16.0824	0.917874	0.458937	−	0.888469i	$-0.348230\pi$
$307$	16.0824	0.917874	0.458937	+	0.888469i	$0.348230\pi$
$308$	0.541890	0.0308770
$309$	16.6014	0.944423
$310$	0	0
$311$	−1.81575	−0.102962	−0.0514809	−	0.998674i	$-0.516394\pi$
$311$	−1.81575	−0.102962	−0.0514809	+	0.998674i	$0.516394\pi$
$312$	−9.85933	−0.558175
$313$	5.57181	0.314937	0.157469	−	0.987524i	$-0.449667\pi$
$313$	5.57181	0.314937	0.157469	+	0.987524i	$0.449667\pi$
$314$	14.4119	0.813308
$315$	0	0
$316$	5.92282	0.333185
$317$	13.1502	0.738587	0.369294	−	0.929313i	$-0.379600\pi$
$317$	13.1502	0.738587	0.369294	+	0.929313i	$0.379600\pi$
$318$	−38.0968	−2.13636
$319$	5.66269	0.317050
$320$	0	0
$321$	−21.1403	−1.17994
$322$	2.83102	0.157766
$323$	−7.89807	−0.439460
$324$	7.52353	0.417974
$325$	0	0
$326$	−7.10256	−0.393374
$327$	−5.38750	−0.297930
$328$	2.85153	0.157449
$329$	12.7556	0.703239
$330$	0	0
$331$	11.7041	0.643316	0.321658	−	0.946856i	$-0.395760\pi$
$331$	11.7041	0.643316	0.321658	+	0.946856i	$0.395760\pi$
$332$	−8.27815	−0.454323
$333$	−0.341567	−0.0187177
$334$	21.6689	1.18567
$335$	0	0
$336$	14.7746	0.806019
$337$	6.44219	0.350928	0.175464	−	0.984486i	$-0.443857\pi$
$337$	6.44219	0.350928	0.175464	+	0.984486i	$0.443857\pi$
$338$	−17.7119	−0.963400
$339$	51.7541	2.81090
$340$	0	0
$341$	−4.69319	−0.254150
$342$	16.0117	0.865814
$343$	1.00000	0.0539949
$344$	8.93628	0.481812
$345$	0	0
$346$	28.0803	1.50961
$347$	−26.2451	−1.40891	−0.704454	−	0.709749i	$-0.748808\pi$
$347$	−26.2451	−1.40891	−0.704454	+	0.709749i	$0.748808\pi$
$348$	−9.46458	−0.507355
$349$	−11.1284	−0.595689	−0.297845	−	0.954614i	$-0.596268\pi$
$349$	−11.1284	−0.595689	−0.297845	+	0.954614i	$0.596268\pi$
$350$	0	0
$351$	14.9006	0.795333
$352$	−2.98764	−0.159242
$353$	−32.1314	−1.71018	−0.855091	−	0.518479i	$-0.826499\pi$
$353$	−32.1314	−1.71018	−0.855091	+	0.518479i	$0.826499\pi$
$354$	−62.3034	−3.31139
$355$	0	0
$356$	−5.47879	−0.290375
$357$	15.7992	0.836182
$358$	−22.3626	−1.18190
$359$	−3.59229	−0.189594	−0.0947970	−	0.995497i	$-0.530220\pi$
$359$	−3.59229	−0.189594	−0.0947970	+	0.995497i	$0.530220\pi$
$360$	0	0
$361$	−16.6226	−0.874872
$362$	−25.0796	−1.31816
$363$	−3.08438	−0.161888
$364$	−0.745113	−0.0390545
$365$	0	0
$366$	−34.5289	−1.80486
$367$	21.2073	1.10701	0.553506	−	0.832845i	$-0.313290\pi$
$367$	21.2073	1.10701	0.553506	+	0.832845i	$0.313290\pi$
$368$	−8.50573	−0.443392
$369$	−7.98941	−0.415912
$370$	0	0
$371$	7.74716	0.402212
$372$	7.84416	0.406701
$373$	31.7346	1.64315	0.821577	−	0.570097i	$-0.193094\pi$
$373$	31.7346	1.64315	0.821577	+	0.570097i	$0.193094\pi$
$374$	−8.16669	−0.422289
$375$	0	0
$376$	−29.6531	−1.52924
$377$	−7.78636	−0.401018
$378$	−17.2770	−0.888635
$379$	27.7243	1.42410	0.712052	−	0.702127i	$-0.247766\pi$
$379$	27.7243	1.42410	0.712052	+	0.702127i	$0.247766\pi$
$380$	0	0
$381$	−33.2984	−1.70593
$382$	15.3078	0.783216
$383$	12.2116	0.623986	0.311993	−	0.950084i	$-0.399003\pi$
$383$	12.2116	0.623986	0.311993	+	0.950084i	$0.399003\pi$
$384$	−42.1176	−2.14930
$385$	0	0
$386$	−9.49658	−0.483363
$387$	−25.0377	−1.27274
$388$	−5.73566	−0.291184
$389$	22.1171	1.12138	0.560691	−	0.828025i	$-0.310535\pi$
$389$	22.1171	1.12138	0.560691	+	0.828025i	$0.310535\pi$
$390$	0	0
$391$	−9.09560	−0.459985
$392$	−2.32471	−0.117416
$393$	15.1825	0.765855
$394$	36.7607	1.85198
$395$	0	0
$396$	3.52953	0.177365
$397$	11.2906	0.566657	0.283328	−	0.959023i	$-0.408561\pi$
$397$	11.2906	0.566657	0.283328	+	0.959023i	$0.408561\pi$
$398$	36.1733	1.81321
$399$	−4.75577	−0.238086
$400$	0	0
$401$	−27.2414	−1.36037	−0.680186	−	0.733039i	$-0.738101\pi$
$401$	−27.2414	−1.36037	−0.680186	+	0.733039i	$0.738101\pi$
$402$	−44.7714	−2.23299
$403$	6.45327	0.321460
$404$	4.68745	0.233209
$405$	0	0
$406$	9.02821	0.448062
$407$	−0.0524408	−0.00259940
$408$	−36.7285	−1.81833
$409$	15.5065	0.766748	0.383374	−	0.923593i	$-0.374762\pi$
$409$	15.5065	0.766748	0.383374	+	0.923593i	$0.374762\pi$
$410$	0	0
$411$	−36.3447	−1.79275
$412$	−2.91668	−0.143695
$413$	12.6697	0.623435
$414$	18.4395	0.906250
$415$	0	0
$416$	4.10809	0.201415
$417$	−46.5393	−2.27904
$418$	2.45828	0.120238
$419$	4.67116	0.228201	0.114100	−	0.993469i	$-0.463601\pi$
$419$	4.67116	0.228201	0.114100	+	0.993469i	$0.463601\pi$
$420$	0	0
$421$	−27.6340	−1.34680	−0.673399	−	0.739279i	$-0.735166\pi$
$421$	−27.6340	−1.34680	−0.673399	+	0.739279i	$0.735166\pi$
$422$	27.5948	1.34329
$423$	83.0819	4.03958
$424$	−18.0099	−0.874638
$425$	0	0
$426$	−60.8779	−2.94954
$427$	7.02163	0.339800
$428$	3.71411	0.179528
$429$	4.24110	0.204762
$430$	0	0
$431$	−5.44040	−0.262055	−0.131027	−	0.991379i	$-0.541828\pi$
$431$	−5.44040	−0.262055	−0.131027	+	0.991379i	$0.541828\pi$
$432$	51.9086	2.49745
$433$	4.50486	0.216490	0.108245	−	0.994124i	$-0.465477\pi$
$433$	4.50486	0.216490	0.108245	+	0.994124i	$0.465477\pi$
$434$	−7.48250	−0.359171
$435$	0	0
$436$	0.946523	0.0453302
$437$	2.73790	0.130971
$438$	−39.9240	−1.90764
$439$	18.9808	0.905904	0.452952	−	0.891535i	$-0.350371\pi$
$439$	18.9808	0.905904	0.452952	+	0.891535i	$0.350371\pi$
$440$	0	0
$441$	6.51337	0.310161
$442$	11.2294	0.534129
$443$	24.1277	1.14634	0.573172	−	0.819435i	$-0.305713\pi$
$443$	24.1277	1.14634	0.573172	+	0.819435i	$0.305713\pi$
$444$	0.0876491	0.00415964
$445$	0	0
$446$	−36.3996	−1.72357
$447$	55.5960	2.62960
$448$	4.81699	0.227581
$449$	−24.4000	−1.15151	−0.575753	−	0.817624i	$-0.695291\pi$
$449$	−24.4000	−1.15151	−0.575753	+	0.817624i	$0.695291\pi$
$450$	0	0
$451$	−1.22662	−0.0577592
$452$	−9.09261	−0.427681
$453$	−50.3016	−2.36338
$454$	−43.9047	−2.06055
$455$	0	0
$456$	11.0558	0.517734
$457$	−21.7913	−1.01936	−0.509678	−	0.860365i	$-0.670235\pi$
$457$	−21.7913	−1.01936	−0.509678	+	0.860365i	$0.670235\pi$
$458$	32.1733	1.50336
$459$	55.5084	2.59091
$460$	0	0
$461$	9.93191	0.462575	0.231288	−	0.972885i	$-0.425706\pi$
$461$	9.93191	0.462575	0.231288	+	0.972885i	$0.425706\pi$
$462$	−4.91751	−0.228783
$463$	2.14301	0.0995940	0.0497970	−	0.998759i	$-0.484143\pi$
$463$	2.14301	0.0995940	0.0497970	+	0.998759i	$0.484143\pi$
$464$	−27.1251	−1.25925
$465$	0	0
$466$	−31.0393	−1.43787
$467$	21.2036	0.981186	0.490593	−	0.871389i	$-0.336780\pi$
$467$	21.2036	0.981186	0.490593	+	0.871389i	$0.336780\pi$
$468$	−4.85320	−0.224339
$469$	9.10448	0.420406
$470$	0	0
$471$	−27.8810	−1.28469
$472$	−29.4534	−1.35570
$473$	−3.84404	−0.176749
$474$	−53.7481	−2.46873
$475$	0	0
$476$	−2.77574	−0.127226
$477$	50.4601	2.31041
$478$	−23.2327	−1.06264
$479$	−36.1249	−1.65059	−0.825294	−	0.564703i	$-0.808991\pi$
$479$	−36.1249	−1.65059	−0.825294	+	0.564703i	$0.808991\pi$
$480$	0	0
$481$	0.0721076	0.00328782
$482$	−19.3506	−0.881397
$483$	−5.47685	−0.249205
$484$	0.541890	0.0246313
$485$	0	0
$486$	−16.4431	−0.745872
$487$	−19.9375	−0.903452	−0.451726	−	0.892157i	$-0.649192\pi$
$487$	−19.9375	−0.903452	−0.451726	+	0.892157i	$0.649192\pi$
$488$	−16.3232	−0.738919
$489$	13.7405	0.621369
$490$	0	0
$491$	22.5808	1.01906	0.509528	−	0.860454i	$-0.329820\pi$
$491$	22.5808	1.01906	0.509528	+	0.860454i	$0.329820\pi$
$492$	2.05016	0.0924282
$493$	−29.0062	−1.30637
$494$	−3.38021	−0.152083
$495$	0	0
$496$	22.4810	1.00943
$497$	12.3798	0.555310
$498$	75.1222	3.36630
$499$	−38.3082	−1.71491	−0.857455	−	0.514559i	$-0.827956\pi$
$499$	−38.3082	−1.71491	−0.857455	+	0.514559i	$0.827956\pi$
$500$	0	0
$501$	−41.9204	−1.87286
$502$	−27.9275	−1.24647
$503$	20.8604	0.930119	0.465060	−	0.885279i	$-0.346033\pi$
$503$	20.8604	0.930119	0.465060	+	0.885279i	$0.346033\pi$
$504$	−15.1417	−0.674465
$505$	0	0
$506$	2.83102	0.125854
$507$	34.2652	1.52177
$508$	5.85016	0.259559
$509$	−6.52736	−0.289320	−0.144660	−	0.989481i	$-0.546209\pi$
$509$	−6.52736	−0.289320	−0.144660	+	0.989481i	$0.546209\pi$
$510$	0	0
$511$	8.11874	0.359152
$512$	−7.96016	−0.351793
$513$	−16.7088	−0.737710
$514$	1.89088	0.0834032
$515$	0	0
$516$	6.42489	0.282840
$517$	12.7556	0.560990
$518$	−0.0836080	−0.00367353
$519$	−54.3238	−2.38455
$520$	0	0
$521$	−19.5823	−0.857918	−0.428959	−	0.903324i	$-0.641119\pi$
$521$	−19.5823	−0.857918	−0.428959	+	0.903324i	$0.641119\pi$
$522$	58.8041	2.57378
$523$	28.2627	1.23584	0.617920	−	0.786241i	$-0.287975\pi$
$523$	28.2627	1.23584	0.617920	+	0.786241i	$0.287975\pi$
$524$	−2.66739	−0.116526
$525$	0	0
$526$	2.77138	0.120838
$527$	24.0401	1.04720
$528$	14.7746	0.642981
$529$	−19.8470	−0.862912
$530$	0	0
$531$	82.5224	3.58117
$532$	0.835534	0.0362250
$533$	1.68663	0.0730562
$534$	49.7186	2.15154
$535$	0	0
$536$	−21.1653	−0.914200
$537$	43.2625	1.86691
$538$	−38.6920	−1.66813
$539$	1.00000	0.0430730
$540$	0	0
$541$	−0.965158	−0.0414954	−0.0207477	−	0.999785i	$-0.506605\pi$
$541$	−0.965158	−0.0414954	−0.0207477	+	0.999785i	$0.506605\pi$
$542$	−12.5711	−0.539977
$543$	48.5188	2.08214
$544$	15.3037	0.656140
$545$	0	0
$546$	6.76172	0.289375
$547$	9.62388	0.411487	0.205744	−	0.978606i	$-0.434039\pi$
$547$	9.62388	0.411487	0.205744	+	0.978606i	$0.434039\pi$
$548$	6.38536	0.272769
$549$	45.7345	1.95190
$550$	0	0
$551$	8.73125	0.371964
$552$	12.7321	0.541914
$553$	10.9299	0.464788
$554$	−40.7847	−1.73278
$555$	0	0
$556$	8.17642	0.346758
$557$	38.2247	1.61963	0.809816	−	0.586684i	$-0.199567\pi$
$557$	38.2247	1.61963	0.809816	+	0.586684i	$0.199567\pi$
$558$	−48.7363	−2.06317
$559$	5.28566	0.223560
$560$	0	0
$561$	15.7992	0.667042
$562$	2.10475	0.0887835
$563$	12.1634	0.512628	0.256314	−	0.966594i	$-0.417492\pi$
$563$	12.1634	0.512628	0.256314	+	0.966594i	$0.417492\pi$
$564$	−21.3196	−0.897716
$565$	0	0
$566$	32.2318	1.35480
$567$	13.8839	0.583068
$568$	−28.7795	−1.20756
$569$	−19.6954	−0.825676	−0.412838	−	0.910805i	$-0.635462\pi$
$569$	−19.6954	−0.825676	−0.412838	+	0.910805i	$0.635462\pi$
$570$	0	0
$571$	26.5682	1.11184	0.555921	−	0.831235i	$-0.312366\pi$
$571$	26.5682	1.11184	0.555921	+	0.831235i	$0.312366\pi$
$572$	−0.745113	−0.0311548
$573$	−29.6144	−1.23716
$574$	−1.95563	−0.0816266
$575$	0	0
$576$	31.3748	1.30728
$577$	2.62474	0.109269	0.0546346	−	0.998506i	$-0.482601\pi$
$577$	2.62474	0.109269	0.0546346	+	0.998506i	$0.482601\pi$
$578$	14.7289	0.612640
$579$	18.3720	0.763514
$580$	0	0
$581$	−15.2765	−0.633774
$582$	52.0497	2.15753
$583$	7.74716	0.320854
$584$	−18.8737	−0.781000
$585$	0	0
$586$	43.6400	1.80275
$587$	37.2262	1.53649	0.768244	−	0.640157i	$-0.221131\pi$
$587$	37.2262	1.53649	0.768244	+	0.640157i	$0.221131\pi$
$588$	−1.67139	−0.0689270
$589$	−7.23638	−0.298170
$590$	0	0
$591$	−71.1168	−2.92536
$592$	0.251199	0.0103242
$593$	6.57022	0.269807	0.134903	−	0.990859i	$-0.456928\pi$
$593$	6.57022	0.269807	0.134903	+	0.990859i	$0.456928\pi$
$594$	−17.2770	−0.708886
$595$	0	0
$596$	−9.76758	−0.400096
$597$	−69.9806	−2.86411
$598$	−3.89272	−0.159185
$599$	29.9245	1.22268	0.611341	−	0.791367i	$-0.290630\pi$
$599$	29.9245	1.22268	0.611341	+	0.791367i	$0.290630\pi$
$600$	0	0
$601$	9.28966	0.378933	0.189467	−	0.981887i	$-0.439324\pi$
$601$	9.28966	0.378933	0.189467	+	0.981887i	$0.439324\pi$
$602$	−6.12867	−0.249786
$603$	59.3008	2.41492
$604$	8.83742	0.359590
$605$	0	0
$606$	−42.5374	−1.72796
$607$	1.95383	0.0793037	0.0396518	−	0.999214i	$-0.487375\pi$
$607$	1.95383	0.0793037	0.0396518	+	0.999214i	$0.487375\pi$
$608$	−4.60661	−0.186823
$609$	−17.4659	−0.707753
$610$	0	0
$611$	−17.5393	−0.709564
$612$	−18.0794	−0.730817
$613$	10.7845	0.435584	0.217792	−	0.975995i	$-0.430115\pi$
$613$	10.7845	0.435584	0.217792	+	0.975995i	$0.430115\pi$
$614$	25.6407	1.03478
$615$	0	0
$616$	−2.32471	−0.0936652
$617$	32.2331	1.29765	0.648827	−	0.760936i	$-0.275260\pi$
$617$	32.2331	1.29765	0.648827	+	0.760936i	$0.275260\pi$
$618$	26.4682	1.06471
$619$	−1.41829	−0.0570061	−0.0285030	−	0.999594i	$-0.509074\pi$
$619$	−1.41829	−0.0570061	−0.0285030	+	0.999594i	$0.509074\pi$
$620$	0	0
$621$	−19.2422	−0.772164
$622$	−2.89491	−0.116075
$623$	−10.1105	−0.405069
$624$	−20.3154	−0.813269
$625$	0	0
$626$	8.88330	0.355048
$627$	−4.75577	−0.189927
$628$	4.89838	0.195467
$629$	0.268619	0.0107106
$630$	0	0
$631$	−4.31580	−0.171809	−0.0859047	−	0.996303i	$-0.527378\pi$
$631$	−4.31580	−0.171809	−0.0859047	+	0.996303i	$0.527378\pi$
$632$	−25.4089	−1.01071
$633$	−53.3846	−2.12185
$634$	20.9657	0.832655
$635$	0	0
$636$	−12.9485	−0.513443
$637$	−1.37503	−0.0544806
$638$	9.02821	0.357430
$639$	80.6343	3.18984
$640$	0	0
$641$	32.5106	1.28409	0.642045	−	0.766667i	$-0.278086\pi$
$641$	32.5106	1.28409	0.642045	+	0.766667i	$0.278086\pi$
$642$	−33.7047	−1.33022
$643$	−11.9632	−0.471784	−0.235892	−	0.971779i	$-0.575801\pi$
$643$	−11.9632	−0.471784	−0.235892	+	0.971779i	$0.575801\pi$
$644$	0.962221	0.0379168
$645$	0	0
$646$	−12.5921	−0.495431
$647$	7.11640	0.279774	0.139887	−	0.990167i	$-0.455326\pi$
$647$	7.11640	0.279774	0.139887	+	0.990167i	$0.455326\pi$
$648$	−32.2760	−1.26792
$649$	12.6697	0.497329
$650$	0	0
$651$	14.4756	0.567342
$652$	−2.41406	−0.0945417
$653$	−22.0303	−0.862110	−0.431055	−	0.902326i	$-0.641859\pi$
$653$	−22.0303	−0.862110	−0.431055	+	0.902326i	$0.641859\pi$
$654$	−8.58946	−0.335874
$655$	0	0
$656$	5.87566	0.229406
$657$	52.8804	2.06306
$658$	20.3366	0.792804
$659$	−29.7872	−1.16034	−0.580172	−	0.814494i	$-0.697015\pi$
$659$	−29.7872	−1.16034	−0.580172	+	0.814494i	$0.697015\pi$
$660$	0	0
$661$	16.6192	0.646410	0.323205	−	0.946329i	$-0.395240\pi$
$661$	16.6192	0.646410	0.323205	+	0.946329i	$0.395240\pi$
$662$	18.6602	0.725250
$663$	−21.7243	−0.843703
$664$	35.5133	1.37818
$665$	0	0
$666$	−0.544570	−0.0211017
$667$	10.0551	0.389336
$668$	7.36493	0.284958
$669$	70.4182	2.72252
$670$	0	0
$671$	7.02163	0.271067
$672$	9.21500	0.355476
$673$	−23.3922	−0.901705	−0.450852	−	0.892599i	$-0.648880\pi$
$673$	−23.3922	−0.901705	−0.450852	+	0.892599i	$0.648880\pi$
$674$	10.2710	0.395623
$675$	0	0
$676$	−6.02001	−0.231539
$677$	−32.2771	−1.24051	−0.620255	−	0.784400i	$-0.712971\pi$
$677$	−32.2771	−1.24051	−0.620255	+	0.784400i	$0.712971\pi$
$678$	82.5132	3.16890
$679$	−10.5846	−0.406198
$680$	0	0
$681$	84.9376	3.25482
$682$	−7.48250	−0.286520
$683$	33.8085	1.29365	0.646823	−	0.762640i	$-0.276097\pi$
$683$	33.8085	1.29365	0.646823	+	0.762640i	$0.276097\pi$
$684$	5.44214	0.208086
$685$	0	0
$686$	1.59433	0.0608718
$687$	−62.2421	−2.37468
$688$	18.4135	0.702007
$689$	−10.6526	−0.405830
$690$	0	0
$691$	11.7874	0.448413	0.224206	−	0.974542i	$-0.428021\pi$
$691$	11.7874	0.448413	0.224206	+	0.974542i	$0.428021\pi$
$692$	9.54408	0.362812
$693$	6.51337	0.247422
$694$	−41.8433	−1.58835
$695$	0	0
$696$	40.6031	1.53906
$697$	6.28314	0.237991
$698$	−17.7423	−0.671557
$699$	60.0483	2.27123
$700$	0	0
$701$	−3.80479	−0.143705	−0.0718525	−	0.997415i	$-0.522891\pi$
$701$	−3.80479	−0.143705	−0.0718525	+	0.997415i	$0.522891\pi$
$702$	23.7564	0.896628
$703$	−0.0808580	−0.00304962
$704$	4.81699	0.181547
$705$	0	0
$706$	−51.2281	−1.92799
$707$	8.65019	0.325324
$708$	−21.1760	−0.795843
$709$	26.3482	0.989526	0.494763	−	0.869028i	$-0.335255\pi$
$709$	26.3482	0.989526	0.494763	+	0.869028i	$0.335255\pi$
$710$	0	0
$711$	71.1907	2.66986
$712$	23.5040	0.880851
$713$	−8.33359	−0.312095
$714$	25.1891	0.942680
$715$	0	0
$716$	−7.60073	−0.284052
$717$	44.9457	1.67853
$718$	−5.72730	−0.213741
$719$	−36.4011	−1.35753	−0.678766	−	0.734354i	$-0.737485\pi$
$719$	−36.4011	−1.35753	−0.678766	+	0.734354i	$0.737485\pi$
$720$	0	0
$721$	−5.38243	−0.200452
$722$	−26.5019	−0.986298
$723$	37.4355	1.39224
$724$	−8.52420	−0.316800
$725$	0	0
$726$	−4.91751	−0.182506
$727$	21.6257	0.802054	0.401027	−	0.916066i	$-0.368653\pi$
$727$	21.6257	0.802054	0.401027	+	0.916066i	$0.368653\pi$
$728$	3.19654	0.118472
$729$	−9.84110	−0.364485
$730$	0	0
$731$	19.6905	0.728278
$732$	−11.7359	−0.433771
$733$	−14.4073	−0.532147	−0.266074	−	0.963953i	$-0.585726\pi$
$733$	−14.4073	−0.532147	−0.266074	+	0.963953i	$0.585726\pi$
$734$	33.8115	1.24800
$735$	0	0
$736$	−5.30508	−0.195548
$737$	9.10448	0.335368
$738$	−12.7378	−0.468884
$739$	−30.5100	−1.12233	−0.561165	−	0.827704i	$-0.689647\pi$
$739$	−30.5100	−1.12233	−0.561165	+	0.827704i	$0.689647\pi$
$740$	0	0
$741$	6.53931	0.240228
$742$	12.3515	0.453439
$743$	−36.1902	−1.32769	−0.663845	−	0.747870i	$-0.731077\pi$
$743$	−36.1902	−1.32769	−0.663845	+	0.747870i	$0.731077\pi$
$744$	−33.6515	−1.23372
$745$	0	0
$746$	50.5954	1.85243
$747$	−99.5012	−3.64056
$748$	−2.77574	−0.101491
$749$	6.85400	0.250440
$750$	0	0
$751$	−29.3111	−1.06958	−0.534789	−	0.844986i	$-0.679609\pi$
$751$	−29.3111	−1.06958	−0.534789	+	0.844986i	$0.679609\pi$
$752$	−61.1010	−2.22812
$753$	54.0283	1.96890
$754$	−12.4140	−0.452092
$755$	0	0
$756$	−5.87222	−0.213570
$757$	−36.3855	−1.32245	−0.661226	−	0.750187i	$-0.729964\pi$
$757$	−36.3855	−1.32245	−0.661226	+	0.750187i	$0.729964\pi$
$758$	44.2018	1.60548
$759$	−5.47685	−0.198797
$760$	0	0
$761$	48.2712	1.74983	0.874916	−	0.484275i	$-0.160917\pi$
$761$	48.2712	1.74983	0.874916	+	0.484275i	$0.160917\pi$
$762$	−53.0887	−1.92320
$763$	1.74671	0.0632351
$764$	5.20291	0.188235
$765$	0	0
$766$	19.4694	0.703458
$767$	−17.4212	−0.629042
$768$	−37.4346	−1.35080
$769$	17.2118	0.620675	0.310338	−	0.950626i	$-0.399558\pi$
$769$	17.2118	0.620675	0.310338	+	0.950626i	$0.399558\pi$
$770$	0	0
$771$	−3.65808	−0.131743
$772$	−3.22775	−0.116169
$773$	12.5890	0.452793	0.226397	−	0.974035i	$-0.427305\pi$
$773$	12.5890	0.452793	0.226397	+	0.974035i	$0.427305\pi$
$774$	−39.9183	−1.43483
$775$	0	0
$776$	24.6060	0.883305
$777$	0.161747	0.00580265
$778$	35.2620	1.26420
$779$	−1.89131	−0.0677632
$780$	0	0
$781$	12.3798	0.442984
$782$	−14.5014	−0.518569
$783$	−61.3641	−2.19297
$784$	−4.79013	−0.171076
$785$	0	0
$786$	24.2059	0.863396
$787$	39.7008	1.41518	0.707591	−	0.706622i	$-0.249782\pi$
$787$	39.7008	1.41518	0.707591	+	0.706622i	$0.249782\pi$
$788$	12.4944	0.445095
$789$	−5.36149	−0.190874
$790$	0	0
$791$	−16.7795	−0.596609
$792$	−15.1417	−0.538037
$793$	−9.65493	−0.342857
$794$	18.0009	0.638827
$795$	0	0
$796$	12.2948	0.435777
$797$	1.17026	0.0414529	0.0207264	−	0.999785i	$-0.493402\pi$
$797$	1.17026	0.0414529	0.0207264	+	0.999785i	$0.493402\pi$
$798$	−7.58226	−0.268409
$799$	−65.3384	−2.31151
$800$	0	0
$801$	−65.8536	−2.32682
$802$	−43.4319	−1.53363
$803$	8.11874	0.286504
$804$	−15.2171	−0.536667
$805$	0	0
$806$	10.2886	0.362402
$807$	74.8532	2.63496
$808$	−20.1092	−0.707438
$809$	−13.8146	−0.485696	−0.242848	−	0.970064i	$-0.578082\pi$
$809$	−13.8146	−0.485696	−0.242848	+	0.970064i	$0.578082\pi$
$810$	0	0
$811$	−27.4448	−0.963717	−0.481858	−	0.876249i	$-0.660038\pi$
$811$	−27.4448	−0.963717	−0.481858	+	0.876249i	$0.660038\pi$
$812$	3.06856	0.107685
$813$	24.3200	0.852940
$814$	−0.0836080	−0.00293046
$815$	0	0
$816$	−75.6803	−2.64934
$817$	−5.92709	−0.207363
$818$	24.7225	0.864403
$819$	−8.95607	−0.312950
$820$	0	0
$821$	−53.9460	−1.88273	−0.941365	−	0.337390i	$-0.890456\pi$
$821$	−53.9460	−1.88273	−0.941365	+	0.337390i	$0.890456\pi$
$822$	−57.9455	−2.02108
$823$	−2.19370	−0.0764677	−0.0382339	−	0.999269i	$-0.512173\pi$
$823$	−2.19370	−0.0764677	−0.0382339	+	0.999269i	$0.512173\pi$
$824$	12.5126	0.435897
$825$	0	0
$826$	20.1997	0.702837
$827$	−19.4389	−0.675956	−0.337978	−	0.941154i	$-0.609743\pi$
$827$	−19.4389	−0.675956	−0.337978	+	0.941154i	$0.609743\pi$
$828$	6.26730	0.217804
$829$	52.1514	1.81129	0.905646	−	0.424035i	$-0.139387\pi$
$829$	52.1514	1.81129	0.905646	+	0.424035i	$0.139387\pi$
$830$	0	0
$831$	78.9017	2.73707
$832$	−6.62349	−0.229628
$833$	−5.12233	−0.177478
$834$	−74.1990	−2.56930
$835$	0	0
$836$	0.835534	0.0288975
$837$	50.8580	1.75791
$838$	7.44737	0.257265
$839$	−7.73762	−0.267132	−0.133566	−	0.991040i	$-0.542643\pi$
$839$	−7.73762	−0.267132	−0.133566	+	0.991040i	$0.542643\pi$
$840$	0	0
$841$	3.06611	0.105728
$842$	−44.0577	−1.51833
$843$	−4.07183	−0.140241
$844$	9.37906	0.322841
$845$	0	0
$846$	132.460	4.55407
$847$	1.00000	0.0343604
$848$	−37.1099	−1.27436
$849$	−62.3553	−2.14003
$850$	0	0
$851$	−0.0931180	−0.00319204
$852$	−20.6915	−0.708879
$853$	4.06256	0.139099	0.0695497	−	0.997578i	$-0.477844\pi$
$853$	4.06256	0.139099	0.0695497	+	0.997578i	$0.477844\pi$
$854$	11.1948	0.383078
$855$	0	0
$856$	−15.9336	−0.544598
$857$	−5.26769	−0.179941	−0.0899705	−	0.995944i	$-0.528677\pi$
$857$	−5.26769	−0.179941	−0.0899705	+	0.995944i	$0.528677\pi$
$858$	6.76172	0.230841
$859$	19.1785	0.654362	0.327181	−	0.944962i	$-0.393901\pi$
$859$	19.1785	0.654362	0.327181	+	0.944962i	$0.393901\pi$
$860$	0	0
$861$	3.78335	0.128936
$862$	−8.67379	−0.295430
$863$	−34.8731	−1.18709	−0.593547	−	0.804800i	$-0.702273\pi$
$863$	−34.8731	−1.18709	−0.593547	+	0.804800i	$0.702273\pi$
$864$	32.3757	1.10144
$865$	0	0
$866$	7.18224	0.244062
$867$	−28.4943	−0.967718
$868$	−2.54319	−0.0863215
$869$	10.9299	0.370773
$870$	0	0
$871$	−12.5189	−0.424187
$872$	−4.06059	−0.137509
$873$	−68.9411	−2.33330
$874$	4.36511	0.147652
$875$	0	0
$876$	−13.5696	−0.458474
$877$	22.8559	0.771789	0.385894	−	0.922543i	$-0.373893\pi$
$877$	22.8559	0.771789	0.385894	+	0.922543i	$0.373893\pi$
$878$	30.2617	1.02128
$879$	−84.4255	−2.84760
$880$	0	0
$881$	−47.5847	−1.60317	−0.801584	−	0.597882i	$-0.796009\pi$
$881$	−47.5847	−1.60317	−0.801584	+	0.597882i	$0.796009\pi$
$882$	10.3845	0.349663
$883$	−19.8333	−0.667445	−0.333722	−	0.942671i	$-0.608305\pi$
$883$	−19.8333	−0.667445	−0.333722	+	0.942671i	$0.608305\pi$
$884$	3.81672	0.128370
$885$	0	0
$886$	38.4676	1.29234
$887$	−1.25682	−0.0422000	−0.0211000	−	0.999777i	$-0.506717\pi$
$887$	−1.25682	−0.0422000	−0.0211000	+	0.999777i	$0.506717\pi$
$888$	−0.376015	−0.0126182
$889$	10.7958	0.362081
$890$	0	0
$891$	13.8839	0.465128
$892$	−12.3717	−0.414234
$893$	19.6677	0.658155
$894$	88.6384	2.96451
$895$	0	0
$896$	13.6551	0.456186
$897$	7.53083	0.251447
$898$	−38.9016	−1.29816
$899$	−26.5761	−0.886363
$900$	0	0
$901$	−39.6835	−1.32205
$902$	−1.95563	−0.0651155
$903$	11.8565	0.394558
$904$	39.0074	1.29737
$905$	0	0
$906$	−80.1974	−2.66438
$907$	40.2035	1.33494	0.667468	−	0.744638i	$-0.267378\pi$
$907$	40.2035	1.33494	0.667468	+	0.744638i	$0.267378\pi$
$908$	−14.9226	−0.495223
$909$	56.3419	1.86874
$910$	0	0
$911$	5.68156	0.188238	0.0941192	−	0.995561i	$-0.469997\pi$
$911$	5.68156	0.188238	0.0941192	+	0.995561i	$0.469997\pi$
$912$	22.7808	0.754346
$913$	−15.2765	−0.505577
$914$	−34.7426	−1.14918
$915$	0	0
$916$	10.9352	0.361310
$917$	−4.92239	−0.162552
$918$	88.4988	2.92089
$919$	4.72985	0.156023	0.0780117	−	0.996952i	$-0.475143\pi$
$919$	4.72985	0.156023	0.0780117	+	0.996952i	$0.475143\pi$
$920$	0	0
$921$	−49.6043	−1.63452
$922$	15.8348	0.521490
$923$	−17.0226	−0.560305
$924$	−1.67139	−0.0549847
$925$	0	0
$926$	3.41666	0.112278
$927$	−35.0578	−1.15145
$928$	−16.9181	−0.555363
$929$	−17.8205	−0.584672	−0.292336	−	0.956316i	$-0.594433\pi$
$929$	−17.8205	−0.584672	−0.292336	+	0.956316i	$0.594433\pi$
$930$	0	0
$931$	1.54189	0.0505334
$932$	−10.5498	−0.345570
$933$	5.60046	0.183351
$934$	33.8056	1.10615
$935$	0	0
$936$	20.8203	0.680531
$937$	15.0090	0.490323	0.245161	−	0.969482i	$-0.421159\pi$
$937$	15.0090	0.490323	0.245161	+	0.969482i	$0.421159\pi$
$938$	14.5155	0.473949
$939$	−17.1855	−0.560829
$940$	0	0
$941$	0.202267	0.00659372	0.00329686	−	0.999995i	$-0.498951\pi$
$941$	0.202267	0.00659372	0.00329686	+	0.999995i	$0.498951\pi$
$942$	−44.4516	−1.44831
$943$	−2.17808	−0.0709279
$944$	−60.6895	−1.97528
$945$	0	0
$946$	−6.12867	−0.199260
$947$	−28.4892	−0.925775	−0.462887	−	0.886417i	$-0.653187\pi$
$947$	−28.4892	−0.925775	−0.462887	+	0.886417i	$0.653187\pi$
$948$	−18.2682	−0.593323
$949$	−11.1635	−0.362383
$950$	0	0
$951$	−40.5601	−1.31525
$952$	11.9079	0.385938
$953$	−22.3346	−0.723489	−0.361745	−	0.932277i	$-0.617819\pi$
$953$	−22.3346	−0.723489	−0.361745	+	0.932277i	$0.617819\pi$
$954$	80.4501	2.60467
$955$	0	0
$956$	−7.89644	−0.255389
$957$	−17.4659	−0.564592
$958$	−57.5950	−1.86081
$959$	11.7835	0.380509
$960$	0	0
$961$	−8.97397	−0.289483
$962$	0.114963	0.00370657
$963$	44.6427	1.43859
$964$	−6.57700	−0.211831
$965$	0	0
$966$	−8.73191	−0.280945
$967$	−54.5233	−1.75335	−0.876675	−	0.481083i	$-0.840244\pi$
$967$	−54.5233	−1.75335	−0.876675	+	0.481083i	$0.840244\pi$
$968$	−2.32471	−0.0747190
$969$	24.3606	0.782576
$970$	0	0
$971$	19.0894	0.612607	0.306304	−	0.951934i	$-0.400908\pi$
$971$	19.0894	0.612607	0.306304	+	0.951934i	$0.400908\pi$
$972$	−5.58875	−0.179259
$973$	15.0887	0.483722
$974$	−31.7869	−1.01852
$975$	0	0
$976$	−33.6345	−1.07662
$977$	36.1337	1.15602	0.578010	−	0.816030i	$-0.303830\pi$
$977$	36.1337	1.15602	0.578010	+	0.816030i	$0.303830\pi$
$978$	21.9070	0.700507
$979$	−10.1105	−0.323134
$980$	0	0
$981$	11.3770	0.363238
$982$	36.0012	1.14884
$983$	37.3701	1.19192	0.595960	−	0.803014i	$-0.296772\pi$
$983$	37.3701	1.19192	0.595960	+	0.803014i	$0.296772\pi$
$984$	−8.79519	−0.280380
$985$	0	0
$986$	−46.2455	−1.47276
$987$	−39.3430	−1.25230
$988$	−1.14888	−0.0365508
$989$	−6.82577	−0.217047
$990$	0	0
$991$	13.0269	0.413814	0.206907	−	0.978361i	$-0.433660\pi$
$991$	13.0269	0.413814	0.206907	+	0.978361i	$0.433660\pi$
$992$	14.0216	0.445185
$993$	−36.0999	−1.14559
$994$	19.7375	0.626036
$995$	0	0
$996$	25.5329	0.809042
$997$	−33.1191	−1.04889	−0.524446	−	0.851444i	$-0.675727\pi$
$997$	−33.1191	−1.04889	−0.524446	+	0.851444i	$0.675727\pi$
$998$	−61.0759	−1.93332
$999$	0.568278	0.0179795

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1925.2.a.bd.1.5	yes	7
5.2	odd	4		1925.2.b.r.1849.10		14
5.3	odd	4		1925.2.b.r.1849.5		14
5.4	even	2		1925.2.a.bb.1.3	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1925.2.a.bb.1.3	✓	7	5.4	even	2
1925.2.a.bd.1.5	yes	7	1.1	even	1	trivial
1925.2.b.r.1849.5		14	5.3	odd	4
1925.2.b.r.1849.10		14	5.2	odd	4

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:	\( N \)	\(=\)	\( 1925 = 5^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1925.a (trivial)

\(f(q)\)	\(=\)	\(q+1.59433 q^{2} -3.08438 q^{3} +0.541890 q^{4} -4.91751 q^{6} +1.00000 q^{7} -2.32471 q^{8} +6.51337 q^{9} +O(q^{10})\)	\(q+1.59433 q^{2} -3.08438 q^{3} +0.541890 q^{4} -4.91751 q^{6} +1.00000 q^{7} -2.32471 q^{8} +6.51337 q^{9} +1.00000 q^{11} -1.67139 q^{12} -1.37503 q^{13} +1.59433 q^{14} -4.79013 q^{16} -5.12233 q^{17} +10.3845 q^{18} +1.54189 q^{19} -3.08438 q^{21} +1.59433 q^{22} +1.77568 q^{23} +7.17028 q^{24} -2.19225 q^{26} -10.8366 q^{27} +0.541890 q^{28} +5.66269 q^{29} -4.69319 q^{31} -2.98764 q^{32} -3.08438 q^{33} -8.16669 q^{34} +3.52953 q^{36} -0.0524408 q^{37} +2.45828 q^{38} +4.24110 q^{39} -1.22662 q^{41} -4.91751 q^{42} -3.84404 q^{43} +0.541890 q^{44} +2.83102 q^{46} +12.7556 q^{47} +14.7746 q^{48} +1.00000 q^{49} +15.7992 q^{51} -0.745113 q^{52} +7.74716 q^{53} -17.2770 q^{54} -2.32471 q^{56} -4.75577 q^{57} +9.02821 q^{58} +12.6697 q^{59} +7.02163 q^{61} -7.48250 q^{62} +6.51337 q^{63} +4.81699 q^{64} -4.91751 q^{66} +9.10448 q^{67} -2.77574 q^{68} -5.47685 q^{69} +12.3798 q^{71} -15.1417 q^{72} +8.11874 q^{73} -0.0836080 q^{74} +0.835534 q^{76} +1.00000 q^{77} +6.76172 q^{78} +10.9299 q^{79} +13.8839 q^{81} -1.95563 q^{82} -15.2765 q^{83} -1.67139 q^{84} -6.12867 q^{86} -17.4659 q^{87} -2.32471 q^{88} -10.1105 q^{89} -1.37503 q^{91} +0.962221 q^{92} +14.4756 q^{93} +20.3366 q^{94} +9.21500 q^{96} -10.5846 q^{97} +1.59433 q^{98} +6.51337 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + q^{2} + 13 q^{4} + 3 q^{6} + 7 q^{7} + 9 q^{9}+O(q^{10}) \) 7 * q + q^2 + 13 * q^4 + 3 * q^6 + 7 * q^7 + 9 * q^9	\( 7 q + q^{2} + 13 q^{4} + 3 q^{6} + 7 q^{7} + 9 q^{9} + 7 q^{11} - 9 q^{12} - 3 q^{13} + q^{14} + 21 q^{16} - 2 q^{17} + 8 q^{18} + 20 q^{19} + q^{22} + 11 q^{23} + 18 q^{24} - 13 q^{26} - 12 q^{27} + 13 q^{28} + 4 q^{29} + 6 q^{31} + q^{32} - 7 q^{34} + 12 q^{36} + 19 q^{37} + 3 q^{38} - 10 q^{39} + 24 q^{41} + 3 q^{42} + 13 q^{44} + 33 q^{46} - q^{47} - 15 q^{48} + 7 q^{49} + 19 q^{51} - 29 q^{52} + 7 q^{53} + 9 q^{54} - 9 q^{57} + 37 q^{58} + 9 q^{59} + 18 q^{61} - 40 q^{62} + 9 q^{63} + 8 q^{64} + 3 q^{66} + 18 q^{68} - 15 q^{69} + 18 q^{71} - 64 q^{72} + 5 q^{73} - 24 q^{74} + 88 q^{76} + 7 q^{77} + 79 q^{78} + 25 q^{79} - q^{81} - 60 q^{82} - 17 q^{83} - 9 q^{84} - 41 q^{86} - 24 q^{87} - 16 q^{89} - 3 q^{91} + 28 q^{92} + 26 q^{93} - 31 q^{94} + 17 q^{96} - 4 q^{97} + q^{98} + 9 q^{99}+O(q^{100}) \) 7 * q + q^2 + 13 * q^4 + 3 * q^6 + 7 * q^7 + 9 * q^9 + 7 * q^11 - 9 * q^12 - 3 * q^13 + q^14 + 21 * q^16 - 2 * q^17 + 8 * q^18 + 20 * q^19 + q^22 + 11 * q^23 + 18 * q^24 - 13 * q^26 - 12 * q^27 + 13 * q^28 + 4 * q^29 + 6 * q^31 + q^32 - 7 * q^34 + 12 * q^36 + 19 * q^37 + 3 * q^38 - 10 * q^39 + 24 * q^41 + 3 * q^42 + 13 * q^44 + 33 * q^46 - q^47 - 15 * q^48 + 7 * q^49 + 19 * q^51 - 29 * q^52 + 7 * q^53 + 9 * q^54 - 9 * q^57 + 37 * q^58 + 9 * q^59 + 18 * q^61 - 40 * q^62 + 9 * q^63 + 8 * q^64 + 3 * q^66 + 18 * q^68 - 15 * q^69 + 18 * q^71 - 64 * q^72 + 5 * q^73 - 24 * q^74 + 88 * q^76 + 7 * q^77 + 79 * q^78 + 25 * q^79 - q^81 - 60 * q^82 - 17 * q^83 - 9 * q^84 - 41 * q^86 - 24 * q^87 - 16 * q^89 - 3 * q^91 + 28 * q^92 + 26 * q^93 - 31 * q^94 + 17 * q^96 - 4 * q^97 + q^98 + 9 * q^99

Introduction

L-functions

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