LMFDB - Embedded newform 315.4.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [315,4,Mod(1,315)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(315, 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("315.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 315 = 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	315.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:	$18.5856016518$
Analytic rank:	$0$
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:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	315.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.56155 q^{2} -5.56155 q^{4} -5.00000 q^{5} -7.00000 q^{7} -21.1771 q^{8} +O(q^{10})$	\(q+1.56155 q^{2} -5.56155 q^{4} -5.00000 q^{5} -7.00000 q^{7} -21.1771 q^{8} -7.80776 q^{10} +10.2462 q^{11} +34.3542 q^{13} -10.9309 q^{14} +11.4233 q^{16} +82.6004 q^{17} +90.7083 q^{19} +27.8078 q^{20} +16.0000 q^{22} +12.1383 q^{23} +25.0000 q^{25} +53.6458 q^{26} +38.9309 q^{28} +105.153 q^{29} -142.108 q^{31} +187.255 q^{32} +128.985 q^{34} +35.0000 q^{35} +64.8466 q^{37} +141.646 q^{38} +105.885 q^{40} +195.201 q^{41} -319.218 q^{43} -56.9848 q^{44} +18.9545 q^{46} +318.847 q^{47} +49.0000 q^{49} +39.0388 q^{50} -191.062 q^{52} +296.799 q^{53} -51.2311 q^{55} +148.240 q^{56} +164.203 q^{58} +284.000 q^{59} -494.540 q^{61} -221.909 q^{62} +201.022 q^{64} -171.771 q^{65} +549.032 q^{67} -459.386 q^{68} +54.6543 q^{70} +740.972 q^{71} -556.850 q^{73} +101.261 q^{74} -504.479 q^{76} -71.7235 q^{77} -376.189 q^{79} -57.1165 q^{80} +304.816 q^{82} -752.466 q^{83} -413.002 q^{85} -498.475 q^{86} -216.985 q^{88} +945.299 q^{89} -240.479 q^{91} -67.5076 q^{92} +497.896 q^{94} -453.542 q^{95} -180.668 q^{97} +76.5161 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - 7 q^{4} - 10 q^{5} - 14 q^{7} + 3 q^{8}+O(q^{10}) $ 2 * q - q^2 - 7 * q^4 - 10 * q^5 - 14 * q^7 + 3 * q^8	$ 2 q - q^{2} - 7 q^{4} - 10 q^{5} - 14 q^{7} + 3 q^{8} + 5 q^{10} + 4 q^{11} - 22 q^{13} + 7 q^{14} - 39 q^{16} + 58 q^{17} + 35 q^{20} + 32 q^{22} + 82 q^{23} + 50 q^{25} + 198 q^{26} + 49 q^{28} + 334 q^{29} - 210 q^{31} + 123 q^{32} + 192 q^{34} + 70 q^{35} + 6 q^{37} + 374 q^{38} - 15 q^{40} + 176 q^{41} + 46 q^{43} - 48 q^{44} - 160 q^{46} + 514 q^{47} + 98 q^{49} - 25 q^{50} - 110 q^{52} + 808 q^{53} - 20 q^{55} - 21 q^{56} - 422 q^{58} + 568 q^{59} - 618 q^{61} - 48 q^{62} + 769 q^{64} + 110 q^{65} + 694 q^{67} - 424 q^{68} - 35 q^{70} + 814 q^{71} + 82 q^{73} + 252 q^{74} - 374 q^{76} - 28 q^{77} + 600 q^{79} + 195 q^{80} + 354 q^{82} - 268 q^{83} - 290 q^{85} - 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 - 10 * q^5 - 14 * q^7 + 3 * q^8 + 5 * q^10 + 4 * q^11 - 22 * q^13 + 7 * q^14 - 39 * q^16 + 58 * q^17 + 35 * q^20 + 32 * q^22 + 82 * q^23 + 50 * q^25 + 198 * q^26 + 49 * q^28 + 334 * q^29 - 210 * q^31 + 123 * q^32 + 192 * q^34 + 70 * q^35 + 6 * q^37 + 374 * q^38 - 15 * q^40 + 176 * q^41 + 46 * q^43 - 48 * q^44 - 160 * q^46 + 514 * q^47 + 98 * q^49 - 25 * q^50 - 110 * q^52 + 808 * q^53 - 20 * q^55 - 21 * q^56 - 422 * q^58 + 568 * q^59 - 618 * q^61 - 48 * q^62 + 769 * q^64 + 110 * q^65 + 694 * q^67 - 424 * q^68 - 35 * q^70 + 814 * q^71 + 82 * q^73 + 252 * q^74 - 374 * q^76 - 28 * q^77 + 600 * q^79 + 195 * q^80 + 354 * q^82 - 268 * q^83 - 290 * q^85 - 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	$-0.410976\pi$
$2$	1.56155	0.552092	0.276046	+	0.961144i	$0.410976\pi$
$3$	0	0
$3$	0	0
$4$	−5.56155	−0.695194
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−21.1771	−0.935904
$9$	0	0
$10$	−7.80776	−0.246903
$11$	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	$-0.380567\pi$
$13$	34.3542	0.732933	0.366467	+	0.930431i	$0.380567\pi$
$14$	−10.9309	−0.208671
$15$	0	0
$16$	11.4233	0.178489
$17$	82.6004	1.17844	0.589222	−	0.807972i	$-0.299435\pi$
$17$	82.6004	1.17844	0.589222	+	0.807972i	$0.299435\pi$
$18$	0	0
$19$	90.7083	1.09526	0.547629	−	0.836721i	$-0.315530\pi$
$19$	90.7083	1.09526	0.547629	+	0.836721i	$0.315530\pi$
$20$	27.8078	0.310900
$21$	0	0
$22$	16.0000	0.155055
$23$	12.1383	0.110044	0.0550218	−	0.998485i	$-0.482477\pi$
$23$	12.1383	0.110044	0.0550218	+	0.998485i	$0.482477\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	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
$35$	35.0000	0.169031
$36$	0	0
$37$	64.8466	0.288127	0.144064	−	0.989568i	$-0.453983\pi$
$37$	64.8466	0.288127	0.144064	+	0.989568i	$0.453983\pi$
$38$	141.646	0.604684
$39$	0	0
$40$	105.885	0.418549
$41$	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	$-0.691529\pi$
$43$	−319.218	−1.13210	−0.566049	+	0.824371i	$0.691529\pi$
$44$	−56.9848	−0.195245
$45$	0	0
$46$	18.9545	0.0607542
$47$	318.847	0.989544	0.494772	−	0.869023i	$-0.335252\pi$
$47$	318.847	0.989544	0.494772	+	0.869023i	$0.335252\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	39.0388	0.110418
$51$	0	0
$52$	−191.062	−0.509531
$53$	296.799	0.769217	0.384609	−	0.923080i	$-0.374336\pi$
$53$	296.799	0.769217	0.384609	+	0.923080i	$0.374336\pi$
$54$	0	0
$55$	−51.2311	−0.125600
$56$	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
$65$	−171.771	−0.327778
$66$	0	0
$67$	549.032	1.00112	0.500559	−	0.865702i	$-0.333128\pi$
$67$	549.032	1.00112	0.500559	+	0.865702i	$0.333128\pi$
$68$	−459.386	−0.819247
$69$	0	0
$70$	54.6543	0.0933206
$71$	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	$-0.647294\pi$
$73$	−556.850	−0.892800	−0.446400	+	0.894834i	$0.647294\pi$
$74$	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	$0.586322\pi$
$80$	−57.1165	−0.0798227
$81$	0	0
$82$	304.816	0.410504
$83$	−752.466	−0.995107	−0.497553	−	0.867433i	$-0.665768\pi$
$83$	−752.466	−0.995107	−0.497553	+	0.867433i	$0.665768\pi$
$84$	0	0
$85$	−413.002	−0.527016
$86$	−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
$95$	−453.542	−0.489815
$96$	0	0
$97$	−180.668	−0.189114	−0.0945572	−	0.995519i	$-0.530144\pi$
$97$	−180.668	−0.189114	−0.0945572	+	0.995519i	$0.530144\pi$
$98$	76.5161	0.0788703
$99$	0	0
$100$	−139.039	−0.139039
$101$	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	$-0.417621\pi$
$103$	535.049	0.511844	0.255922	+	0.966697i	$0.417621\pi$
$104$	−727.521	−0.685955
$105$	0	0
$106$	463.468	0.424679
$107$	−327.153	−0.295581	−0.147790	−	0.989019i	$-0.547216\pi$
$107$	−327.153	−0.295581	−0.147790	+	0.989019i	$0.547216\pi$
$108$	0	0
$109$	1621.70	1.42506	0.712528	−	0.701644i	$-0.247550\pi$
$109$	1621.70	1.42506	0.712528	+	0.701644i	$0.247550\pi$
$110$	−80.0000	−0.0693427
$111$	0	0
$112$	−79.9630	−0.0674625
$113$	−450.716	−0.375219	−0.187610	−	0.982244i	$-0.560074\pi$
$113$	−450.716	−0.375219	−0.187610	+	0.982244i	$0.560074\pi$
$114$	0	0
$115$	−60.6913	−0.0492130
$116$	−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
$125$	−125.000	−0.0894427
$126$	0	0
$127$	1710.10	1.19486	0.597428	−	0.801923i	$-0.296189\pi$
$127$	1710.10	1.19486	0.597428	+	0.801923i	$0.296189\pi$
$128$	−1184.13	−0.817683
$129$	0	0
$130$	−268.229	−0.180964
$131$	−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	$-0.273377\pi$
$137$	2095.24	1.30663	0.653316	+	0.757085i	$0.273377\pi$
$138$	0	0
$139$	−1385.00	−0.845135	−0.422568	−	0.906331i	$-0.638871\pi$
$139$	−1385.00	−0.845135	−0.422568	+	0.906331i	$0.638871\pi$
$140$	−194.654	−0.117509
$141$	0	0
$142$	1157.07	0.683795
$143$	352.000	0.205844
$144$	0	0
$145$	−525.767	−0.301121
$146$	−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
$155$	710.540	0.368206
$156$	0	0
$157$	1732.35	0.880615	0.440308	−	0.897847i	$-0.354869\pi$
$157$	1732.35	0.880615	0.440308	+	0.897847i	$0.354869\pi$
$158$	−587.439	−0.295786
$159$	0	0
$160$	−936.274	−0.462618
$161$	−84.9678	−0.0415926
$162$	0	0
$163$	3303.26	1.58731	0.793654	−	0.608369i	$-0.208176\pi$
$163$	3303.26	1.58731	0.793654	+	0.608369i	$0.208176\pi$
$164$	−1085.62	−0.516906
$165$	0	0
$166$	−1175.02	−0.549391
$167$	−601.941	−0.278920	−0.139460	−	0.990228i	$-0.544537\pi$
$167$	−601.941	−0.278920	−0.139460	+	0.990228i	$0.544537\pi$
$168$	0	0
$169$	−1016.79	−0.462809
$170$	−644.924	−0.290961
$171$	0	0
$172$	1775.35	0.787028
$173$	−3741.17	−1.64414	−0.822068	−	0.569389i	$-0.807180\pi$
$173$	−3741.17	−1.64414	−0.822068	+	0.569389i	$0.807180\pi$
$174$	0	0
$175$	−175.000	−0.0755929
$176$	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
$185$	−324.233	−0.128854
$186$	0	0
$187$	846.341	0.330966
$188$	−1773.28	−0.687925
$189$	0	0
$190$	−708.229	−0.270423
$191$	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	$-0.412201\pi$
$193$	1460.44	0.544689	0.272345	+	0.962200i	$0.412201\pi$
$194$	−282.123	−0.104409
$195$	0	0
$196$	−272.516	−0.0993134
$197$	4831.42	1.74733	0.873665	−	0.486527i	$-0.161737\pi$
$197$	4831.42	1.74733	0.873665	+	0.486527i	$0.161737\pi$
$198$	0	0
$199$	−873.070	−0.311006	−0.155503	−	0.987835i	$-0.549700\pi$
$199$	−873.070	−0.311006	−0.155503	+	0.987835i	$0.549700\pi$
$200$	−529.427	−0.187181
$201$	0	0
$202$	2284.63	0.795774
$203$	−736.074	−0.254494
$204$	0	0
$205$	−976.004	−0.332522
$206$	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
$215$	1596.09	0.506290
$216$	0	0
$217$	994.756	0.311191
$218$	2532.38	0.786762
$219$	0	0
$220$	284.924	0.0873163
$221$	2837.67	0.863720
$222$	0	0
$223$	−2435.31	−0.731304	−0.365652	−	0.930752i	$-0.619154\pi$
$223$	−2435.31	−0.731304	−0.365652	+	0.930752i	$0.619154\pi$
$224$	−1310.78	−0.390984
$225$	0	0
$226$	−703.817	−0.207156
$227$	3508.47	1.02584	0.512919	−	0.858437i	$-0.328564\pi$
$227$	3508.47	1.02584	0.512919	+	0.858437i	$0.328564\pi$
$228$	0	0
$229$	3429.54	0.989653	0.494827	−	0.868992i	$-0.335232\pi$
$229$	3429.54	0.989653	0.494827	+	0.868992i	$0.335232\pi$
$230$	−94.7727	−0.0271701
$231$	0	0
$232$	−2226.84	−0.630170
$233$	3989.71	1.12178	0.560890	−	0.827890i	$-0.310459\pi$
$233$	3989.71	1.12178	0.560890	+	0.827890i	$0.310459\pi$
$234$	0	0
$235$	−1594.23	−0.442537
$236$	−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
$245$	−245.000	−0.0638877
$246$	0	0
$247$	3116.21	0.802751
$248$	3009.43	0.770561
$249$	0	0
$250$	−195.194	−0.0493806
$251$	−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	$-0.666154\pi$
$257$	−4108.52	−0.997209	−0.498604	+	0.866830i	$0.666154\pi$
$258$	0	0
$259$	−453.926	−0.108902
$260$	955.312	0.227869
$261$	0	0
$262$	−2474.27	−0.583438
$263$	6236.89	1.46229	0.731146	−	0.682221i	$-0.238986\pi$
$263$	6236.89	1.46229	0.731146	+	0.682221i	$0.238986\pi$
$264$	0	0
$265$	−1484.00	−0.344004
$266$	−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
$275$	256.155	0.0561700
$276$	0	0
$277$	5804.64	1.25909	0.629543	−	0.776966i	$-0.283242\pi$
$277$	5804.64	1.25909	0.629543	+	0.776966i	$0.283242\pi$
$278$	−2162.74	−0.466593
$279$	0	0
$280$	−741.198	−0.158197
$281$	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	$-0.215812\pi$
$283$	7415.73	1.55767	0.778833	+	0.627232i	$0.215812\pi$
$284$	−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
$290$	−821.013	−0.166247
$291$	0	0
$292$	3096.95	0.620669
$293$	482.022	0.0961094	0.0480547	−	0.998845i	$-0.484698\pi$
$293$	482.022	0.0961094	0.0480547	+	0.998845i	$0.484698\pi$
$294$	0	0
$295$	−1420.00	−0.280256
$296$	−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
$305$	2472.70	0.464217
$306$	0	0
$307$	2757.90	0.512709	0.256354	−	0.966583i	$-0.417479\pi$
$307$	2757.90	0.512709	0.256354	+	0.966583i	$0.417479\pi$
$308$	398.894	0.0737957
$309$	0	0
$310$	1109.55	0.203284
$311$	−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	$-0.435383\pi$
$313$	2232.83	0.403218	0.201609	+	0.979466i	$0.435383\pi$
$314$	2705.16	0.486181
$315$	0	0
$316$	2092.20	0.372453
$317$	−81.4013	−0.0144226	−0.00721128	−	0.999974i	$-0.502295\pi$
$317$	−81.4013	−0.0144226	−0.00721128	+	0.999974i	$0.502295\pi$
$318$	0	0
$319$	1077.42	0.189104
$320$	−1005.11	−0.175585
$321$	0	0
$322$	−132.682	−0.0229629
$323$	7492.54	1.29070
$324$	0	0
$325$	858.854	0.146587
$326$	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
$335$	−2745.16	−0.447714
$336$	0	0
$337$	4124.63	0.666715	0.333358	−	0.942800i	$-0.391818\pi$
$337$	4124.63	0.666715	0.333358	+	0.942800i	$0.391818\pi$
$338$	−1587.77	−0.255513
$339$	0	0
$340$	2296.93	0.366378
$341$	−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	$-0.530293\pi$
$347$	−1228.44	−0.190047	−0.0950233	+	0.995475i	$0.530293\pi$
$348$	0	0
$349$	3320.93	0.509357	0.254678	−	0.967026i	$-0.418030\pi$
$349$	3320.93	0.509357	0.254678	+	0.967026i	$0.418030\pi$
$350$	−273.272	−0.0417343
$351$	0	0
$352$	1918.65	0.290524
$353$	4589.25	0.691958	0.345979	−	0.938242i	$-0.387547\pi$
$353$	4589.25	0.691958	0.345979	+	0.938242i	$0.387547\pi$
$354$	0	0
$355$	−3704.86	−0.553897
$356$	−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
$365$	2784.25	0.399272
$366$	0	0
$367$	5612.33	0.798259	0.399130	−	0.916894i	$-0.369312\pi$
$367$	5612.33	0.798259	0.399130	+	0.916894i	$0.369312\pi$
$368$	138.659	0.0196416
$369$	0	0
$370$	−506.307	−0.0711396
$371$	−2077.59	−0.290737
$372$	0	0
$373$	−3691.05	−0.512374	−0.256187	−	0.966627i	$-0.582466\pi$
$373$	−3691.05	−0.512374	−0.256187	+	0.966627i	$0.582466\pi$
$374$	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	$0.381903\pi$
$380$	2522.40	0.340516
$381$	0	0
$382$	4262.85	0.570959
$383$	−7300.09	−0.973936	−0.486968	−	0.873420i	$-0.661897\pi$
$383$	−7300.09	−0.973936	−0.486968	+	0.873420i	$0.661897\pi$
$384$	0	0
$385$	358.617	0.0474723
$386$	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
$395$	1880.95	0.239597
$396$	0	0
$397$	−11520.1	−1.45637	−0.728185	−	0.685381i	$-0.759636\pi$
$397$	−11520.1	−1.45637	−0.728185	+	0.685381i	$0.759636\pi$
$398$	−1363.34	−0.171704
$399$	0	0
$400$	285.582	0.0356978
$401$	−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	$0.579697\pi$
$410$	−1524.08	−0.183583
$411$	0	0
$412$	−2975.70	−0.355831
$413$	−1988.00	−0.236860
$414$	0	0
$415$	3762.33	0.445025
$416$	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
$425$	2065.01	0.235689
$426$	0	0
$427$	3461.78	0.392335
$428$	1819.48	0.205486
$429$	0	0
$430$	2492.38	0.279519
$431$	−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	$-0.442384\pi$
$433$	3243.99	0.360037	0.180019	+	0.983663i	$0.442384\pi$
$434$	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	$0.547661\pi$
$440$	1084.92	0.117549
$441$	0	0
$442$	4431.17	0.476853
$443$	−8180.27	−0.877328	−0.438664	−	0.898651i	$-0.644548\pi$
$443$	−8180.27	−0.877328	−0.438664	+	0.898651i	$0.644548\pi$
$444$	0	0
$445$	−4726.50	−0.503500
$446$	−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
$455$	1202.40	0.123888
$456$	0	0
$457$	−10807.0	−1.10619	−0.553095	−	0.833118i	$-0.686553\pi$
$457$	−10807.0	−1.10619	−0.553095	+	0.833118i	$0.686553\pi$
$458$	5355.41	0.546380
$459$	0	0
$460$	337.538	0.0342126
$461$	−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	$-0.571186\pi$
$463$	−4418.93	−0.443554	−0.221777	+	0.975097i	$0.571186\pi$
$464$	1201.20	0.120182
$465$	0	0
$466$	6230.15	0.619326
$467$	−3373.08	−0.334235	−0.167117	−	0.985937i	$-0.553446\pi$
$467$	−3373.08	−0.334235	−0.167117	+	0.985937i	$0.553446\pi$
$468$	0	0
$469$	−3843.23	−0.378387
$470$	−2489.48	−0.244321
$471$	0	0
$472$	−6014.29	−0.586505
$473$	−3270.77	−0.317950
$474$	0	0
$475$	2267.71	0.219052
$476$	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
$485$	903.342	0.0845746
$486$	0	0
$487$	−9108.62	−0.847538	−0.423769	−	0.905770i	$-0.639293\pi$
$487$	−9108.62	−0.847538	−0.423769	+	0.905770i	$0.639293\pi$
$488$	10472.9	0.971488
$489$	0	0
$490$	−382.580	−0.0352719
$491$	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	$0.856104\pi$
$500$	695.194	0.0621801
$501$	0	0
$502$	−8210.17	−0.729956
$503$	−8106.49	−0.718590	−0.359295	−	0.933224i	$-0.616983\pi$
$503$	−8106.49	−0.718590	−0.359295	+	0.933224i	$0.616983\pi$
$504$	0	0
$505$	−7315.26	−0.644604
$506$	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
$515$	−2675.25	−0.228904
$516$	0	0
$517$	3266.97	0.277913
$518$	−708.830	−0.0601239
$519$	0	0
$520$	3637.60	0.306768
$521$	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	$-0.275548\pi$
$523$	15504.2	1.29628	0.648138	+	0.761523i	$0.275548\pi$
$524$	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
$530$	−2317.34	−0.189922
$531$	0	0
$532$	3531.35	0.287789
$533$	6705.96	0.544967
$534$	0	0
$535$	1635.77	0.132188
$536$	−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
$545$	−8108.52	−0.637304
$546$	0	0
$547$	−9177.75	−0.717390	−0.358695	−	0.933455i	$-0.616778\pi$
$547$	−9177.75	−0.717390	−0.358695	+	0.933455i	$0.616778\pi$
$548$	−11652.8	−0.908363
$549$	0	0
$550$	400.000	0.0310110
$551$	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	$-0.0989896\pi$
$557$	25030.2	1.90407	0.952033	+	0.305996i	$0.0989896\pi$
$558$	0	0
$559$	−10966.5	−0.829753
$560$	399.815	0.0301701
$561$	0	0
$562$	13165.3	0.988157
$563$	−19486.1	−1.45869	−0.729345	−	0.684146i	$-0.760175\pi$
$563$	−19486.1	−1.45869	−0.729345	+	0.684146i	$0.760175\pi$
$564$	0	0
$565$	2253.58	0.167803
$566$	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
$575$	303.457	0.0220087
$576$	0	0
$577$	8283.57	0.597660	0.298830	−	0.954306i	$-0.403404\pi$
$577$	8283.57	0.597660	0.298830	+	0.954306i	$0.403404\pi$
$578$	2982.29	0.214614
$579$	0	0
$580$	2924.08	0.209338
$581$	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	$-0.424634\pi$
$587$	6671.87	0.469127	0.234563	+	0.972101i	$0.424634\pi$
$588$	0	0
$589$	−12890.4	−0.901763
$590$	−2217.40	−0.154727
$591$	0	0
$592$	740.761	0.0514275
$593$	11409.0	0.790071	0.395035	−	0.918666i	$-0.370732\pi$
$593$	11409.0	0.790071	0.395035	+	0.918666i	$0.370732\pi$
$594$	0	0
$595$	2891.01	0.199193
$596$	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
$605$	6130.08	0.411939
$606$	0	0
$607$	−21115.1	−1.41192	−0.705959	−	0.708253i	$-0.749484\pi$
$607$	−21115.1	−1.41192	−0.705959	+	0.708253i	$0.749484\pi$
$608$	16985.6	1.13299
$609$	0	0
$610$	3861.25	0.256291
$611$	10953.7	0.725269
$612$	0	0
$613$	−1963.61	−0.129379	−0.0646895	−	0.997905i	$-0.520606\pi$
$613$	−1963.61	−0.129379	−0.0646895	+	0.997905i	$0.520606\pi$
$614$	4306.60	0.283062
$615$	0	0
$616$	1518.89	0.0993474
$617$	−17352.1	−1.13221	−0.566103	−	0.824335i	$-0.691550\pi$
$617$	−17352.1	−1.13221	−0.566103	+	0.824335i	$0.691550\pi$
$618$	0	0
$619$	−3614.68	−0.234711	−0.117356	−	0.993090i	$-0.537442\pi$
$619$	−3614.68	−0.234711	−0.117356	+	0.993090i	$0.537442\pi$
$620$	−3951.70	−0.255975
$621$	0	0
$622$	−9083.94	−0.585583
$623$	−6617.09	−0.425535
$624$	0	0
$625$	625.000	0.0400000
$626$	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
$635$	−8550.49	−0.534356
$636$	0	0
$637$	1683.35	0.104705
$638$	1682.45	0.104403
$639$	0	0
$640$	5920.66	0.365679
$641$	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	$-0.513036\pi$
$643$	−1335.12	−0.0818850	−0.0409425	+	0.999162i	$0.513036\pi$
$644$	472.553	0.0289149
$645$	0	0
$646$	11700.0	0.712586
$647$	5944.39	0.361202	0.180601	−	0.983556i	$-0.442196\pi$
$647$	5944.39	0.361202	0.180601	+	0.983556i	$0.442196\pi$
$648$	0	0
$649$	2909.92	0.176001
$650$	1341.15	0.0809293
$651$	0	0
$652$	−18371.3	−1.10349
$653$	5561.29	0.333277	0.166639	−	0.986018i	$-0.446709\pi$
$653$	5561.29	0.333277	0.166639	+	0.986018i	$0.446709\pi$
$654$	0	0
$655$	7922.46	0.472605
$656$	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
$665$	3174.79	0.185133
$666$	0	0
$667$	1276.38	0.0740954
$668$	3347.73	0.193903
$669$	0	0
$670$	−4286.71	−0.247179
$671$	−5067.16	−0.291528
$672$	0	0
$673$	11427.3	0.654515	0.327257	−	0.944935i	$-0.393876\pi$
$673$	11427.3	0.654515	0.327257	+	0.944935i	$0.393876\pi$
$674$	6440.83	0.368088
$675$	0	0
$676$	5654.94	0.321742
$677$	16259.3	0.923035	0.461518	−	0.887131i	$-0.347305\pi$
$677$	16259.3	0.923035	0.461518	+	0.887131i	$0.347305\pi$
$678$	0	0
$679$	1264.68	0.0714785
$680$	8746.17	0.493236
$681$	0	0
$682$	−2273.73	−0.127662
$683$	−10873.9	−0.609192	−0.304596	−	0.952482i	$-0.598521\pi$
$683$	−10873.9	−0.609192	−0.304596	+	0.952482i	$0.598521\pi$
$684$	0	0
$685$	−10476.2	−0.584344
$686$	−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
$695$	6924.98	0.377956
$696$	0	0
$697$	16123.7	0.876222
$698$	5185.81	0.281212
$699$	0	0
$700$	973.272	0.0525517
$701$	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	$0.706512\pi$
$710$	−5785.33	−0.305802
$711$	0	0
$712$	−20018.7	−1.05370
$713$	−1724.94	−0.0906026
$714$	0	0
$715$	−1760.00	−0.0920563
$716$	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
$725$	2628.84	0.134666
$726$	0	0
$727$	−30003.0	−1.53060	−0.765302	−	0.643672i	$-0.777410\pi$
$727$	−30003.0	−1.53060	−0.765302	+	0.643672i	$0.777410\pi$
$728$	5092.65	0.259267
$729$	0	0
$730$	4347.76	0.220435
$731$	−26367.5	−1.33411
$732$	0	0
$733$	16863.7	0.849762	0.424881	−	0.905249i	$-0.360316\pi$
$733$	16863.7	0.849762	0.424881	+	0.905249i	$0.360316\pi$
$734$	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	$0.777780\pi$
$740$	1803.24	0.0895789
$741$	0	0
$742$	−3244.27	−0.160514
$743$	−28510.9	−1.40776	−0.703879	−	0.710320i	$-0.748550\pi$
$743$	−28510.9	−1.40776	−0.703879	+	0.710320i	$0.748550\pi$
$744$	0	0
$745$	12514.4	0.615428
$746$	−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
$755$	−1976.06	−0.0952532
$756$	0	0
$757$	−11272.1	−0.541203	−0.270601	−	0.962691i	$-0.587223\pi$
$757$	−11272.1	−0.541203	−0.270601	+	0.962691i	$0.587223\pi$
$758$	8354.56	0.400332
$759$	0	0
$760$	9604.69	0.458419
$761$	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	$0.632933\pi$
$770$	560.000	0.0262091
$771$	0	0
$772$	−8122.33	−0.378665
$773$	−5348.18	−0.248849	−0.124425	−	0.992229i	$-0.539709\pi$
$773$	−5348.18	−0.248849	−0.124425	+	0.992229i	$0.539709\pi$
$774$	0	0
$775$	−3552.70	−0.164667
$776$	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
$785$	−8661.75	−0.393823
$786$	0	0
$787$	−21615.1	−0.979030	−0.489515	−	0.871995i	$-0.662826\pi$
$787$	−21615.1	−0.979030	−0.489515	+	0.871995i	$0.662826\pi$
$788$	−26870.2	−1.21473
$789$	0	0
$790$	2937.20	0.132279
$791$	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	$-0.176393\pi$
$797$	38266.0	1.70069	0.850345	+	0.526225i	$0.176393\pi$
$798$	0	0
$799$	26336.8	1.16612
$800$	4681.37	0.206889
$801$	0	0
$802$	−5954.09	−0.262152
$803$	−5705.61	−0.250743
$804$	0	0
$805$	424.839	0.0186008
$806$	−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
$815$	−16516.3	−0.709866
$816$	0	0
$817$	−28955.7	−1.23994
$818$	−6400.53	−0.273581
$819$	0	0
$820$	5428.10	0.231167
$821$	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	$-0.460403\pi$
$823$	5858.93	0.248153	0.124076	+	0.992273i	$0.460403\pi$
$824$	−11330.8	−0.479037
$825$	0	0
$826$	−3104.37	−0.130768
$827$	−16375.7	−0.688560	−0.344280	−	0.938867i	$-0.611877\pi$
$827$	−16375.7	−0.688560	−0.344280	+	0.938867i	$0.611877\pi$
$828$	0	0
$829$	−15152.9	−0.634841	−0.317421	−	0.948285i	$-0.602817\pi$
$829$	−15152.9	−0.634841	−0.317421	+	0.948285i	$0.602817\pi$
$830$	5875.08	0.245695
$831$	0	0
$832$	6905.94	0.287765
$833$	4047.42	0.168349
$834$	0	0
$835$	3009.71	0.124737
$836$	−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
$845$	5083.96	0.206975
$846$	0	0
$847$	8582.11	0.348152
$848$	3390.42	0.137297
$849$	0	0
$850$	3224.62	0.130122
$851$	787.125	0.0317066
$852$	0	0
$853$	−4227.64	−0.169697	−0.0848486	−	0.996394i	$-0.527041\pi$
$853$	−4227.64	−0.169697	−0.0848486	+	0.996394i	$0.527041\pi$
$854$	5405.75	0.216605
$855$	0	0
$856$	6928.15	0.276635
$857$	−32233.1	−1.28479	−0.642394	−	0.766375i	$-0.722059\pi$
$857$	−32233.1	−1.28479	−0.642394	+	0.766375i	$0.722059\pi$
$858$	0	0
$859$	37392.3	1.48523	0.742613	−	0.669720i	$-0.233586\pi$
$859$	37392.3	1.48523	0.742613	+	0.669720i	$0.233586\pi$
$860$	−8876.73	−0.351970
$861$	0	0
$862$	−17963.8	−0.709801
$863$	−8372.73	−0.330256	−0.165128	−	0.986272i	$-0.552804\pi$
$863$	−8372.73	−0.330256	−0.165128	+	0.986272i	$0.552804\pi$
$864$	0	0
$865$	18705.8	0.735280
$866$	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
$875$	875.000	0.0338062
$876$	0	0
$877$	38583.0	1.48558	0.742790	−	0.669524i	$-0.233502\pi$
$877$	38583.0	1.48558	0.742790	+	0.669524i	$0.233502\pi$
$878$	−4285.23	−0.164715
$879$	0	0
$880$	−585.227	−0.0224182
$881$	−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	$-0.189944\pi$
$883$	43408.1	1.65436	0.827180	+	0.561937i	$0.189944\pi$
$884$	−15781.8	−0.600453
$885$	0	0
$886$	−12773.9	−0.484366
$887$	−19523.6	−0.739050	−0.369525	−	0.929221i	$-0.620480\pi$
$887$	−19523.6	−0.739050	−0.369525	+	0.929221i	$0.620480\pi$
$888$	0	0
$889$	−11970.7	−0.451613
$890$	−7380.67	−0.277978
$891$	0	0
$892$	13544.1	0.508398
$893$	28922.0	1.08381
$894$	0	0
$895$	4841.76	0.180829
$896$	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
$905$	9478.23	0.348140
$906$	0	0
$907$	44114.7	1.61500	0.807499	−	0.589869i	$-0.200820\pi$
$907$	44114.7	1.61500	0.807499	+	0.589869i	$0.200820\pi$
$908$	−19512.5	−0.713156
$909$	0	0
$910$	1877.60	0.0683978
$911$	−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	$0.819451\pi$
$920$	1285.26	0.0460586
$921$	0	0
$922$	−27757.9	−0.991494
$923$	25455.5	0.907775
$924$	0	0
$925$	1621.16	0.0576255
$926$	−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
$935$	−4231.70	−0.148012
$936$	0	0
$937$	−23431.0	−0.816924	−0.408462	−	0.912775i	$-0.633935\pi$
$937$	−23431.0	−0.816924	−0.408462	+	0.912775i	$0.633935\pi$
$938$	−6001.40	−0.208905
$939$	0	0
$940$	8866.41	0.307649
$941$	−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	$-0.704241\pi$
$947$	−34884.0	−1.19702	−0.598511	+	0.801115i	$0.704241\pi$
$948$	0	0
$949$	−19130.1	−0.654363
$950$	3541.15	0.120937
$951$	0	0
$952$	12244.6	0.416860
$953$	−34486.4	−1.17222	−0.586108	−	0.810233i	$-0.699341\pi$
$953$	−34486.4	−1.17222	−0.586108	+	0.810233i	$0.699341\pi$
$954$	0	0
$955$	−13649.4	−0.462496
$956$	−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
$965$	−7302.21	−0.243592
$966$	0	0
$967$	−46341.6	−1.54110	−0.770550	−	0.637379i	$-0.780019\pi$
$967$	−46341.6	−1.54110	−0.770550	+	0.637379i	$0.780019\pi$
$968$	25963.4	0.862083
$969$	0	0
$970$	1410.62	0.0466930
$971$	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	$-0.505056\pi$
$977$	−970.127	−0.0317678	−0.0158839	+	0.999874i	$0.505056\pi$
$978$	0	0
$979$	9685.73	0.316198
$980$	1362.58	0.0444143
$981$	0	0
$982$	4211.26	0.136850
$983$	−18342.2	−0.595142	−0.297571	−	0.954700i	$-0.596176\pi$
$983$	−18342.2	−0.595142	−0.297571	+	0.954700i	$0.596176\pi$
$984$	0	0
$985$	−24157.1	−0.781430
$986$	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
$995$	4365.35	0.139086
$996$	0	0
$997$	31492.9	1.00039	0.500196	−	0.865912i	$-0.333261\pi$
$997$	31492.9	1.00039	0.500196	+	0.865912i	$0.333261\pi$
$998$	−31315.7	−0.993268
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.4.a.h.1.2	✓	2
3.2	odd	2		315.4.a.j.1.1	yes	2
5.4	even	2		1575.4.a.u.1.1		2
7.6	odd	2		2205.4.a.y.1.2		2
15.14	odd	2		1575.4.a.r.1.2		2
21.20	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	1.1	even	1	trivial
315.4.a.j.1.1	yes	2	3.2	odd	2
1575.4.a.r.1.2		2	15.14	odd	2
1575.4.a.u.1.1		2	5.4	even	2
2205.4.a.y.1.2		2	7.6	odd	2
2205.4.a.ba.1.1		2	21.20	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:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	315.a (trivial)

\(f(q)\)	\(=\)	\(q+1.56155 q^{2} -5.56155 q^{4} -5.00000 q^{5} -7.00000 q^{7} -21.1771 q^{8} +O(q^{10})\)	\(q+1.56155 q^{2} -5.56155 q^{4} -5.00000 q^{5} -7.00000 q^{7} -21.1771 q^{8} -7.80776 q^{10} +10.2462 q^{11} +34.3542 q^{13} -10.9309 q^{14} +11.4233 q^{16} +82.6004 q^{17} +90.7083 q^{19} +27.8078 q^{20} +16.0000 q^{22} +12.1383 q^{23} +25.0000 q^{25} +53.6458 q^{26} +38.9309 q^{28} +105.153 q^{29} -142.108 q^{31} +187.255 q^{32} +128.985 q^{34} +35.0000 q^{35} +64.8466 q^{37} +141.646 q^{38} +105.885 q^{40} +195.201 q^{41} -319.218 q^{43} -56.9848 q^{44} +18.9545 q^{46} +318.847 q^{47} +49.0000 q^{49} +39.0388 q^{50} -191.062 q^{52} +296.799 q^{53} -51.2311 q^{55} +148.240 q^{56} +164.203 q^{58} +284.000 q^{59} -494.540 q^{61} -221.909 q^{62} +201.022 q^{64} -171.771 q^{65} +549.032 q^{67} -459.386 q^{68} +54.6543 q^{70} +740.972 q^{71} -556.850 q^{73} +101.261 q^{74} -504.479 q^{76} -71.7235 q^{77} -376.189 q^{79} -57.1165 q^{80} +304.816 q^{82} -752.466 q^{83} -413.002 q^{85} -498.475 q^{86} -216.985 q^{88} +945.299 q^{89} -240.479 q^{91} -67.5076 q^{92} +497.896 q^{94} -453.542 q^{95} -180.668 q^{97} +76.5161 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - 7 q^{4} - 10 q^{5} - 14 q^{7} + 3 q^{8}+O(q^{10}) \) 2 * q - q^2 - 7 * q^4 - 10 * q^5 - 14 * q^7 + 3 * q^8	\( 2 q - q^{2} - 7 q^{4} - 10 q^{5} - 14 q^{7} + 3 q^{8} + 5 q^{10} + 4 q^{11} - 22 q^{13} + 7 q^{14} - 39 q^{16} + 58 q^{17} + 35 q^{20} + 32 q^{22} + 82 q^{23} + 50 q^{25} + 198 q^{26} + 49 q^{28} + 334 q^{29} - 210 q^{31} + 123 q^{32} + 192 q^{34} + 70 q^{35} + 6 q^{37} + 374 q^{38} - 15 q^{40} + 176 q^{41} + 46 q^{43} - 48 q^{44} - 160 q^{46} + 514 q^{47} + 98 q^{49} - 25 q^{50} - 110 q^{52} + 808 q^{53} - 20 q^{55} - 21 q^{56} - 422 q^{58} + 568 q^{59} - 618 q^{61} - 48 q^{62} + 769 q^{64} + 110 q^{65} + 694 q^{67} - 424 q^{68} - 35 q^{70} + 814 q^{71} + 82 q^{73} + 252 q^{74} - 374 q^{76} - 28 q^{77} + 600 q^{79} + 195 q^{80} + 354 q^{82} - 268 q^{83} - 290 q^{85} - 1434 q^{86} - 368 q^{88} - 72 q^{89} + 154 q^{91} - 168 q^{92} - 2 q^{94} + 1626 q^{97} - 49 q^{98}+O(q^{100}) \) 2 * q - q^2 - 7 * q^4 - 10 * q^5 - 14 * q^7 + 3 * q^8 + 5 * q^10 + 4 * q^11 - 22 * q^13 + 7 * q^14 - 39 * q^16 + 58 * q^17 + 35 * q^20 + 32 * q^22 + 82 * q^23 + 50 * q^25 + 198 * q^26 + 49 * q^28 + 334 * q^29 - 210 * q^31 + 123 * q^32 + 192 * q^34 + 70 * q^35 + 6 * q^37 + 374 * q^38 - 15 * q^40 + 176 * q^41 + 46 * q^43 - 48 * q^44 - 160 * q^46 + 514 * q^47 + 98 * q^49 - 25 * q^50 - 110 * q^52 + 808 * q^53 - 20 * q^55 - 21 * q^56 - 422 * q^58 + 568 * q^59 - 618 * q^61 - 48 * q^62 + 769 * q^64 + 110 * q^65 + 694 * q^67 - 424 * q^68 - 35 * q^70 + 814 * q^71 + 82 * q^73 + 252 * q^74 - 374 * q^76 - 28 * q^77 + 600 * q^79 + 195 * q^80 + 354 * q^82 - 268 * q^83 - 290 * q^85 - 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

L-functions

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