LMFDB - Embedded newform 304.4.a.i.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,-1,0,14] 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:	3.3.3144.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 16x - 8 $ x^3 - x^2 - 16*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 19)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.20905$ 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)

$f(q)$	$=$	$q-6.71610 q^{3} +18.1342 q^{5} +25.8362 q^{7} +18.1060 q^{9} +8.13420 q^{11} -4.56640 q^{13} -121.791 q^{15} -62.5850 q^{17} +19.0000 q^{19} -173.518 q^{21} +52.7502 q^{23} +203.849 q^{25} +59.7330 q^{27} +171.620 q^{29} -168.749 q^{31} -54.6301 q^{33} +468.519 q^{35} -147.534 q^{37} +30.6684 q^{39} +308.774 q^{41} +448.950 q^{43} +328.338 q^{45} -113.335 q^{47} +324.509 q^{49} +420.327 q^{51} +155.402 q^{53} +147.507 q^{55} -127.606 q^{57} -182.347 q^{59} +404.080 q^{61} +467.790 q^{63} -82.8080 q^{65} +106.400 q^{67} -354.276 q^{69} -472.079 q^{71} +843.821 q^{73} -1369.07 q^{75} +210.157 q^{77} +591.036 q^{79} -890.035 q^{81} -290.388 q^{83} -1134.93 q^{85} -1152.62 q^{87} -964.896 q^{89} -117.978 q^{91} +1133.34 q^{93} +344.550 q^{95} -219.495 q^{97} +147.278 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{3} + 14 q^{5} + 35 q^{7} + 48 q^{9} - 16 q^{11} + 65 q^{13} - 140 q^{15} + 29 q^{17} + 57 q^{19} - 25 q^{21} + 101 q^{23} - 37 q^{25} + 377 q^{27} + 377 q^{29} + 140 q^{31} - 130 q^{33} + 438 q^{35}+ \cdots - 250 q^{99}+O(q^{100}) $ 3 * q - q^3 + 14 * q^5 + 35 * q^7 + 48 * q^9 - 16 * q^11 + 65 * q^13 - 140 * q^15 + 29 * q^17 + 57 * q^19 - 25 * q^21 + 101 * q^23 - 37 * q^25 + 377 * q^27 + 377 * q^29 + 140 * q^31 - 130 * q^33 + 438 * q^35 - 290 * q^37 + 371 * q^39 + 956 * q^41 + 570 * q^43 + 230 * q^45 - 66 * q^47 - 98 * q^49 - 187 * q^51 + 817 * q^53 + 198 * q^55 - 19 * q^57 - 265 * q^59 + 988 * q^61 + 1304 * q^63 - 240 * q^65 + 207 * q^67 + 807 * q^69 - 846 * q^71 + 627 * q^73 - 2031 * q^75 + 88 * q^77 - 382 * q^79 - 501 * q^81 + 766 * q^83 - 1242 * q^85 + 1671 * q^87 - 172 * q^89 + 457 * q^91 + 892 * q^93 + 266 * q^95 - 2450 * q^97 - 250 * 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
$3$	−6.71610	−1.29251	−0.646257	−	0.763120i	$-0.723667\pi$
$3$	−6.71610	−1.29251	−0.646257	+	0.763120i	$0.723667\pi$
$4$	0	0
$5$	18.1342	1.62197	0.810986	−	0.585065i	$-0.198931\pi$
$5$	18.1342	1.62197	0.810986	+	0.585065i	$0.198931\pi$
$6$	0	0
$7$	25.8362	1.39502	0.697512	−	0.716573i	$-0.254290\pi$
$7$	25.8362	1.39502	0.697512	+	0.716573i	$0.254290\pi$
$8$	0	0
$9$	18.1060	0.670593
$10$	0	0
$11$	8.13420	0.222959	0.111480	−	0.993767i	$-0.464441\pi$
$11$	8.13420	0.222959	0.111480	+	0.993767i	$0.464441\pi$
$12$	0	0
$13$	−4.56640	−0.0974224	−0.0487112	−	0.998813i	$-0.515511\pi$
$13$	−4.56640	−0.0974224	−0.0487112	+	0.998813i	$0.515511\pi$
$14$	0	0
$15$	−121.791	−2.09642
$16$	0	0
$17$	−62.5850	−0.892888	−0.446444	−	0.894812i	$-0.647310\pi$
$17$	−62.5850	−0.892888	−0.446444	+	0.894812i	$0.647310\pi$
$18$	0	0
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	0	0
$21$	−173.518	−1.80309
$22$	0	0
$23$	52.7502	0.478225	0.239113	−	0.970992i	$-0.423144\pi$
$23$	52.7502	0.478225	0.239113	+	0.970992i	$0.423144\pi$
$24$	0	0
$25$	203.849	1.63079
$26$	0	0
$27$	59.7330	0.425764
$28$	0	0
$29$	171.620	1.09894	0.549468	−	0.835515i	$-0.314831\pi$
$29$	171.620	1.09894	0.549468	+	0.835515i	$0.314831\pi$
$30$	0	0
$31$	−168.749	−0.977685	−0.488842	−	0.872372i	$-0.662581\pi$
$31$	−168.749	−0.977685	−0.488842	+	0.872372i	$0.662581\pi$
$32$	0	0
$33$	−54.6301	−0.288178
$34$	0	0
$35$	468.519	2.26269
$36$	0	0
$37$	−147.534	−0.655528	−0.327764	−	0.944760i	$-0.606295\pi$
$37$	−147.534	−0.655528	−0.327764	+	0.944760i	$0.606295\pi$
$38$	0	0
$39$	30.6684	0.125920
$40$	0	0
$41$	308.774	1.17616	0.588078	−	0.808804i	$-0.299885\pi$
$41$	308.774	1.17616	0.588078	+	0.808804i	$0.299885\pi$
$42$	0	0
$43$	448.950	1.59219	0.796096	−	0.605170i	$-0.206895\pi$
$43$	448.950	1.59219	0.796096	+	0.605170i	$0.206895\pi$
$44$	0	0
$45$	328.338	1.08768
$46$	0	0
$47$	−113.335	−0.351737	−0.175868	−	0.984414i	$-0.556273\pi$
$47$	−113.335	−0.351737	−0.175868	+	0.984414i	$0.556273\pi$
$48$	0	0
$49$	324.509	0.946091
$50$	0	0
$51$	420.327	1.15407
$52$	0	0
$53$	155.402	0.402758	0.201379	−	0.979513i	$-0.435458\pi$
$53$	155.402	0.402758	0.201379	+	0.979513i	$0.435458\pi$
$54$	0	0
$55$	147.507	0.361634
$56$	0	0
$57$	−127.606	−0.296523
$58$	0	0
$59$	−182.347	−0.402365	−0.201183	−	0.979554i	$-0.564478\pi$
$59$	−182.347	−0.402365	−0.201183	+	0.979554i	$0.564478\pi$
$60$	0	0
$61$	404.080	0.848149	0.424075	−	0.905627i	$-0.360599\pi$
$61$	404.080	0.848149	0.424075	+	0.905627i	$0.360599\pi$
$62$	0	0
$63$	467.790	0.935492
$64$	0	0
$65$	−82.8080	−0.158016
$66$	0	0
$67$	106.400	0.194013	0.0970064	−	0.995284i	$-0.469073\pi$
$67$	106.400	0.194013	0.0970064	+	0.995284i	$0.469073\pi$
$68$	0	0
$69$	−354.276	−0.618113
$70$	0	0
$71$	−472.079	−0.789091	−0.394546	−	0.918876i	$-0.629098\pi$
$71$	−472.079	−0.789091	−0.394546	+	0.918876i	$0.629098\pi$
$72$	0	0
$73$	843.821	1.35290	0.676451	−	0.736488i	$-0.263517\pi$
$73$	843.821	1.35290	0.676451	+	0.736488i	$0.263517\pi$
$74$	0	0
$75$	−1369.07	−2.10782
$76$	0	0
$77$	210.157	0.311034
$78$	0	0
$79$	591.036	0.841731	0.420866	−	0.907123i	$-0.361726\pi$
$79$	591.036	0.841731	0.420866	+	0.907123i	$0.361726\pi$
$80$	0	0
$81$	−890.035	−1.22090
$82$	0	0
$83$	−290.388	−0.384027	−0.192013	−	0.981392i	$-0.561502\pi$
$83$	−290.388	−0.384027	−0.192013	+	0.981392i	$0.561502\pi$
$84$	0	0
$85$	−1134.93	−1.44824
$86$	0	0
$87$	−1152.62	−1.42039
$88$	0	0
$89$	−964.896	−1.14920	−0.574600	−	0.818435i	$-0.694842\pi$
$89$	−964.896	−1.14920	−0.574600	+	0.818435i	$0.694842\pi$
$90$	0	0
$91$	−117.978	−0.135907
$92$	0	0
$93$	1133.34	1.26367
$94$	0	0
$95$	344.550	0.372106
$96$	0	0
$97$	−219.495	−0.229756	−0.114878	−	0.993380i	$-0.536648\pi$
$97$	−219.495	−0.229756	−0.114878	+	0.993380i	$0.536648\pi$
$98$	0	0
$99$	147.278	0.149515
$100$	0	0
$101$	1447.94	1.42649	0.713247	−	0.700913i	$-0.247224\pi$
$101$	1447.94	1.42649	0.713247	+	0.700913i	$0.247224\pi$
$102$	0	0
$103$	−883.567	−0.845247	−0.422623	−	0.906305i	$-0.638891\pi$
$103$	−883.567	−0.845247	−0.422623	+	0.906305i	$0.638891\pi$
$104$	0	0
$105$	−3146.62	−2.92456
$106$	0	0
$107$	1307.82	1.18160	0.590801	−	0.806817i	$-0.298812\pi$
$107$	1307.82	1.18160	0.590801	+	0.806817i	$0.298812\pi$
$108$	0	0
$109$	870.507	0.764949	0.382475	−	0.923966i	$-0.375072\pi$
$109$	870.507	0.764949	0.382475	+	0.923966i	$0.375072\pi$
$110$	0	0
$111$	990.856	0.847279
$112$	0	0
$113$	−1181.41	−0.983521	−0.491761	−	0.870730i	$-0.663646\pi$
$113$	−1181.41	−0.983521	−0.491761	+	0.870730i	$0.663646\pi$
$114$	0	0
$115$	956.583	0.775668
$116$	0	0
$117$	−82.6792	−0.0653307
$118$	0	0
$119$	−1616.96	−1.24560
$120$	0	0
$121$	−1264.83	−0.950289
$122$	0	0
$123$	−2073.76	−1.52020
$124$	0	0
$125$	1429.87	1.02313
$126$	0	0
$127$	−887.509	−0.620108	−0.310054	−	0.950719i	$-0.600347\pi$
$127$	−887.509	−0.620108	−0.310054	+	0.950719i	$0.600347\pi$
$128$	0	0
$129$	−3015.19	−2.05793
$130$	0	0
$131$	2344.76	1.56384	0.781920	−	0.623379i	$-0.214241\pi$
$131$	2344.76	1.56384	0.781920	+	0.623379i	$0.214241\pi$
$132$	0	0
$133$	490.888	0.320040
$134$	0	0
$135$	1083.21	0.690577
$136$	0	0
$137$	2244.82	1.39991	0.699956	−	0.714186i	$-0.253203\pi$
$137$	2244.82	1.39991	0.699956	+	0.714186i	$0.253203\pi$
$138$	0	0
$139$	296.146	0.180711	0.0903554	−	0.995910i	$-0.471200\pi$
$139$	296.146	0.180711	0.0903554	+	0.995910i	$0.471200\pi$
$140$	0	0
$141$	761.170	0.454625
$142$	0	0
$143$	−37.1440	−0.0217212
$144$	0	0
$145$	3112.20	1.78244
$146$	0	0
$147$	−2179.44	−1.22284
$148$	0	0
$149$	1791.09	0.984780	0.492390	−	0.870375i	$-0.336123\pi$
$149$	1791.09	0.984780	0.492390	+	0.870375i	$0.336123\pi$
$150$	0	0
$151$	2352.65	1.26792	0.633960	−	0.773366i	$-0.281429\pi$
$151$	2352.65	1.26792	0.633960	+	0.773366i	$0.281429\pi$
$152$	0	0
$153$	−1133.16	−0.598764
$154$	0	0
$155$	−3060.13	−1.58578
$156$	0	0
$157$	−1438.26	−0.731118	−0.365559	−	0.930788i	$-0.619122\pi$
$157$	−1438.26	−0.731118	−0.365559	+	0.930788i	$0.619122\pi$
$158$	0	0
$159$	−1043.70	−0.520570
$160$	0	0
$161$	1362.86	0.667135
$162$	0	0
$163$	−127.493	−0.0612640	−0.0306320	−	0.999531i	$-0.509752\pi$
$163$	−127.493	−0.0612640	−0.0306320	+	0.999531i	$0.509752\pi$
$164$	0	0
$165$	−990.673	−0.467417
$166$	0	0
$167$	−3419.05	−1.58428	−0.792139	−	0.610341i	$-0.791033\pi$
$167$	−3419.05	−1.58428	−0.792139	+	0.610341i	$0.791033\pi$
$168$	0	0
$169$	−2176.15	−0.990509
$170$	0	0
$171$	344.014	0.153844
$172$	0	0
$173$	−362.598	−0.159352	−0.0796758	−	0.996821i	$-0.525389\pi$
$173$	−362.598	−0.159352	−0.0796758	+	0.996821i	$0.525389\pi$
$174$	0	0
$175$	5266.69	2.27500
$176$	0	0
$177$	1224.66	0.520063
$178$	0	0
$179$	−2417.89	−1.00962	−0.504809	−	0.863231i	$-0.668437\pi$
$179$	−2417.89	−1.00962	−0.504809	+	0.863231i	$0.668437\pi$
$180$	0	0
$181$	−2444.64	−1.00391	−0.501957	−	0.864892i	$-0.667387\pi$
$181$	−2444.64	−1.00391	−0.501957	+	0.864892i	$0.667387\pi$
$182$	0	0
$183$	−2713.84	−1.09624
$184$	0	0
$185$	−2675.42	−1.06325
$186$	0	0
$187$	−509.079	−0.199078
$188$	0	0
$189$	1543.27	0.593951
$190$	0	0
$191$	1387.66	0.525693	0.262846	−	0.964838i	$-0.415339\pi$
$191$	1387.66	0.525693	0.262846	+	0.964838i	$0.415339\pi$
$192$	0	0
$193$	−3208.03	−1.19647	−0.598237	−	0.801319i	$-0.704132\pi$
$193$	−3208.03	−1.19647	−0.598237	+	0.801319i	$0.704132\pi$
$194$	0	0
$195$	556.147	0.204238
$196$	0	0
$197$	−3445.36	−1.24605	−0.623025	−	0.782202i	$-0.714097\pi$
$197$	−3445.36	−1.24605	−0.623025	+	0.782202i	$0.714097\pi$
$198$	0	0
$199$	−2025.71	−0.721602	−0.360801	−	0.932643i	$-0.617497\pi$
$199$	−2025.71	−0.721602	−0.360801	+	0.932643i	$0.617497\pi$
$200$	0	0
$201$	−714.595	−0.250764
$202$	0	0
$203$	4434.02	1.53304
$204$	0	0
$205$	5599.37	1.90769
$206$	0	0
$207$	955.095	0.320694
$208$	0	0
$209$	154.550	0.0511504
$210$	0	0
$211$	−4309.54	−1.40607	−0.703036	−	0.711155i	$-0.748173\pi$
$211$	−4309.54	−1.40607	−0.703036	+	0.711155i	$0.748173\pi$
$212$	0	0
$213$	3170.53	1.01991
$214$	0	0
$215$	8141.35	2.58249
$216$	0	0
$217$	−4359.84	−1.36389
$218$	0	0
$219$	−5667.19	−1.74864
$220$	0	0
$221$	285.788	0.0869873
$222$	0	0
$223$	−825.648	−0.247935	−0.123968	−	0.992286i	$-0.539562\pi$
$223$	−825.648	−0.247935	−0.123968	+	0.992286i	$0.539562\pi$
$224$	0	0
$225$	3690.89	1.09360
$226$	0	0
$227$	1501.19	0.438931	0.219466	−	0.975620i	$-0.429569\pi$
$227$	1501.19	0.438931	0.219466	+	0.975620i	$0.429569\pi$
$228$	0	0
$229$	−5250.40	−1.51509	−0.757547	−	0.652781i	$-0.773602\pi$
$229$	−5250.40	−1.51509	−0.757547	+	0.652781i	$0.773602\pi$
$230$	0	0
$231$	−1411.43	−0.402015
$232$	0	0
$233$	2139.06	0.601435	0.300717	−	0.953713i	$-0.402774\pi$
$233$	2139.06	0.601435	0.300717	+	0.953713i	$0.402774\pi$
$234$	0	0
$235$	−2055.24	−0.570508
$236$	0	0
$237$	−3969.46	−1.08795
$238$	0	0
$239$	−3772.70	−1.02107	−0.510534	−	0.859857i	$-0.670552\pi$
$239$	−3772.70	−1.02107	−0.510534	+	0.859857i	$0.670552\pi$
$240$	0	0
$241$	6415.39	1.71474	0.857369	−	0.514702i	$-0.172097\pi$
$241$	6415.39	1.71474	0.857369	+	0.514702i	$0.172097\pi$
$242$	0	0
$243$	4364.77	1.15226
$244$	0	0
$245$	5884.71	1.53453
$246$	0	0
$247$	−86.7616	−0.0223502
$248$	0	0
$249$	1950.27	0.496360
$250$	0	0
$251$	6277.31	1.57857	0.789283	−	0.614029i	$-0.210452\pi$
$251$	6277.31	1.57857	0.789283	+	0.614029i	$0.210452\pi$
$252$	0	0
$253$	429.081	0.106625
$254$	0	0
$255$	7622.30	1.87187
$256$	0	0
$257$	−3183.98	−0.772807	−0.386404	−	0.922330i	$-0.626283\pi$
$257$	−3183.98	−0.772807	−0.386404	+	0.922330i	$0.626283\pi$
$258$	0	0
$259$	−3811.73	−0.914476
$260$	0	0
$261$	3107.36	0.736938
$262$	0	0
$263$	2624.18	0.615261	0.307630	−	0.951506i	$-0.400464\pi$
$263$	2624.18	0.615261	0.307630	+	0.951506i	$0.400464\pi$
$264$	0	0
$265$	2818.10	0.653261
$266$	0	0
$267$	6480.34	1.48536
$268$	0	0
$269$	7444.76	1.68742	0.843708	−	0.536803i	$-0.180368\pi$
$269$	7444.76	1.68742	0.843708	+	0.536803i	$0.180368\pi$
$270$	0	0
$271$	4004.49	0.897621	0.448810	−	0.893627i	$-0.351848\pi$
$271$	4004.49	0.897621	0.448810	+	0.893627i	$0.351848\pi$
$272$	0	0
$273$	792.355	0.175661
$274$	0	0
$275$	1658.15	0.363601
$276$	0	0
$277$	−5830.66	−1.26473	−0.632365	−	0.774671i	$-0.717916\pi$
$277$	−5830.66	−1.26473	−0.632365	+	0.774671i	$0.717916\pi$
$278$	0	0
$279$	−3055.37	−0.655628
$280$	0	0
$281$	−7504.37	−1.59314	−0.796572	−	0.604544i	$-0.793355\pi$
$281$	−7504.37	−1.59314	−0.796572	+	0.604544i	$0.793355\pi$
$282$	0	0
$283$	−5910.87	−1.24157	−0.620785	−	0.783980i	$-0.713186\pi$
$283$	−5910.87	−1.24157	−0.620785	+	0.783980i	$0.713186\pi$
$284$	0	0
$285$	−2314.03	−0.480952
$286$	0	0
$287$	7977.54	1.64076
$288$	0	0
$289$	−996.118	−0.202752
$290$	0	0
$291$	1474.15	0.296962
$292$	0	0
$293$	3245.59	0.647131	0.323566	−	0.946206i	$-0.395118\pi$
$293$	3245.59	0.647131	0.323566	+	0.946206i	$0.395118\pi$
$294$	0	0
$295$	−3306.72	−0.652625
$296$	0	0
$297$	485.880	0.0949280
$298$	0	0
$299$	−240.878	−0.0465898
$300$	0	0
$301$	11599.2	2.22115
$302$	0	0
$303$	−9724.54	−1.84376
$304$	0	0
$305$	7327.66	1.37567
$306$	0	0
$307$	−7489.14	−1.39227	−0.696137	−	0.717909i	$-0.745099\pi$
$307$	−7489.14	−1.39227	−0.696137	+	0.717909i	$0.745099\pi$
$308$	0	0
$309$	5934.12	1.09249
$310$	0	0
$311$	−2136.71	−0.389588	−0.194794	−	0.980844i	$-0.562404\pi$
$311$	−2136.71	−0.389588	−0.194794	+	0.980844i	$0.562404\pi$
$312$	0	0
$313$	2212.15	0.399483	0.199742	−	0.979849i	$-0.435990\pi$
$313$	2212.15	0.399483	0.199742	+	0.979849i	$0.435990\pi$
$314$	0	0
$315$	8483.00	1.51734
$316$	0	0
$317$	429.326	0.0760674	0.0380337	−	0.999276i	$-0.487891\pi$
$317$	429.326	0.0760674	0.0380337	+	0.999276i	$0.487891\pi$
$318$	0	0
$319$	1396.00	0.245018
$320$	0	0
$321$	−8783.43	−1.52724
$322$	0	0
$323$	−1189.11	−0.204842
$324$	0	0
$325$	−930.857	−0.158876
$326$	0	0
$327$	−5846.42	−0.988708
$328$	0	0
$329$	−2928.15	−0.490681
$330$	0	0
$331$	765.454	0.127109	0.0635546	−	0.997978i	$-0.479756\pi$
$331$	765.454	0.127109	0.0635546	+	0.997978i	$0.479756\pi$
$332$	0	0
$333$	−2671.26	−0.439592
$334$	0	0
$335$	1929.48	0.314684
$336$	0	0
$337$	−3049.81	−0.492978	−0.246489	−	0.969146i	$-0.579277\pi$
$337$	−3049.81	−0.492978	−0.246489	+	0.969146i	$0.579277\pi$
$338$	0	0
$339$	7934.48	1.27121
$340$	0	0
$341$	−1372.64	−0.217984
$342$	0	0
$343$	−477.732	−0.0752045
$344$	0	0
$345$	−6424.50	−1.00256
$346$	0	0
$347$	5907.00	0.913845	0.456922	−	0.889507i	$-0.348952\pi$
$347$	5907.00	0.913845	0.456922	+	0.889507i	$0.348952\pi$
$348$	0	0
$349$	−12107.4	−1.85700	−0.928502	−	0.371327i	$-0.878903\pi$
$349$	−12107.4	−1.85700	−0.928502	+	0.371327i	$0.878903\pi$
$350$	0	0
$351$	−272.765	−0.0414789
$352$	0	0
$353$	−2420.40	−0.364943	−0.182471	−	0.983211i	$-0.558410\pi$
$353$	−2420.40	−0.364943	−0.182471	+	0.983211i	$0.558410\pi$
$354$	0	0
$355$	−8560.77	−1.27988
$356$	0	0
$357$	10859.7	1.60995
$358$	0	0
$359$	1455.80	0.214023	0.107011	−	0.994258i	$-0.465872\pi$
$359$	1455.80	0.214023	0.107011	+	0.994258i	$0.465872\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	8494.76	1.22826
$364$	0	0
$365$	15302.0	2.19437
$366$	0	0
$367$	−8783.80	−1.24935	−0.624674	−	0.780886i	$-0.714768\pi$
$367$	−8783.80	−1.24935	−0.624674	+	0.780886i	$0.714768\pi$
$368$	0	0
$369$	5590.66	0.788721
$370$	0	0
$371$	4015.00	0.561856
$372$	0	0
$373$	−9199.84	−1.27708	−0.638538	−	0.769590i	$-0.720461\pi$
$373$	−9199.84	−1.27708	−0.638538	+	0.769590i	$0.720461\pi$
$374$	0	0
$375$	−9603.13	−1.32241
$376$	0	0
$377$	−783.688	−0.107061
$378$	0	0
$379$	6161.38	0.835063	0.417531	−	0.908662i	$-0.362895\pi$
$379$	6161.38	0.835063	0.417531	+	0.908662i	$0.362895\pi$
$380$	0	0
$381$	5960.60	0.801498
$382$	0	0
$383$	−2630.79	−0.350985	−0.175492	−	0.984481i	$-0.556152\pi$
$383$	−2630.79	−0.350985	−0.175492	+	0.984481i	$0.556152\pi$
$384$	0	0
$385$	3811.03	0.504488
$386$	0	0
$387$	8128.69	1.06771
$388$	0	0
$389$	−5866.48	−0.764633	−0.382317	−	0.924031i	$-0.624874\pi$
$389$	−5866.48	−0.764633	−0.382317	+	0.924031i	$0.624874\pi$
$390$	0	0
$391$	−3301.37	−0.427001
$392$	0	0
$393$	−15747.7	−2.02129
$394$	0	0
$395$	10718.0	1.36526
$396$	0	0
$397$	−14254.0	−1.80199	−0.900993	−	0.433833i	$-0.857161\pi$
$397$	−14254.0	−1.80199	−0.900993	+	0.433833i	$0.857161\pi$
$398$	0	0
$399$	−3296.85	−0.413657
$400$	0	0
$401$	9909.27	1.23403	0.617014	−	0.786952i	$-0.288342\pi$
$401$	9909.27	1.23403	0.617014	+	0.786952i	$0.288342\pi$
$402$	0	0
$403$	770.576	0.0952484
$404$	0	0
$405$	−16140.1	−1.98026
$406$	0	0
$407$	−1200.07	−0.146156
$408$	0	0
$409$	5805.87	0.701912	0.350956	−	0.936392i	$-0.385857\pi$
$409$	5805.87	0.701912	0.350956	+	0.936392i	$0.385857\pi$
$410$	0	0
$411$	−15076.4	−1.80941
$412$	0	0
$413$	−4711.15	−0.561309
$414$	0	0
$415$	−5265.95	−0.622881
$416$	0	0
$417$	−1988.95	−0.233571
$418$	0	0
$419$	−12260.9	−1.42955	−0.714777	−	0.699353i	$-0.753472\pi$
$419$	−12260.9	−1.42955	−0.714777	+	0.699353i	$0.753472\pi$
$420$	0	0
$421$	5837.85	0.675818	0.337909	−	0.941179i	$-0.390280\pi$
$421$	5837.85	0.675818	0.337909	+	0.941179i	$0.390280\pi$
$422$	0	0
$423$	−2052.05	−0.235872
$424$	0	0
$425$	−12757.9	−1.45612
$426$	0	0
$427$	10439.9	1.18319
$428$	0	0
$429$	249.463	0.0280750
$430$	0	0
$431$	2770.16	0.309591	0.154796	−	0.987946i	$-0.450528\pi$
$431$	2770.16	0.309591	0.154796	+	0.987946i	$0.450528\pi$
$432$	0	0
$433$	5663.00	0.628513	0.314257	−	0.949338i	$-0.398245\pi$
$433$	5663.00	0.628513	0.314257	+	0.949338i	$0.398245\pi$
$434$	0	0
$435$	−20901.8	−2.30383
$436$	0	0
$437$	1002.25	0.109712
$438$	0	0
$439$	8399.20	0.913148	0.456574	−	0.889685i	$-0.349076\pi$
$439$	8399.20	0.913148	0.456574	+	0.889685i	$0.349076\pi$
$440$	0	0
$441$	5875.56	0.634442
$442$	0	0
$443$	−6154.68	−0.660085	−0.330043	−	0.943966i	$-0.607063\pi$
$443$	−6154.68	−0.660085	−0.330043	+	0.943966i	$0.607063\pi$
$444$	0	0
$445$	−17497.6	−1.86397
$446$	0	0
$447$	−12029.2	−1.27284
$448$	0	0
$449$	3445.03	0.362095	0.181048	−	0.983474i	$-0.442051\pi$
$449$	3445.03	0.362095	0.181048	+	0.983474i	$0.442051\pi$
$450$	0	0
$451$	2511.63	0.262235
$452$	0	0
$453$	−15800.6	−1.63880
$454$	0	0
$455$	−2139.44	−0.220437
$456$	0	0
$457$	−502.346	−0.0514196	−0.0257098	−	0.999669i	$-0.508185\pi$
$457$	−502.346	−0.0514196	−0.0257098	+	0.999669i	$0.508185\pi$
$458$	0	0
$459$	−3738.39	−0.380159
$460$	0	0
$461$	546.259	0.0551883	0.0275942	−	0.999619i	$-0.491215\pi$
$461$	546.259	0.0551883	0.0275942	+	0.999619i	$0.491215\pi$
$462$	0	0
$463$	−18540.2	−1.86098	−0.930490	−	0.366316i	$-0.880619\pi$
$463$	−18540.2	−1.86098	−0.930490	+	0.366316i	$0.880619\pi$
$464$	0	0
$465$	20552.1	2.04964
$466$	0	0
$467$	−12475.1	−1.23614	−0.618070	−	0.786123i	$-0.712085\pi$
$467$	−12475.1	−1.23614	−0.618070	+	0.786123i	$0.712085\pi$
$468$	0	0
$469$	2748.98	0.270653
$470$	0	0
$471$	9659.49	0.944981
$472$	0	0
$473$	3651.85	0.354994
$474$	0	0
$475$	3873.13	0.374130
$476$	0	0
$477$	2813.71	0.270086
$478$	0	0
$479$	10569.2	1.00818	0.504091	−	0.863651i	$-0.331828\pi$
$479$	10569.2	1.00818	0.504091	+	0.863651i	$0.331828\pi$
$480$	0	0
$481$	673.701	0.0638631
$482$	0	0
$483$	−9153.13	−0.862282
$484$	0	0
$485$	−3980.36	−0.372657
$486$	0	0
$487$	11227.9	1.04473	0.522366	−	0.852721i	$-0.325049\pi$
$487$	11227.9	1.04473	0.522366	+	0.852721i	$0.325049\pi$
$488$	0	0
$489$	856.256	0.0791846
$490$	0	0
$491$	536.840	0.0493427	0.0246713	−	0.999696i	$-0.492146\pi$
$491$	536.840	0.0493427	0.0246713	+	0.999696i	$0.492146\pi$
$492$	0	0
$493$	−10740.9	−0.981226
$494$	0	0
$495$	2670.77	0.242509
$496$	0	0
$497$	−12196.7	−1.10080
$498$	0	0
$499$	−1319.91	−0.118412	−0.0592058	−	0.998246i	$-0.518857\pi$
$499$	−1319.91	−0.118412	−0.0592058	+	0.998246i	$0.518857\pi$
$500$	0	0
$501$	22962.7	2.04770
$502$	0	0
$503$	−1749.27	−0.155062	−0.0775310	−	0.996990i	$-0.524704\pi$
$503$	−1749.27	−0.155062	−0.0775310	+	0.996990i	$0.524704\pi$
$504$	0	0
$505$	26257.3	2.31373
$506$	0	0
$507$	14615.2	1.28025
$508$	0	0
$509$	1882.19	0.163903	0.0819516	−	0.996636i	$-0.473885\pi$
$509$	1882.19	0.163903	0.0819516	+	0.996636i	$0.473885\pi$
$510$	0	0
$511$	21801.1	1.88733
$512$	0	0
$513$	1134.93	0.0976769
$514$	0	0
$515$	−16022.8	−1.37097
$516$	0	0
$517$	−921.891	−0.0784231
$518$	0	0
$519$	2435.25	0.205964
$520$	0	0
$521$	−3238.50	−0.272325	−0.136163	−	0.990686i	$-0.543477\pi$
$521$	−3238.50	−0.272325	−0.136163	+	0.990686i	$0.543477\pi$
$522$	0	0
$523$	−99.0144	−0.00827839	−0.00413919	−	0.999991i	$-0.501318\pi$
$523$	−99.0144	−0.00827839	−0.00413919	+	0.999991i	$0.501318\pi$
$524$	0	0
$525$	−35371.6	−2.94046
$526$	0	0
$527$	10561.2	0.872963
$528$	0	0
$529$	−9384.42	−0.771301
$530$	0	0
$531$	−3301.57	−0.269823
$532$	0	0
$533$	−1409.98	−0.114584
$534$	0	0
$535$	23716.2	1.91653
$536$	0	0
$537$	16238.8	1.30495
$538$	0	0
$539$	2639.62	0.210940
$540$	0	0
$541$	17183.7	1.36559	0.682794	−	0.730611i	$-0.260765\pi$
$541$	17183.7	1.36559	0.682794	+	0.730611i	$0.260765\pi$
$542$	0	0
$543$	16418.4	1.29757
$544$	0	0
$545$	15786.0	1.24073
$546$	0	0
$547$	1965.86	0.153664	0.0768319	−	0.997044i	$-0.475520\pi$
$547$	1965.86	0.153664	0.0768319	+	0.997044i	$0.475520\pi$
$548$	0	0
$549$	7316.27	0.568762
$550$	0	0
$551$	3260.79	0.252113
$552$	0	0
$553$	15270.1	1.17423
$554$	0	0
$555$	17968.4	1.37426
$556$	0	0
$557$	6039.93	0.459461	0.229731	−	0.973254i	$-0.426215\pi$
$557$	6039.93	0.459461	0.229731	+	0.973254i	$0.426215\pi$
$558$	0	0
$559$	−2050.09	−0.155115
$560$	0	0
$561$	3419.02	0.257311
$562$	0	0
$563$	−5260.06	−0.393757	−0.196878	−	0.980428i	$-0.563080\pi$
$563$	−5260.06	−0.393757	−0.196878	+	0.980428i	$0.563080\pi$
$564$	0	0
$565$	−21424.0	−1.59524
$566$	0	0
$567$	−22995.1	−1.70318
$568$	0	0
$569$	20567.4	1.51534	0.757672	−	0.652635i	$-0.226337\pi$
$569$	20567.4	1.51534	0.757672	+	0.652635i	$0.226337\pi$
$570$	0	0
$571$	−11462.4	−0.840080	−0.420040	−	0.907506i	$-0.637984\pi$
$571$	−11462.4	−0.840080	−0.420040	+	0.907506i	$0.637984\pi$
$572$	0	0
$573$	−9319.64	−0.679466
$574$	0	0
$575$	10753.1	0.779886
$576$	0	0
$577$	−27029.6	−1.95019	−0.975094	−	0.221790i	$-0.928810\pi$
$577$	−27029.6	−1.95019	−0.975094	+	0.221790i	$0.928810\pi$
$578$	0	0
$579$	21545.5	1.54646
$580$	0	0
$581$	−7502.52	−0.535726
$582$	0	0
$583$	1264.07	0.0897986
$584$	0	0
$585$	−1499.32	−0.105965
$586$	0	0
$587$	−15200.4	−1.06881	−0.534403	−	0.845230i	$-0.679464\pi$
$587$	−15200.4	−1.06881	−0.534403	+	0.845230i	$0.679464\pi$
$588$	0	0
$589$	−3206.23	−0.224296
$590$	0	0
$591$	23139.4	1.61054
$592$	0	0
$593$	−19026.7	−1.31759	−0.658796	−	0.752322i	$-0.728934\pi$
$593$	−19026.7	−1.31759	−0.658796	+	0.752322i	$0.728934\pi$
$594$	0	0
$595$	−29322.2	−2.02033
$596$	0	0
$597$	13604.9	0.932681
$598$	0	0
$599$	−3927.31	−0.267889	−0.133945	−	0.990989i	$-0.542764\pi$
$599$	−3927.31	−0.267889	−0.133945	+	0.990989i	$0.542764\pi$
$600$	0	0
$601$	13718.1	0.931069	0.465534	−	0.885030i	$-0.345862\pi$
$601$	13718.1	0.931069	0.465534	+	0.885030i	$0.345862\pi$
$602$	0	0
$603$	1926.48	0.130104
$604$	0	0
$605$	−22936.8	−1.54134
$606$	0	0
$607$	−26461.5	−1.76942	−0.884712	−	0.466138i	$-0.845645\pi$
$607$	−26461.5	−1.76942	−0.884712	+	0.466138i	$0.845645\pi$
$608$	0	0
$609$	−29779.3	−1.98148
$610$	0	0
$611$	517.534	0.0342671
$612$	0	0
$613$	−233.384	−0.0153773	−0.00768865	−	0.999970i	$-0.502447\pi$
$613$	−233.384	−0.0153773	−0.00768865	+	0.999970i	$0.502447\pi$
$614$	0	0
$615$	−37605.9	−2.46572
$616$	0	0
$617$	4202.77	0.274225	0.137113	−	0.990555i	$-0.456218\pi$
$617$	4202.77	0.274225	0.137113	+	0.990555i	$0.456218\pi$
$618$	0	0
$619$	23009.4	1.49407	0.747033	−	0.664787i	$-0.231478\pi$
$619$	23009.4	1.49407	0.747033	+	0.664787i	$0.231478\pi$
$620$	0	0
$621$	3150.93	0.203611
$622$	0	0
$623$	−24929.2	−1.60316
$624$	0	0
$625$	448.344	0.0286940
$626$	0	0
$627$	−1037.97	−0.0661126
$628$	0	0
$629$	9233.45	0.585313
$630$	0	0
$631$	−18819.3	−1.18730	−0.593650	−	0.804723i	$-0.702314\pi$
$631$	−18819.3	−1.18730	−0.593650	+	0.804723i	$0.702314\pi$
$632$	0	0
$633$	28943.3	1.81737
$634$	0	0
$635$	−16094.3	−1.00580
$636$	0	0
$637$	−1481.84	−0.0921705
$638$	0	0
$639$	−8547.46	−0.529159
$640$	0	0
$641$	25253.7	1.55610	0.778052	−	0.628200i	$-0.216208\pi$
$641$	25253.7	1.55610	0.778052	+	0.628200i	$0.216208\pi$
$642$	0	0
$643$	−11712.1	−0.718324	−0.359162	−	0.933275i	$-0.616937\pi$
$643$	−11712.1	−0.718324	−0.359162	+	0.933275i	$0.616937\pi$
$644$	0	0
$645$	−54678.1	−3.33791
$646$	0	0
$647$	26533.3	1.61226	0.806131	−	0.591737i	$-0.201558\pi$
$647$	26533.3	1.61226	0.806131	+	0.591737i	$0.201558\pi$
$648$	0	0
$649$	−1483.25	−0.0897111
$650$	0	0
$651$	29281.1	1.76285
$652$	0	0
$653$	27898.9	1.67193	0.835964	−	0.548785i	$-0.184909\pi$
$653$	27898.9	1.67193	0.835964	+	0.548785i	$0.184909\pi$
$654$	0	0
$655$	42520.4	2.53650
$656$	0	0
$657$	15278.2	0.907245
$658$	0	0
$659$	1274.66	0.0753468	0.0376734	−	0.999290i	$-0.488005\pi$
$659$	1274.66	0.0753468	0.0376734	+	0.999290i	$0.488005\pi$
$660$	0	0
$661$	−5049.52	−0.297131	−0.148565	−	0.988903i	$-0.547466\pi$
$661$	−5049.52	−0.297131	−0.148565	+	0.988903i	$0.547466\pi$
$662$	0	0
$663$	−1919.38	−0.112432
$664$	0	0
$665$	8901.86	0.519097
$666$	0	0
$667$	9053.01	0.525538
$668$	0	0
$669$	5545.14	0.320460
$670$	0	0
$671$	3286.86	0.189103
$672$	0	0
$673$	8398.64	0.481045	0.240523	−	0.970644i	$-0.422681\pi$
$673$	8398.64	0.481045	0.240523	+	0.970644i	$0.422681\pi$
$674$	0	0
$675$	12176.5	0.694333
$676$	0	0
$677$	9875.31	0.560619	0.280309	−	0.959910i	$-0.409563\pi$
$677$	9875.31	0.560619	0.280309	+	0.959910i	$0.409563\pi$
$678$	0	0
$679$	−5670.91	−0.320515
$680$	0	0
$681$	−10082.1	−0.567325
$682$	0	0
$683$	8653.78	0.484814	0.242407	−	0.970175i	$-0.422063\pi$
$683$	8653.78	0.484814	0.242407	+	0.970175i	$0.422063\pi$
$684$	0	0
$685$	40708.0	2.27062
$686$	0	0
$687$	35262.2	1.95828
$688$	0	0
$689$	−709.629	−0.0392376
$690$	0	0
$691$	−2916.50	−0.160563	−0.0802813	−	0.996772i	$-0.525582\pi$
$691$	−2916.50	−0.160563	−0.0802813	+	0.996772i	$0.525582\pi$
$692$	0	0
$693$	3805.10	0.208577
$694$	0	0
$695$	5370.37	0.293108
$696$	0	0
$697$	−19324.6	−1.05017
$698$	0	0
$699$	−14366.1	−0.777363
$700$	0	0
$701$	−9070.78	−0.488729	−0.244364	−	0.969683i	$-0.578579\pi$
$701$	−9070.78	−0.488729	−0.244364	+	0.969683i	$0.578579\pi$
$702$	0	0
$703$	−2803.16	−0.150388
$704$	0	0
$705$	13803.2	0.737389
$706$	0	0
$707$	37409.4	1.98999
$708$	0	0
$709$	−5957.11	−0.315549	−0.157774	−	0.987475i	$-0.550432\pi$
$709$	−5957.11	−0.315549	−0.157774	+	0.987475i	$0.550432\pi$
$710$	0	0
$711$	10701.3	0.564459
$712$	0	0
$713$	−8901.55	−0.467553
$714$	0	0
$715$	−673.577	−0.0352312
$716$	0	0
$717$	25337.8	1.31975
$718$	0	0
$719$	31140.8	1.61524	0.807620	−	0.589703i	$-0.200755\pi$
$719$	31140.8	1.61524	0.807620	+	0.589703i	$0.200755\pi$
$720$	0	0
$721$	−22828.0	−1.17914
$722$	0	0
$723$	−43086.4	−2.21632
$724$	0	0
$725$	34984.7	1.79214
$726$	0	0
$727$	−14969.7	−0.763682	−0.381841	−	0.924228i	$-0.624710\pi$
$727$	−14969.7	−0.763682	−0.381841	+	0.924228i	$0.624710\pi$
$728$	0	0
$729$	−5283.30	−0.268420
$730$	0	0
$731$	−28097.5	−1.42165
$732$	0	0
$733$	−12414.1	−0.625545	−0.312772	−	0.949828i	$-0.601258\pi$
$733$	−12414.1	−0.625545	−0.312772	+	0.949828i	$0.601258\pi$
$734$	0	0
$735$	−39522.3	−1.98341
$736$	0	0
$737$	865.481	0.0432570
$738$	0	0
$739$	−1324.11	−0.0659111	−0.0329555	−	0.999457i	$-0.510492\pi$
$739$	−1324.11	−0.0659111	−0.0329555	+	0.999457i	$0.510492\pi$
$740$	0	0
$741$	582.700	0.0288880
$742$	0	0
$743$	4391.55	0.216838	0.108419	−	0.994105i	$-0.465421\pi$
$743$	4391.55	0.216838	0.108419	+	0.994105i	$0.465421\pi$
$744$	0	0
$745$	32480.1	1.59729
$746$	0	0
$747$	−5257.76	−0.257525
$748$	0	0
$749$	33789.0	1.64836
$750$	0	0
$751$	31947.5	1.55230	0.776152	−	0.630546i	$-0.217169\pi$
$751$	31947.5	1.55230	0.776152	+	0.630546i	$0.217169\pi$
$752$	0	0
$753$	−42159.0	−2.04032
$754$	0	0
$755$	42663.4	2.05653
$756$	0	0
$757$	18569.8	0.891585	0.445793	−	0.895136i	$-0.352922\pi$
$757$	18569.8	0.891585	0.445793	+	0.895136i	$0.352922\pi$
$758$	0	0
$759$	−2881.75	−0.137814
$760$	0	0
$761$	5507.32	0.262339	0.131170	−	0.991360i	$-0.458127\pi$
$761$	5507.32	0.262339	0.131170	+	0.991360i	$0.458127\pi$
$762$	0	0
$763$	22490.6	1.06712
$764$	0	0
$765$	−20549.0	−0.971178
$766$	0	0
$767$	832.669	0.0391994
$768$	0	0
$769$	−14977.9	−0.702362	−0.351181	−	0.936308i	$-0.614220\pi$
$769$	−14977.9	−0.702362	−0.351181	+	0.936308i	$0.614220\pi$
$770$	0	0
$771$	21384.0	0.998864
$772$	0	0
$773$	19545.6	0.909450	0.454725	−	0.890632i	$-0.349738\pi$
$773$	19545.6	0.909450	0.454725	+	0.890632i	$0.349738\pi$
$774$	0	0
$775$	−34399.4	−1.59440
$776$	0	0
$777$	25600.0	1.18197
$778$	0	0
$779$	5866.70	0.269829
$780$	0	0
$781$	−3839.98	−0.175935
$782$	0	0
$783$	10251.4	0.467887
$784$	0	0
$785$	−26081.7	−1.18585
$786$	0	0
$787$	4274.62	0.193613	0.0968067	−	0.995303i	$-0.469137\pi$
$787$	4274.62	0.193613	0.0968067	+	0.995303i	$0.469137\pi$
$788$	0	0
$789$	−17624.2	−0.795233
$790$	0	0
$791$	−30523.2	−1.37204
$792$	0	0
$793$	−1845.19	−0.0826287
$794$	0	0
$795$	−18926.6	−0.844350
$796$	0	0
$797$	−25450.6	−1.13112	−0.565562	−	0.824706i	$-0.691341\pi$
$797$	−25450.6	−1.13112	−0.565562	+	0.824706i	$0.691341\pi$
$798$	0	0
$799$	7093.08	0.314062
$800$	0	0
$801$	−17470.4	−0.770645
$802$	0	0
$803$	6863.81	0.301642
$804$	0	0
$805$	24714.5	1.08207
$806$	0	0
$807$	−49999.7	−2.18101
$808$	0	0
$809$	4002.04	0.173924	0.0869619	−	0.996212i	$-0.472284\pi$
$809$	4002.04	0.173924	0.0869619	+	0.996212i	$0.472284\pi$
$810$	0	0
$811$	37915.1	1.64165	0.820826	−	0.571179i	$-0.193514\pi$
$811$	37915.1	1.64165	0.820826	+	0.571179i	$0.193514\pi$
$812$	0	0
$813$	−26894.5	−1.16019
$814$	0	0
$815$	−2311.99	−0.0993685
$816$	0	0
$817$	8530.05	0.365274
$818$	0	0
$819$	−2136.12	−0.0911379
$820$	0	0
$821$	4739.43	0.201470	0.100735	−	0.994913i	$-0.467881\pi$
$821$	4739.43	0.201470	0.100735	+	0.994913i	$0.467881\pi$
$822$	0	0
$823$	−20752.2	−0.878952	−0.439476	−	0.898254i	$-0.644836\pi$
$823$	−20752.2	−0.878952	−0.439476	+	0.898254i	$0.644836\pi$
$824$	0	0
$825$	−11136.3	−0.469959
$826$	0	0
$827$	−34264.8	−1.44075	−0.720377	−	0.693583i	$-0.756031\pi$
$827$	−34264.8	−1.44075	−0.720377	+	0.693583i	$0.756031\pi$
$828$	0	0
$829$	−39707.5	−1.66357	−0.831784	−	0.555100i	$-0.812680\pi$
$829$	−39707.5	−1.66357	−0.831784	+	0.555100i	$0.812680\pi$
$830$	0	0
$831$	39159.3	1.63468
$832$	0	0
$833$	−20309.4	−0.844753
$834$	0	0
$835$	−62001.8	−2.56965
$836$	0	0
$837$	−10079.9	−0.416263
$838$	0	0
$839$	4524.04	0.186159	0.0930794	−	0.995659i	$-0.470329\pi$
$839$	4524.04	0.186159	0.0930794	+	0.995659i	$0.470329\pi$
$840$	0	0
$841$	5064.59	0.207659
$842$	0	0
$843$	50400.1	2.05916
$844$	0	0
$845$	−39462.7	−1.60658
$846$	0	0
$847$	−32678.5	−1.32568
$848$	0	0
$849$	39698.0	1.60475
$850$	0	0
$851$	−7782.47	−0.313490
$852$	0	0
$853$	−7595.54	−0.304884	−0.152442	−	0.988312i	$-0.548714\pi$
$853$	−7595.54	−0.304884	−0.152442	+	0.988312i	$0.548714\pi$
$854$	0	0
$855$	6238.42	0.249531
$856$	0	0
$857$	−19528.9	−0.778405	−0.389203	−	0.921152i	$-0.627249\pi$
$857$	−19528.9	−0.778405	−0.389203	+	0.921152i	$0.627249\pi$
$858$	0	0
$859$	−25980.8	−1.03196	−0.515979	−	0.856601i	$-0.672572\pi$
$859$	−25980.8	−1.03196	−0.515979	+	0.856601i	$0.672572\pi$
$860$	0	0
$861$	−53578.0	−2.12071
$862$	0	0
$863$	48294.6	1.90494	0.952472	−	0.304625i	$-0.0985311\pi$
$863$	48294.6	1.90494	0.952472	+	0.304625i	$0.0985311\pi$
$864$	0	0
$865$	−6575.43	−0.258464
$866$	0	0
$867$	6690.03	0.262059
$868$	0	0
$869$	4807.61	0.187672
$870$	0	0
$871$	−485.866	−0.0189012
$872$	0	0
$873$	−3974.17	−0.154072
$874$	0	0
$875$	36942.3	1.42729
$876$	0	0
$877$	44377.5	1.70869	0.854346	−	0.519705i	$-0.173958\pi$
$877$	44377.5	1.70869	0.854346	+	0.519705i	$0.173958\pi$
$878$	0	0
$879$	−21797.7	−0.836426
$880$	0	0
$881$	−12139.4	−0.464231	−0.232116	−	0.972688i	$-0.574565\pi$
$881$	−12139.4	−0.464231	−0.232116	+	0.972688i	$0.574565\pi$
$882$	0	0
$883$	−5048.07	−0.192391	−0.0961954	−	0.995362i	$-0.530667\pi$
$883$	−5048.07	−0.192391	−0.0961954	+	0.995362i	$0.530667\pi$
$884$	0	0
$885$	22208.2	0.843527
$886$	0	0
$887$	−20373.4	−0.771221	−0.385610	−	0.922662i	$-0.626009\pi$
$887$	−20373.4	−0.771221	−0.385610	+	0.922662i	$0.626009\pi$
$888$	0	0
$889$	−22929.9	−0.865065
$890$	0	0
$891$	−7239.72	−0.272211
$892$	0	0
$893$	−2153.37	−0.0806940
$894$	0	0
$895$	−43846.5	−1.63757
$896$	0	0
$897$	1617.76	0.0602180
$898$	0	0
$899$	−28960.8	−1.07441
$900$	0	0
$901$	−9725.85	−0.359617
$902$	0	0
$903$	−77901.2	−2.87086
$904$	0	0
$905$	−44331.6	−1.62832
$906$	0	0
$907$	7456.13	0.272962	0.136481	−	0.990643i	$-0.456421\pi$
$907$	7456.13	0.272962	0.136481	+	0.990643i	$0.456421\pi$
$908$	0	0
$909$	26216.5	0.956596
$910$	0	0
$911$	10653.2	0.387440	0.193720	−	0.981057i	$-0.437945\pi$
$911$	10653.2	0.387440	0.193720	+	0.981057i	$0.437945\pi$
$912$	0	0
$913$	−2362.07	−0.0856224
$914$	0	0
$915$	−49213.3	−1.77808
$916$	0	0
$917$	60579.8	2.18159
$918$	0	0
$919$	12569.7	0.451183	0.225591	−	0.974222i	$-0.427569\pi$
$919$	12569.7	0.451183	0.225591	+	0.974222i	$0.427569\pi$
$920$	0	0
$921$	50297.8	1.79953
$922$	0	0
$923$	2155.70	0.0768752
$924$	0	0
$925$	−30074.8	−1.06903
$926$	0	0
$927$	−15997.9	−0.566816
$928$	0	0
$929$	4920.06	0.173759	0.0868795	−	0.996219i	$-0.472311\pi$
$929$	4920.06	0.173759	0.0868795	+	0.996219i	$0.472311\pi$
$930$	0	0
$931$	6165.67	0.217048
$932$	0	0
$933$	14350.4	0.503548
$934$	0	0
$935$	−9231.74	−0.322898
$936$	0	0
$937$	1991.87	0.0694465	0.0347233	−	0.999397i	$-0.488945\pi$
$937$	1991.87	0.0694465	0.0347233	+	0.999397i	$0.488945\pi$
$938$	0	0
$939$	−14857.0	−0.516337
$940$	0	0
$941$	−7640.33	−0.264684	−0.132342	−	0.991204i	$-0.542250\pi$
$941$	−7640.33	−0.264684	−0.132342	+	0.991204i	$0.542250\pi$
$942$	0	0
$943$	16287.9	0.562467
$944$	0	0
$945$	27986.0	0.963371
$946$	0	0
$947$	6521.15	0.223769	0.111884	−	0.993721i	$-0.464311\pi$
$947$	6521.15	0.223769	0.111884	+	0.993721i	$0.464311\pi$
$948$	0	0
$949$	−3853.22	−0.131803
$950$	0	0
$951$	−2883.40	−0.0983182
$952$	0	0
$953$	−35757.9	−1.21544	−0.607719	−	0.794152i	$-0.707915\pi$
$953$	−35757.9	−1.21544	−0.607719	+	0.794152i	$0.707915\pi$
$954$	0	0
$955$	25164.1	0.852659
$956$	0	0
$957$	−9375.64	−0.316689
$958$	0	0
$959$	57997.6	1.95291
$960$	0	0
$961$	−1314.75	−0.0441323
$962$	0	0
$963$	23679.3	0.792374
$964$	0	0
$965$	−58175.1	−1.94065
$966$	0	0
$967$	6342.30	0.210915	0.105457	−	0.994424i	$-0.466369\pi$
$967$	6342.30	0.210915	0.105457	+	0.994424i	$0.466369\pi$
$968$	0	0
$969$	7986.21	0.264762
$970$	0	0
$971$	−30351.2	−1.00311	−0.501553	−	0.865127i	$-0.667238\pi$
$971$	−30351.2	−1.00311	−0.501553	+	0.865127i	$0.667238\pi$
$972$	0	0
$973$	7651.29	0.252096
$974$	0	0
$975$	6251.73	0.205349
$976$	0	0
$977$	39843.6	1.30472	0.652359	−	0.757910i	$-0.273780\pi$
$977$	39843.6	1.30472	0.652359	+	0.757910i	$0.273780\pi$
$978$	0	0
$979$	−7848.66	−0.256225
$980$	0	0
$981$	15761.4	0.512969
$982$	0	0
$983$	−24068.7	−0.780949	−0.390475	−	0.920614i	$-0.627689\pi$
$983$	−24068.7	−0.780949	−0.390475	+	0.920614i	$0.627689\pi$
$984$	0	0
$985$	−62478.9	−2.02106
$986$	0	0
$987$	19665.8	0.634213
$988$	0	0
$989$	23682.2	0.761426
$990$	0	0
$991$	−3235.83	−0.103723	−0.0518615	−	0.998654i	$-0.516515\pi$
$991$	−3235.83	−0.103723	−0.0518615	+	0.998654i	$0.516515\pi$
$992$	0	0
$993$	−5140.86	−0.164290
$994$	0	0
$995$	−36734.6	−1.17042
$996$	0	0
$997$	19444.5	0.617665	0.308833	−	0.951116i	$-0.400062\pi$
$997$	19444.5	0.617665	0.308833	+	0.951116i	$0.400062\pi$
$998$	0	0
$999$	−8812.68	−0.279100

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.4.a.i.1.1		3
4.3	odd	2		19.4.a.b.1.1	✓	3
8.3	odd	2		1216.4.a.s.1.1		3
8.5	even	2		1216.4.a.u.1.3		3
12.11	even	2		171.4.a.f.1.3		3
20.3	even	4		475.4.b.f.324.5		6
20.7	even	4		475.4.b.f.324.2		6
20.19	odd	2		475.4.a.f.1.3		3
28.27	even	2		931.4.a.c.1.1		3
44.43	even	2		2299.4.a.h.1.3		3
76.75	even	2		361.4.a.i.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.4.a.b.1.1	✓	3	4.3	odd	2
171.4.a.f.1.3		3	12.11	even	2
304.4.a.i.1.1		3	1.1	even	1	trivial
361.4.a.i.1.3		3	76.75	even	2
475.4.a.f.1.3		3	20.19	odd	2
475.4.b.f.324.2		6	20.7	even	4
475.4.b.f.324.5		6	20.3	even	4
931.4.a.c.1.1		3	28.27	even	2
1216.4.a.s.1.1		3	8.3	odd	2
1216.4.a.u.1.3		3	8.5	even	2
2299.4.a.h.1.3		3	44.43	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)

\(f(q)\)	\(=\)	\(q-6.71610 q^{3} +18.1342 q^{5} +25.8362 q^{7} +18.1060 q^{9} +8.13420 q^{11} -4.56640 q^{13} -121.791 q^{15} -62.5850 q^{17} +19.0000 q^{19} -173.518 q^{21} +52.7502 q^{23} +203.849 q^{25} +59.7330 q^{27} +171.620 q^{29} -168.749 q^{31} -54.6301 q^{33} +468.519 q^{35} -147.534 q^{37} +30.6684 q^{39} +308.774 q^{41} +448.950 q^{43} +328.338 q^{45} -113.335 q^{47} +324.509 q^{49} +420.327 q^{51} +155.402 q^{53} +147.507 q^{55} -127.606 q^{57} -182.347 q^{59} +404.080 q^{61} +467.790 q^{63} -82.8080 q^{65} +106.400 q^{67} -354.276 q^{69} -472.079 q^{71} +843.821 q^{73} -1369.07 q^{75} +210.157 q^{77} +591.036 q^{79} -890.035 q^{81} -290.388 q^{83} -1134.93 q^{85} -1152.62 q^{87} -964.896 q^{89} -117.978 q^{91} +1133.34 q^{93} +344.550 q^{95} -219.495 q^{97} +147.278 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{3} + 14 q^{5} + 35 q^{7} + 48 q^{9} - 16 q^{11} + 65 q^{13} - 140 q^{15} + 29 q^{17} + 57 q^{19} - 25 q^{21} + 101 q^{23} - 37 q^{25} + 377 q^{27} + 377 q^{29} + 140 q^{31} - 130 q^{33} + 438 q^{35}+ \cdots - 250 q^{99}+O(q^{100}) \) 3 * q - q^3 + 14 * q^5 + 35 * q^7 + 48 * q^9 - 16 * q^11 + 65 * q^13 - 140 * q^15 + 29 * q^17 + 57 * q^19 - 25 * q^21 + 101 * q^23 - 37 * q^25 + 377 * q^27 + 377 * q^29 + 140 * q^31 - 130 * q^33 + 438 * q^35 - 290 * q^37 + 371 * q^39 + 956 * q^41 + 570 * q^43 + 230 * q^45 - 66 * q^47 - 98 * q^49 - 187 * q^51 + 817 * q^53 + 198 * q^55 - 19 * q^57 - 265 * q^59 + 988 * q^61 + 1304 * q^63 - 240 * q^65 + 207 * q^67 + 807 * q^69 - 846 * q^71 + 627 * q^73 - 2031 * q^75 + 88 * q^77 - 382 * q^79 - 501 * q^81 + 766 * q^83 - 1242 * q^85 + 1671 * q^87 - 172 * q^89 + 457 * q^91 + 892 * q^93 + 266 * q^95 - 2450 * q^97 - 250 * q^99

Introduction

L-functions

Modular forms

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