LMFDB - Embedded newform 1575.4.a.u.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1575,4,Mod(1,1575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1575, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1575.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1575 = 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1575.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:	$92.9280082590$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 315)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	1575.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.56155 q^{2} -5.56155 q^{4} +7.00000 q^{7} +21.1771 q^{8} +O(q^{10})$	$q-1.56155 q^{2} -5.56155 q^{4} +7.00000 q^{7} +21.1771 q^{8} +10.2462 q^{11} -34.3542 q^{13} -10.9309 q^{14} +11.4233 q^{16} -82.6004 q^{17} +90.7083 q^{19} -16.0000 q^{22} -12.1383 q^{23} +53.6458 q^{26} -38.9309 q^{28} +105.153 q^{29} -142.108 q^{31} -187.255 q^{32} +128.985 q^{34} -64.8466 q^{37} -141.646 q^{38} +195.201 q^{41} +319.218 q^{43} -56.9848 q^{44} +18.9545 q^{46} -318.847 q^{47} +49.0000 q^{49} +191.062 q^{52} -296.799 q^{53} +148.240 q^{56} -164.203 q^{58} +284.000 q^{59} -494.540 q^{61} +221.909 q^{62} +201.022 q^{64} -549.032 q^{67} +459.386 q^{68} +740.972 q^{71} +556.850 q^{73} +101.261 q^{74} -504.479 q^{76} +71.7235 q^{77} -376.189 q^{79} -304.816 q^{82} +752.466 q^{83} -498.475 q^{86} +216.985 q^{88} +945.299 q^{89} -240.479 q^{91} +67.5076 q^{92} +497.896 q^{94} +180.668 q^{97} -76.5161 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - 7 q^{4} + 14 q^{7} - 3 q^{8}+O(q^{10}) $ 2 * q + q^2 - 7 * q^4 + 14 * q^7 - 3 * q^8	$ 2 q + q^{2} - 7 q^{4} + 14 q^{7} - 3 q^{8} + 4 q^{11} + 22 q^{13} + 7 q^{14} - 39 q^{16} - 58 q^{17} - 32 q^{22} - 82 q^{23} + 198 q^{26} - 49 q^{28} + 334 q^{29} - 210 q^{31} - 123 q^{32} + 192 q^{34} - 6 q^{37} - 374 q^{38} + 176 q^{41} - 46 q^{43} - 48 q^{44} - 160 q^{46} - 514 q^{47} + 98 q^{49} + 110 q^{52} - 808 q^{53} - 21 q^{56} + 422 q^{58} + 568 q^{59} - 618 q^{61} + 48 q^{62} + 769 q^{64} - 694 q^{67} + 424 q^{68} + 814 q^{71} - 82 q^{73} + 252 q^{74} - 374 q^{76} + 28 q^{77} + 600 q^{79} - 354 q^{82} + 268 q^{83} - 1434 q^{86} + 368 q^{88} - 72 q^{89} + 154 q^{91} + 168 q^{92} - 2 q^{94} - 1626 q^{97} + 49 q^{98}+O(q^{100}) $ 2 * q + q^2 - 7 * q^4 + 14 * q^7 - 3 * q^8 + 4 * q^11 + 22 * q^13 + 7 * q^14 - 39 * q^16 - 58 * q^17 - 32 * q^22 - 82 * q^23 + 198 * q^26 - 49 * q^28 + 334 * q^29 - 210 * q^31 - 123 * q^32 + 192 * q^34 - 6 * q^37 - 374 * q^38 + 176 * q^41 - 46 * q^43 - 48 * q^44 - 160 * q^46 - 514 * q^47 + 98 * q^49 + 110 * q^52 - 808 * q^53 - 21 * q^56 + 422 * q^58 + 568 * q^59 - 618 * q^61 + 48 * q^62 + 769 * q^64 - 694 * q^67 + 424 * q^68 + 814 * q^71 - 82 * q^73 + 252 * q^74 - 374 * q^76 + 28 * q^77 + 600 * q^79 - 354 * q^82 + 268 * q^83 - 1434 * q^86 + 368 * q^88 - 72 * q^89 + 154 * q^91 + 168 * q^92 - 2 * q^94 - 1626 * q^97 + 49 * q^98

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 $

$2$	−1.56155	−0.552092	−0.276046	−	0.961144i	$-0.589024\pi$
$2$	−1.56155	−0.552092	−0.276046	+	0.961144i	$0.589024\pi$
$3$	0	0
$3$	0	0
$4$	−5.56155	−0.695194
$5$	0	0
$5$	0	0
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	21.1771	0.935904
$9$	0	0
$10$	0	0
$11$	10.2462	0.280850	0.140425	−	0.990091i	$-0.455153\pi$
$11$	10.2462	0.280850	0.140425	+	0.990091i	$0.455153\pi$
$12$	0	0
$13$	−34.3542	−0.732933	−0.366467	−	0.930431i	$-0.619433\pi$
$13$	−34.3542	−0.732933	−0.366467	+	0.930431i	$0.619433\pi$
$14$	−10.9309	−0.208671
$15$	0	0
$16$	11.4233	0.178489
$17$	−82.6004	−1.17844	−0.589222	−	0.807972i	$-0.700565\pi$
$17$	−82.6004	−1.17844	−0.589222	+	0.807972i	$0.700565\pi$
$18$	0	0
$19$	90.7083	1.09526	0.547629	−	0.836721i	$-0.315530\pi$
$19$	90.7083	1.09526	0.547629	+	0.836721i	$0.315530\pi$
$20$	0	0
$21$	0	0
$22$	−16.0000	−0.155055
$23$	−12.1383	−0.110044	−0.0550218	−	0.998485i	$-0.517523\pi$
$23$	−12.1383	−0.110044	−0.0550218	+	0.998485i	$0.517523\pi$
$24$	0	0
$25$	0	0
$26$	53.6458	0.404647
$27$	0	0
$28$	−38.9309	−0.262759
$29$	105.153	0.673328	0.336664	−	0.941625i	$-0.390701\pi$
$29$	105.153	0.673328	0.336664	+	0.941625i	$0.390701\pi$
$30$	0	0
$31$	−142.108	−0.823334	−0.411667	−	0.911334i	$-0.635053\pi$
$31$	−142.108	−0.823334	−0.411667	+	0.911334i	$0.635053\pi$
$32$	−187.255	−1.03445
$33$	0	0
$34$	128.985	0.650609
$35$	0	0
$36$	0	0
$37$	−64.8466	−0.288127	−0.144064	−	0.989568i	$-0.546017\pi$
$37$	−64.8466	−0.288127	−0.144064	+	0.989568i	$0.546017\pi$
$38$	−141.646	−0.604684
$39$	0	0
$40$	0	0
$41$	195.201	0.743542	0.371771	−	0.928324i	$-0.378751\pi$
$41$	195.201	0.743542	0.371771	+	0.928324i	$0.378751\pi$
$42$	0	0
$43$	319.218	1.13210	0.566049	−	0.824371i	$-0.308471\pi$
$43$	319.218	1.13210	0.566049	+	0.824371i	$0.308471\pi$
$44$	−56.9848	−0.195245
$45$	0	0
$46$	18.9545	0.0607542
$47$	−318.847	−0.989544	−0.494772	−	0.869023i	$-0.664748\pi$
$47$	−318.847	−0.989544	−0.494772	+	0.869023i	$0.664748\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	191.062	0.509531
$53$	−296.799	−0.769217	−0.384609	−	0.923080i	$-0.625664\pi$
$53$	−296.799	−0.769217	−0.384609	+	0.923080i	$0.625664\pi$
$54$	0	0
$55$	0	0
$56$	148.240	0.353738
$57$	0	0
$58$	−164.203	−0.371739
$59$	284.000	0.626672	0.313336	−	0.949642i	$-0.398553\pi$
$59$	284.000	0.626672	0.313336	+	0.949642i	$0.398553\pi$
$60$	0	0
$61$	−494.540	−1.03802	−0.519011	−	0.854768i	$-0.673700\pi$
$61$	−494.540	−1.03802	−0.519011	+	0.854768i	$0.673700\pi$
$62$	221.909	0.454556
$63$	0	0
$64$	201.022	0.392621
$65$	0	0
$66$	0	0
$67$	−549.032	−1.00112	−0.500559	−	0.865702i	$-0.666872\pi$
$67$	−549.032	−1.00112	−0.500559	+	0.865702i	$0.666872\pi$
$68$	459.386	0.819247
$69$	0	0
$70$	0	0
$71$	740.972	1.23855	0.619276	−	0.785174i	$-0.287426\pi$
$71$	740.972	1.23855	0.619276	+	0.785174i	$0.287426\pi$
$72$	0	0
$73$	556.850	0.892800	0.446400	−	0.894834i	$-0.352706\pi$
$73$	556.850	0.892800	0.446400	+	0.894834i	$0.352706\pi$
$74$	101.261	0.159073
$75$	0	0
$76$	−504.479	−0.761417
$77$	71.7235	0.106151
$78$	0	0
$79$	−376.189	−0.535755	−0.267877	−	0.963453i	$-0.586322\pi$
$79$	−376.189	−0.535755	−0.267877	+	0.963453i	$0.586322\pi$
$80$	0	0
$81$	0	0
$82$	−304.816	−0.410504
$83$	752.466	0.995107	0.497553	−	0.867433i	$-0.334232\pi$
$83$	752.466	0.995107	0.497553	+	0.867433i	$0.334232\pi$
$84$	0	0
$85$	0	0
$86$	−498.475	−0.625023
$87$	0	0
$88$	216.985	0.262848
$89$	945.299	1.12586	0.562930	−	0.826505i	$-0.309674\pi$
$89$	945.299	1.12586	0.562930	+	0.826505i	$0.309674\pi$
$90$	0	0
$91$	−240.479	−0.277023
$92$	67.5076	0.0765016
$93$	0	0
$94$	497.896	0.546319
$95$	0	0
$96$	0	0
$97$	180.668	0.189114	0.0945572	−	0.995519i	$-0.469856\pi$
$97$	180.668	0.189114	0.0945572	+	0.995519i	$0.469856\pi$
$98$	−76.5161	−0.0788703
$99$	0	0
$100$	0	0
$101$	1463.05	1.44138	0.720689	−	0.693258i	$-0.243826\pi$
$101$	1463.05	1.44138	0.720689	+	0.693258i	$0.243826\pi$
$102$	0	0
$103$	−535.049	−0.511844	−0.255922	−	0.966697i	$-0.582379\pi$
$103$	−535.049	−0.511844	−0.255922	+	0.966697i	$0.582379\pi$
$104$	−727.521	−0.685955
$105$	0	0
$106$	463.468	0.424679
$107$	327.153	0.295581	0.147790	−	0.989019i	$-0.452784\pi$
$107$	327.153	0.295581	0.147790	+	0.989019i	$0.452784\pi$
$108$	0	0
$109$	1621.70	1.42506	0.712528	−	0.701644i	$-0.247550\pi$
$109$	1621.70	1.42506	0.712528	+	0.701644i	$0.247550\pi$
$110$	0	0
$111$	0	0
$112$	79.9630	0.0674625
$113$	450.716	0.375219	0.187610	−	0.982244i	$-0.439926\pi$
$113$	450.716	0.375219	0.187610	+	0.982244i	$0.439926\pi$
$114$	0	0
$115$	0	0
$116$	−584.816	−0.468093
$117$	0	0
$118$	−443.481	−0.345981
$119$	−578.203	−0.445410
$120$	0	0
$121$	−1226.02	−0.921123
$122$	772.250	0.573084
$123$	0	0
$124$	790.341	0.572377
$125$	0	0
$126$	0	0
$127$	−1710.10	−1.19486	−0.597428	−	0.801923i	$-0.703811\pi$
$127$	−1710.10	−1.19486	−0.597428	+	0.801923i	$0.703811\pi$
$128$	1184.13	0.817683
$129$	0	0
$130$	0	0
$131$	−1584.49	−1.05678	−0.528388	−	0.849003i	$-0.677203\pi$
$131$	−1584.49	−1.05678	−0.528388	+	0.849003i	$0.677203\pi$
$132$	0	0
$133$	634.958	0.413969
$134$	857.343	0.552710
$135$	0	0
$136$	−1749.23	−1.10291
$137$	−2095.24	−1.30663	−0.653316	−	0.757085i	$-0.726623\pi$
$137$	−2095.24	−1.30663	−0.653316	+	0.757085i	$0.726623\pi$
$138$	0	0
$139$	−1385.00	−0.845135	−0.422568	−	0.906331i	$-0.638871\pi$
$139$	−1385.00	−0.845135	−0.422568	+	0.906331i	$0.638871\pi$
$140$	0	0
$141$	0	0
$142$	−1157.07	−0.683795
$143$	−352.000	−0.205844
$144$	0	0
$145$	0	0
$146$	−869.551	−0.492908
$147$	0	0
$148$	360.648	0.200304
$149$	−2502.89	−1.37614	−0.688069	−	0.725645i	$-0.741541\pi$
$149$	−2502.89	−1.37614	−0.688069	+	0.725645i	$0.741541\pi$
$150$	0	0
$151$	395.212	0.212993	0.106496	−	0.994313i	$-0.466037\pi$
$151$	395.212	0.212993	0.106496	+	0.994313i	$0.466037\pi$
$152$	1920.94	1.02506
$153$	0	0
$154$	−112.000	−0.0586053
$155$	0	0
$156$	0	0
$157$	−1732.35	−0.880615	−0.440308	−	0.897847i	$-0.645131\pi$
$157$	−1732.35	−0.880615	−0.440308	+	0.897847i	$0.645131\pi$
$158$	587.439	0.295786
$159$	0	0
$160$	0	0
$161$	−84.9678	−0.0415926
$162$	0	0
$163$	−3303.26	−1.58731	−0.793654	−	0.608369i	$-0.791824\pi$
$163$	−3303.26	−1.58731	−0.793654	+	0.608369i	$0.791824\pi$
$164$	−1085.62	−0.516906
$165$	0	0
$166$	−1175.02	−0.549391
$167$	601.941	0.278920	0.139460	−	0.990228i	$-0.455463\pi$
$167$	601.941	0.278920	0.139460	+	0.990228i	$0.455463\pi$
$168$	0	0
$169$	−1016.79	−0.462809
$170$	0	0
$171$	0	0
$172$	−1775.35	−0.787028
$173$	3741.17	1.64414	0.822068	−	0.569389i	$-0.192820\pi$
$173$	3741.17	1.64414	0.822068	+	0.569389i	$0.192820\pi$
$174$	0	0
$175$	0	0
$176$	117.045	0.0501286
$177$	0	0
$178$	−1476.13	−0.621578
$179$	−968.352	−0.404347	−0.202173	−	0.979350i	$-0.564800\pi$
$179$	−968.352	−0.404347	−0.202173	+	0.979350i	$0.564800\pi$
$180$	0	0
$181$	−1895.65	−0.778465	−0.389233	−	0.921139i	$-0.627260\pi$
$181$	−1895.65	−0.778465	−0.389233	+	0.921139i	$0.627260\pi$
$182$	375.521	0.152942
$183$	0	0
$184$	−257.053	−0.102990
$185$	0	0
$186$	0	0
$187$	−846.341	−0.330966
$188$	1773.28	0.687925
$189$	0	0
$190$	0	0
$191$	2729.88	1.03417	0.517086	−	0.855933i	$-0.327017\pi$
$191$	2729.88	1.03417	0.517086	+	0.855933i	$0.327017\pi$
$192$	0	0
$193$	−1460.44	−0.544689	−0.272345	−	0.962200i	$-0.587799\pi$
$193$	−1460.44	−0.544689	−0.272345	+	0.962200i	$0.587799\pi$
$194$	−282.123	−0.104409
$195$	0	0
$196$	−272.516	−0.0993134
$197$	−4831.42	−1.74733	−0.873665	−	0.486527i	$-0.838263\pi$
$197$	−4831.42	−1.74733	−0.873665	+	0.486527i	$0.838263\pi$
$198$	0	0
$199$	−873.070	−0.311006	−0.155503	−	0.987835i	$-0.549700\pi$
$199$	−873.070	−0.311006	−0.155503	+	0.987835i	$0.549700\pi$
$200$	0	0
$201$	0	0
$202$	−2284.63	−0.795774
$203$	736.074	0.254494
$204$	0	0
$205$	0	0
$206$	835.508	0.282585
$207$	0	0
$208$	−392.438	−0.130820
$209$	929.417	0.307603
$210$	0	0
$211$	−5148.02	−1.67964	−0.839821	−	0.542863i	$-0.817340\pi$
$211$	−5148.02	−1.67964	−0.839821	+	0.542863i	$0.817340\pi$
$212$	1650.66	0.534755
$213$	0	0
$214$	−510.867	−0.163188
$215$	0	0
$216$	0	0
$217$	−994.756	−0.311191
$218$	−2532.38	−0.786762
$219$	0	0
$220$	0	0
$221$	2837.67	0.863720
$222$	0	0
$223$	2435.31	0.731304	0.365652	−	0.930752i	$-0.380846\pi$
$223$	2435.31	0.731304	0.365652	+	0.930752i	$0.380846\pi$
$224$	−1310.78	−0.390984
$225$	0	0
$226$	−703.817	−0.207156
$227$	−3508.47	−1.02584	−0.512919	−	0.858437i	$-0.671436\pi$
$227$	−3508.47	−1.02584	−0.512919	+	0.858437i	$0.671436\pi$
$228$	0	0
$229$	3429.54	0.989653	0.494827	−	0.868992i	$-0.335232\pi$
$229$	3429.54	0.989653	0.494827	+	0.868992i	$0.335232\pi$
$230$	0	0
$231$	0	0
$232$	2226.84	0.630170
$233$	−3989.71	−1.12178	−0.560890	−	0.827890i	$-0.689541\pi$
$233$	−3989.71	−1.12178	−0.560890	+	0.827890i	$0.689541\pi$
$234$	0	0
$235$	0	0
$236$	−1579.48	−0.435659
$237$	0	0
$238$	902.894	0.245907
$239$	5934.71	1.60621	0.803105	−	0.595837i	$-0.203180\pi$
$239$	5934.71	1.60621	0.803105	+	0.595837i	$0.203180\pi$
$240$	0	0
$241$	−6481.90	−1.73251	−0.866257	−	0.499598i	$-0.833481\pi$
$241$	−6481.90	−1.73251	−0.866257	+	0.499598i	$0.833481\pi$
$242$	1914.49	0.508545
$243$	0	0
$244$	2750.41	0.721627
$245$	0	0
$246$	0	0
$247$	−3116.21	−0.802751
$248$	−3009.43	−0.770561
$249$	0	0
$250$	0	0
$251$	−5257.70	−1.32216	−0.661081	−	0.750314i	$-0.729902\pi$
$251$	−5257.70	−1.32216	−0.661081	+	0.750314i	$0.729902\pi$
$252$	0	0
$253$	−124.371	−0.0309057
$254$	2670.41	0.659671
$255$	0	0
$256$	−3457.26	−0.844057
$257$	4108.52	0.997209	0.498604	−	0.866830i	$-0.333846\pi$
$257$	4108.52	0.997209	0.498604	+	0.866830i	$0.333846\pi$
$258$	0	0
$259$	−453.926	−0.108902
$260$	0	0
$261$	0	0
$262$	2474.27	0.583438
$263$	−6236.89	−1.46229	−0.731146	−	0.682221i	$-0.761014\pi$
$263$	−6236.89	−1.46229	−0.731146	+	0.682221i	$0.761014\pi$
$264$	0	0
$265$	0	0
$266$	−991.521	−0.228549
$267$	0	0
$268$	3053.47	0.695972
$269$	4356.67	0.987475	0.493738	−	0.869611i	$-0.335630\pi$
$269$	4356.67	0.987475	0.493738	+	0.869611i	$0.335630\pi$
$270$	0	0
$271$	−63.0970	−0.0141434	−0.00707171	−	0.999975i	$-0.502251\pi$
$271$	−63.0970	−0.0141434	−0.00707171	+	0.999975i	$0.502251\pi$
$272$	−943.568	−0.210339
$273$	0	0
$274$	3271.83	0.721382
$275$	0	0
$276$	0	0
$277$	−5804.64	−1.25909	−0.629543	−	0.776966i	$-0.716758\pi$
$277$	−5804.64	−1.25909	−0.629543	+	0.776966i	$0.716758\pi$
$278$	2162.74	0.466593
$279$	0	0
$280$	0	0
$281$	8430.89	1.78984	0.894920	−	0.446227i	$-0.147233\pi$
$281$	8430.89	1.78984	0.894920	+	0.446227i	$0.147233\pi$
$282$	0	0
$283$	−7415.73	−1.55767	−0.778833	−	0.627232i	$-0.784188\pi$
$283$	−7415.73	−1.55767	−0.778833	+	0.627232i	$0.784188\pi$
$284$	−4120.95	−0.861034
$285$	0	0
$286$	549.667	0.113645
$287$	1366.41	0.281033
$288$	0	0
$289$	1909.82	0.388728
$290$	0	0
$291$	0	0
$292$	−3096.95	−0.620669
$293$	−482.022	−0.0961094	−0.0480547	−	0.998845i	$-0.515302\pi$
$293$	−482.022	−0.0961094	−0.0480547	+	0.998845i	$0.515302\pi$
$294$	0	0
$295$	0	0
$296$	−1373.26	−0.269659
$297$	0	0
$298$	3908.39	0.759755
$299$	417.000	0.0806546
$300$	0	0
$301$	2234.52	0.427893
$302$	−617.144	−0.117592
$303$	0	0
$304$	1036.19	0.195492
$305$	0	0
$306$	0	0
$307$	−2757.90	−0.512709	−0.256354	−	0.966583i	$-0.582521\pi$
$307$	−2757.90	−0.512709	−0.256354	+	0.966583i	$0.582521\pi$
$308$	−398.894	−0.0737957
$309$	0	0
$310$	0	0
$311$	−5817.25	−1.06066	−0.530331	−	0.847791i	$-0.677932\pi$
$311$	−5817.25	−1.06066	−0.530331	+	0.847791i	$0.677932\pi$
$312$	0	0
$313$	−2232.83	−0.403218	−0.201609	−	0.979466i	$-0.564617\pi$
$313$	−2232.83	−0.403218	−0.201609	+	0.979466i	$0.564617\pi$
$314$	2705.16	0.486181
$315$	0	0
$316$	2092.20	0.372453
$317$	81.4013	0.0144226	0.00721128	−	0.999974i	$-0.497705\pi$
$317$	81.4013	0.0144226	0.00721128	+	0.999974i	$0.497705\pi$
$318$	0	0
$319$	1077.42	0.189104
$320$	0	0
$321$	0	0
$322$	132.682	0.0229629
$323$	−7492.54	−1.29070
$324$	0	0
$325$	0	0
$326$	5158.21	0.876341
$327$	0	0
$328$	4133.78	0.695884
$329$	−2231.93	−0.374012
$330$	0	0
$331$	−6550.18	−1.08771	−0.543853	−	0.839181i	$-0.683035\pi$
$331$	−6550.18	−1.08771	−0.543853	+	0.839181i	$0.683035\pi$
$332$	−4184.88	−0.691792
$333$	0	0
$334$	−939.963	−0.153990
$335$	0	0
$336$	0	0
$337$	−4124.63	−0.666715	−0.333358	−	0.942800i	$-0.608182\pi$
$337$	−4124.63	−0.666715	−0.333358	+	0.942800i	$0.608182\pi$
$338$	1587.77	0.255513
$339$	0	0
$340$	0	0
$341$	−1456.07	−0.231233
$342$	0	0
$343$	343.000	0.0539949
$344$	6760.10	1.05954
$345$	0	0
$346$	−5842.03	−0.907715
$347$	1228.44	0.190047	0.0950233	−	0.995475i	$-0.469707\pi$
$347$	1228.44	0.190047	0.0950233	+	0.995475i	$0.469707\pi$
$348$	0	0
$349$	3320.93	0.509357	0.254678	−	0.967026i	$-0.418030\pi$
$349$	3320.93	0.509357	0.254678	+	0.967026i	$0.418030\pi$
$350$	0	0
$351$	0	0
$352$	−1918.65	−0.290524
$353$	−4589.25	−0.691958	−0.345979	−	0.938242i	$-0.612453\pi$
$353$	−4589.25	−0.691958	−0.345979	+	0.938242i	$0.612453\pi$
$354$	0	0
$355$	0	0
$356$	−5257.33	−0.782691
$357$	0	0
$358$	1512.13	0.223237
$359$	−12381.6	−1.82026	−0.910130	−	0.414323i	$-0.864018\pi$
$359$	−12381.6	−1.82026	−0.910130	+	0.414323i	$0.864018\pi$
$360$	0	0
$361$	1369.00	0.199592
$362$	2960.15	0.429785
$363$	0	0
$364$	1337.44	0.192585
$365$	0	0
$366$	0	0
$367$	−5612.33	−0.798259	−0.399130	−	0.916894i	$-0.630688\pi$
$367$	−5612.33	−0.798259	−0.399130	+	0.916894i	$0.630688\pi$
$368$	−138.659	−0.0196416
$369$	0	0
$370$	0	0
$371$	−2077.59	−0.290737
$372$	0	0
$373$	3691.05	0.512374	0.256187	−	0.966627i	$-0.417534\pi$
$373$	3691.05	0.512374	0.256187	+	0.966627i	$0.417534\pi$
$374$	1321.61	0.182724
$375$	0	0
$376$	−6752.24	−0.926118
$377$	−3612.46	−0.493504
$378$	0	0
$379$	5350.16	0.725117	0.362559	−	0.931961i	$-0.381903\pi$
$379$	5350.16	0.725117	0.362559	+	0.931961i	$0.381903\pi$
$380$	0	0
$381$	0	0
$382$	−4262.85	−0.570959
$383$	7300.09	0.973936	0.486968	−	0.873420i	$-0.338103\pi$
$383$	7300.09	0.973936	0.486968	+	0.873420i	$0.338103\pi$
$384$	0	0
$385$	0	0
$386$	2280.56	0.300719
$387$	0	0
$388$	−1004.80	−0.131471
$389$	9102.27	1.18638	0.593192	−	0.805061i	$-0.297867\pi$
$389$	9102.27	1.18638	0.593192	+	0.805061i	$0.297867\pi$
$390$	0	0
$391$	1002.62	0.129680
$392$	1037.68	0.133701
$393$	0	0
$394$	7544.51	0.964688
$395$	0	0
$396$	0	0
$397$	11520.1	1.45637	0.728185	−	0.685381i	$-0.240364\pi$
$397$	11520.1	1.45637	0.728185	+	0.685381i	$0.240364\pi$
$398$	1363.34	0.171704
$399$	0	0
$400$	0	0
$401$	−3812.93	−0.474834	−0.237417	−	0.971408i	$-0.576301\pi$
$401$	−3812.93	−0.474834	−0.237417	+	0.971408i	$0.576301\pi$
$402$	0	0
$403$	4882.00	0.603448
$404$	−8136.85	−1.00204
$405$	0	0
$406$	−1149.42	−0.140504
$407$	−664.432	−0.0809205
$408$	0	0
$409$	−4098.82	−0.495535	−0.247767	−	0.968820i	$-0.579697\pi$
$409$	−4098.82	−0.495535	−0.247767	+	0.968820i	$0.579697\pi$
$410$	0	0
$411$	0	0
$412$	2975.70	0.355831
$413$	1988.00	0.236860
$414$	0	0
$415$	0	0
$416$	6432.98	0.758180
$417$	0	0
$418$	−1451.33	−0.169825
$419$	−14312.4	−1.66876	−0.834378	−	0.551193i	$-0.814173\pi$
$419$	−14312.4	−1.66876	−0.834378	+	0.551193i	$0.814173\pi$
$420$	0	0
$421$	6679.57	0.773259	0.386630	−	0.922235i	$-0.373639\pi$
$421$	6679.57	0.773259	0.386630	+	0.922235i	$0.373639\pi$
$422$	8038.91	0.927317
$423$	0	0
$424$	−6285.34	−0.719913
$425$	0	0
$426$	0	0
$427$	−3461.78	−0.392335
$428$	−1819.48	−0.205486
$429$	0	0
$430$	0	0
$431$	−11503.8	−1.28566	−0.642828	−	0.766010i	$-0.722239\pi$
$431$	−11503.8	−1.28566	−0.642828	+	0.766010i	$0.722239\pi$
$432$	0	0
$433$	−3243.99	−0.360037	−0.180019	−	0.983663i	$-0.557616\pi$
$433$	−3243.99	−0.360037	−0.180019	+	0.983663i	$0.557616\pi$
$434$	1553.36	0.171806
$435$	0	0
$436$	−9019.19	−0.990691
$437$	−1101.04	−0.120526
$438$	0	0
$439$	−2744.21	−0.298346	−0.149173	−	0.988811i	$-0.547661\pi$
$439$	−2744.21	−0.298346	−0.149173	+	0.988811i	$0.547661\pi$
$440$	0	0
$441$	0	0
$442$	−4431.17	−0.476853
$443$	8180.27	0.877328	0.438664	−	0.898651i	$-0.355452\pi$
$443$	8180.27	0.877328	0.438664	+	0.898651i	$0.355452\pi$
$444$	0	0
$445$	0	0
$446$	−3802.87	−0.403747
$447$	0	0
$448$	1407.15	0.148397
$449$	−6725.70	−0.706916	−0.353458	−	0.935450i	$-0.614994\pi$
$449$	−6725.70	−0.706916	−0.353458	+	0.935450i	$0.614994\pi$
$450$	0	0
$451$	2000.07	0.208824
$452$	−2506.68	−0.260850
$453$	0	0
$454$	5478.65	0.566357
$455$	0	0
$456$	0	0
$457$	10807.0	1.10619	0.553095	−	0.833118i	$-0.313447\pi$
$457$	10807.0	1.10619	0.553095	+	0.833118i	$0.313447\pi$
$458$	−5355.41	−0.546380
$459$	0	0
$460$	0	0
$461$	−17775.8	−1.79588	−0.897942	−	0.440113i	$-0.854938\pi$
$461$	−17775.8	−1.79588	−0.897942	+	0.440113i	$0.854938\pi$
$462$	0	0
$463$	4418.93	0.443554	0.221777	−	0.975097i	$-0.428814\pi$
$463$	4418.93	0.443554	0.221777	+	0.975097i	$0.428814\pi$
$464$	1201.20	0.120182
$465$	0	0
$466$	6230.15	0.619326
$467$	3373.08	0.334235	0.167117	−	0.985937i	$-0.446554\pi$
$467$	3373.08	0.334235	0.167117	+	0.985937i	$0.446554\pi$
$468$	0	0
$469$	−3843.23	−0.378387
$470$	0	0
$471$	0	0
$472$	6014.29	0.586505
$473$	3270.77	0.317950
$474$	0	0
$475$	0	0
$476$	3215.70	0.309646
$477$	0	0
$478$	−9267.36	−0.886776
$479$	−4620.02	−0.440697	−0.220349	−	0.975421i	$-0.570720\pi$
$479$	−4620.02	−0.440697	−0.220349	+	0.975421i	$0.570720\pi$
$480$	0	0
$481$	2227.75	0.211178
$482$	10121.8	0.956508
$483$	0	0
$484$	6818.55	0.640360
$485$	0	0
$486$	0	0
$487$	9108.62	0.847538	0.423769	−	0.905770i	$-0.360707\pi$
$487$	9108.62	0.847538	0.423769	+	0.905770i	$0.360707\pi$
$488$	−10472.9	−0.971488
$489$	0	0
$490$	0	0
$491$	2696.84	0.247875	0.123938	−	0.992290i	$-0.460448\pi$
$491$	2696.84	0.247875	0.123938	+	0.992290i	$0.460448\pi$
$492$	0	0
$493$	−8685.71	−0.793478
$494$	4866.12	0.443193
$495$	0	0
$496$	−1623.34	−0.146956
$497$	5186.80	0.468128
$498$	0	0
$499$	−20054.2	−1.79910	−0.899549	−	0.436820i	$-0.856104\pi$
$499$	−20054.2	−1.79910	−0.899549	+	0.436820i	$0.856104\pi$
$500$	0	0
$501$	0	0
$502$	8210.17	0.729956
$503$	8106.49	0.718590	0.359295	−	0.933224i	$-0.383017\pi$
$503$	8106.49	0.718590	0.359295	+	0.933224i	$0.383017\pi$
$504$	0	0
$505$	0	0
$506$	194.212	0.0170628
$507$	0	0
$508$	9510.80	0.830657
$509$	−12455.8	−1.08466	−0.542332	−	0.840164i	$-0.682458\pi$
$509$	−12455.8	−1.08466	−0.542332	+	0.840164i	$0.682458\pi$
$510$	0	0
$511$	3897.95	0.337447
$512$	−4074.36	−0.351686
$513$	0	0
$514$	−6415.68	−0.550551
$515$	0	0
$516$	0	0
$517$	−3266.97	−0.277913
$518$	708.830	0.0601239
$519$	0	0
$520$	0	0
$521$	5539.61	0.465825	0.232913	−	0.972498i	$-0.425174\pi$
$521$	5539.61	0.465825	0.232913	+	0.972498i	$0.425174\pi$
$522$	0	0
$523$	−15504.2	−1.29628	−0.648138	−	0.761523i	$-0.724452\pi$
$523$	−15504.2	−1.29628	−0.648138	+	0.761523i	$0.724452\pi$
$524$	8812.24	0.734665
$525$	0	0
$526$	9739.23	0.807321
$527$	11738.2	0.970252
$528$	0	0
$529$	−12019.7	−0.987890
$530$	0	0
$531$	0	0
$532$	−3531.35	−0.287789
$533$	−6705.96	−0.544967
$534$	0	0
$535$	0	0
$536$	−11626.9	−0.936951
$537$	0	0
$538$	−6803.17	−0.545178
$539$	502.064	0.0401214
$540$	0	0
$541$	7853.16	0.624092	0.312046	−	0.950067i	$-0.398986\pi$
$541$	7853.16	0.624092	0.312046	+	0.950067i	$0.398986\pi$
$542$	98.5293	0.00780848
$543$	0	0
$544$	15467.3	1.21904
$545$	0	0
$546$	0	0
$547$	9177.75	0.717390	0.358695	−	0.933455i	$-0.383222\pi$
$547$	9177.75	0.717390	0.358695	+	0.933455i	$0.383222\pi$
$548$	11652.8	0.908363
$549$	0	0
$550$	0	0
$551$	9538.29	0.737468
$552$	0	0
$553$	−2633.33	−0.202496
$554$	9064.25	0.695132
$555$	0	0
$556$	7702.73	0.587533
$557$	−25030.2	−1.90407	−0.952033	−	0.305996i	$-0.901010\pi$
$557$	−25030.2	−1.90407	−0.952033	+	0.305996i	$0.901010\pi$
$558$	0	0
$559$	−10966.5	−0.829753
$560$	0	0
$561$	0	0
$562$	−13165.3	−0.988157
$563$	19486.1	1.45869	0.729345	−	0.684146i	$-0.239825\pi$
$563$	19486.1	1.45869	0.729345	+	0.684146i	$0.239825\pi$
$564$	0	0
$565$	0	0
$566$	11580.0	0.859975
$567$	0	0
$568$	15691.6	1.15916
$569$	5813.63	0.428331	0.214165	−	0.976797i	$-0.431297\pi$
$569$	5813.63	0.428331	0.214165	+	0.976797i	$0.431297\pi$
$570$	0	0
$571$	268.770	0.0196982	0.00984910	−	0.999951i	$-0.496865\pi$
$571$	268.770	0.0196982	0.00984910	+	0.999951i	$0.496865\pi$
$572$	1957.67	0.143102
$573$	0	0
$574$	−2133.71	−0.155156
$575$	0	0
$576$	0	0
$577$	−8283.57	−0.597660	−0.298830	−	0.954306i	$-0.596596\pi$
$577$	−8283.57	−0.597660	−0.298830	+	0.954306i	$0.596596\pi$
$578$	−2982.29	−0.214614
$579$	0	0
$580$	0	0
$581$	5267.26	0.376115
$582$	0	0
$583$	−3041.07	−0.216035
$584$	11792.5	0.835575
$585$	0	0
$586$	752.703	0.0530612
$587$	−6671.87	−0.469127	−0.234563	−	0.972101i	$-0.575366\pi$
$587$	−6671.87	−0.469127	−0.234563	+	0.972101i	$0.575366\pi$
$588$	0	0
$589$	−12890.4	−0.901763
$590$	0	0
$591$	0	0
$592$	−740.761	−0.0514275
$593$	−11409.0	−0.790071	−0.395035	−	0.918666i	$-0.629268\pi$
$593$	−11409.0	−0.790071	−0.395035	+	0.918666i	$0.629268\pi$
$594$	0	0
$595$	0	0
$596$	13919.9	0.956683
$597$	0	0
$598$	−651.167	−0.0445288
$599$	13118.3	0.894826	0.447413	−	0.894327i	$-0.352345\pi$
$599$	13118.3	0.894826	0.447413	+	0.894327i	$0.352345\pi$
$600$	0	0
$601$	21970.2	1.49115	0.745577	−	0.666420i	$-0.232174\pi$
$601$	21970.2	1.49115	0.745577	+	0.666420i	$0.232174\pi$
$602$	−3489.33	−0.236237
$603$	0	0
$604$	−2197.99	−0.148071
$605$	0	0
$606$	0	0
$607$	21115.1	1.41192	0.705959	−	0.708253i	$-0.250516\pi$
$607$	21115.1	1.41192	0.705959	+	0.708253i	$0.250516\pi$
$608$	−16985.6	−1.13299
$609$	0	0
$610$	0	0
$611$	10953.7	0.725269
$612$	0	0
$613$	1963.61	0.129379	0.0646895	−	0.997905i	$-0.479394\pi$
$613$	1963.61	0.129379	0.0646895	+	0.997905i	$0.479394\pi$
$614$	4306.60	0.283062
$615$	0	0
$616$	1518.89	0.0993474
$617$	17352.1	1.13221	0.566103	−	0.824335i	$-0.308450\pi$
$617$	17352.1	1.13221	0.566103	+	0.824335i	$0.308450\pi$
$618$	0	0
$619$	−3614.68	−0.234711	−0.117356	−	0.993090i	$-0.537442\pi$
$619$	−3614.68	−0.234711	−0.117356	+	0.993090i	$0.537442\pi$
$620$	0	0
$621$	0	0
$622$	9083.94	0.585583
$623$	6617.09	0.425535
$624$	0	0
$625$	0	0
$626$	3486.68	0.222613
$627$	0	0
$628$	9634.56	0.612199
$629$	5356.35	0.339542
$630$	0	0
$631$	16368.8	1.03269	0.516347	−	0.856379i	$-0.327291\pi$
$631$	16368.8	1.03269	0.516347	+	0.856379i	$0.327291\pi$
$632$	−7966.59	−0.501415
$633$	0	0
$634$	−127.112	−0.00796258
$635$	0	0
$636$	0	0
$637$	−1683.35	−0.104705
$638$	−1682.45	−0.104403
$639$	0	0
$640$	0	0
$641$	849.649	0.0523543	0.0261772	−	0.999657i	$-0.491667\pi$
$641$	849.649	0.0523543	0.0261772	+	0.999657i	$0.491667\pi$
$642$	0	0
$643$	1335.12	0.0818850	0.0409425	−	0.999162i	$-0.486964\pi$
$643$	1335.12	0.0818850	0.0409425	+	0.999162i	$0.486964\pi$
$644$	472.553	0.0289149
$645$	0	0
$646$	11700.0	0.712586
$647$	−5944.39	−0.361202	−0.180601	−	0.983556i	$-0.557804\pi$
$647$	−5944.39	−0.361202	−0.180601	+	0.983556i	$0.557804\pi$
$648$	0	0
$649$	2909.92	0.176001
$650$	0	0
$651$	0	0
$652$	18371.3	1.10349
$653$	−5561.29	−0.333277	−0.166639	−	0.986018i	$-0.553291\pi$
$653$	−5561.29	−0.333277	−0.166639	+	0.986018i	$0.553291\pi$
$654$	0	0
$655$	0	0
$656$	2229.84	0.132714
$657$	0	0
$658$	3485.27	0.206489
$659$	−27399.5	−1.61963	−0.809813	−	0.586688i	$-0.800431\pi$
$659$	−27399.5	−1.61963	−0.809813	+	0.586688i	$0.800431\pi$
$660$	0	0
$661$	3931.52	0.231344	0.115672	−	0.993287i	$-0.463098\pi$
$661$	3931.52	0.231344	0.115672	+	0.993287i	$0.463098\pi$
$662$	10228.5	0.600514
$663$	0	0
$664$	15935.0	0.931324
$665$	0	0
$666$	0	0
$667$	−1276.38	−0.0740954
$668$	−3347.73	−0.193903
$669$	0	0
$670$	0	0
$671$	−5067.16	−0.291528
$672$	0	0
$673$	−11427.3	−0.654515	−0.327257	−	0.944935i	$-0.606124\pi$
$673$	−11427.3	−0.654515	−0.327257	+	0.944935i	$0.606124\pi$
$674$	6440.83	0.368088
$675$	0	0
$676$	5654.94	0.321742
$677$	−16259.3	−0.923035	−0.461518	−	0.887131i	$-0.652695\pi$
$677$	−16259.3	−0.923035	−0.461518	+	0.887131i	$0.652695\pi$
$678$	0	0
$679$	1264.68	0.0714785
$680$	0	0
$681$	0	0
$682$	2273.73	0.127662
$683$	10873.9	0.609192	0.304596	−	0.952482i	$-0.401479\pi$
$683$	10873.9	0.609192	0.304596	+	0.952482i	$0.401479\pi$
$684$	0	0
$685$	0	0
$686$	−535.613	−0.0298102
$687$	0	0
$688$	3646.52	0.202067
$689$	10196.3	0.563785
$690$	0	0
$691$	21613.9	1.18991	0.594957	−	0.803758i	$-0.297169\pi$
$691$	21613.9	1.18991	0.594957	+	0.803758i	$0.297169\pi$
$692$	−20806.7	−1.14299
$693$	0	0
$694$	−1918.28	−0.104923
$695$	0	0
$696$	0	0
$697$	−16123.7	−0.876222
$698$	−5185.81	−0.281212
$699$	0	0
$700$	0	0
$701$	20667.7	1.11357	0.556783	−	0.830658i	$-0.312036\pi$
$701$	20667.7	1.11357	0.556783	+	0.830658i	$0.312036\pi$
$702$	0	0
$703$	−5882.12	−0.315574
$704$	2059.71	0.110267
$705$	0	0
$706$	7166.36	0.382025
$707$	10241.4	0.544790
$708$	0	0
$709$	−22813.3	−1.20842	−0.604212	−	0.796824i	$-0.706512\pi$
$709$	−22813.3	−1.20842	−0.604212	+	0.796824i	$0.706512\pi$
$710$	0	0
$711$	0	0
$712$	20018.7	1.05370
$713$	1724.94	0.0906026
$714$	0	0
$715$	0	0
$716$	5385.54	0.281099
$717$	0	0
$718$	19334.4	1.00495
$719$	−9551.14	−0.495407	−0.247703	−	0.968836i	$-0.579676\pi$
$719$	−9551.14	−0.495407	−0.247703	+	0.968836i	$0.579676\pi$
$720$	0	0
$721$	−3745.34	−0.193459
$722$	−2137.77	−0.110193
$723$	0	0
$724$	10542.7	0.541185
$725$	0	0
$726$	0	0
$727$	30003.0	1.53060	0.765302	−	0.643672i	$-0.222590\pi$
$727$	30003.0	1.53060	0.765302	+	0.643672i	$0.222590\pi$
$728$	−5092.65	−0.259267
$729$	0	0
$730$	0	0
$731$	−26367.5	−1.33411
$732$	0	0
$733$	−16863.7	−0.849762	−0.424881	−	0.905249i	$-0.639684\pi$
$733$	−16863.7	−0.849762	−0.424881	+	0.905249i	$0.639684\pi$
$734$	8763.95	0.440713
$735$	0	0
$736$	2272.95	0.113834
$737$	−5625.50	−0.281164
$738$	0	0
$739$	−30778.9	−1.53210	−0.766048	−	0.642783i	$-0.777780\pi$
$739$	−30778.9	−1.53210	−0.766048	+	0.642783i	$0.777780\pi$
$740$	0	0
$741$	0	0
$742$	3244.27	0.160514
$743$	28510.9	1.40776	0.703879	−	0.710320i	$-0.251450\pi$
$743$	28510.9	1.40776	0.703879	+	0.710320i	$0.251450\pi$
$744$	0	0
$745$	0	0
$746$	−5763.77	−0.282878
$747$	0	0
$748$	4706.97	0.230085
$749$	2290.07	0.111719
$750$	0	0
$751$	12480.0	0.606395	0.303197	−	0.952928i	$-0.401946\pi$
$751$	12480.0	0.606395	0.303197	+	0.952928i	$0.401946\pi$
$752$	−3642.28	−0.176623
$753$	0	0
$754$	5641.04	0.272460
$755$	0	0
$756$	0	0
$757$	11272.1	0.541203	0.270601	−	0.962691i	$-0.412777\pi$
$757$	11272.1	0.541203	0.270601	+	0.962691i	$0.412777\pi$
$758$	−8354.56	−0.400332
$759$	0	0
$760$	0	0
$761$	28775.4	1.37071	0.685354	−	0.728210i	$-0.259647\pi$
$761$	28775.4	1.37071	0.685354	+	0.728210i	$0.259647\pi$
$762$	0	0
$763$	11351.9	0.538621
$764$	−15182.4	−0.718951
$765$	0	0
$766$	−11399.5	−0.537702
$767$	−9756.58	−0.459309
$768$	0	0
$769$	−17298.3	−0.811175	−0.405587	−	0.914056i	$-0.632933\pi$
$769$	−17298.3	−0.811175	−0.405587	+	0.914056i	$0.632933\pi$
$770$	0	0
$771$	0	0
$772$	8122.33	0.378665
$773$	5348.18	0.248849	0.124425	−	0.992229i	$-0.460291\pi$
$773$	5348.18	0.248849	0.124425	+	0.992229i	$0.460291\pi$
$774$	0	0
$775$	0	0
$776$	3826.03	0.176993
$777$	0	0
$778$	−14213.7	−0.654994
$779$	17706.3	0.814371
$780$	0	0
$781$	7592.15	0.347847
$782$	−1565.65	−0.0715954
$783$	0	0
$784$	559.741	0.0254984
$785$	0	0
$786$	0	0
$787$	21615.1	0.979030	0.489515	−	0.871995i	$-0.337174\pi$
$787$	21615.1	0.979030	0.489515	+	0.871995i	$0.337174\pi$
$788$	26870.2	1.21473
$789$	0	0
$790$	0	0
$791$	3155.01	0.141820
$792$	0	0
$793$	16989.5	0.760800
$794$	−17989.3	−0.804050
$795$	0	0
$796$	4855.62	0.216210
$797$	−38266.0	−1.70069	−0.850345	−	0.526225i	$-0.823607\pi$
$797$	−38266.0	−1.70069	−0.850345	+	0.526225i	$0.823607\pi$
$798$	0	0
$799$	26336.8	1.16612
$800$	0	0
$801$	0	0
$802$	5954.09	0.262152
$803$	5705.61	0.250743
$804$	0	0
$805$	0	0
$806$	−7623.50	−0.333159
$807$	0	0
$808$	30983.2	1.34899
$809$	−4724.05	−0.205301	−0.102651	−	0.994717i	$-0.532732\pi$
$809$	−4724.05	−0.205301	−0.102651	+	0.994717i	$0.532732\pi$
$810$	0	0
$811$	9687.39	0.419446	0.209723	−	0.977761i	$-0.432744\pi$
$811$	9687.39	0.419446	0.209723	+	0.977761i	$0.432744\pi$
$812$	−4093.71	−0.176923
$813$	0	0
$814$	1037.55	0.0446756
$815$	0	0
$816$	0	0
$817$	28955.7	1.23994
$818$	6400.53	0.273581
$819$	0	0
$820$	0	0
$821$	32250.8	1.37096	0.685482	−	0.728090i	$-0.259592\pi$
$821$	32250.8	1.37096	0.685482	+	0.728090i	$0.259592\pi$
$822$	0	0
$823$	−5858.93	−0.248153	−0.124076	−	0.992273i	$-0.539597\pi$
$823$	−5858.93	−0.248153	−0.124076	+	0.992273i	$0.539597\pi$
$824$	−11330.8	−0.479037
$825$	0	0
$826$	−3104.37	−0.130768
$827$	16375.7	0.688560	0.344280	−	0.938867i	$-0.388123\pi$
$827$	16375.7	0.688560	0.344280	+	0.938867i	$0.388123\pi$
$828$	0	0
$829$	−15152.9	−0.634841	−0.317421	−	0.948285i	$-0.602817\pi$
$829$	−15152.9	−0.634841	−0.317421	+	0.948285i	$0.602817\pi$
$830$	0	0
$831$	0	0
$832$	−6905.94	−0.287765
$833$	−4047.42	−0.168349
$834$	0	0
$835$	0	0
$836$	−5169.00	−0.213844
$837$	0	0
$838$	22349.6	0.921307
$839$	8829.87	0.363338	0.181669	−	0.983360i	$-0.441850\pi$
$839$	8829.87	0.363338	0.181669	+	0.983360i	$0.441850\pi$
$840$	0	0
$841$	−13331.8	−0.546630
$842$	−10430.5	−0.426911
$843$	0	0
$844$	28631.0	1.16768
$845$	0	0
$846$	0	0
$847$	−8582.11	−0.348152
$848$	−3390.42	−0.137297
$849$	0	0
$850$	0	0
$851$	787.125	0.0317066
$852$	0	0
$853$	4227.64	0.169697	0.0848486	−	0.996394i	$-0.472959\pi$
$853$	4227.64	0.169697	0.0848486	+	0.996394i	$0.472959\pi$
$854$	5405.75	0.216605
$855$	0	0
$856$	6928.15	0.276635
$857$	32233.1	1.28479	0.642394	−	0.766375i	$-0.277941\pi$
$857$	32233.1	1.28479	0.642394	+	0.766375i	$0.277941\pi$
$858$	0	0
$859$	37392.3	1.48523	0.742613	−	0.669720i	$-0.233586\pi$
$859$	37392.3	1.48523	0.742613	+	0.669720i	$0.233586\pi$
$860$	0	0
$861$	0	0
$862$	17963.8	0.709801
$863$	8372.73	0.330256	0.165128	−	0.986272i	$-0.447196\pi$
$863$	8372.73	0.330256	0.165128	+	0.986272i	$0.447196\pi$
$864$	0	0
$865$	0	0
$866$	5065.66	0.198774
$867$	0	0
$868$	5532.39	0.216338
$869$	−3854.52	−0.150467
$870$	0	0
$871$	18861.5	0.733753
$872$	34343.0	1.33372
$873$	0	0
$874$	1719.33	0.0665416
$875$	0	0
$876$	0	0
$877$	−38583.0	−1.48558	−0.742790	−	0.669524i	$-0.766498\pi$
$877$	−38583.0	−1.48558	−0.742790	+	0.669524i	$0.766498\pi$
$878$	4285.23	0.164715
$879$	0	0
$880$	0	0
$881$	−41611.7	−1.59130	−0.795650	−	0.605757i	$-0.792870\pi$
$881$	−41611.7	−1.59130	−0.795650	+	0.605757i	$0.792870\pi$
$882$	0	0
$883$	−43408.1	−1.65436	−0.827180	−	0.561937i	$-0.810056\pi$
$883$	−43408.1	−1.65436	−0.827180	+	0.561937i	$0.810056\pi$
$884$	−15781.8	−0.600453
$885$	0	0
$886$	−12773.9	−0.484366
$887$	19523.6	0.739050	0.369525	−	0.929221i	$-0.379520\pi$
$887$	19523.6	0.739050	0.369525	+	0.929221i	$0.379520\pi$
$888$	0	0
$889$	−11970.7	−0.451613
$890$	0	0
$891$	0	0
$892$	−13544.1	−0.508398
$893$	−28922.0	−1.08381
$894$	0	0
$895$	0	0
$896$	8288.92	0.309055
$897$	0	0
$898$	10502.5	0.390283
$899$	−14943.1	−0.554373
$900$	0	0
$901$	24515.7	0.906479
$902$	−3123.21	−0.115290
$903$	0	0
$904$	9544.84	0.351169
$905$	0	0
$906$	0	0
$907$	−44114.7	−1.61500	−0.807499	−	0.589869i	$-0.799180\pi$
$907$	−44114.7	−1.61500	−0.807499	+	0.589869i	$0.799180\pi$
$908$	19512.5	0.713156
$909$	0	0
$910$	0	0
$911$	−49231.9	−1.79048	−0.895240	−	0.445585i	$-0.852996\pi$
$911$	−49231.9	−1.79048	−0.895240	+	0.445585i	$0.852996\pi$
$912$	0	0
$913$	7709.92	0.279476
$914$	−16875.7	−0.610719
$915$	0	0
$916$	−19073.6	−0.688001
$917$	−11091.4	−0.399424
$918$	0	0
$919$	−46993.6	−1.68681	−0.843403	−	0.537282i	$-0.819451\pi$
$919$	−46993.6	−1.68681	−0.843403	+	0.537282i	$0.819451\pi$
$920$	0	0
$921$	0	0
$922$	27757.9	0.991494
$923$	−25455.5	−0.907775
$924$	0	0
$925$	0	0
$926$	−6900.40	−0.244883
$927$	0	0
$928$	−19690.5	−0.696521
$929$	33023.5	1.16627	0.583135	−	0.812375i	$-0.301826\pi$
$929$	33023.5	1.16627	0.583135	+	0.812375i	$0.301826\pi$
$930$	0	0
$931$	4444.71	0.156466
$932$	22189.0	0.779855
$933$	0	0
$934$	−5267.25	−0.184529
$935$	0	0
$936$	0	0
$937$	23431.0	0.816924	0.408462	−	0.912775i	$-0.366065\pi$
$937$	23431.0	0.816924	0.408462	+	0.912775i	$0.366065\pi$
$938$	6001.40	0.208905
$939$	0	0
$940$	0	0
$941$	−43688.0	−1.51348	−0.756742	−	0.653714i	$-0.773210\pi$
$941$	−43688.0	−1.51348	−0.756742	+	0.653714i	$0.773210\pi$
$942$	0	0
$943$	−2369.40	−0.0818221
$944$	3244.21	0.111854
$945$	0	0
$946$	−5107.48	−0.175538
$947$	34884.0	1.19702	0.598511	−	0.801115i	$-0.295759\pi$
$947$	34884.0	1.19702	0.598511	+	0.801115i	$0.295759\pi$
$948$	0	0
$949$	−19130.1	−0.654363
$950$	0	0
$951$	0	0
$952$	−12244.6	−0.416860
$953$	34486.4	1.17222	0.586108	−	0.810233i	$-0.300659\pi$
$953$	34486.4	1.17222	0.586108	+	0.810233i	$0.300659\pi$
$954$	0	0
$955$	0	0
$956$	−33006.2	−1.11663
$957$	0	0
$958$	7214.40	0.243306
$959$	−14666.7	−0.493861
$960$	0	0
$961$	−9596.33	−0.322122
$962$	−3478.75	−0.116590
$963$	0	0
$964$	36049.4	1.20443
$965$	0	0
$966$	0	0
$967$	46341.6	1.54110	0.770550	−	0.637379i	$-0.219981\pi$
$967$	46341.6	1.54110	0.770550	+	0.637379i	$0.219981\pi$
$968$	−25963.4	−0.862083
$969$	0	0
$970$	0	0
$971$	34551.4	1.14192	0.570962	−	0.820977i	$-0.306571\pi$
$971$	34551.4	1.14192	0.570962	+	0.820977i	$0.306571\pi$
$972$	0	0
$973$	−9694.97	−0.319431
$974$	−14223.6	−0.467919
$975$	0	0
$976$	−5649.27	−0.185275
$977$	970.127	0.0317678	0.0158839	−	0.999874i	$-0.494944\pi$
$977$	970.127	0.0317678	0.0158839	+	0.999874i	$0.494944\pi$
$978$	0	0
$979$	9685.73	0.316198
$980$	0	0
$981$	0	0
$982$	−4211.26	−0.136850
$983$	18342.2	0.595142	0.297571	−	0.954700i	$-0.403824\pi$
$983$	18342.2	0.595142	0.297571	+	0.954700i	$0.403824\pi$
$984$	0	0
$985$	0	0
$986$	13563.2	0.438073
$987$	0	0
$988$	17331.0	0.558068
$989$	−3874.75	−0.124580
$990$	0	0
$991$	48979.6	1.57002	0.785010	−	0.619484i	$-0.212658\pi$
$991$	48979.6	1.57002	0.785010	+	0.619484i	$0.212658\pi$
$992$	26610.4	0.851694
$993$	0	0
$994$	−8099.46	−0.258450
$995$	0	0
$996$	0	0
$997$	−31492.9	−1.00039	−0.500196	−	0.865912i	$-0.666739\pi$
$997$	−31492.9	−1.00039	−0.500196	+	0.865912i	$0.666739\pi$
$998$	31315.7	0.993268
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.4.a.u.1.1		2
3.2	odd	2		1575.4.a.r.1.2		2
5.4	even	2		315.4.a.h.1.2	✓	2
15.14	odd	2		315.4.a.j.1.1	yes	2
35.34	odd	2		2205.4.a.y.1.2		2
105.104	even	2		2205.4.a.ba.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.4.a.h.1.2	✓	2	5.4	even	2
315.4.a.j.1.1	yes	2	15.14	odd	2
1575.4.a.r.1.2		2	3.2	odd	2
1575.4.a.u.1.1		2	1.1	even	1	trivial
2205.4.a.y.1.2		2	35.34	odd	2
2205.4.a.ba.1.1		2	105.104	even	2

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:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1575.a (trivial)

\(f(q)\)	\(=\)	\(q-1.56155 q^{2} -5.56155 q^{4} +7.00000 q^{7} +21.1771 q^{8} +O(q^{10})\)	\(q-1.56155 q^{2} -5.56155 q^{4} +7.00000 q^{7} +21.1771 q^{8} +10.2462 q^{11} -34.3542 q^{13} -10.9309 q^{14} +11.4233 q^{16} -82.6004 q^{17} +90.7083 q^{19} -16.0000 q^{22} -12.1383 q^{23} +53.6458 q^{26} -38.9309 q^{28} +105.153 q^{29} -142.108 q^{31} -187.255 q^{32} +128.985 q^{34} -64.8466 q^{37} -141.646 q^{38} +195.201 q^{41} +319.218 q^{43} -56.9848 q^{44} +18.9545 q^{46} -318.847 q^{47} +49.0000 q^{49} +191.062 q^{52} -296.799 q^{53} +148.240 q^{56} -164.203 q^{58} +284.000 q^{59} -494.540 q^{61} +221.909 q^{62} +201.022 q^{64} -549.032 q^{67} +459.386 q^{68} +740.972 q^{71} +556.850 q^{73} +101.261 q^{74} -504.479 q^{76} +71.7235 q^{77} -376.189 q^{79} -304.816 q^{82} +752.466 q^{83} -498.475 q^{86} +216.985 q^{88} +945.299 q^{89} -240.479 q^{91} +67.5076 q^{92} +497.896 q^{94} +180.668 q^{97} -76.5161 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - 7 q^{4} + 14 q^{7} - 3 q^{8}+O(q^{10}) \) 2 * q + q^2 - 7 * q^4 + 14 * q^7 - 3 * q^8	\( 2 q + q^{2} - 7 q^{4} + 14 q^{7} - 3 q^{8} + 4 q^{11} + 22 q^{13} + 7 q^{14} - 39 q^{16} - 58 q^{17} - 32 q^{22} - 82 q^{23} + 198 q^{26} - 49 q^{28} + 334 q^{29} - 210 q^{31} - 123 q^{32} + 192 q^{34} - 6 q^{37} - 374 q^{38} + 176 q^{41} - 46 q^{43} - 48 q^{44} - 160 q^{46} - 514 q^{47} + 98 q^{49} + 110 q^{52} - 808 q^{53} - 21 q^{56} + 422 q^{58} + 568 q^{59} - 618 q^{61} + 48 q^{62} + 769 q^{64} - 694 q^{67} + 424 q^{68} + 814 q^{71} - 82 q^{73} + 252 q^{74} - 374 q^{76} + 28 q^{77} + 600 q^{79} - 354 q^{82} + 268 q^{83} - 1434 q^{86} + 368 q^{88} - 72 q^{89} + 154 q^{91} + 168 q^{92} - 2 q^{94} - 1626 q^{97} + 49 q^{98}+O(q^{100}) \) 2 * q + q^2 - 7 * q^4 + 14 * q^7 - 3 * q^8 + 4 * q^11 + 22 * q^13 + 7 * q^14 - 39 * q^16 - 58 * q^17 - 32 * q^22 - 82 * q^23 + 198 * q^26 - 49 * q^28 + 334 * q^29 - 210 * q^31 - 123 * q^32 + 192 * q^34 - 6 * q^37 - 374 * q^38 + 176 * q^41 - 46 * q^43 - 48 * q^44 - 160 * q^46 - 514 * q^47 + 98 * q^49 + 110 * q^52 - 808 * q^53 - 21 * q^56 + 422 * q^58 + 568 * q^59 - 618 * q^61 + 48 * q^62 + 769 * q^64 - 694 * q^67 + 424 * q^68 + 814 * q^71 - 82 * q^73 + 252 * q^74 - 374 * q^76 + 28 * q^77 + 600 * q^79 - 354 * q^82 + 268 * q^83 - 1434 * q^86 + 368 * q^88 - 72 * q^89 + 154 * q^91 + 168 * q^92 - 2 * q^94 - 1626 * q^97 + 49 * q^98

Introduction

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