LMFDB - Embedded newform 1925.2.a.z.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:	$6$
Coefficient field:	6.6.9921856.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 7x^{4} + 11x^{2} - 2x - 1 $ x^6 - 7x^4 + 11x^2 - 2*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 385)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.28581$ 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)

$f(q)$	$=$	$q+1.28581 q^{2} -2.01740 q^{3} -0.346698 q^{4} -2.59399 q^{6} -1.00000 q^{7} -3.01740 q^{8} +1.06991 q^{9} +O(q^{10})$	\(q+1.28581 q^{2} -2.01740 q^{3} -0.346698 q^{4} -2.59399 q^{6} -1.00000 q^{7} -3.01740 q^{8} +1.06991 q^{9} -1.00000 q^{11} +0.699429 q^{12} +1.52250 q^{13} -1.28581 q^{14} -3.18641 q^{16} +0.461961 q^{17} +1.37570 q^{18} -5.59663 q^{19} +2.01740 q^{21} -1.28581 q^{22} +5.33461 q^{23} +6.08732 q^{24} +1.95764 q^{26} +3.89376 q^{27} +0.346698 q^{28} -2.46161 q^{29} -3.92656 q^{31} +1.93770 q^{32} +2.01740 q^{33} +0.593993 q^{34} -0.370937 q^{36} -8.62527 q^{37} -7.19619 q^{38} -3.07149 q^{39} +4.95796 q^{41} +2.59399 q^{42} +8.80450 q^{43} +0.346698 q^{44} +6.85929 q^{46} +13.6318 q^{47} +6.42826 q^{48} +1.00000 q^{49} -0.931961 q^{51} -0.527847 q^{52} -0.386308 q^{53} +5.00663 q^{54} +3.01740 q^{56} +11.2907 q^{57} -3.16516 q^{58} +8.74963 q^{59} +8.97533 q^{61} -5.04880 q^{62} -1.06991 q^{63} +8.86432 q^{64} +2.59399 q^{66} +7.72244 q^{67} -0.160161 q^{68} -10.7621 q^{69} +3.38553 q^{71} -3.22836 q^{72} -12.9545 q^{73} -11.0904 q^{74} +1.94034 q^{76} +1.00000 q^{77} -3.94935 q^{78} -7.16847 q^{79} -11.0650 q^{81} +6.37499 q^{82} +9.78304 q^{83} -0.699429 q^{84} +11.3209 q^{86} +4.96606 q^{87} +3.01740 q^{88} -8.43620 q^{89} -1.52250 q^{91} -1.84950 q^{92} +7.92145 q^{93} +17.5279 q^{94} -3.90912 q^{96} +13.7066 q^{97} +1.28581 q^{98} -1.06991 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{3} + 2 q^{4} - 2 q^{6} - 6 q^{7} + 10 q^{9}+O(q^{10}) $ 6 * q + 6 * q^3 + 2 * q^4 - 2 * q^6 - 6 * q^7 + 10 * q^9	$ 6 q + 6 q^{3} + 2 q^{4} - 2 q^{6} - 6 q^{7} + 10 q^{9} - 6 q^{11} + 12 q^{12} + 6 q^{13} - 6 q^{16} - 2 q^{17} + 14 q^{18} - 2 q^{19} - 6 q^{21} + 8 q^{23} + 22 q^{24} + 24 q^{27} - 2 q^{28} - 8 q^{29} + 4 q^{31} + 10 q^{32} - 6 q^{33} - 10 q^{34} + 26 q^{36} + 22 q^{37} - 10 q^{38} - 8 q^{39} + 4 q^{41} + 2 q^{42} + 30 q^{43} - 2 q^{44} - 8 q^{46} + 16 q^{47} - 8 q^{48} + 6 q^{49} - 4 q^{51} + 22 q^{52} + 6 q^{53} + 38 q^{54} + 18 q^{57} - 14 q^{58} + 14 q^{59} + 12 q^{61} - 14 q^{62} - 10 q^{63} - 22 q^{64} + 2 q^{66} + 46 q^{67} - 20 q^{68} + 12 q^{69} + 4 q^{71} + 32 q^{72} - 4 q^{73} - 8 q^{74} - 8 q^{76} + 6 q^{77} + 24 q^{78} - 8 q^{79} + 26 q^{81} - 10 q^{82} + 14 q^{83} - 12 q^{84} + 12 q^{86} - 2 q^{87} + 8 q^{89} - 6 q^{91} + 18 q^{92} + 12 q^{93} - 10 q^{94} - 16 q^{96} + 42 q^{97} - 10 q^{99}+O(q^{100}) $ 6 * q + 6 * q^3 + 2 * q^4 - 2 * q^6 - 6 * q^7 + 10 * q^9 - 6 * q^11 + 12 * q^12 + 6 * q^13 - 6 * q^16 - 2 * q^17 + 14 * q^18 - 2 * q^19 - 6 * q^21 + 8 * q^23 + 22 * q^24 + 24 * q^27 - 2 * q^28 - 8 * q^29 + 4 * q^31 + 10 * q^32 - 6 * q^33 - 10 * q^34 + 26 * q^36 + 22 * q^37 - 10 * q^38 - 8 * q^39 + 4 * q^41 + 2 * q^42 + 30 * q^43 - 2 * q^44 - 8 * q^46 + 16 * q^47 - 8 * q^48 + 6 * q^49 - 4 * q^51 + 22 * q^52 + 6 * q^53 + 38 * q^54 + 18 * q^57 - 14 * q^58 + 14 * q^59 + 12 * q^61 - 14 * q^62 - 10 * q^63 - 22 * q^64 + 2 * q^66 + 46 * q^67 - 20 * q^68 + 12 * q^69 + 4 * q^71 + 32 * q^72 - 4 * q^73 - 8 * q^74 - 8 * q^76 + 6 * q^77 + 24 * q^78 - 8 * q^79 + 26 * q^81 - 10 * q^82 + 14 * q^83 - 12 * q^84 + 12 * q^86 - 2 * q^87 + 8 * q^89 - 6 * q^91 + 18 * q^92 + 12 * q^93 - 10 * q^94 - 16 * q^96 + 42 * q^97 - 10 * 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 $

$2$	1.28581	0.909204	0.454602	−	0.890695i	$-0.349782\pi$
$2$	1.28581	0.909204	0.454602	+	0.890695i	$0.349782\pi$
$3$	−2.01740	−1.16475	−0.582374	−	0.812921i	$-0.697876\pi$
$3$	−2.01740	−1.16475	−0.582374	+	0.812921i	$0.697876\pi$
$4$	−0.346698	−0.173349
$5$	0	0
$5$	0	0
$6$	−2.59399	−1.05899
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−3.01740	−1.06681
$9$	1.06991	0.356638
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0.699429	0.201908
$13$	1.52250	0.422265	0.211133	−	0.977457i	$-0.432285\pi$
$13$	1.52250	0.422265	0.211133	+	0.977457i	$0.432285\pi$
$14$	−1.28581	−0.343647
$15$	0	0
$16$	−3.18641	−0.796601
$17$	0.461961	0.112042	0.0560210	−	0.998430i	$-0.482159\pi$
$17$	0.461961	0.112042	0.0560210	+	0.998430i	$0.482159\pi$
$18$	1.37570	0.324256
$19$	−5.59663	−1.28396	−0.641978	−	0.766723i	$-0.721886\pi$
$19$	−5.59663	−1.28396	−0.641978	+	0.766723i	$0.721886\pi$
$20$	0	0
$21$	2.01740	0.440233
$22$	−1.28581	−0.274135
$23$	5.33461	1.11234	0.556172	−	0.831067i	$-0.312270\pi$
$23$	5.33461	1.11234	0.556172	+	0.831067i	$0.312270\pi$
$24$	6.08732	1.24257
$25$	0	0
$26$	1.95764	0.383925
$27$	3.89376	0.749355
$28$	0.346698	0.0655197
$29$	−2.46161	−0.457109	−0.228555	−	0.973531i	$-0.573400\pi$
$29$	−2.46161	−0.457109	−0.228555	+	0.973531i	$0.573400\pi$
$30$	0	0
$31$	−3.92656	−0.705231	−0.352616	−	0.935768i	$-0.614708\pi$
$31$	−3.92656	−0.705231	−0.352616	+	0.935768i	$0.614708\pi$
$32$	1.93770	0.342540
$33$	2.01740	0.351185
$34$	0.593993	0.101869
$35$	0	0
$36$	−0.370937	−0.0618228
$37$	−8.62527	−1.41799	−0.708993	−	0.705215i	$-0.750850\pi$
$37$	−8.62527	−1.41799	−0.708993	+	0.705215i	$0.750850\pi$
$38$	−7.19619	−1.16738
$39$	−3.07149	−0.491833
$40$	0	0
$41$	4.95796	0.774304	0.387152	−	0.922016i	$-0.373459\pi$
$41$	4.95796	0.774304	0.387152	+	0.922016i	$0.373459\pi$
$42$	2.59399	0.400262
$43$	8.80450	1.34267	0.671337	−	0.741152i	$-0.265720\pi$
$43$	8.80450	1.34267	0.671337	+	0.741152i	$0.265720\pi$
$44$	0.346698	0.0522666
$45$	0	0
$46$	6.85929	1.01135
$47$	13.6318	1.98840	0.994200	−	0.107543i	$-0.0342985\pi$
$47$	13.6318	1.98840	0.994200	+	0.107543i	$0.0342985\pi$
$48$	6.42826	0.927840
$49$	1.00000	0.142857
$50$	0	0
$51$	−0.931961	−0.130501
$52$	−0.527847	−0.0731992
$53$	−0.386308	−0.0530635	−0.0265317	−	0.999648i	$-0.508446\pi$
$53$	−0.386308	−0.0530635	−0.0265317	+	0.999648i	$0.508446\pi$
$54$	5.00663	0.681316
$55$	0	0
$56$	3.01740	0.403217
$57$	11.2907	1.49548
$58$	−3.16516	−0.415606
$59$	8.74963	1.13910	0.569552	−	0.821955i	$-0.307117\pi$
$59$	8.74963	1.13910	0.569552	+	0.821955i	$0.307117\pi$
$60$	0	0
$61$	8.97533	1.14917	0.574586	−	0.818444i	$-0.305163\pi$
$61$	8.97533	1.14917	0.574586	+	0.818444i	$0.305163\pi$
$62$	−5.04880	−0.641199
$63$	−1.06991	−0.134796
$64$	8.86432	1.10804
$65$	0	0
$66$	2.59399	0.319298
$67$	7.72244	0.943446	0.471723	−	0.881747i	$-0.343632\pi$
$67$	7.72244	0.943446	0.471723	+	0.881747i	$0.343632\pi$
$68$	−0.160161	−0.0194223
$69$	−10.7621	−1.29560
$70$	0	0
$71$	3.38553	0.401789	0.200894	−	0.979613i	$-0.435615\pi$
$71$	3.38553	0.401789	0.200894	+	0.979613i	$0.435615\pi$
$72$	−3.22836	−0.380466
$73$	−12.9545	−1.51621	−0.758105	−	0.652133i	$-0.773874\pi$
$73$	−12.9545	−1.51621	−0.758105	+	0.652133i	$0.773874\pi$
$74$	−11.0904	−1.28924
$75$	0	0
$76$	1.94034	0.222572
$77$	1.00000	0.113961
$78$	−3.94935	−0.447176
$79$	−7.16847	−0.806516	−0.403258	−	0.915086i	$-0.632122\pi$
$79$	−7.16847	−0.806516	−0.403258	+	0.915086i	$0.632122\pi$
$80$	0	0
$81$	−11.0650	−1.22945
$82$	6.37499	0.704000
$83$	9.78304	1.07383	0.536914	−	0.843637i	$-0.319590\pi$
$83$	9.78304	1.07383	0.536914	+	0.843637i	$0.319590\pi$
$84$	−0.699429	−0.0763139
$85$	0	0
$86$	11.3209	1.22076
$87$	4.96606	0.532417
$88$	3.01740	0.321656
$89$	−8.43620	−0.894235	−0.447118	−	0.894475i	$-0.647549\pi$
$89$	−8.43620	−0.894235	−0.447118	+	0.894475i	$0.647549\pi$
$90$	0	0
$91$	−1.52250	−0.159601
$92$	−1.84950	−0.192823
$93$	7.92145	0.821417
$94$	17.5279	1.80786
$95$	0	0
$96$	−3.90912	−0.398973
$97$	13.7066	1.39170	0.695849	−	0.718188i	$-0.255028\pi$
$97$	13.7066	1.39170	0.695849	+	0.718188i	$0.255028\pi$
$98$	1.28581	0.129886
$99$	−1.06991	−0.107530
$100$	0	0
$101$	9.28320	0.923713	0.461856	−	0.886955i	$-0.347184\pi$
$101$	9.28320	0.923713	0.461856	+	0.886955i	$0.347184\pi$
$102$	−1.19832	−0.118652
$103$	14.2301	1.40213	0.701066	−	0.713096i	$-0.252708\pi$
$103$	14.2301	1.40213	0.701066	+	0.713096i	$0.252708\pi$
$104$	−4.59399	−0.450478
$105$	0	0
$106$	−0.496718	−0.0482455
$107$	1.09038	0.105411	0.0527054	−	0.998610i	$-0.483216\pi$
$107$	1.09038	0.105411	0.0527054	+	0.998610i	$0.483216\pi$
$108$	−1.34996	−0.129900
$109$	−10.5868	−1.01404	−0.507018	−	0.861936i	$-0.669252\pi$
$109$	−10.5868	−1.01404	−0.507018	+	0.861936i	$0.669252\pi$
$110$	0	0
$111$	17.4006	1.65160
$112$	3.18641	0.301087
$113$	−4.41431	−0.415264	−0.207632	−	0.978207i	$-0.566576\pi$
$113$	−4.41431	−0.415264	−0.207632	+	0.978207i	$0.566576\pi$
$114$	14.5176	1.35970
$115$	0	0
$116$	0.853434	0.0792394
$117$	1.62894	0.150596
$118$	11.2503	1.03568
$119$	−0.461961	−0.0423479
$120$	0	0
$121$	1.00000	0.0909091
$122$	11.5405	1.04483
$123$	−10.0022	−0.901868
$124$	1.36133	0.122251
$125$	0	0
$126$	−1.37570	−0.122557
$127$	−1.46002	−0.129555	−0.0647777	−	0.997900i	$-0.520634\pi$
$127$	−1.46002	−0.129555	−0.0647777	+	0.997900i	$0.520634\pi$
$128$	7.52242	0.664894
$129$	−17.7622	−1.56388
$130$	0	0
$131$	−2.12968	−0.186071	−0.0930353	−	0.995663i	$-0.529657\pi$
$131$	−2.12968	−0.186071	−0.0930353	+	0.995663i	$0.529657\pi$
$132$	−0.699429	−0.0608775
$133$	5.59663	0.485290
$134$	9.92957	0.857785
$135$	0	0
$136$	−1.39392	−0.119528
$137$	−9.53143	−0.814325	−0.407163	−	0.913356i	$-0.633482\pi$
$137$	−9.53143	−0.814325	−0.407163	+	0.913356i	$0.633482\pi$
$138$	−13.8379	−1.17796
$139$	19.1380	1.62326	0.811632	−	0.584168i	$-0.198579\pi$
$139$	19.1380	1.62326	0.811632	+	0.584168i	$0.198579\pi$
$140$	0	0
$141$	−27.5008	−2.31599
$142$	4.35314	0.365308
$143$	−1.52250	−0.127318
$144$	−3.40918	−0.284098
$145$	0	0
$146$	−16.6570	−1.37854
$147$	−2.01740	−0.166393
$148$	2.99036	0.245806
$149$	7.76143	0.635841	0.317920	−	0.948117i	$-0.397016\pi$
$149$	7.76143	0.635841	0.317920	+	0.948117i	$0.397016\pi$
$150$	0	0
$151$	−22.0862	−1.79735	−0.898673	−	0.438619i	$-0.855468\pi$
$151$	−22.0862	−1.79735	−0.898673	+	0.438619i	$0.855468\pi$
$152$	16.8873	1.36974
$153$	0.494258	0.0399584
$154$	1.28581	0.103613
$155$	0	0
$156$	1.06488	0.0852586
$157$	−0.0674671	−0.00538446	−0.00269223	−	0.999996i	$-0.500857\pi$
$157$	−0.0674671	−0.00538446	−0.00269223	+	0.999996i	$0.500857\pi$
$158$	−9.21727	−0.733287
$159$	0.779339	0.0618056
$160$	0	0
$161$	−5.33461	−0.420426
$162$	−14.2275	−1.11782
$163$	20.9616	1.64184	0.820922	−	0.571041i	$-0.193460\pi$
$163$	20.9616	1.64184	0.820922	+	0.571041i	$0.193460\pi$
$164$	−1.71891	−0.134225
$165$	0	0
$166$	12.5791	0.976328
$167$	−20.7810	−1.60808	−0.804042	−	0.594573i	$-0.797321\pi$
$167$	−20.7810	−1.60808	−0.804042	+	0.594573i	$0.797321\pi$
$168$	−6.08732	−0.469647
$169$	−10.6820	−0.821692
$170$	0	0
$171$	−5.98791	−0.457907
$172$	−3.05250	−0.232751
$173$	7.87496	0.598722	0.299361	−	0.954140i	$-0.403227\pi$
$173$	7.87496	0.598722	0.299361	+	0.954140i	$0.403227\pi$
$174$	6.38540	0.484076
$175$	0	0
$176$	3.18641	0.240184
$177$	−17.6515	−1.32677
$178$	−10.8473	−0.813042
$179$	9.87117	0.737806	0.368903	−	0.929468i	$-0.379733\pi$
$179$	9.87117	0.737806	0.368903	+	0.929468i	$0.379733\pi$
$180$	0	0
$181$	−1.97839	−0.147052	−0.0735262	−	0.997293i	$-0.523425\pi$
$181$	−1.97839	−0.147052	−0.0735262	+	0.997293i	$0.523425\pi$
$182$	−1.95764	−0.145110
$183$	−18.1068	−1.33850
$184$	−16.0967	−1.18666
$185$	0	0
$186$	10.1855	0.746835
$187$	−0.461961	−0.0337819
$188$	−4.72611	−0.344687
$189$	−3.89376	−0.283229
$190$	0	0
$191$	−7.19200	−0.520395	−0.260198	−	0.965555i	$-0.583788\pi$
$191$	−7.19200	−0.520395	−0.260198	+	0.965555i	$0.583788\pi$
$192$	−17.8829	−1.29059
$193$	0.421905	0.0303694	0.0151847	−	0.999885i	$-0.495166\pi$
$193$	0.421905	0.0303694	0.0151847	+	0.999885i	$0.495166\pi$
$194$	17.6241	1.26534
$195$	0	0
$196$	−0.346698	−0.0247641
$197$	2.70618	0.192807	0.0964035	−	0.995342i	$-0.469266\pi$
$197$	2.70618	0.192807	0.0964035	+	0.995342i	$0.469266\pi$
$198$	−1.37570	−0.0977670
$199$	10.1815	0.721748	0.360874	−	0.932615i	$-0.382479\pi$
$199$	10.1815	0.721748	0.360874	+	0.932615i	$0.382479\pi$
$200$	0	0
$201$	−15.5793	−1.09888
$202$	11.9364	0.839843
$203$	2.46161	0.172771
$204$	0.323109	0.0226221
$205$	0	0
$206$	18.2972	1.27482
$207$	5.70757	0.396704
$208$	−4.85130	−0.336377
$209$	5.59663	0.387127
$210$	0	0
$211$	15.0464	1.03584	0.517918	−	0.855430i	$-0.326707\pi$
$211$	15.0464	1.03584	0.517918	+	0.855430i	$0.326707\pi$
$212$	0.133932	0.00919849
$213$	−6.82998	−0.467982
$214$	1.40202	0.0958399
$215$	0	0
$216$	−11.7490	−0.799421
$217$	3.92656	0.266552
$218$	−13.6126	−0.921965
$219$	26.1344	1.76600
$220$	0	0
$221$	0.703335	0.0473114
$222$	22.3739	1.50164
$223$	27.6742	1.85320	0.926600	−	0.376049i	$-0.122717\pi$
$223$	27.6742	1.85320	0.926600	+	0.376049i	$0.122717\pi$
$224$	−1.93770	−0.129468
$225$	0	0
$226$	−5.67596	−0.377559
$227$	−8.09640	−0.537377	−0.268688	−	0.963227i	$-0.586590\pi$
$227$	−8.09640	−0.537377	−0.268688	+	0.963227i	$0.586590\pi$
$228$	−3.91445	−0.259240
$229$	6.20190	0.409833	0.204917	−	0.978779i	$-0.434308\pi$
$229$	6.20190	0.409833	0.204917	+	0.978779i	$0.434308\pi$
$230$	0	0
$231$	−2.01740	−0.132735
$232$	7.42767	0.487650
$233$	4.69314	0.307458	0.153729	−	0.988113i	$-0.450872\pi$
$233$	4.69314	0.307458	0.153729	+	0.988113i	$0.450872\pi$
$234$	2.09451	0.136922
$235$	0	0
$236$	−3.03348	−0.197462
$237$	14.4617	0.939388
$238$	−0.593993	−0.0385028
$239$	−5.00440	−0.323708	−0.161854	−	0.986815i	$-0.551747\pi$
$239$	−5.00440	−0.323708	−0.161854	+	0.986815i	$0.551747\pi$
$240$	0	0
$241$	−2.68494	−0.172952	−0.0864759	−	0.996254i	$-0.527561\pi$
$241$	−2.68494	−0.172952	−0.0864759	+	0.996254i	$0.527561\pi$
$242$	1.28581	0.0826549
$243$	10.6413	0.682642
$244$	−3.11172	−0.199208
$245$	0	0
$246$	−12.8609	−0.819982
$247$	−8.52087	−0.542170
$248$	11.8480	0.752350
$249$	−19.7363	−1.25074
$250$	0	0
$251$	−26.2690	−1.65809	−0.829044	−	0.559184i	$-0.811114\pi$
$251$	−26.2690	−1.65809	−0.829044	+	0.559184i	$0.811114\pi$
$252$	0.370937	0.0233668
$253$	−5.33461	−0.335384
$254$	−1.87730	−0.117792
$255$	0	0
$256$	−8.05626	−0.503516
$257$	6.89020	0.429799	0.214899	−	0.976636i	$-0.431058\pi$
$257$	6.89020	0.429799	0.214899	+	0.976636i	$0.431058\pi$
$258$	−22.8388	−1.42188
$259$	8.62527	0.535948
$260$	0	0
$261$	−2.63371	−0.163023
$262$	−2.73835	−0.169176
$263$	−20.7169	−1.27746	−0.638730	−	0.769431i	$-0.720540\pi$
$263$	−20.7169	−1.27746	−0.638730	+	0.769431i	$0.720540\pi$
$264$	−6.08732	−0.374648
$265$	0	0
$266$	7.19619	0.441227
$267$	17.0192	1.04156
$268$	−2.67735	−0.163545
$269$	−5.20076	−0.317096	−0.158548	−	0.987351i	$-0.550681\pi$
$269$	−5.20076	−0.317096	−0.158548	+	0.987351i	$0.550681\pi$
$270$	0	0
$271$	−5.31287	−0.322734	−0.161367	−	0.986894i	$-0.551590\pi$
$271$	−5.31287	−0.322734	−0.161367	+	0.986894i	$0.551590\pi$
$272$	−1.47199	−0.0892527
$273$	3.07149	0.185895
$274$	−12.2556	−0.740387
$275$	0	0
$276$	3.73118	0.224591
$277$	−25.7738	−1.54860	−0.774298	−	0.632821i	$-0.781897\pi$
$277$	−25.7738	−1.54860	−0.774298	+	0.632821i	$0.781897\pi$
$278$	24.6078	1.47588
$279$	−4.20108	−0.251512
$280$	0	0
$281$	20.1954	1.20476	0.602379	−	0.798211i	$-0.294220\pi$
$281$	20.1954	1.20476	0.602379	+	0.798211i	$0.294220\pi$
$282$	−35.3608	−2.10570
$283$	3.68401	0.218992	0.109496	−	0.993987i	$-0.465076\pi$
$283$	3.68401	0.218992	0.109496	+	0.993987i	$0.465076\pi$
$284$	−1.17376	−0.0696496
$285$	0	0
$286$	−1.95764	−0.115758
$287$	−4.95796	−0.292659
$288$	2.07317	0.122163
$289$	−16.7866	−0.987447
$290$	0	0
$291$	−27.6518	−1.62098
$292$	4.49129	0.262833
$293$	9.86412	0.576268	0.288134	−	0.957590i	$-0.406965\pi$
$293$	9.86412	0.576268	0.288134	+	0.957590i	$0.406965\pi$
$294$	−2.59399	−0.151285
$295$	0	0
$296$	26.0259	1.51273
$297$	−3.89376	−0.225939
$298$	9.97970	0.578109
$299$	8.12194	0.469704
$300$	0	0
$301$	−8.80450	−0.507483
$302$	−28.3986	−1.63415
$303$	−18.7279	−1.07589
$304$	17.8331	1.02280
$305$	0	0
$306$	0.635521	0.0363303
$307$	12.5167	0.714364	0.357182	−	0.934035i	$-0.383738\pi$
$307$	12.5167	0.714364	0.357182	+	0.934035i	$0.383738\pi$
$308$	−0.346698	−0.0197549
$309$	−28.7078	−1.63313
$310$	0	0
$311$	13.9286	0.789819	0.394910	−	0.918720i	$-0.370776\pi$
$311$	13.9286	0.789819	0.394910	+	0.918720i	$0.370776\pi$
$312$	9.26793	0.524693
$313$	0.596995	0.0337442	0.0168721	−	0.999858i	$-0.494629\pi$
$313$	0.596995	0.0337442	0.0168721	+	0.999858i	$0.494629\pi$
$314$	−0.0867497	−0.00489557
$315$	0	0
$316$	2.48529	0.139809
$317$	11.9926	0.673571	0.336785	−	0.941581i	$-0.390660\pi$
$317$	11.9926	0.673571	0.336785	+	0.941581i	$0.390660\pi$
$318$	1.00208	0.0561938
$319$	2.46161	0.137824
$320$	0	0
$321$	−2.19973	−0.122777
$322$	−6.85929	−0.382253
$323$	−2.58542	−0.143857
$324$	3.83622	0.213123
$325$	0	0
$326$	26.9527	1.49277
$327$	21.3579	1.18110
$328$	−14.9602	−0.826037
$329$	−13.6318	−0.751545
$330$	0	0
$331$	3.40511	0.187162	0.0935809	−	0.995612i	$-0.470169\pi$
$331$	3.40511	0.187162	0.0935809	+	0.995612i	$0.470169\pi$
$332$	−3.39176	−0.186147
$333$	−9.22830	−0.505708
$334$	−26.7204	−1.46208
$335$	0	0
$336$	−6.42826	−0.350691
$337$	−1.81193	−0.0987022	−0.0493511	−	0.998781i	$-0.515715\pi$
$337$	−1.81193	−0.0987022	−0.0493511	+	0.998781i	$0.515715\pi$
$338$	−13.7350	−0.747085
$339$	8.90545	0.483678
$340$	0	0
$341$	3.92656	0.212635
$342$	−7.69931	−0.416331
$343$	−1.00000	−0.0539949
$344$	−26.5667	−1.43238
$345$	0	0
$346$	10.1257	0.544360
$347$	−20.6118	−1.10650	−0.553250	−	0.833015i	$-0.686612\pi$
$347$	−20.6118	−1.10650	−0.553250	+	0.833015i	$0.686612\pi$
$348$	−1.72172	−0.0922939
$349$	−28.5720	−1.52943	−0.764713	−	0.644371i	$-0.777119\pi$
$349$	−28.5720	−1.52943	−0.764713	+	0.644371i	$0.777119\pi$
$350$	0	0
$351$	5.92825	0.316426
$352$	−1.93770	−0.103280
$353$	−0.762881	−0.0406040	−0.0203020	−	0.999794i	$-0.506463\pi$
$353$	−0.762881	−0.0406040	−0.0203020	+	0.999794i	$0.506463\pi$
$354$	−22.6965	−1.20630
$355$	0	0
$356$	2.92481	0.155015
$357$	0.931961	0.0493246
$358$	12.6924	0.670816
$359$	3.09932	0.163576	0.0817879	−	0.996650i	$-0.473937\pi$
$359$	3.09932	0.163576	0.0817879	+	0.996650i	$0.473937\pi$
$360$	0	0
$361$	12.3223	0.648542
$362$	−2.54383	−0.133701
$363$	−2.01740	−0.105886
$364$	0.527847	0.0276667
$365$	0	0
$366$	−23.2819	−1.21697
$367$	26.0930	1.36205	0.681023	−	0.732262i	$-0.261535\pi$
$367$	26.0930	1.36205	0.681023	+	0.732262i	$0.261535\pi$
$368$	−16.9982	−0.886094
$369$	5.30459	0.276146
$370$	0	0
$371$	0.386308	0.0200561
$372$	−2.74635	−0.142392
$373$	8.59461	0.445012	0.222506	−	0.974931i	$-0.428576\pi$
$373$	8.59461	0.445012	0.222506	+	0.974931i	$0.428576\pi$
$374$	−0.593993	−0.0307146
$375$	0	0
$376$	−41.1326	−2.12125
$377$	−3.74780	−0.193021
$378$	−5.00663	−0.257513
$379$	−38.0675	−1.95540	−0.977699	−	0.210010i	$-0.932650\pi$
$379$	−38.0675	−1.95540	−0.977699	+	0.210010i	$0.932650\pi$
$380$	0	0
$381$	2.94544	0.150899
$382$	−9.24754	−0.473145
$383$	5.14216	0.262752	0.131376	−	0.991333i	$-0.458060\pi$
$383$	5.14216	0.262752	0.131376	+	0.991333i	$0.458060\pi$
$384$	−15.1757	−0.774434
$385$	0	0
$386$	0.542489	0.0276120
$387$	9.42005	0.478848
$388$	−4.75206	−0.241249
$389$	22.3888	1.13516	0.567578	−	0.823319i	$-0.307880\pi$
$389$	22.3888	1.13516	0.567578	+	0.823319i	$0.307880\pi$
$390$	0	0
$391$	2.46438	0.124629
$392$	−3.01740	−0.152402
$393$	4.29641	0.216725
$394$	3.47962	0.175301
$395$	0	0
$396$	0.370937	0.0186403
$397$	−9.47613	−0.475593	−0.237797	−	0.971315i	$-0.576425\pi$
$397$	−9.47613	−0.475593	−0.237797	+	0.971315i	$0.576425\pi$
$398$	13.0915	0.656216
$399$	−11.2907	−0.565240
$400$	0	0
$401$	−6.79535	−0.339344	−0.169672	−	0.985501i	$-0.554271\pi$
$401$	−6.79535	−0.339344	−0.169672	+	0.985501i	$0.554271\pi$
$402$	−20.0320	−0.999103
$403$	−5.97818	−0.297795
$404$	−3.21846	−0.160125
$405$	0	0
$406$	3.16516	0.157084
$407$	8.62527	0.427539
$408$	2.81210	0.139220
$409$	27.4705	1.35833	0.679163	−	0.733987i	$-0.262343\pi$
$409$	27.4705	1.35833	0.679163	+	0.733987i	$0.262343\pi$
$410$	0	0
$411$	19.2287	0.948484
$412$	−4.93354	−0.243058
$413$	−8.74963	−0.430541
$414$	7.33885	0.360685
$415$	0	0
$416$	2.95015	0.144643
$417$	−38.6091	−1.89069
$418$	7.19619	0.351977
$419$	4.72063	0.230618	0.115309	−	0.993330i	$-0.463214\pi$
$419$	4.72063	0.230618	0.115309	+	0.993330i	$0.463214\pi$
$420$	0	0
$421$	21.7598	1.06051	0.530254	−	0.847839i	$-0.322097\pi$
$421$	21.7598	1.06051	0.530254	+	0.847839i	$0.322097\pi$
$422$	19.3468	0.941785
$423$	14.5848	0.709139
$424$	1.16565	0.0566088
$425$	0	0
$426$	−8.78204	−0.425491
$427$	−8.97533	−0.434347
$428$	−0.378032	−0.0182728
$429$	3.07149	0.148293
$430$	0	0
$431$	25.5581	1.23109	0.615545	−	0.788102i	$-0.288936\pi$
$431$	25.5581	1.23109	0.615545	+	0.788102i	$0.288936\pi$
$432$	−12.4071	−0.596937
$433$	−3.80718	−0.182961	−0.0914806	−	0.995807i	$-0.529160\pi$
$433$	−3.80718	−0.182961	−0.0914806	+	0.995807i	$0.529160\pi$
$434$	5.04880	0.242350
$435$	0	0
$436$	3.67043	0.175782
$437$	−29.8559	−1.42820
$438$	33.6039	1.60566
$439$	18.4690	0.881477	0.440738	−	0.897636i	$-0.354717\pi$
$439$	18.4690	0.881477	0.440738	+	0.897636i	$0.354717\pi$
$440$	0	0
$441$	1.06991	0.0509483
$442$	0.904353	0.0430157
$443$	−16.5595	−0.786766	−0.393383	−	0.919375i	$-0.628695\pi$
$443$	−16.5595	−0.786766	−0.393383	+	0.919375i	$0.628695\pi$
$444$	−6.03276	−0.286302
$445$	0	0
$446$	35.5837	1.68494
$447$	−15.6579	−0.740594
$448$	−8.86432	−0.418800
$449$	16.6419	0.785381	0.392690	−	0.919671i	$-0.371544\pi$
$449$	16.6419	0.785381	0.392690	+	0.919671i	$0.371544\pi$
$450$	0	0
$451$	−4.95796	−0.233461
$452$	1.53043	0.0719855
$453$	44.5567	2.09346
$454$	−10.4104	−0.488585
$455$	0	0
$456$	−34.0685	−1.59540
$457$	7.86437	0.367880	0.183940	−	0.982937i	$-0.441115\pi$
$457$	7.86437	0.367880	0.183940	+	0.982937i	$0.441115\pi$
$458$	7.97446	0.372622
$459$	1.79876	0.0839591
$460$	0	0
$461$	11.0462	0.514474	0.257237	−	0.966348i	$-0.417188\pi$
$461$	11.0462	0.514474	0.257237	+	0.966348i	$0.417188\pi$
$462$	−2.59399	−0.120683
$463$	−38.6222	−1.79493	−0.897464	−	0.441088i	$-0.854593\pi$
$463$	−38.6222	−1.79493	−0.897464	+	0.441088i	$0.854593\pi$
$464$	7.84369	0.364134
$465$	0	0
$466$	6.03448	0.279542
$467$	−11.1611	−0.516472	−0.258236	−	0.966082i	$-0.583141\pi$
$467$	−11.1611	−0.516472	−0.258236	+	0.966082i	$0.583141\pi$
$468$	−0.564751	−0.0261056
$469$	−7.72244	−0.356589
$470$	0	0
$471$	0.136108	0.00627154
$472$	−26.4012	−1.21521
$473$	−8.80450	−0.404831
$474$	18.5950	0.854095
$475$	0	0
$476$	0.160161	0.00734095
$477$	−0.413316	−0.0189244
$478$	−6.43470	−0.294316
$479$	42.9399	1.96197	0.980987	−	0.194073i	$-0.0621700\pi$
$479$	42.9399	1.96197	0.980987	+	0.194073i	$0.0621700\pi$
$480$	0	0
$481$	−13.1320	−0.598766
$482$	−3.45231	−0.157248
$483$	10.7621	0.489691
$484$	−0.346698	−0.0157590
$485$	0	0
$486$	13.6827	0.620660
$487$	−2.51082	−0.113776	−0.0568880	−	0.998381i	$-0.518118\pi$
$487$	−2.51082	−0.113776	−0.0568880	+	0.998381i	$0.518118\pi$
$488$	−27.0822	−1.22595
$489$	−42.2881	−1.91233
$490$	0	0
$491$	−20.7752	−0.937572	−0.468786	−	0.883312i	$-0.655308\pi$
$491$	−20.7752	−0.937572	−0.468786	+	0.883312i	$0.655308\pi$
$492$	3.46774	0.156338
$493$	−1.13717	−0.0512154
$494$	−10.9562	−0.492943
$495$	0	0
$496$	12.5116	0.561788
$497$	−3.38553	−0.151862
$498$	−25.3771	−1.13718
$499$	36.3068	1.62531	0.812657	−	0.582743i	$-0.198020\pi$
$499$	36.3068	1.62531	0.812657	+	0.582743i	$0.198020\pi$
$500$	0	0
$501$	41.9237	1.87301
$502$	−33.7769	−1.50754
$503$	−34.1755	−1.52381	−0.761905	−	0.647689i	$-0.775735\pi$
$503$	−34.1755	−1.52381	−0.761905	+	0.647689i	$0.775735\pi$
$504$	3.22836	0.143803
$505$	0	0
$506$	−6.85929	−0.304932
$507$	21.5499	0.957064
$508$	0.506184	0.0224583
$509$	41.0462	1.81934	0.909671	−	0.415330i	$-0.136334\pi$
$509$	41.0462	1.81934	0.909671	+	0.415330i	$0.136334\pi$
$510$	0	0
$511$	12.9545	0.573073
$512$	−25.4036	−1.12269
$513$	−21.7919	−0.962138
$514$	8.85947	0.390775
$515$	0	0
$516$	6.15812	0.271096
$517$	−13.6318	−0.599525
$518$	11.0904	0.487286
$519$	−15.8870	−0.697360
$520$	0	0
$521$	−21.6790	−0.949772	−0.474886	−	0.880047i	$-0.657511\pi$
$521$	−21.6790	−0.949772	−0.474886	+	0.880047i	$0.657511\pi$
$522$	−3.38645	−0.148221
$523$	18.6389	0.815021	0.407511	−	0.913200i	$-0.366397\pi$
$523$	18.6389	0.815021	0.407511	+	0.913200i	$0.366397\pi$
$524$	0.738353	0.0322551
$525$	0	0
$526$	−26.6380	−1.16147
$527$	−1.81392	−0.0790154
$528$	−6.42826	−0.279754
$529$	5.45808	0.237308
$530$	0	0
$531$	9.36135	0.406248
$532$	−1.94034	−0.0841244
$533$	7.54849	0.326961
$534$	21.8834	0.946989
$535$	0	0
$536$	−23.3017	−1.00648
$537$	−19.9141	−0.859358
$538$	−6.68718	−0.288305
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	26.6920	1.14758	0.573790	−	0.819003i	$-0.305473\pi$
$541$	26.6920	1.14758	0.573790	+	0.819003i	$0.305473\pi$
$542$	−6.83133	−0.293431
$543$	3.99121	0.171279
$544$	0.895141	0.0383789
$545$	0	0
$546$	3.94935	0.169017
$547$	36.9991	1.58197	0.790983	−	0.611838i	$-0.209569\pi$
$547$	36.9991	1.58197	0.790983	+	0.611838i	$0.209569\pi$
$548$	3.30453	0.141162
$549$	9.60283	0.409839
$550$	0	0
$551$	13.7767	0.586908
$552$	32.4735	1.38216
$553$	7.16847	0.304834
$554$	−33.1401	−1.40799
$555$	0	0
$556$	−6.63510	−0.281391
$557$	−8.70244	−0.368734	−0.184367	−	0.982857i	$-0.559024\pi$
$557$	−8.70244	−0.368734	−0.184367	+	0.982857i	$0.559024\pi$
$558$	−5.40178	−0.228676
$559$	13.4048	0.566964
$560$	0	0
$561$	0.931961	0.0393474
$562$	25.9674	1.09537
$563$	32.9970	1.39066	0.695330	−	0.718691i	$-0.255258\pi$
$563$	32.9970	1.39066	0.695330	+	0.718691i	$0.255258\pi$
$564$	9.53447	0.401473
$565$	0	0
$566$	4.73693	0.199108
$567$	11.0650	0.464687
$568$	−10.2155	−0.428633
$569$	41.6339	1.74538	0.872691	−	0.488272i	$-0.162373\pi$
$569$	41.6339	1.74538	0.872691	+	0.488272i	$0.162373\pi$
$570$	0	0
$571$	−19.9013	−0.832842	−0.416421	−	0.909172i	$-0.636716\pi$
$571$	−19.9013	−0.832842	−0.416421	+	0.909172i	$0.636716\pi$
$572$	0.527847	0.0220704
$573$	14.5092	0.606129
$574$	−6.37499	−0.266087
$575$	0	0
$576$	9.48406	0.395169
$577$	−10.0044	−0.416489	−0.208244	−	0.978077i	$-0.566775\pi$
$577$	−10.0044	−0.416489	−0.208244	+	0.978077i	$0.566775\pi$
$578$	−21.5843	−0.897790
$579$	−0.851153	−0.0353727
$580$	0	0
$581$	−9.78304	−0.405869
$582$	−35.5549	−1.47380
$583$	0.386308	0.0159992
$584$	39.0889	1.61751
$585$	0	0
$586$	12.6834	0.523945
$587$	21.9774	0.907103	0.453551	−	0.891230i	$-0.350157\pi$
$587$	21.9774	0.907103	0.453551	+	0.891230i	$0.350157\pi$
$588$	0.699429	0.0288440
$589$	21.9755	0.905485
$590$	0	0
$591$	−5.45945	−0.224572
$592$	27.4836	1.12957
$593$	−39.7669	−1.63303	−0.816515	−	0.577325i	$-0.804097\pi$
$593$	−39.7669	−1.63303	−0.816515	+	0.577325i	$0.804097\pi$
$594$	−5.00663	−0.205424
$595$	0	0
$596$	−2.69087	−0.110222
$597$	−20.5402	−0.840654
$598$	10.4433	0.427057
$599$	20.0835	0.820588	0.410294	−	0.911953i	$-0.365426\pi$
$599$	20.0835	0.820588	0.410294	+	0.911953i	$0.365426\pi$
$600$	0	0
$601$	26.4569	1.07920	0.539601	−	0.841921i	$-0.318575\pi$
$601$	26.4569	1.07920	0.539601	+	0.841921i	$0.318575\pi$
$602$	−11.3209	−0.461405
$603$	8.26234	0.336469
$604$	7.65722	0.311568
$605$	0	0
$606$	−24.0805	−0.978205
$607$	25.0986	1.01872	0.509360	−	0.860553i	$-0.329882\pi$
$607$	25.0986	1.01872	0.509360	+	0.860553i	$0.329882\pi$
$608$	−10.8446	−0.439806
$609$	−4.96606	−0.201235
$610$	0	0
$611$	20.7544	0.839633
$612$	−0.171358	−0.00692674
$613$	43.6609	1.76345	0.881723	−	0.471767i	$-0.156384\pi$
$613$	43.6609	1.76345	0.881723	+	0.471767i	$0.156384\pi$
$614$	16.0940	0.649503
$615$	0	0
$616$	−3.01740	−0.121575
$617$	49.4485	1.99072	0.995362	−	0.0962032i	$-0.0306699\pi$
$617$	49.4485	1.99072	0.995362	+	0.0962032i	$0.0306699\pi$
$618$	−36.9127	−1.48485
$619$	−18.3592	−0.737918	−0.368959	−	0.929446i	$-0.620286\pi$
$619$	−18.3592	−0.737918	−0.368959	+	0.929446i	$0.620286\pi$
$620$	0	0
$621$	20.7717	0.833540
$622$	17.9095	0.718107
$623$	8.43620	0.337989
$624$	9.78702	0.391795
$625$	0	0
$626$	0.767622	0.0306803
$627$	−11.2907	−0.450906
$628$	0.0233907	0.000933390 0
$629$	−3.98454	−0.158874
$630$	0	0
$631$	36.3055	1.44530	0.722651	−	0.691214i	$-0.242924\pi$
$631$	36.3055	1.44530	0.722651	+	0.691214i	$0.242924\pi$
$632$	21.6302	0.860401
$633$	−30.3546	−1.20649
$634$	15.4202	0.612413
$635$	0	0
$636$	−0.270195	−0.0107139
$637$	1.52250	0.0603236
$638$	3.16516	0.125310
$639$	3.62223	0.143293
$640$	0	0
$641$	−21.9189	−0.865743	−0.432872	−	0.901456i	$-0.642500\pi$
$641$	−21.9189	−0.865743	−0.432872	+	0.901456i	$0.642500\pi$
$642$	−2.82843	−0.111629
$643$	−19.4956	−0.768833	−0.384416	−	0.923160i	$-0.625597\pi$
$643$	−19.4956	−0.768833	−0.384416	+	0.923160i	$0.625597\pi$
$644$	1.84950	0.0728804
$645$	0	0
$646$	−3.32436	−0.130795
$647$	5.35681	0.210598	0.105299	−	0.994441i	$-0.466420\pi$
$647$	5.35681	0.210598	0.105299	+	0.994441i	$0.466420\pi$
$648$	33.3876	1.31159
$649$	−8.74963	−0.343453
$650$	0	0
$651$	−7.92145	−0.310466
$652$	−7.26735	−0.284612
$653$	39.0621	1.52862	0.764309	−	0.644850i	$-0.223080\pi$
$653$	39.0621	1.52862	0.764309	+	0.644850i	$0.223080\pi$
$654$	27.4622	1.07386
$655$	0	0
$656$	−15.7981	−0.616811
$657$	−13.8602	−0.540738
$658$	−17.5279	−0.683307
$659$	46.8045	1.82325	0.911623	−	0.411027i	$-0.134830\pi$
$659$	46.8045	1.82325	0.911623	+	0.411027i	$0.134830\pi$
$660$	0	0
$661$	5.87018	0.228324	0.114162	−	0.993462i	$-0.463582\pi$
$661$	5.87018	0.228324	0.114162	+	0.993462i	$0.463582\pi$
$662$	4.37832	0.170168
$663$	−1.41891	−0.0551059
$664$	−29.5194	−1.14557
$665$	0	0
$666$	−11.8658	−0.459791
$667$	−13.1317	−0.508463
$668$	7.20473	0.278759
$669$	−55.8300	−2.15851
$670$	0	0
$671$	−8.97533	−0.346489
$672$	3.90912	0.150798
$673$	37.2683	1.43659	0.718294	−	0.695740i	$-0.244923\pi$
$673$	37.2683	1.43659	0.718294	+	0.695740i	$0.244923\pi$
$674$	−2.32980	−0.0897404
$675$	0	0
$676$	3.70342	0.142439
$677$	14.2954	0.549417	0.274708	−	0.961528i	$-0.411419\pi$
$677$	14.2954	0.549417	0.274708	+	0.961528i	$0.411419\pi$
$678$	11.4507	0.439761
$679$	−13.7066	−0.526013
$680$	0	0
$681$	16.3337	0.625909
$682$	5.04880	0.193329
$683$	−48.6433	−1.86128	−0.930642	−	0.365931i	$-0.880751\pi$
$683$	−48.6433	−1.86128	−0.930642	+	0.365931i	$0.880751\pi$
$684$	2.07600	0.0793777
$685$	0	0
$686$	−1.28581	−0.0490924
$687$	−12.5117	−0.477352
$688$	−28.0547	−1.06958
$689$	−0.588153	−0.0224069
$690$	0	0
$691$	−14.1547	−0.538470	−0.269235	−	0.963075i	$-0.586771\pi$
$691$	−14.1547	−0.538470	−0.269235	+	0.963075i	$0.586771\pi$
$692$	−2.73023	−0.103788
$693$	1.06991	0.0406427
$694$	−26.5028	−1.00603
$695$	0	0
$696$	−14.9846	−0.567990
$697$	2.29038	0.0867544
$698$	−36.7382	−1.39056
$699$	−9.46796	−0.358111
$700$	0	0
$701$	−25.4581	−0.961539	−0.480769	−	0.876847i	$-0.659643\pi$
$701$	−25.4581	−0.961539	−0.480769	+	0.876847i	$0.659643\pi$
$702$	7.62259	0.287696
$703$	48.2725	1.82063
$704$	−8.86432	−0.334087
$705$	0	0
$706$	−0.980918	−0.0369173
$707$	−9.28320	−0.349131
$708$	6.11974	0.229994
$709$	−44.0885	−1.65578	−0.827889	−	0.560892i	$-0.810458\pi$
$709$	−44.0885	−1.65578	−0.827889	+	0.560892i	$0.810458\pi$
$710$	0	0
$711$	−7.66964	−0.287634
$712$	25.4554	0.953982
$713$	−20.9467	−0.784459
$714$	1.19832	0.0448461
$715$	0	0
$716$	−3.42231	−0.127898
$717$	10.0959	0.377038
$718$	3.98513	0.148724
$719$	−40.6844	−1.51727	−0.758636	−	0.651515i	$-0.774134\pi$
$719$	−40.6844	−1.51727	−0.758636	+	0.651515i	$0.774134\pi$
$720$	0	0
$721$	−14.2301	−0.529956
$722$	15.8441	0.589656
$723$	5.41660	0.201445
$724$	0.685903	0.0254914
$725$	0	0
$726$	−2.59399	−0.0962721
$727$	−15.7930	−0.585729	−0.292864	−	0.956154i	$-0.594608\pi$
$727$	−15.7930	−0.585729	−0.292864	+	0.956154i	$0.594608\pi$
$728$	4.59399	0.170265
$729$	11.7272	0.434342
$730$	0	0
$731$	4.06733	0.150436
$732$	6.27760	0.232027
$733$	1.61762	0.0597480	0.0298740	−	0.999554i	$-0.490489\pi$
$733$	1.61762	0.0597480	0.0298740	+	0.999554i	$0.490489\pi$
$734$	33.5506	1.23838
$735$	0	0
$736$	10.3369	0.381022
$737$	−7.72244	−0.284460
$738$	6.82069	0.251073
$739$	18.9487	0.697039	0.348520	−	0.937301i	$-0.386684\pi$
$739$	18.9487	0.697039	0.348520	+	0.937301i	$0.386684\pi$
$740$	0	0
$741$	17.1900	0.631491
$742$	0.496718	0.0182351
$743$	41.4420	1.52036	0.760180	−	0.649713i	$-0.225111\pi$
$743$	41.4420	1.52036	0.760180	+	0.649713i	$0.225111\pi$
$744$	−23.9022	−0.876298
$745$	0	0
$746$	11.0510	0.404606
$747$	10.4670	0.382968
$748$	0.160161	0.00585605
$749$	−1.09038	−0.0398416
$750$	0	0
$751$	26.9736	0.984282	0.492141	−	0.870515i	$-0.336214\pi$
$751$	26.9736	0.984282	0.492141	+	0.870515i	$0.336214\pi$
$752$	−43.4364	−1.58396
$753$	52.9952	1.93125
$754$	−4.81895	−0.175496
$755$	0	0
$756$	1.34996	0.0490975
$757$	33.9266	1.23308	0.616542	−	0.787322i	$-0.288533\pi$
$757$	33.9266	1.23308	0.616542	+	0.787322i	$0.288533\pi$
$758$	−48.9476	−1.77786
$759$	10.7621	0.390638
$760$	0	0
$761$	−5.66634	−0.205405	−0.102702	−	0.994712i	$-0.532749\pi$
$761$	−5.66634	−0.205405	−0.102702	+	0.994712i	$0.532749\pi$
$762$	3.78727	0.137198
$763$	10.5868	0.383269
$764$	2.49345	0.0902099
$765$	0	0
$766$	6.61184	0.238895
$767$	13.3213	0.481004
$768$	16.2527	0.586469
$769$	31.5022	1.13600	0.567999	−	0.823029i	$-0.307718\pi$
$769$	31.5022	1.13600	0.567999	+	0.823029i	$0.307718\pi$
$770$	0	0
$771$	−13.9003	−0.500607
$772$	−0.146274	−0.00526450
$773$	−15.3865	−0.553415	−0.276708	−	0.960954i	$-0.589243\pi$
$773$	−15.3865	−0.553415	−0.276708	+	0.960954i	$0.589243\pi$
$774$	12.1124	0.435371
$775$	0	0
$776$	−41.3584	−1.48468
$777$	−17.4006	−0.624245
$778$	28.7877	1.03209
$779$	−27.7479	−0.994171
$780$	0	0
$781$	−3.38553	−0.121144
$782$	3.16872	0.113313
$783$	−9.58492	−0.342537
$784$	−3.18641	−0.113800
$785$	0	0
$786$	5.52436	0.197047
$787$	3.21842	0.114724	0.0573622	−	0.998353i	$-0.481731\pi$
$787$	3.21842	0.114724	0.0573622	+	0.998353i	$0.481731\pi$
$788$	−0.938225	−0.0334229
$789$	41.7943	1.48792
$790$	0	0
$791$	4.41431	0.156955
$792$	3.22836	0.114715
$793$	13.6649	0.485256
$794$	−12.1845	−0.432411
$795$	0	0
$796$	−3.52990	−0.125114
$797$	−32.8541	−1.16375	−0.581877	−	0.813277i	$-0.697681\pi$
$797$	−32.8541	−1.16375	−0.581877	+	0.813277i	$0.697681\pi$
$798$	−14.5176	−0.513918
$799$	6.29735	0.222784
$800$	0	0
$801$	−9.02600	−0.318918
$802$	−8.73752	−0.308533
$803$	12.9545	0.457154
$804$	5.40130	0.190489
$805$	0	0
$806$	−7.68680	−0.270756
$807$	10.4920	0.369337
$808$	−28.0111	−0.985429
$809$	−36.3050	−1.27642	−0.638209	−	0.769863i	$-0.720324\pi$
$809$	−36.3050	−1.27642	−0.638209	+	0.769863i	$0.720324\pi$
$810$	0	0
$811$	19.9568	0.700776	0.350388	−	0.936605i	$-0.386050\pi$
$811$	19.9568	0.700776	0.350388	+	0.936605i	$0.386050\pi$
$812$	−0.853434	−0.0299497
$813$	10.7182	0.375903
$814$	11.0904	0.388720
$815$	0	0
$816$	2.96960	0.103957
$817$	−49.2755	−1.72393
$818$	35.3217	1.23500
$819$	−1.62894	−0.0569199
$820$	0	0
$821$	9.29322	0.324335	0.162168	−	0.986763i	$-0.448151\pi$
$821$	9.29322	0.324335	0.162168	+	0.986763i	$0.448151\pi$
$822$	24.7245	0.862365
$823$	47.9034	1.66981	0.834904	−	0.550396i	$-0.185523\pi$
$823$	47.9034	1.66981	0.834904	+	0.550396i	$0.185523\pi$
$824$	−42.9379	−1.49581
$825$	0	0
$826$	−11.2503	−0.391450
$827$	18.9150	0.657740	0.328870	−	0.944375i	$-0.393332\pi$
$827$	18.9150	0.657740	0.328870	+	0.944375i	$0.393332\pi$
$828$	−1.97880	−0.0687681
$829$	−25.3538	−0.880574	−0.440287	−	0.897857i	$-0.645123\pi$
$829$	−25.3538	−0.880574	−0.440287	+	0.897857i	$0.645123\pi$
$830$	0	0
$831$	51.9961	1.80372
$832$	13.4959	0.467887
$833$	0.461961	0.0160060
$834$	−49.6439	−1.71903
$835$	0	0
$836$	−1.94034	−0.0671080
$837$	−15.2891	−0.528468
$838$	6.06982	0.209679
$839$	−22.9031	−0.790703	−0.395352	−	0.918530i	$-0.629377\pi$
$839$	−22.9031	−0.790703	−0.395352	+	0.918530i	$0.629377\pi$
$840$	0	0
$841$	−22.9405	−0.791051
$842$	27.9789	0.964217
$843$	−40.7423	−1.40324
$844$	−5.21654	−0.179561
$845$	0	0
$846$	18.7533	0.644752
$847$	−1.00000	−0.0343604
$848$	1.23093	0.0422704
$849$	−7.43213	−0.255070
$850$	0	0
$851$	−46.0125	−1.57729
$852$	2.36794	0.0811242
$853$	−23.6051	−0.808223	−0.404112	−	0.914710i	$-0.632419\pi$
$853$	−23.6051	−0.808223	−0.404112	+	0.914710i	$0.632419\pi$
$854$	−11.5405	−0.394909
$855$	0	0
$856$	−3.29011	−0.112454
$857$	−15.9522	−0.544917	−0.272459	−	0.962167i	$-0.587837\pi$
$857$	−15.9522	−0.544917	−0.272459	+	0.962167i	$0.587837\pi$
$858$	3.94935	0.134829
$859$	30.7145	1.04797	0.523983	−	0.851729i	$-0.324446\pi$
$859$	30.7145	1.04797	0.523983	+	0.851729i	$0.324446\pi$
$860$	0	0
$861$	10.0022	0.340874
$862$	32.8628	1.11931
$863$	−15.6856	−0.533943	−0.266972	−	0.963704i	$-0.586023\pi$
$863$	−15.6856	−0.533943	−0.266972	+	0.963704i	$0.586023\pi$
$864$	7.54494	0.256684
$865$	0	0
$866$	−4.89530	−0.166349
$867$	33.8653	1.15013
$868$	−1.36133	−0.0462065
$869$	7.16847	0.243174
$870$	0	0
$871$	11.7574	0.398384
$872$	31.9448	1.08179
$873$	14.6649	0.496332
$874$	−38.3889	−1.29852
$875$	0	0
$876$	−9.06075	−0.306134
$877$	23.5128	0.793971	0.396985	−	0.917825i	$-0.370056\pi$
$877$	23.5128	0.793971	0.396985	+	0.917825i	$0.370056\pi$
$878$	23.7476	0.801442
$879$	−19.8999	−0.671207
$880$	0	0
$881$	−44.7503	−1.50767	−0.753837	−	0.657061i	$-0.771799\pi$
$881$	−44.7503	−1.50767	−0.753837	+	0.657061i	$0.771799\pi$
$882$	1.37570	0.0463224
$883$	47.2821	1.59117	0.795585	−	0.605842i	$-0.207164\pi$
$883$	47.2821	1.59117	0.795585	+	0.605842i	$0.207164\pi$
$884$	−0.243844	−0.00820138
$885$	0	0
$886$	−21.2923	−0.715330
$887$	−53.6155	−1.80023	−0.900116	−	0.435650i	$-0.856519\pi$
$887$	−53.6155	−1.80023	−0.900116	+	0.435650i	$0.856519\pi$
$888$	−52.5048	−1.76194
$889$	1.46002	0.0489673
$890$	0	0
$891$	11.0650	0.370692
$892$	−9.59457	−0.321250
$893$	−76.2921	−2.55302
$894$	−20.1331	−0.673351
$895$	0	0
$896$	−7.52242	−0.251306
$897$	−16.3852	−0.547087
$898$	21.3983	0.714071
$899$	9.66566	0.322368
$900$	0	0
$901$	−0.178459	−0.00594533
$902$	−6.37499	−0.212264
$903$	17.7622	0.591090
$904$	13.3198	0.443009
$905$	0	0
$906$	57.2913	1.90338
$907$	13.5122	0.448666	0.224333	−	0.974513i	$-0.427980\pi$
$907$	13.5122	0.448666	0.224333	+	0.974513i	$0.427980\pi$
$908$	2.80700	0.0931537
$909$	9.93222	0.329431
$910$	0	0
$911$	50.4653	1.67199	0.835994	−	0.548738i	$-0.184892\pi$
$911$	50.4653	1.67199	0.835994	+	0.548738i	$0.184892\pi$
$912$	−35.9766	−1.19131
$913$	−9.78304	−0.323771
$914$	10.1121	0.334478
$915$	0	0
$916$	−2.15018	−0.0710441
$917$	2.12968	0.0703281
$918$	2.31287	0.0763359
$919$	−50.8377	−1.67698	−0.838490	−	0.544917i	$-0.816561\pi$
$919$	−50.8377	−1.67698	−0.838490	+	0.544917i	$0.816561\pi$
$920$	0	0
$921$	−25.2512	−0.832055
$922$	14.2033	0.467762
$923$	5.15447	0.169661
$924$	0.699429	0.0230095
$925$	0	0
$926$	−49.6608	−1.63196
$927$	15.2250	0.500053
$928$	−4.76986	−0.156578
$929$	−8.58672	−0.281721	−0.140861	−	0.990029i	$-0.544987\pi$
$929$	−8.58672	−0.281721	−0.140861	+	0.990029i	$0.544987\pi$
$930$	0	0
$931$	−5.59663	−0.183422
$932$	−1.62710	−0.0532975
$933$	−28.0996	−0.919940
$934$	−14.3510	−0.469578
$935$	0	0
$936$	−4.91518	−0.160658
$937$	47.4182	1.54908	0.774542	−	0.632522i	$-0.217980\pi$
$937$	47.4182	1.54908	0.774542	+	0.632522i	$0.217980\pi$
$938$	−9.92957	−0.324212
$939$	−1.20438	−0.0393035
$940$	0	0
$941$	27.2565	0.888536	0.444268	−	0.895894i	$-0.353464\pi$
$941$	27.2565	0.888536	0.444268	+	0.895894i	$0.353464\pi$
$942$	0.175009	0.00570211
$943$	26.4488	0.861291
$944$	−27.8799	−0.907413
$945$	0	0
$946$	−11.3209	−0.368074
$947$	37.6529	1.22355	0.611777	−	0.791030i	$-0.290455\pi$
$947$	37.6529	1.22355	0.611777	+	0.791030i	$0.290455\pi$
$948$	−5.01383	−0.162842
$949$	−19.7232	−0.640243
$950$	0	0
$951$	−24.1939	−0.784540
$952$	1.39392	0.0451772
$953$	−24.6383	−0.798113	−0.399057	−	0.916926i	$-0.630662\pi$
$953$	−24.6383	−0.798113	−0.399057	+	0.916926i	$0.630662\pi$
$954$	−0.531445	−0.0172062
$955$	0	0
$956$	1.73501	0.0561144
$957$	−4.96606	−0.160530
$958$	55.2125	1.78383
$959$	9.53143	0.307786
$960$	0	0
$961$	−15.5821	−0.502649
$962$	−16.8852	−0.544400
$963$	1.16661	0.0375935
$964$	0.930861	0.0299810
$965$	0	0
$966$	13.8379	0.445229
$967$	−42.0665	−1.35277	−0.676383	−	0.736550i	$-0.736454\pi$
$967$	−42.0665	−1.35277	−0.676383	+	0.736550i	$0.736454\pi$
$968$	−3.01740	−0.0969830
$969$	5.21584	0.167557
$970$	0	0
$971$	−9.27510	−0.297652	−0.148826	−	0.988863i	$-0.547549\pi$
$971$	−9.27510	−0.297652	−0.148826	+	0.988863i	$0.547549\pi$
$972$	−3.68932	−0.118335
$973$	−19.1380	−0.613536
$974$	−3.22843	−0.103446
$975$	0	0
$976$	−28.5990	−0.915433
$977$	21.9870	0.703426	0.351713	−	0.936108i	$-0.385599\pi$
$977$	21.9870	0.703426	0.351713	+	0.936108i	$0.385599\pi$
$978$	−54.3744	−1.73870
$979$	8.43620	0.269622
$980$	0	0
$981$	−11.3270	−0.361644
$982$	−26.7129	−0.852444
$983$	11.7989	0.376327	0.188164	−	0.982138i	$-0.439746\pi$
$983$	11.7989	0.376327	0.188164	+	0.982138i	$0.439746\pi$
$984$	30.1807	0.962125
$985$	0	0
$986$	−1.46218	−0.0465652
$987$	27.5008	0.875360
$988$	2.95416	0.0939845
$989$	46.9686	1.49351
$990$	0	0
$991$	32.6351	1.03669	0.518344	−	0.855172i	$-0.326549\pi$
$991$	32.6351	1.03669	0.518344	+	0.855172i	$0.326549\pi$
$992$	−7.60850	−0.241570
$993$	−6.86948	−0.217996
$994$	−4.35314	−0.138073
$995$	0	0
$996$	6.84254	0.216814
$997$	38.2161	1.21032	0.605158	−	0.796105i	$-0.293110\pi$
$997$	38.2161	1.21032	0.605158	+	0.796105i	$0.293110\pi$
$998$	46.6835	1.47774
$999$	−33.5848	−1.06257

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1925.2.a.z.1.5		6
5.2	odd	4		385.2.b.c.309.9	yes	12
5.3	odd	4		385.2.b.c.309.4	✓	12
5.4	even	2		1925.2.a.y.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
385.2.b.c.309.4	✓	12	5.3	odd	4
385.2.b.c.309.9	yes	12	5.2	odd	4
1925.2.a.y.1.2		6	5.4	even	2
1925.2.a.z.1.5		6	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:	\( N \)	\(=\)	\( 1925 = 5^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1925.a (trivial)

\(f(q)\)	\(=\)	\(q+1.28581 q^{2} -2.01740 q^{3} -0.346698 q^{4} -2.59399 q^{6} -1.00000 q^{7} -3.01740 q^{8} +1.06991 q^{9} +O(q^{10})\)	\(q+1.28581 q^{2} -2.01740 q^{3} -0.346698 q^{4} -2.59399 q^{6} -1.00000 q^{7} -3.01740 q^{8} +1.06991 q^{9} -1.00000 q^{11} +0.699429 q^{12} +1.52250 q^{13} -1.28581 q^{14} -3.18641 q^{16} +0.461961 q^{17} +1.37570 q^{18} -5.59663 q^{19} +2.01740 q^{21} -1.28581 q^{22} +5.33461 q^{23} +6.08732 q^{24} +1.95764 q^{26} +3.89376 q^{27} +0.346698 q^{28} -2.46161 q^{29} -3.92656 q^{31} +1.93770 q^{32} +2.01740 q^{33} +0.593993 q^{34} -0.370937 q^{36} -8.62527 q^{37} -7.19619 q^{38} -3.07149 q^{39} +4.95796 q^{41} +2.59399 q^{42} +8.80450 q^{43} +0.346698 q^{44} +6.85929 q^{46} +13.6318 q^{47} +6.42826 q^{48} +1.00000 q^{49} -0.931961 q^{51} -0.527847 q^{52} -0.386308 q^{53} +5.00663 q^{54} +3.01740 q^{56} +11.2907 q^{57} -3.16516 q^{58} +8.74963 q^{59} +8.97533 q^{61} -5.04880 q^{62} -1.06991 q^{63} +8.86432 q^{64} +2.59399 q^{66} +7.72244 q^{67} -0.160161 q^{68} -10.7621 q^{69} +3.38553 q^{71} -3.22836 q^{72} -12.9545 q^{73} -11.0904 q^{74} +1.94034 q^{76} +1.00000 q^{77} -3.94935 q^{78} -7.16847 q^{79} -11.0650 q^{81} +6.37499 q^{82} +9.78304 q^{83} -0.699429 q^{84} +11.3209 q^{86} +4.96606 q^{87} +3.01740 q^{88} -8.43620 q^{89} -1.52250 q^{91} -1.84950 q^{92} +7.92145 q^{93} +17.5279 q^{94} -3.90912 q^{96} +13.7066 q^{97} +1.28581 q^{98} -1.06991 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{3} + 2 q^{4} - 2 q^{6} - 6 q^{7} + 10 q^{9}+O(q^{10}) \) 6 * q + 6 * q^3 + 2 * q^4 - 2 * q^6 - 6 * q^7 + 10 * q^9	\( 6 q + 6 q^{3} + 2 q^{4} - 2 q^{6} - 6 q^{7} + 10 q^{9} - 6 q^{11} + 12 q^{12} + 6 q^{13} - 6 q^{16} - 2 q^{17} + 14 q^{18} - 2 q^{19} - 6 q^{21} + 8 q^{23} + 22 q^{24} + 24 q^{27} - 2 q^{28} - 8 q^{29} + 4 q^{31} + 10 q^{32} - 6 q^{33} - 10 q^{34} + 26 q^{36} + 22 q^{37} - 10 q^{38} - 8 q^{39} + 4 q^{41} + 2 q^{42} + 30 q^{43} - 2 q^{44} - 8 q^{46} + 16 q^{47} - 8 q^{48} + 6 q^{49} - 4 q^{51} + 22 q^{52} + 6 q^{53} + 38 q^{54} + 18 q^{57} - 14 q^{58} + 14 q^{59} + 12 q^{61} - 14 q^{62} - 10 q^{63} - 22 q^{64} + 2 q^{66} + 46 q^{67} - 20 q^{68} + 12 q^{69} + 4 q^{71} + 32 q^{72} - 4 q^{73} - 8 q^{74} - 8 q^{76} + 6 q^{77} + 24 q^{78} - 8 q^{79} + 26 q^{81} - 10 q^{82} + 14 q^{83} - 12 q^{84} + 12 q^{86} - 2 q^{87} + 8 q^{89} - 6 q^{91} + 18 q^{92} + 12 q^{93} - 10 q^{94} - 16 q^{96} + 42 q^{97} - 10 q^{99}+O(q^{100}) \) 6 * q + 6 * q^3 + 2 * q^4 - 2 * q^6 - 6 * q^7 + 10 * q^9 - 6 * q^11 + 12 * q^12 + 6 * q^13 - 6 * q^16 - 2 * q^17 + 14 * q^18 - 2 * q^19 - 6 * q^21 + 8 * q^23 + 22 * q^24 + 24 * q^27 - 2 * q^28 - 8 * q^29 + 4 * q^31 + 10 * q^32 - 6 * q^33 - 10 * q^34 + 26 * q^36 + 22 * q^37 - 10 * q^38 - 8 * q^39 + 4 * q^41 + 2 * q^42 + 30 * q^43 - 2 * q^44 - 8 * q^46 + 16 * q^47 - 8 * q^48 + 6 * q^49 - 4 * q^51 + 22 * q^52 + 6 * q^53 + 38 * q^54 + 18 * q^57 - 14 * q^58 + 14 * q^59 + 12 * q^61 - 14 * q^62 - 10 * q^63 - 22 * q^64 + 2 * q^66 + 46 * q^67 - 20 * q^68 + 12 * q^69 + 4 * q^71 + 32 * q^72 - 4 * q^73 - 8 * q^74 - 8 * q^76 + 6 * q^77 + 24 * q^78 - 8 * q^79 + 26 * q^81 - 10 * q^82 + 14 * q^83 - 12 * q^84 + 12 * q^86 - 2 * q^87 + 8 * q^89 - 6 * q^91 + 18 * q^92 + 12 * q^93 - 10 * q^94 - 16 * q^96 + 42 * q^97 - 10 * q^99

Introduction

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