LMFDB - Embedded newform 304.4.a.j.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [304,4,Mod(1,304)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("304.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(304, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 304 = 2^{4} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	304.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,5,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$17.9365806417$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.3221.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 9x + 2 $ x^3 - x^2 - 9*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 152)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.218090$ of defining polynomial
Character	$\chi$	$=$	304.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-4.73435 q^{3} -18.0754 q^{5} -0.213413 q^{7} -4.58596 q^{9} +3.39329 q^{11} -90.7156 q^{13} +85.5752 q^{15} -2.59392 q^{17} -19.0000 q^{19} +1.01037 q^{21} -26.6112 q^{23} +201.720 q^{25} +149.539 q^{27} +60.1034 q^{29} +176.070 q^{31} -16.0650 q^{33} +3.85753 q^{35} -154.115 q^{37} +429.479 q^{39} +434.137 q^{41} +365.511 q^{43} +82.8931 q^{45} -204.021 q^{47} -342.954 q^{49} +12.2805 q^{51} -135.726 q^{53} -61.3351 q^{55} +89.9526 q^{57} -759.895 q^{59} +284.941 q^{61} +0.978705 q^{63} +1639.72 q^{65} -590.922 q^{67} +125.986 q^{69} +972.291 q^{71} +368.462 q^{73} -955.013 q^{75} -0.724174 q^{77} -204.854 q^{79} -584.148 q^{81} +782.229 q^{83} +46.8861 q^{85} -284.550 q^{87} +213.620 q^{89} +19.3599 q^{91} -833.576 q^{93} +343.433 q^{95} -1219.54 q^{97} -15.5615 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 5 q^{3} + 2 q^{5} + 35 q^{7} + 48 q^{9} + 28 q^{11} - 109 q^{13} + 228 q^{15} - 123 q^{17} - 57 q^{19} + 25 q^{21} + 193 q^{23} + 187 q^{25} + 719 q^{27} - 297 q^{29} + 140 q^{31} + 30 q^{33} + 246 q^{35}+ \cdots - 86 q^{99}+O(q^{100}) $ 3 * q + 5 * q^3 + 2 * q^5 + 35 * q^7 + 48 * q^9 + 28 * q^11 - 109 * q^13 + 228 * q^15 - 123 * q^17 - 57 * q^19 + 25 * q^21 + 193 * q^23 + 187 * q^25 + 719 * q^27 - 297 * q^29 + 140 * q^31 + 30 * q^33 + 246 * q^35 + 38 * q^37 - 57 * q^39 + 736 * q^41 + 514 * q^43 + 1046 * q^45 - 134 * q^47 - 42 * q^49 - 929 * q^51 + 311 * q^53 + 130 * q^55 - 95 * q^57 - 199 * q^59 + 56 * q^61 - 508 * q^63 + 1156 * q^65 + 509 * q^67 + 621 * q^69 + 874 * q^71 + 203 * q^73 - 129 * q^75 + 616 * q^77 - 242 * q^79 + 3531 * q^81 + 62 * q^83 - 1430 * q^85 - 3237 * q^87 + 1764 * q^89 + 687 * q^91 - 2592 * q^93 - 38 * q^95 - 2178 * q^97 - 86 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	−4.73435	−0.911125	−0.455563	−	0.890204i	$-0.650562\pi$
$3$	−4.73435	−0.911125	−0.455563	+	0.890204i	$0.650562\pi$
$4$	0	0
$5$	−18.0754	−1.61671	−0.808356	−	0.588693i	$-0.799643\pi$
$5$	−18.0754	−1.61671	−0.808356	+	0.588693i	$0.799643\pi$
$6$	0	0
$7$	−0.213413	−0.0115232	−0.00576162	−	0.999983i	$-0.501834\pi$
$7$	−0.213413	−0.0115232	−0.00576162	+	0.999983i	$0.501834\pi$
$8$	0	0
$9$	−4.58596	−0.169850
$10$	0	0
$11$	3.39329	0.0930106	0.0465053	−	0.998918i	$-0.485192\pi$
$11$	3.39329	0.0930106	0.0465053	+	0.998918i	$0.485192\pi$
$12$	0	0
$13$	−90.7156	−1.93538	−0.967692	−	0.252135i	$-0.918867\pi$
$13$	−90.7156	−1.93538	−0.967692	+	0.252135i	$0.918867\pi$
$14$	0	0
$15$	85.5752	1.47303
$16$	0	0
$17$	−2.59392	−0.0370069	−0.0185035	−	0.999829i	$-0.505890\pi$
$17$	−2.59392	−0.0370069	−0.0185035	+	0.999829i	$0.505890\pi$
$18$	0	0
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	0	0
$21$	1.01037	0.0104991
$22$	0	0
$23$	−26.6112	−0.241253	−0.120626	−	0.992698i	$-0.538490\pi$
$23$	−26.6112	−0.241253	−0.120626	+	0.992698i	$0.538490\pi$
$24$	0	0
$25$	201.720	1.61376
$26$	0	0
$27$	149.539	1.06588
$28$	0	0
$29$	60.1034	0.384859	0.192430	−	0.981311i	$-0.438363\pi$
$29$	60.1034	0.384859	0.192430	+	0.981311i	$0.438363\pi$
$30$	0	0
$31$	176.070	1.02010	0.510050	−	0.860145i	$-0.329627\pi$
$31$	176.070	1.02010	0.510050	+	0.860145i	$0.329627\pi$
$32$	0	0
$33$	−16.0650	−0.0847443
$34$	0	0
$35$	3.85753	0.0186298
$36$	0	0
$37$	−154.115	−0.684768	−0.342384	−	0.939560i	$-0.611234\pi$
$37$	−154.115	−0.684768	−0.342384	+	0.939560i	$0.611234\pi$
$38$	0	0
$39$	429.479	1.76338
$40$	0	0
$41$	434.137	1.65368	0.826839	−	0.562439i	$-0.190137\pi$
$41$	434.137	1.65368	0.826839	+	0.562439i	$0.190137\pi$
$42$	0	0
$43$	365.511	1.29628	0.648138	−	0.761523i	$-0.275548\pi$
$43$	365.511	1.29628	0.648138	+	0.761523i	$0.275548\pi$
$44$	0	0
$45$	82.8931	0.274599
$46$	0	0
$47$	−204.021	−0.633180	−0.316590	−	0.948562i	$-0.602538\pi$
$47$	−204.021	−0.633180	−0.316590	+	0.948562i	$0.602538\pi$
$48$	0	0
$49$	−342.954	−0.999867
$50$	0	0
$51$	12.2805	0.0337180
$52$	0	0
$53$	−135.726	−0.351763	−0.175881	−	0.984411i	$-0.556278\pi$
$53$	−135.726	−0.351763	−0.175881	+	0.984411i	$0.556278\pi$
$54$	0	0
$55$	−61.3351	−0.150371
$56$	0	0
$57$	89.9526	0.209027
$58$	0	0
$59$	−759.895	−1.67678	−0.838389	−	0.545072i	$-0.816502\pi$
$59$	−759.895	−1.67678	−0.838389	+	0.545072i	$0.816502\pi$
$60$	0	0
$61$	284.941	0.598082	0.299041	−	0.954240i	$-0.403333\pi$
$61$	284.941	0.598082	0.299041	+	0.954240i	$0.403333\pi$
$62$	0	0
$63$	0.978705	0.00195723
$64$	0	0
$65$	1639.72	3.12896
$66$	0	0
$67$	−590.922	−1.07750	−0.538750	−	0.842465i	$-0.681103\pi$
$67$	−590.922	−1.07750	−0.538750	+	0.842465i	$0.681103\pi$
$68$	0	0
$69$	125.986	0.219811
$70$	0	0
$71$	972.291	1.62521	0.812604	−	0.582817i	$-0.198050\pi$
$71$	972.291	1.62521	0.812604	+	0.582817i	$0.198050\pi$
$72$	0	0
$73$	368.462	0.590756	0.295378	−	0.955380i	$-0.404554\pi$
$73$	368.462	0.590756	0.295378	+	0.955380i	$0.404554\pi$
$74$	0	0
$75$	−955.013	−1.47034
$76$	0	0
$77$	−0.724174	−0.00107178
$78$	0	0
$79$	−204.854	−0.291745	−0.145873	−	0.989303i	$-0.546599\pi$
$79$	−204.854	−0.291745	−0.145873	+	0.989303i	$0.546599\pi$
$80$	0	0
$81$	−584.148	−0.801300
$82$	0	0
$83$	782.229	1.03447	0.517234	−	0.855844i	$-0.326962\pi$
$83$	782.229	1.03447	0.517234	+	0.855844i	$0.326962\pi$
$84$	0	0
$85$	46.8861	0.0598296
$86$	0	0
$87$	−284.550	−0.350655
$88$	0	0
$89$	213.620	0.254423	0.127211	−	0.991876i	$-0.459397\pi$
$89$	213.620	0.254423	0.127211	+	0.991876i	$0.459397\pi$
$90$	0	0
$91$	19.3599	0.0223019
$92$	0	0
$93$	−833.576	−0.929438
$94$	0	0
$95$	343.433	0.370899
$96$	0	0
$97$	−1219.54	−1.27655	−0.638274	−	0.769809i	$-0.720351\pi$
$97$	−1219.54	−1.27655	−0.638274	+	0.769809i	$0.720351\pi$
$98$	0	0
$99$	−15.5615	−0.0157979
$100$	0	0
$101$	−53.9766	−0.0531770	−0.0265885	−	0.999646i	$-0.508464\pi$
$101$	−53.9766	−0.0531770	−0.0265885	+	0.999646i	$0.508464\pi$
$102$	0	0
$103$	−1987.50	−1.90130	−0.950652	−	0.310260i	$-0.899584\pi$
$103$	−1987.50	−1.90130	−0.950652	+	0.310260i	$0.899584\pi$
$104$	0	0
$105$	−18.2629	−0.0169741
$106$	0	0
$107$	1076.92	0.972991	0.486496	−	0.873683i	$-0.338275\pi$
$107$	1076.92	0.972991	0.486496	+	0.873683i	$0.338275\pi$
$108$	0	0
$109$	−1200.57	−1.05499	−0.527494	−	0.849559i	$-0.676868\pi$
$109$	−1200.57	−1.05499	−0.527494	+	0.849559i	$0.676868\pi$
$110$	0	0
$111$	729.636	0.623910
$112$	0	0
$113$	1402.72	1.16776	0.583880	−	0.811840i	$-0.301534\pi$
$113$	1402.72	1.16776	0.583880	+	0.811840i	$0.301534\pi$
$114$	0	0
$115$	481.007	0.390036
$116$	0	0
$117$	416.018	0.328726
$118$	0	0
$119$	0.553577	0.000426440 0
$120$	0	0
$121$	−1319.49	−0.991349
$122$	0	0
$123$	−2055.35	−1.50671
$124$	0	0
$125$	−1386.75	−0.992275
$126$	0	0
$127$	1055.22	0.737289	0.368644	−	0.929570i	$-0.379822\pi$
$127$	1055.22	0.737289	0.368644	+	0.929570i	$0.379822\pi$
$128$	0	0
$129$	−1730.45	−1.18107
$130$	0	0
$131$	1026.98	0.684944	0.342472	−	0.939528i	$-0.388736\pi$
$131$	1026.98	0.684944	0.342472	+	0.939528i	$0.388736\pi$
$132$	0	0
$133$	4.05485	0.00264361
$134$	0	0
$135$	−2702.98	−1.72322
$136$	0	0
$137$	−1728.02	−1.07763	−0.538813	−	0.842425i	$-0.681127\pi$
$137$	−1728.02	−1.07763	−0.538813	+	0.842425i	$0.681127\pi$
$138$	0	0
$139$	−624.188	−0.380885	−0.190442	−	0.981698i	$-0.560992\pi$
$139$	−624.188	−0.380885	−0.190442	+	0.981698i	$0.560992\pi$
$140$	0	0
$141$	965.904	0.576906
$142$	0	0
$143$	−307.825	−0.180011
$144$	0	0
$145$	−1086.39	−0.622207
$146$	0	0
$147$	1623.67	0.911004
$148$	0	0
$149$	57.5590	0.0316471	0.0158235	−	0.999875i	$-0.494963\pi$
$149$	57.5590	0.0316471	0.0158235	+	0.999875i	$0.494963\pi$
$150$	0	0
$151$	2567.35	1.38363	0.691814	−	0.722076i	$-0.256812\pi$
$151$	2567.35	1.38363	0.691814	+	0.722076i	$0.256812\pi$
$152$	0	0
$153$	11.8956	0.00628564
$154$	0	0
$155$	−3182.53	−1.64921
$156$	0	0
$157$	−3015.67	−1.53297	−0.766486	−	0.642261i	$-0.777997\pi$
$157$	−3015.67	−1.53297	−0.766486	+	0.642261i	$0.777997\pi$
$158$	0	0
$159$	642.575	0.320500
$160$	0	0
$161$	5.67918	0.00278001
$162$	0	0
$163$	3331.49	1.60088	0.800438	−	0.599416i	$-0.204600\pi$
$163$	3331.49	1.60088	0.800438	+	0.599416i	$0.204600\pi$
$164$	0	0
$165$	290.382	0.137007
$166$	0	0
$167$	−2465.56	−1.14246	−0.571229	−	0.820791i	$-0.693533\pi$
$167$	−2465.56	−1.14246	−0.571229	+	0.820791i	$0.693533\pi$
$168$	0	0
$169$	6032.33	2.74571
$170$	0	0
$171$	87.1333	0.0389664
$172$	0	0
$173$	−549.111	−0.241319	−0.120659	−	0.992694i	$-0.538501\pi$
$173$	−549.111	−0.241319	−0.120659	+	0.992694i	$0.538501\pi$
$174$	0	0
$175$	−43.0498	−0.0185958
$176$	0	0
$177$	3597.61	1.52776
$178$	0	0
$179$	4186.74	1.74822	0.874110	−	0.485728i	$-0.161445\pi$
$179$	4186.74	1.74822	0.874110	+	0.485728i	$0.161445\pi$
$180$	0	0
$181$	−3954.07	−1.62378	−0.811890	−	0.583811i	$-0.801561\pi$
$181$	−3954.07	−1.62378	−0.811890	+	0.583811i	$0.801561\pi$
$182$	0	0
$183$	−1349.01	−0.544928
$184$	0	0
$185$	2785.70	1.10707
$186$	0	0
$187$	−8.80193	−0.00344204
$188$	0	0
$189$	−31.9136	−0.0122824
$190$	0	0
$191$	1623.62	0.615084	0.307542	−	0.951535i	$-0.400494\pi$
$191$	1623.62	0.615084	0.307542	+	0.951535i	$0.400494\pi$
$192$	0	0
$193$	1817.44	0.677833	0.338917	−	0.940816i	$-0.389940\pi$
$193$	1817.44	0.677833	0.338917	+	0.940816i	$0.389940\pi$
$194$	0	0
$195$	−7763.01	−2.85088
$196$	0	0
$197$	151.771	0.0548894	0.0274447	−	0.999623i	$-0.491263\pi$
$197$	151.771	0.0548894	0.0274447	+	0.999623i	$0.491263\pi$
$198$	0	0
$199$	−1229.91	−0.438119	−0.219060	−	0.975711i	$-0.570299\pi$
$199$	−1229.91	−0.438119	−0.219060	+	0.975711i	$0.570299\pi$
$200$	0	0
$201$	2797.63	0.981738
$202$	0	0
$203$	−12.8269	−0.00443483
$204$	0	0
$205$	−7847.20	−2.67352
$206$	0	0
$207$	122.038	0.0409769
$208$	0	0
$209$	−64.4726	−0.0213381
$210$	0	0
$211$	2998.12	0.978194	0.489097	−	0.872229i	$-0.337326\pi$
$211$	2998.12	0.978194	0.489097	+	0.872229i	$0.337326\pi$
$212$	0	0
$213$	−4603.16	−1.48077
$214$	0	0
$215$	−6606.75	−2.09571
$216$	0	0
$217$	−37.5757	−0.0117549
$218$	0	0
$219$	−1744.43	−0.538253
$220$	0	0
$221$	235.309	0.0716226
$222$	0	0
$223$	781.297	0.234617	0.117308	−	0.993096i	$-0.462573\pi$
$223$	781.297	0.234617	0.117308	+	0.993096i	$0.462573\pi$
$224$	0	0
$225$	−925.080	−0.274098
$226$	0	0
$227$	2695.16	0.788035	0.394017	−	0.919103i	$-0.371085\pi$
$227$	2695.16	0.788035	0.394017	+	0.919103i	$0.371085\pi$
$228$	0	0
$229$	3952.95	1.14069	0.570346	−	0.821405i	$-0.306809\pi$
$229$	3952.95	1.14069	0.570346	+	0.821405i	$0.306809\pi$
$230$	0	0
$231$	3.42849	0.000976529 0
$232$	0	0
$233$	4315.28	1.21332	0.606659	−	0.794962i	$-0.292509\pi$
$233$	4315.28	1.21332	0.606659	+	0.794962i	$0.292509\pi$
$234$	0	0
$235$	3687.75	1.02367
$236$	0	0
$237$	969.849	0.265816
$238$	0	0
$239$	−3808.38	−1.03073	−0.515363	−	0.856972i	$-0.672343\pi$
$239$	−3808.38	−1.03073	−0.515363	+	0.856972i	$0.672343\pi$
$240$	0	0
$241$	4274.22	1.14243	0.571217	−	0.820799i	$-0.306471\pi$
$241$	4274.22	1.14243	0.571217	+	0.820799i	$0.306471\pi$
$242$	0	0
$243$	−1271.99	−0.335795
$244$	0	0
$245$	6199.04	1.61650
$246$	0	0
$247$	1723.60	0.444008
$248$	0	0
$249$	−3703.35	−0.942530
$250$	0	0
$251$	−2552.20	−0.641806	−0.320903	−	0.947112i	$-0.603986\pi$
$251$	−2552.20	−0.641806	−0.320903	+	0.947112i	$0.603986\pi$
$252$	0	0
$253$	−90.2995	−0.0224390
$254$	0	0
$255$	−221.975	−0.0545123
$256$	0	0
$257$	2099.29	0.509533	0.254766	−	0.967003i	$-0.418001\pi$
$257$	2099.29	0.509533	0.254766	+	0.967003i	$0.418001\pi$
$258$	0	0
$259$	32.8903	0.00789075
$260$	0	0
$261$	−275.632	−0.0653685
$262$	0	0
$263$	7691.71	1.80339	0.901695	−	0.432374i	$-0.142324\pi$
$263$	7691.71	1.80339	0.901695	+	0.432374i	$0.142324\pi$
$264$	0	0
$265$	2453.31	0.568700
$266$	0	0
$267$	−1011.35	−0.231811
$268$	0	0
$269$	1460.77	0.331096	0.165548	−	0.986202i	$-0.447061\pi$
$269$	1460.77	0.331096	0.165548	+	0.986202i	$0.447061\pi$
$270$	0	0
$271$	−3180.10	−0.712832	−0.356416	−	0.934327i	$-0.616001\pi$
$271$	−3180.10	−0.712832	−0.356416	+	0.934327i	$0.616001\pi$
$272$	0	0
$273$	−91.6566	−0.0203198
$274$	0	0
$275$	684.495	0.150097
$276$	0	0
$277$	−1227.66	−0.266293	−0.133146	−	0.991096i	$-0.542508\pi$
$277$	−1227.66	−0.266293	−0.133146	+	0.991096i	$0.542508\pi$
$278$	0	0
$279$	−807.449	−0.173264
$280$	0	0
$281$	6497.13	1.37931	0.689655	−	0.724138i	$-0.257762\pi$
$281$	6497.13	1.37931	0.689655	+	0.724138i	$0.257762\pi$
$282$	0	0
$283$	48.1723	0.0101185	0.00505927	−	0.999987i	$-0.498390\pi$
$283$	48.1723	0.0101185	0.00505927	+	0.999987i	$0.498390\pi$
$284$	0	0
$285$	−1625.93	−0.337936
$286$	0	0
$287$	−92.6506	−0.0190557
$288$	0	0
$289$	−4906.27	−0.998630
$290$	0	0
$291$	5773.71	1.16310
$292$	0	0
$293$	1140.74	0.227449	0.113724	−	0.993512i	$-0.463722\pi$
$293$	1140.74	0.227449	0.113724	+	0.993512i	$0.463722\pi$
$294$	0	0
$295$	13735.4	2.71087
$296$	0	0
$297$	507.429	0.0991382
$298$	0	0
$299$	2414.05	0.466916
$300$	0	0
$301$	−78.0049	−0.0149373
$302$	0	0
$303$	255.544	0.0484509
$304$	0	0
$305$	−5150.43	−0.966927
$306$	0	0
$307$	6728.53	1.25087	0.625435	−	0.780276i	$-0.284921\pi$
$307$	6728.53	1.25087	0.625435	+	0.780276i	$0.284921\pi$
$308$	0	0
$309$	9409.52	1.73233
$310$	0	0
$311$	9758.17	1.77921	0.889606	−	0.456728i	$-0.150979\pi$
$311$	9758.17	1.77921	0.889606	+	0.456728i	$0.150979\pi$
$312$	0	0
$313$	−1660.82	−0.299920	−0.149960	−	0.988692i	$-0.547915\pi$
$313$	−1660.82	−0.299920	−0.149960	+	0.988692i	$0.547915\pi$
$314$	0	0
$315$	−17.6905	−0.00316427
$316$	0	0
$317$	6578.84	1.16563	0.582814	−	0.812605i	$-0.301951\pi$
$317$	6578.84	1.16563	0.582814	+	0.812605i	$0.301951\pi$
$318$	0	0
$319$	203.948	0.0357960
$320$	0	0
$321$	−5098.53	−0.886517
$322$	0	0
$323$	49.2845	0.00848997
$324$	0	0
$325$	−18299.2	−3.12325
$326$	0	0
$327$	5683.91	0.961226
$328$	0	0
$329$	43.5407	0.00729629
$330$	0	0
$331$	5522.33	0.917024	0.458512	−	0.888688i	$-0.348383\pi$
$331$	5522.33	0.917024	0.458512	+	0.888688i	$0.348383\pi$
$332$	0	0
$333$	706.767	0.116308
$334$	0	0
$335$	10681.1	1.74201
$336$	0	0
$337$	1350.63	0.218319	0.109159	−	0.994024i	$-0.465184\pi$
$337$	1350.63	0.218319	0.109159	+	0.994024i	$0.465184\pi$
$338$	0	0
$339$	−6640.97	−1.06398
$340$	0	0
$341$	597.457	0.0948800
$342$	0	0
$343$	146.392	0.0230450
$344$	0	0
$345$	−2277.26	−0.355372
$346$	0	0
$347$	−7266.15	−1.12411	−0.562057	−	0.827099i	$-0.689990\pi$
$347$	−7266.15	−1.12411	−0.562057	+	0.827099i	$0.689990\pi$
$348$	0	0
$349$	10360.4	1.58905	0.794524	−	0.607233i	$-0.207721\pi$
$349$	10360.4	1.58905	0.794524	+	0.607233i	$0.207721\pi$
$350$	0	0
$351$	−13565.5	−2.06289
$352$	0	0
$353$	−4014.20	−0.605253	−0.302626	−	0.953109i	$-0.597863\pi$
$353$	−4014.20	−0.605253	−0.302626	+	0.953109i	$0.597863\pi$
$354$	0	0
$355$	−17574.5	−2.62749
$356$	0	0
$357$	−2.62083	−0.000388540 0
$358$	0	0
$359$	−7120.01	−1.04674	−0.523370	−	0.852106i	$-0.675325\pi$
$359$	−7120.01	−1.04674	−0.523370	+	0.852106i	$0.675325\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	6246.90	0.903243
$364$	0	0
$365$	−6660.09	−0.955083
$366$	0	0
$367$	−12447.9	−1.77050	−0.885251	−	0.465114i	$-0.846013\pi$
$367$	−12447.9	−1.77050	−0.885251	+	0.465114i	$0.846013\pi$
$368$	0	0
$369$	−1990.93	−0.280878
$370$	0	0
$371$	28.9658	0.00405345
$372$	0	0
$373$	−2379.11	−0.330257	−0.165128	−	0.986272i	$-0.552804\pi$
$373$	−2379.11	−0.330257	−0.165128	+	0.986272i	$0.552804\pi$
$374$	0	0
$375$	6565.34	0.904087
$376$	0	0
$377$	−5452.32	−0.744850
$378$	0	0
$379$	−10559.0	−1.43108	−0.715541	−	0.698571i	$-0.753820\pi$
$379$	−10559.0	−1.43108	−0.715541	+	0.698571i	$0.753820\pi$
$380$	0	0
$381$	−4995.78	−0.671763
$382$	0	0
$383$	−3074.48	−0.410180	−0.205090	−	0.978743i	$-0.565749\pi$
$383$	−3074.48	−0.410180	−0.205090	+	0.978743i	$0.565749\pi$
$384$	0	0
$385$	13.0897	0.00173277
$386$	0	0
$387$	−1676.22	−0.220173
$388$	0	0
$389$	7437.27	0.969369	0.484684	−	0.874689i	$-0.338934\pi$
$389$	7437.27	0.969369	0.484684	+	0.874689i	$0.338934\pi$
$390$	0	0
$391$	69.0272	0.00892802
$392$	0	0
$393$	−4862.08	−0.624070
$394$	0	0
$395$	3702.82	0.471668
$396$	0	0
$397$	−10690.7	−1.35151	−0.675756	−	0.737125i	$-0.736183\pi$
$397$	−10690.7	−1.35151	−0.675756	+	0.737125i	$0.736183\pi$
$398$	0	0
$399$	−19.1971	−0.00240866
$400$	0	0
$401$	−8536.44	−1.06307	−0.531533	−	0.847038i	$-0.678384\pi$
$401$	−8536.44	−1.06307	−0.531533	+	0.847038i	$0.678384\pi$
$402$	0	0
$403$	−15972.3	−1.97428
$404$	0	0
$405$	10558.7	1.29547
$406$	0	0
$407$	−522.959	−0.0636907
$408$	0	0
$409$	5072.21	0.613214	0.306607	−	0.951836i	$-0.400806\pi$
$409$	5072.21	0.613214	0.306607	+	0.951836i	$0.400806\pi$
$410$	0	0
$411$	8181.05	0.981852
$412$	0	0
$413$	162.172	0.0193219
$414$	0	0
$415$	−14139.1	−1.67244
$416$	0	0
$417$	2955.12	0.347034
$418$	0	0
$419$	13968.1	1.62861	0.814306	−	0.580436i	$-0.197118\pi$
$419$	13968.1	1.62861	0.814306	+	0.580436i	$0.197118\pi$
$420$	0	0
$421$	14983.2	1.73453	0.867264	−	0.497849i	$-0.165877\pi$
$421$	14983.2	1.73453	0.867264	+	0.497849i	$0.165877\pi$
$422$	0	0
$423$	935.630	0.107546
$424$	0	0
$425$	−523.246	−0.0597203
$426$	0	0
$427$	−60.8103	−0.00689184
$428$	0	0
$429$	1457.35	0.164013
$430$	0	0
$431$	9101.07	1.01713	0.508565	−	0.861023i	$-0.330176\pi$
$431$	9101.07	1.01713	0.508565	+	0.861023i	$0.330176\pi$
$432$	0	0
$433$	−8406.65	−0.933020	−0.466510	−	0.884516i	$-0.654489\pi$
$433$	−8406.65	−0.933020	−0.466510	+	0.884516i	$0.654489\pi$
$434$	0	0
$435$	5143.36	0.566909
$436$	0	0
$437$	505.612	0.0553472
$438$	0	0
$439$	−4687.92	−0.509664	−0.254832	−	0.966985i	$-0.582020\pi$
$439$	−4687.92	−0.509664	−0.254832	+	0.966985i	$0.582020\pi$
$440$	0	0
$441$	1572.78	0.169828
$442$	0	0
$443$	−9503.30	−1.01922	−0.509611	−	0.860405i	$-0.670211\pi$
$443$	−9503.30	−1.01922	−0.509611	+	0.860405i	$0.670211\pi$
$444$	0	0
$445$	−3861.26	−0.411329
$446$	0	0
$447$	−272.504	−0.0288345
$448$	0	0
$449$	−2589.35	−0.272159	−0.136079	−	0.990698i	$-0.543450\pi$
$449$	−2589.35	−0.272159	−0.136079	+	0.990698i	$0.543450\pi$
$450$	0	0
$451$	1473.15	0.153810
$452$	0	0
$453$	−12154.7	−1.26066
$454$	0	0
$455$	−349.939	−0.0360558
$456$	0	0
$457$	−11145.8	−1.14087	−0.570436	−	0.821342i	$-0.693226\pi$
$457$	−11145.8	−1.14087	−0.570436	+	0.821342i	$0.693226\pi$
$458$	0	0
$459$	−387.892	−0.0394450
$460$	0	0
$461$	4634.89	0.468261	0.234131	−	0.972205i	$-0.424776\pi$
$461$	4634.89	0.468261	0.234131	+	0.972205i	$0.424776\pi$
$462$	0	0
$463$	−8329.82	−0.836112	−0.418056	−	0.908421i	$-0.637288\pi$
$463$	−8329.82	−0.836112	−0.418056	+	0.908421i	$0.637288\pi$
$464$	0	0
$465$	15067.2	1.50264
$466$	0	0
$467$	−13878.1	−1.37516	−0.687582	−	0.726107i	$-0.741328\pi$
$467$	−13878.1	−1.37516	−0.687582	+	0.726107i	$0.741328\pi$
$468$	0	0
$469$	126.111	0.0124163
$470$	0	0
$471$	14277.2	1.39673
$472$	0	0
$473$	1240.29	0.120567
$474$	0	0
$475$	−3832.68	−0.370222
$476$	0	0
$477$	622.435	0.0597471
$478$	0	0
$479$	−15896.9	−1.51639	−0.758194	−	0.652029i	$-0.773918\pi$
$479$	−15896.9	−1.51639	−0.758194	+	0.652029i	$0.773918\pi$
$480$	0	0
$481$	13980.7	1.32529
$482$	0	0
$483$	−26.8872	−0.00253294
$484$	0	0
$485$	22043.6	2.06381
$486$	0	0
$487$	10987.2	1.02233	0.511166	−	0.859482i	$-0.329214\pi$
$487$	10987.2	1.02233	0.511166	+	0.859482i	$0.329214\pi$
$488$	0	0
$489$	−15772.4	−1.45860
$490$	0	0
$491$	−4422.00	−0.406440	−0.203220	−	0.979133i	$-0.565141\pi$
$491$	−4422.00	−0.406440	−0.203220	+	0.979133i	$0.565141\pi$
$492$	0	0
$493$	−155.903	−0.0142425
$494$	0	0
$495$	281.281	0.0255406
$496$	0	0
$497$	−207.500	−0.0187277
$498$	0	0
$499$	−9203.68	−0.825678	−0.412839	−	0.910804i	$-0.635463\pi$
$499$	−9203.68	−0.825678	−0.412839	+	0.910804i	$0.635463\pi$
$500$	0	0
$501$	11672.8	1.04092
$502$	0	0
$503$	5394.92	0.478226	0.239113	−	0.970992i	$-0.423143\pi$
$503$	5394.92	0.478226	0.239113	+	0.970992i	$0.423143\pi$
$504$	0	0
$505$	975.649	0.0859719
$506$	0	0
$507$	−28559.1	−2.50169
$508$	0	0
$509$	14409.8	1.25482	0.627410	−	0.778689i	$-0.284115\pi$
$509$	14409.8	1.25482	0.627410	+	0.778689i	$0.284115\pi$
$510$	0	0
$511$	−78.6347	−0.00680742
$512$	0	0
$513$	−2841.24	−0.244530
$514$	0	0
$515$	35924.9	3.07386
$516$	0	0
$517$	−692.302	−0.0588924
$518$	0	0
$519$	2599.68	0.219872
$520$	0	0
$521$	−2249.98	−0.189201	−0.0946003	−	0.995515i	$-0.530157\pi$
$521$	−2249.98	−0.189201	−0.0946003	+	0.995515i	$0.530157\pi$
$522$	0	0
$523$	19298.1	1.61348	0.806738	−	0.590909i	$-0.201231\pi$
$523$	19298.1	1.61348	0.806738	+	0.590909i	$0.201231\pi$
$524$	0	0
$525$	203.813	0.0169431
$526$	0	0
$527$	−456.711	−0.0377507
$528$	0	0
$529$	−11458.8	−0.941797
$530$	0	0
$531$	3484.85	0.284801
$532$	0	0
$533$	−39383.0	−3.20050
$534$	0	0
$535$	−19465.8	−1.57305
$536$	0	0
$537$	−19821.5	−1.59285
$538$	0	0
$539$	−1163.75	−0.0929982
$540$	0	0
$541$	−307.192	−0.0244126	−0.0122063	−	0.999926i	$-0.503885\pi$
$541$	−307.192	−0.0244126	−0.0122063	+	0.999926i	$0.503885\pi$
$542$	0	0
$543$	18720.0	1.47947
$544$	0	0
$545$	21700.8	1.70561
$546$	0	0
$547$	13468.0	1.05274	0.526371	−	0.850255i	$-0.323552\pi$
$547$	13468.0	1.05274	0.526371	+	0.850255i	$0.323552\pi$
$548$	0	0
$549$	−1306.73	−0.101584
$550$	0	0
$551$	−1141.96	−0.0882928
$552$	0	0
$553$	43.7186	0.00336185
$554$	0	0
$555$	−13188.5	−1.00868
$556$	0	0
$557$	3808.21	0.289693	0.144846	−	0.989454i	$-0.453731\pi$
$557$	3808.21	0.289693	0.144846	+	0.989454i	$0.453731\pi$
$558$	0	0
$559$	−33157.5	−2.50879
$560$	0	0
$561$	41.6714	0.00313613
$562$	0	0
$563$	22451.9	1.68070	0.840351	−	0.542042i	$-0.182349\pi$
$563$	22451.9	1.68070	0.840351	+	0.542042i	$0.182349\pi$
$564$	0	0
$565$	−25354.7	−1.88793
$566$	0	0
$567$	124.665	0.00923358
$568$	0	0
$569$	21111.7	1.55545	0.777723	−	0.628607i	$-0.216375\pi$
$569$	21111.7	1.55545	0.777723	+	0.628607i	$0.216375\pi$
$570$	0	0
$571$	7839.51	0.574559	0.287279	−	0.957847i	$-0.407249\pi$
$571$	7839.51	0.574559	0.287279	+	0.957847i	$0.407249\pi$
$572$	0	0
$573$	−7686.78	−0.560419
$574$	0	0
$575$	−5368.01	−0.389324
$576$	0	0
$577$	−2092.09	−0.150944	−0.0754720	−	0.997148i	$-0.524046\pi$
$577$	−2092.09	−0.150944	−0.0754720	+	0.997148i	$0.524046\pi$
$578$	0	0
$579$	−8604.37	−0.617591
$580$	0	0
$581$	−166.938	−0.0119204
$582$	0	0
$583$	−460.559	−0.0327177
$584$	0	0
$585$	−7519.70	−0.531455
$586$	0	0
$587$	19301.8	1.35719	0.678595	−	0.734513i	$-0.262589\pi$
$587$	19301.8	1.35719	0.678595	+	0.734513i	$0.262589\pi$
$588$	0	0
$589$	−3345.33	−0.234027
$590$	0	0
$591$	−718.535	−0.0500111
$592$	0	0
$593$	−22422.7	−1.55276	−0.776381	−	0.630264i	$-0.782947\pi$
$593$	−22422.7	−1.55276	−0.776381	+	0.630264i	$0.782947\pi$
$594$	0	0
$595$	−10.0061	−0.000689431 0
$596$	0	0
$597$	5822.80	0.399182
$598$	0	0
$599$	6779.52	0.462443	0.231222	−	0.972901i	$-0.425728\pi$
$599$	6779.52	0.462443	0.231222	+	0.972901i	$0.425728\pi$
$600$	0	0
$601$	13065.1	0.886746	0.443373	−	0.896337i	$-0.353782\pi$
$601$	13065.1	0.886746	0.443373	+	0.896337i	$0.353782\pi$
$602$	0	0
$603$	2709.94	0.183014
$604$	0	0
$605$	23850.2	1.60273
$606$	0	0
$607$	17590.3	1.17622	0.588112	−	0.808780i	$-0.299872\pi$
$607$	17590.3	1.17622	0.588112	+	0.808780i	$0.299872\pi$
$608$	0	0
$609$	60.7268	0.00404068
$610$	0	0
$611$	18507.9	1.22545
$612$	0	0
$613$	10723.1	0.706530	0.353265	−	0.935523i	$-0.385071\pi$
$613$	10723.1	0.706530	0.353265	+	0.935523i	$0.385071\pi$
$614$	0	0
$615$	37151.4	2.43591
$616$	0	0
$617$	−2406.12	−0.156997	−0.0784983	−	0.996914i	$-0.525013\pi$
$617$	−2406.12	−0.156997	−0.0784983	+	0.996914i	$0.525013\pi$
$618$	0	0
$619$	−5375.50	−0.349046	−0.174523	−	0.984653i	$-0.555838\pi$
$619$	−5375.50	−0.349046	−0.174523	+	0.984653i	$0.555838\pi$
$620$	0	0
$621$	−3979.40	−0.257146
$622$	0	0
$623$	−45.5893	−0.00293178
$624$	0	0
$625$	−149.015	−0.00953698
$626$	0	0
$627$	305.236	0.0194417
$628$	0	0
$629$	399.763	0.0253412
$630$	0	0
$631$	16557.2	1.04458	0.522291	−	0.852767i	$-0.325077\pi$
$631$	16557.2	1.04458	0.522291	+	0.852767i	$0.325077\pi$
$632$	0	0
$633$	−14194.1	−0.891258
$634$	0	0
$635$	−19073.5	−1.19198
$636$	0	0
$637$	31111.3	1.93513
$638$	0	0
$639$	−4458.89	−0.276042
$640$	0	0
$641$	−1184.92	−0.0730131	−0.0365066	−	0.999333i	$-0.511623\pi$
$641$	−1184.92	−0.0730131	−0.0365066	+	0.999333i	$0.511623\pi$
$642$	0	0
$643$	8126.38	0.498403	0.249202	−	0.968452i	$-0.419832\pi$
$643$	8126.38	0.498403	0.249202	+	0.968452i	$0.419832\pi$
$644$	0	0
$645$	31278.7	1.90945
$646$	0	0
$647$	−16438.6	−0.998868	−0.499434	−	0.866352i	$-0.666459\pi$
$647$	−16438.6	−0.998868	−0.499434	+	0.866352i	$0.666459\pi$
$648$	0	0
$649$	−2578.55	−0.155958
$650$	0	0
$651$	177.896	0.0107101
$652$	0	0
$653$	−10561.8	−0.632947	−0.316473	−	0.948601i	$-0.602499\pi$
$653$	−10561.8	−0.632947	−0.316473	+	0.948601i	$0.602499\pi$
$654$	0	0
$655$	−18563.1	−1.10736
$656$	0	0
$657$	−1689.75	−0.100340
$658$	0	0
$659$	28057.4	1.65851	0.829257	−	0.558868i	$-0.188764\pi$
$659$	28057.4	1.65851	0.829257	+	0.558868i	$0.188764\pi$
$660$	0	0
$661$	16123.6	0.948769	0.474385	−	0.880318i	$-0.342671\pi$
$661$	16123.6	0.948769	0.474385	+	0.880318i	$0.342671\pi$
$662$	0	0
$663$	−1114.03	−0.0652572
$664$	0	0
$665$	−73.2931	−0.00427396
$666$	0	0
$667$	−1599.42	−0.0928483
$668$	0	0
$669$	−3698.93	−0.213765
$670$	0	0
$671$	966.889	0.0556279
$672$	0	0
$673$	−31833.0	−1.82329	−0.911643	−	0.410982i	$-0.865186\pi$
$673$	−31833.0	−1.82329	−0.911643	+	0.410982i	$0.865186\pi$
$674$	0	0
$675$	30165.0	1.72008
$676$	0	0
$677$	−18505.4	−1.05055	−0.525273	−	0.850934i	$-0.676037\pi$
$677$	−18505.4	−1.05055	−0.525273	+	0.850934i	$0.676037\pi$
$678$	0	0
$679$	260.266	0.0147100
$680$	0	0
$681$	−12759.8	−0.717999
$682$	0	0
$683$	−2660.37	−0.149043	−0.0745213	−	0.997219i	$-0.523743\pi$
$683$	−2660.37	−0.149043	−0.0745213	+	0.997219i	$0.523743\pi$
$684$	0	0
$685$	31234.7	1.74221
$686$	0	0
$687$	−18714.6	−1.03931
$688$	0	0
$689$	12312.5	0.680796
$690$	0	0
$691$	−23883.5	−1.31487	−0.657433	−	0.753513i	$-0.728358\pi$
$691$	−23883.5	−1.31487	−0.657433	+	0.753513i	$0.728358\pi$
$692$	0	0
$693$	3.32103	0.000182043 0
$694$	0	0
$695$	11282.5	0.615781
$696$	0	0
$697$	−1126.12	−0.0611975
$698$	0	0
$699$	−20430.0	−1.10549
$700$	0	0
$701$	25949.8	1.39816	0.699079	−	0.715044i	$-0.253594\pi$
$701$	25949.8	1.39816	0.699079	+	0.715044i	$0.253594\pi$
$702$	0	0
$703$	2928.19	0.157097
$704$	0	0
$705$	−17459.1	−0.932692
$706$	0	0
$707$	11.5193	0.000612771 0
$708$	0	0
$709$	12320.8	0.652633	0.326316	−	0.945261i	$-0.394193\pi$
$709$	12320.8	0.652633	0.326316	+	0.945261i	$0.394193\pi$
$710$	0	0
$711$	939.452	0.0495530
$712$	0	0
$713$	−4685.42	−0.246102
$714$	0	0
$715$	5564.06	0.291026
$716$	0	0
$717$	18030.2	0.939120
$718$	0	0
$719$	27252.6	1.41356	0.706781	−	0.707432i	$-0.250146\pi$
$719$	27252.6	1.41356	0.706781	+	0.707432i	$0.250146\pi$
$720$	0	0
$721$	424.159	0.0219092
$722$	0	0
$723$	−20235.6	−1.04090
$724$	0	0
$725$	12124.1	0.621071
$726$	0	0
$727$	−24073.8	−1.22813	−0.614063	−	0.789257i	$-0.710466\pi$
$727$	−24073.8	−1.22813	−0.614063	+	0.789257i	$0.710466\pi$
$728$	0	0
$729$	21794.0	1.10725
$730$	0	0
$731$	−948.105	−0.0479712
$732$	0	0
$733$	10232.2	0.515600	0.257800	−	0.966198i	$-0.417002\pi$
$733$	10232.2	0.515600	0.257800	+	0.966198i	$0.417002\pi$
$734$	0	0
$735$	−29348.4	−1.47283
$736$	0	0
$737$	−2005.17	−0.100219
$738$	0	0
$739$	−5327.77	−0.265203	−0.132602	−	0.991169i	$-0.542333\pi$
$739$	−5327.77	−0.265203	−0.132602	+	0.991169i	$0.542333\pi$
$740$	0	0
$741$	−8160.11	−0.404547
$742$	0	0
$743$	−20416.6	−1.00809	−0.504046	−	0.863677i	$-0.668156\pi$
$743$	−20416.6	−1.00809	−0.504046	+	0.863677i	$0.668156\pi$
$744$	0	0
$745$	−1040.40	−0.0511642
$746$	0	0
$747$	−3587.27	−0.175705
$748$	0	0
$749$	−229.830	−0.0112120
$750$	0	0
$751$	−11444.8	−0.556092	−0.278046	−	0.960568i	$-0.589687\pi$
$751$	−11444.8	−0.556092	−0.278046	+	0.960568i	$0.589687\pi$
$752$	0	0
$753$	12083.0	0.584766
$754$	0	0
$755$	−46405.8	−2.23693
$756$	0	0
$757$	31158.8	1.49602	0.748009	−	0.663689i	$-0.231010\pi$
$757$	31158.8	1.49602	0.748009	+	0.663689i	$0.231010\pi$
$758$	0	0
$759$	427.509	0.0204448
$760$	0	0
$761$	−8933.92	−0.425564	−0.212782	−	0.977100i	$-0.568252\pi$
$761$	−8933.92	−0.425564	−0.212782	+	0.977100i	$0.568252\pi$
$762$	0	0
$763$	256.217	0.0121569
$764$	0	0
$765$	−215.018	−0.0101621
$766$	0	0
$767$	68934.4	3.24521
$768$	0	0
$769$	29688.6	1.39220	0.696099	−	0.717946i	$-0.254918\pi$
$769$	29688.6	1.39220	0.696099	+	0.717946i	$0.254918\pi$
$770$	0	0
$771$	−9938.75	−0.464248
$772$	0	0
$773$	−35704.1	−1.66130	−0.830651	−	0.556793i	$-0.812032\pi$
$773$	−35704.1	−1.66130	−0.830651	+	0.556793i	$0.812032\pi$
$774$	0	0
$775$	35516.8	1.64620
$776$	0	0
$777$	−155.714	−0.00718946
$778$	0	0
$779$	−8248.60	−0.379380
$780$	0	0
$781$	3299.27	0.151161
$782$	0	0
$783$	8987.79	0.410214
$784$	0	0
$785$	54509.4	2.47838
$786$	0	0
$787$	26828.3	1.21515	0.607577	−	0.794261i	$-0.292142\pi$
$787$	26828.3	1.21515	0.607577	+	0.794261i	$0.292142\pi$
$788$	0	0
$789$	−36415.2	−1.64311
$790$	0	0
$791$	−299.360	−0.0134564
$792$	0	0
$793$	−25848.6	−1.15752
$794$	0	0
$795$	−11614.8	−0.518157
$796$	0	0
$797$	−42689.8	−1.89730	−0.948650	−	0.316327i	$-0.897550\pi$
$797$	−42689.8	−1.89730	−0.948650	+	0.316327i	$0.897550\pi$
$798$	0	0
$799$	529.213	0.0234320
$800$	0	0
$801$	−979.651	−0.0432138
$802$	0	0
$803$	1250.30	0.0549465
$804$	0	0
$805$	−102.653	−0.00449448
$806$	0	0
$807$	−6915.81	−0.301670
$808$	0	0
$809$	−21486.1	−0.933759	−0.466880	−	0.884321i	$-0.654622\pi$
$809$	−21486.1	−0.933759	−0.466880	+	0.884321i	$0.654622\pi$
$810$	0	0
$811$	−3074.32	−0.133112	−0.0665560	−	0.997783i	$-0.521201\pi$
$811$	−3074.32	−0.133112	−0.0665560	+	0.997783i	$0.521201\pi$
$812$	0	0
$813$	15055.7	0.649480
$814$	0	0
$815$	−60218.1	−2.58816
$816$	0	0
$817$	−6944.70	−0.297386
$818$	0	0
$819$	−88.7839	−0.00378799
$820$	0	0
$821$	−32460.2	−1.37987	−0.689933	−	0.723873i	$-0.742360\pi$
$821$	−32460.2	−1.37987	−0.689933	+	0.723873i	$0.742360\pi$
$822$	0	0
$823$	25382.1	1.07505	0.537523	−	0.843249i	$-0.319360\pi$
$823$	25382.1	1.07505	0.537523	+	0.843249i	$0.319360\pi$
$824$	0	0
$825$	−3240.64	−0.136757
$826$	0	0
$827$	9830.51	0.413350	0.206675	−	0.978410i	$-0.433736\pi$
$827$	9830.51	0.413350	0.206675	+	0.978410i	$0.433736\pi$
$828$	0	0
$829$	47474.0	1.98895	0.994476	−	0.104968i	$-0.0334739\pi$
$829$	47474.0	1.98895	0.994476	+	0.104968i	$0.0334739\pi$
$830$	0	0
$831$	5812.18	0.242626
$832$	0	0
$833$	889.596	0.0370020
$834$	0	0
$835$	44565.9	1.84703
$836$	0	0
$837$	26329.3	1.08730
$838$	0	0
$839$	−29050.3	−1.19539	−0.597693	−	0.801725i	$-0.703916\pi$
$839$	−29050.3	−1.19539	−0.597693	+	0.801725i	$0.703916\pi$
$840$	0	0
$841$	−20776.6	−0.851883
$842$	0	0
$843$	−30759.7	−1.25673
$844$	0	0
$845$	−109037.	−4.43903
$846$	0	0
$847$	281.596	0.0114236
$848$	0	0
$849$	−228.064	−0.00921926
$850$	0	0
$851$	4101.19	0.165202
$852$	0	0
$853$	21804.2	0.875217	0.437609	−	0.899166i	$-0.355826\pi$
$853$	21804.2	0.875217	0.437609	+	0.899166i	$0.355826\pi$
$854$	0	0
$855$	−1574.97	−0.0629974
$856$	0	0
$857$	30879.5	1.23083	0.615416	−	0.788203i	$-0.288988\pi$
$857$	30879.5	1.23083	0.615416	+	0.788203i	$0.288988\pi$
$858$	0	0
$859$	−6065.94	−0.240940	−0.120470	−	0.992717i	$-0.538440\pi$
$859$	−6065.94	−0.240940	−0.120470	+	0.992717i	$0.538440\pi$
$860$	0	0
$861$	438.640	0.0173622
$862$	0	0
$863$	1527.39	0.0602467	0.0301233	−	0.999546i	$-0.490410\pi$
$863$	1527.39	0.0602467	0.0301233	+	0.999546i	$0.490410\pi$
$864$	0	0
$865$	9925.40	0.390143
$866$	0	0
$867$	23228.0	0.909878
$868$	0	0
$869$	−695.129	−0.0271354
$870$	0	0
$871$	53605.8	2.08538
$872$	0	0
$873$	5592.75	0.216822
$874$	0	0
$875$	295.950	0.0114342
$876$	0	0
$877$	−565.469	−0.0217725	−0.0108863	−	0.999941i	$-0.503465\pi$
$877$	−565.469	−0.0217725	−0.0108863	+	0.999941i	$0.503465\pi$
$878$	0	0
$879$	−5400.64	−0.207234
$880$	0	0
$881$	−28897.8	−1.10510	−0.552549	−	0.833481i	$-0.686345\pi$
$881$	−28897.8	−1.10510	−0.552549	+	0.833481i	$0.686345\pi$
$882$	0	0
$883$	−26073.0	−0.993687	−0.496844	−	0.867840i	$-0.665508\pi$
$883$	−26073.0	−0.993687	−0.496844	+	0.867840i	$0.665508\pi$
$884$	0	0
$885$	−65028.2	−2.46994
$886$	0	0
$887$	859.344	0.0325298	0.0162649	−	0.999868i	$-0.494822\pi$
$887$	859.344	0.0325298	0.0162649	+	0.999868i	$0.494822\pi$
$888$	0	0
$889$	−225.198	−0.00849596
$890$	0	0
$891$	−1982.19	−0.0745294
$892$	0	0
$893$	3876.39	0.145261
$894$	0	0
$895$	−75676.9	−2.82637
$896$	0	0
$897$	−11428.9	−0.425419
$898$	0	0
$899$	10582.4	0.392595
$900$	0	0
$901$	352.063	0.0130177
$902$	0	0
$903$	369.302	0.0136098
$904$	0	0
$905$	71471.5	2.62519
$906$	0	0
$907$	−36272.4	−1.32790	−0.663951	−	0.747776i	$-0.731122\pi$
$907$	−36272.4	−1.32790	−0.663951	+	0.747776i	$0.731122\pi$
$908$	0	0
$909$	247.535	0.00903213
$910$	0	0
$911$	20484.7	0.744993	0.372497	−	0.928034i	$-0.378502\pi$
$911$	20484.7	0.744993	0.372497	+	0.928034i	$0.378502\pi$
$912$	0	0
$913$	2654.33	0.0962165
$914$	0	0
$915$	24383.9	0.880992
$916$	0	0
$917$	−219.171	−0.00789277
$918$	0	0
$919$	−6354.25	−0.228082	−0.114041	−	0.993476i	$-0.536380\pi$
$919$	−6354.25	−0.228082	−0.114041	+	0.993476i	$0.536380\pi$
$920$	0	0
$921$	−31855.2	−1.13970
$922$	0	0
$923$	−88202.0	−3.14540
$924$	0	0
$925$	−31088.2	−1.10505
$926$	0	0
$927$	9114.60	0.322937
$928$	0	0
$929$	33316.3	1.17661	0.588307	−	0.808638i	$-0.299795\pi$
$929$	33316.3	1.17661	0.588307	+	0.808638i	$0.299795\pi$
$930$	0	0
$931$	6516.13	0.229385
$932$	0	0
$933$	−46198.6	−1.62109
$934$	0	0
$935$	159.098	0.00556478
$936$	0	0
$937$	−6181.13	−0.215506	−0.107753	−	0.994178i	$-0.534365\pi$
$937$	−6181.13	−0.215506	−0.107753	+	0.994178i	$0.534365\pi$
$938$	0	0
$939$	7862.89	0.273265
$940$	0	0
$941$	609.315	0.0211085	0.0105543	−	0.999944i	$-0.496640\pi$
$941$	609.315	0.0211085	0.0105543	+	0.999944i	$0.496640\pi$
$942$	0	0
$943$	−11552.9	−0.398954
$944$	0	0
$945$	576.851	0.0198571
$946$	0	0
$947$	9348.71	0.320794	0.160397	−	0.987053i	$-0.448722\pi$
$947$	9348.71	0.320794	0.160397	+	0.987053i	$0.448722\pi$
$948$	0	0
$949$	−33425.2	−1.14334
$950$	0	0
$951$	−31146.5	−1.06203
$952$	0	0
$953$	−36990.5	−1.25734	−0.628668	−	0.777674i	$-0.716399\pi$
$953$	−36990.5	−1.25734	−0.628668	+	0.777674i	$0.716399\pi$
$954$	0	0
$955$	−29347.6	−0.994414
$956$	0	0
$957$	−965.563	−0.0326146
$958$	0	0
$959$	368.783	0.0124177
$960$	0	0
$961$	1209.59	0.0406027
$962$	0	0
$963$	−4938.73	−0.165263
$964$	0	0
$965$	−32850.9	−1.09586
$966$	0	0
$967$	48806.2	1.62306	0.811532	−	0.584308i	$-0.198634\pi$
$967$	48806.2	1.62306	0.811532	+	0.584308i	$0.198634\pi$
$968$	0	0
$969$	−233.330	−0.00773543
$970$	0	0
$971$	−401.490	−0.0132692	−0.00663462	−	0.999978i	$-0.502112\pi$
$971$	−401.490	−0.0132692	−0.00663462	+	0.999978i	$0.502112\pi$
$972$	0	0
$973$	133.210	0.00438903
$974$	0	0
$975$	86634.6	2.84567
$976$	0	0
$977$	6786.27	0.222223	0.111112	−	0.993808i	$-0.464559\pi$
$977$	6786.27	0.222223	0.111112	+	0.993808i	$0.464559\pi$
$978$	0	0
$979$	724.874	0.0236640
$980$	0	0
$981$	5505.76	0.179190
$982$	0	0
$983$	17163.1	0.556884	0.278442	−	0.960453i	$-0.410182\pi$
$983$	17163.1	0.556884	0.278442	+	0.960453i	$0.410182\pi$
$984$	0	0
$985$	−2743.31	−0.0887404
$986$	0	0
$987$	−206.137	−0.00664783
$988$	0	0
$989$	−9726.67	−0.312730
$990$	0	0
$991$	−9323.61	−0.298864	−0.149432	−	0.988772i	$-0.547745\pi$
$991$	−9323.61	−0.298864	−0.149432	+	0.988772i	$0.547745\pi$
$992$	0	0
$993$	−26144.6	−0.835524
$994$	0	0
$995$	22231.0	0.708313
$996$	0	0
$997$	14764.7	0.469011	0.234506	−	0.972115i	$-0.424653\pi$
$997$	14764.7	0.469011	0.234506	+	0.972115i	$0.424653\pi$
$998$	0	0
$999$	−23046.3	−0.729881

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.4.a.j.1.1		3
4.3	odd	2		152.4.a.b.1.3	✓	3
8.3	odd	2		1216.4.a.x.1.1		3
8.5	even	2		1216.4.a.q.1.3		3
12.11	even	2		1368.4.a.e.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.4.a.b.1.3	✓	3	4.3	odd	2
304.4.a.j.1.1		3	1.1	even	1	trivial
1216.4.a.q.1.3		3	8.5	even	2
1216.4.a.x.1.1		3	8.3	odd	2
1368.4.a.e.1.3		3	12.11	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 304 = 2^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	304.a (trivial)

\(f(q)\)	\(=\)	\(q-4.73435 q^{3} -18.0754 q^{5} -0.213413 q^{7} -4.58596 q^{9} +3.39329 q^{11} -90.7156 q^{13} +85.5752 q^{15} -2.59392 q^{17} -19.0000 q^{19} +1.01037 q^{21} -26.6112 q^{23} +201.720 q^{25} +149.539 q^{27} +60.1034 q^{29} +176.070 q^{31} -16.0650 q^{33} +3.85753 q^{35} -154.115 q^{37} +429.479 q^{39} +434.137 q^{41} +365.511 q^{43} +82.8931 q^{45} -204.021 q^{47} -342.954 q^{49} +12.2805 q^{51} -135.726 q^{53} -61.3351 q^{55} +89.9526 q^{57} -759.895 q^{59} +284.941 q^{61} +0.978705 q^{63} +1639.72 q^{65} -590.922 q^{67} +125.986 q^{69} +972.291 q^{71} +368.462 q^{73} -955.013 q^{75} -0.724174 q^{77} -204.854 q^{79} -584.148 q^{81} +782.229 q^{83} +46.8861 q^{85} -284.550 q^{87} +213.620 q^{89} +19.3599 q^{91} -833.576 q^{93} +343.433 q^{95} -1219.54 q^{97} -15.5615 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 5 q^{3} + 2 q^{5} + 35 q^{7} + 48 q^{9} + 28 q^{11} - 109 q^{13} + 228 q^{15} - 123 q^{17} - 57 q^{19} + 25 q^{21} + 193 q^{23} + 187 q^{25} + 719 q^{27} - 297 q^{29} + 140 q^{31} + 30 q^{33} + 246 q^{35}+ \cdots - 86 q^{99}+O(q^{100}) \) 3 * q + 5 * q^3 + 2 * q^5 + 35 * q^7 + 48 * q^9 + 28 * q^11 - 109 * q^13 + 228 * q^15 - 123 * q^17 - 57 * q^19 + 25 * q^21 + 193 * q^23 + 187 * q^25 + 719 * q^27 - 297 * q^29 + 140 * q^31 + 30 * q^33 + 246 * q^35 + 38 * q^37 - 57 * q^39 + 736 * q^41 + 514 * q^43 + 1046 * q^45 - 134 * q^47 - 42 * q^49 - 929 * q^51 + 311 * q^53 + 130 * q^55 - 95 * q^57 - 199 * q^59 + 56 * q^61 - 508 * q^63 + 1156 * q^65 + 509 * q^67 + 621 * q^69 + 874 * q^71 + 203 * q^73 - 129 * q^75 + 616 * q^77 - 242 * q^79 + 3531 * q^81 + 62 * q^83 - 1430 * q^85 - 3237 * q^87 + 1764 * q^89 + 687 * q^91 - 2592 * q^93 - 38 * q^95 - 2178 * q^97 - 86 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more