LMFDB - Embedded newform 1216.4.a.y.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,4,Mod(1,1216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, 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("1216.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1216.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:	$71.7463225670$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 28x^{3} - 8x^{2} + 73x - 12 $ x^5 - 2x^4 - 28x^3 - 8x^2 + 73x - 12
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 608)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.35870$ of defining polynomial
Character	$\chi$	$=$	1216.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-7.68562 q^{3} +15.2627 q^{5} -2.79440 q^{7} +32.0687 q^{9} +0.456205 q^{11} +0.618515 q^{13} -117.303 q^{15} +12.2449 q^{17} +19.0000 q^{19} +21.4767 q^{21} -116.067 q^{23} +107.949 q^{25} -38.9560 q^{27} -100.701 q^{29} -206.884 q^{31} -3.50622 q^{33} -42.6500 q^{35} +212.785 q^{37} -4.75367 q^{39} -42.1719 q^{41} +34.2379 q^{43} +489.453 q^{45} -393.638 q^{47} -335.191 q^{49} -94.1097 q^{51} +497.486 q^{53} +6.96291 q^{55} -146.027 q^{57} +82.9196 q^{59} +655.685 q^{61} -89.6128 q^{63} +9.44018 q^{65} +88.1809 q^{67} +892.045 q^{69} +430.377 q^{71} +65.8355 q^{73} -829.651 q^{75} -1.27482 q^{77} +528.469 q^{79} -566.454 q^{81} -592.763 q^{83} +186.890 q^{85} +773.950 q^{87} +1151.89 q^{89} -1.72838 q^{91} +1590.03 q^{93} +289.990 q^{95} -51.7168 q^{97} +14.6299 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 6 q^{3} - 5 q^{5} - 7 q^{7} - 5 q^{9} - 13 q^{11} + 72 q^{13} - 72 q^{15} - 59 q^{17} + 95 q^{19} + 224 q^{21} - 52 q^{23} - 86 q^{25} - 54 q^{27} + 128 q^{29} + 110 q^{31} - 68 q^{33} + 45 q^{35}+ \cdots - 601 q^{99}+O(q^{100}) $ 5 * q - 6 * q^3 - 5 * q^5 - 7 * q^7 - 5 * q^9 - 13 * q^11 + 72 * q^13 - 72 * q^15 - 59 * q^17 + 95 * q^19 + 224 * q^21 - 52 * q^23 - 86 * q^25 - 54 * q^27 + 128 * q^29 + 110 * q^31 - 68 * q^33 + 45 * q^35 + 436 * q^37 - 356 * q^39 - 804 * q^41 + 143 * q^43 + 579 * q^45 - 661 * q^47 - 406 * q^49 - 570 * q^51 + 898 * q^53 - 17 * q^55 - 114 * q^57 - 196 * q^59 + 1079 * q^61 - 143 * q^63 - 1632 * q^65 - 832 * q^67 + 1644 * q^69 - 834 * q^71 - 1375 * q^73 - 1054 * q^75 + 1473 * q^77 - 154 * q^79 - 1859 * q^81 - 2224 * q^83 + 1807 * q^85 - 2004 * q^87 - 542 * q^89 - 1890 * q^91 + 2788 * q^93 - 95 * q^95 - 1528 * q^97 - 601 * 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
$3$	−7.68562	−1.47910	−0.739549	−	0.673103i	$-0.764961\pi$
$3$	−7.68562	−1.47910	−0.739549	+	0.673103i	$0.764961\pi$
$4$	0	0
$5$	15.2627	1.36513	0.682567	−	0.730823i	$-0.260864\pi$
$5$	15.2627	1.36513	0.682567	+	0.730823i	$0.260864\pi$
$6$	0	0
$7$	−2.79440	−0.150883	−0.0754417	−	0.997150i	$-0.524037\pi$
$7$	−2.79440	−0.150883	−0.0754417	+	0.997150i	$0.524037\pi$
$8$	0	0
$9$	32.0687	1.18773
$10$	0	0
$11$	0.456205	0.0125046	0.00625232	−	0.999980i	$-0.498010\pi$
$11$	0.456205	0.0125046	0.00625232	+	0.999980i	$0.498010\pi$
$12$	0	0
$13$	0.618515	0.0131958	0.00659789	−	0.999978i	$-0.497900\pi$
$13$	0.618515	0.0131958	0.00659789	+	0.999978i	$0.497900\pi$
$14$	0	0
$15$	−117.303	−2.01916
$16$	0	0
$17$	12.2449	0.174696	0.0873479	−	0.996178i	$-0.472161\pi$
$17$	12.2449	0.174696	0.0873479	+	0.996178i	$0.472161\pi$
$18$	0	0
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	0	0
$21$	21.4767	0.223171
$22$	0	0
$23$	−116.067	−1.05224	−0.526122	−	0.850409i	$-0.676355\pi$
$23$	−116.067	−1.05224	−0.526122	+	0.850409i	$0.676355\pi$
$24$	0	0
$25$	107.949	0.863588
$26$	0	0
$27$	−38.9560	−0.277670
$28$	0	0
$29$	−100.701	−0.644818	−0.322409	−	0.946600i	$-0.604493\pi$
$29$	−100.701	−0.644818	−0.322409	+	0.946600i	$0.604493\pi$
$30$	0	0
$31$	−206.884	−1.19863	−0.599315	−	0.800513i	$-0.704560\pi$
$31$	−206.884	−1.19863	−0.599315	+	0.800513i	$0.704560\pi$
$32$	0	0
$33$	−3.50622	−0.0184956
$34$	0	0
$35$	−42.6500	−0.205976
$36$	0	0
$37$	212.785	0.945450	0.472725	−	0.881210i	$-0.343270\pi$
$37$	212.785	0.945450	0.472725	+	0.881210i	$0.343270\pi$
$38$	0	0
$39$	−4.75367	−0.0195179
$40$	0	0
$41$	−42.1719	−0.160638	−0.0803189	−	0.996769i	$-0.525594\pi$
$41$	−42.1719	−0.160638	−0.0803189	+	0.996769i	$0.525594\pi$
$42$	0	0
$43$	34.2379	0.121424	0.0607120	−	0.998155i	$-0.480663\pi$
$43$	34.2379	0.121424	0.0607120	+	0.998155i	$0.480663\pi$
$44$	0	0
$45$	489.453	1.62141
$46$	0	0
$47$	−393.638	−1.22166	−0.610829	−	0.791762i	$-0.709164\pi$
$47$	−393.638	−1.22166	−0.610829	+	0.791762i	$0.709164\pi$
$48$	0	0
$49$	−335.191	−0.977234
$50$	0	0
$51$	−94.1097	−0.258392
$52$	0	0
$53$	497.486	1.28934	0.644670	−	0.764461i	$-0.276995\pi$
$53$	497.486	1.28934	0.644670	+	0.764461i	$0.276995\pi$
$54$	0	0
$55$	6.96291	0.0170705
$56$	0	0
$57$	−146.027	−0.339328
$58$	0	0
$59$	82.9196	0.182970	0.0914849	−	0.995806i	$-0.470839\pi$
$59$	82.9196	0.182970	0.0914849	+	0.995806i	$0.470839\pi$
$60$	0	0
$61$	655.685	1.37626	0.688130	−	0.725588i	$-0.258432\pi$
$61$	655.685	1.37626	0.688130	+	0.725588i	$0.258432\pi$
$62$	0	0
$63$	−89.6128	−0.179209
$64$	0	0
$65$	9.44018	0.0180140
$66$	0	0
$67$	88.1809	0.160791	0.0803956	−	0.996763i	$-0.474382\pi$
$67$	88.1809	0.160791	0.0803956	+	0.996763i	$0.474382\pi$
$68$	0	0
$69$	892.045	1.55637
$70$	0	0
$71$	430.377	0.719386	0.359693	−	0.933071i	$-0.382881\pi$
$71$	430.377	0.719386	0.359693	+	0.933071i	$0.382881\pi$
$72$	0	0
$73$	65.8355	0.105554	0.0527771	−	0.998606i	$-0.483193\pi$
$73$	65.8355	0.105554	0.0527771	+	0.998606i	$0.483193\pi$
$74$	0	0
$75$	−829.651	−1.27733
$76$	0	0
$77$	−1.27482	−0.00188675
$78$	0	0
$79$	528.469	0.752626	0.376313	−	0.926493i	$-0.377192\pi$
$79$	528.469	0.752626	0.376313	+	0.926493i	$0.377192\pi$
$80$	0	0
$81$	−566.454	−0.777029
$82$	0	0
$83$	−592.763	−0.783906	−0.391953	−	0.919985i	$-0.628200\pi$
$83$	−592.763	−0.783906	−0.391953	+	0.919985i	$0.628200\pi$
$84$	0	0
$85$	186.890	0.238483
$86$	0	0
$87$	773.950	0.953749
$88$	0	0
$89$	1151.89	1.37191	0.685955	−	0.727644i	$-0.259385\pi$
$89$	1151.89	1.37191	0.685955	+	0.727644i	$0.259385\pi$
$90$	0	0
$91$	−1.72838	−0.00199103
$92$	0	0
$93$	1590.03	1.77289
$94$	0	0
$95$	289.990	0.313183
$96$	0	0
$97$	−51.7168	−0.0541345	−0.0270673	−	0.999634i	$-0.508617\pi$
$97$	−51.7168	−0.0541345	−0.0270673	+	0.999634i	$0.508617\pi$
$98$	0	0
$99$	14.6299	0.0148521
$100$	0	0
$101$	−1656.63	−1.63209	−0.816044	−	0.577990i	$-0.803837\pi$
$101$	−1656.63	−1.63209	−0.816044	+	0.577990i	$0.803837\pi$
$102$	0	0
$103$	157.395	0.150569	0.0752843	−	0.997162i	$-0.476014\pi$
$103$	157.395	0.150569	0.0752843	+	0.997162i	$0.476014\pi$
$104$	0	0
$105$	327.791	0.304659
$106$	0	0
$107$	−943.505	−0.852449	−0.426225	−	0.904617i	$-0.640157\pi$
$107$	−943.505	−0.852449	−0.426225	+	0.904617i	$0.640157\pi$
$108$	0	0
$109$	−978.577	−0.859915	−0.429957	−	0.902849i	$-0.641471\pi$
$109$	−978.577	−0.859915	−0.429957	+	0.902849i	$0.641471\pi$
$110$	0	0
$111$	−1635.38	−1.39841
$112$	0	0
$113$	−979.602	−0.815515	−0.407758	−	0.913090i	$-0.633689\pi$
$113$	−979.602	−0.815515	−0.407758	+	0.913090i	$0.633689\pi$
$114$	0	0
$115$	−1771.49	−1.43645
$116$	0	0
$117$	19.8350	0.0156730
$118$	0	0
$119$	−34.2172	−0.0263587
$120$	0	0
$121$	−1330.79	−0.999844
$122$	0	0
$123$	324.117	0.237599
$124$	0	0
$125$	−260.251	−0.186220
$126$	0	0
$127$	−1327.42	−0.927473	−0.463737	−	0.885973i	$-0.653492\pi$
$127$	−1327.42	−0.927473	−0.463737	+	0.885973i	$0.653492\pi$
$128$	0	0
$129$	−263.139	−0.179598
$130$	0	0
$131$	2034.30	1.35678	0.678388	−	0.734704i	$-0.262679\pi$
$131$	2034.30	1.35678	0.678388	+	0.734704i	$0.262679\pi$
$132$	0	0
$133$	−53.0936	−0.0346150
$134$	0	0
$135$	−594.571	−0.379056
$136$	0	0
$137$	−1715.79	−1.07000	−0.535001	−	0.844852i	$-0.679689\pi$
$137$	−1715.79	−1.07000	−0.535001	+	0.844852i	$0.679689\pi$
$138$	0	0
$139$	−1762.30	−1.07537	−0.537683	−	0.843147i	$-0.680700\pi$
$139$	−1762.30	−1.07537	−0.537683	+	0.843147i	$0.680700\pi$
$140$	0	0
$141$	3025.35	1.80695
$142$	0	0
$143$	0.282170	0.000165009 0
$144$	0	0
$145$	−1536.97	−0.880263
$146$	0	0
$147$	2576.15	1.44542
$148$	0	0
$149$	−2065.65	−1.13574	−0.567869	−	0.823119i	$-0.692232\pi$
$149$	−2065.65	−1.13574	−0.567869	+	0.823119i	$0.692232\pi$
$150$	0	0
$151$	1177.09	0.634372	0.317186	−	0.948363i	$-0.397262\pi$
$151$	1177.09	0.634372	0.317186	+	0.948363i	$0.397262\pi$
$152$	0	0
$153$	392.678	0.207491
$154$	0	0
$155$	−3157.61	−1.63629
$156$	0	0
$157$	1802.51	0.916279	0.458140	−	0.888880i	$-0.348516\pi$
$157$	1802.51	0.916279	0.458140	+	0.888880i	$0.348516\pi$
$158$	0	0
$159$	−3823.49	−1.90706
$160$	0	0
$161$	324.337	0.158766
$162$	0	0
$163$	−2084.81	−1.00181	−0.500904	−	0.865503i	$-0.666999\pi$
$163$	−2084.81	−1.00181	−0.500904	+	0.865503i	$0.666999\pi$
$164$	0	0
$165$	−53.5142	−0.0252489
$166$	0	0
$167$	939.440	0.435306	0.217653	−	0.976026i	$-0.430160\pi$
$167$	939.440	0.435306	0.217653	+	0.976026i	$0.430160\pi$
$168$	0	0
$169$	−2196.62	−0.999826
$170$	0	0
$171$	609.305	0.272484
$172$	0	0
$173$	214.271	0.0941662	0.0470831	−	0.998891i	$-0.485007\pi$
$173$	214.271	0.0941662	0.0470831	+	0.998891i	$0.485007\pi$
$174$	0	0
$175$	−301.651	−0.130301
$176$	0	0
$177$	−637.288	−0.270630
$178$	0	0
$179$	2644.23	1.10413	0.552064	−	0.833802i	$-0.313840\pi$
$179$	2644.23	1.10413	0.552064	+	0.833802i	$0.313840\pi$
$180$	0	0
$181$	−1405.73	−0.577278	−0.288639	−	0.957438i	$-0.593203\pi$
$181$	−1405.73	−0.577278	−0.288639	+	0.957438i	$0.593203\pi$
$182$	0	0
$183$	−5039.34	−2.03562
$184$	0	0
$185$	3247.66	1.29066
$186$	0	0
$187$	5.58620	0.00218451
$188$	0	0
$189$	108.859	0.0418957
$190$	0	0
$191$	159.777	0.0605292	0.0302646	−	0.999542i	$-0.490365\pi$
$191$	159.777	0.0605292	0.0302646	+	0.999542i	$0.490365\pi$
$192$	0	0
$193$	−1991.99	−0.742936	−0.371468	−	0.928446i	$-0.621145\pi$
$193$	−1991.99	−0.742936	−0.371468	+	0.928446i	$0.621145\pi$
$194$	0	0
$195$	−72.5536	−0.0266445
$196$	0	0
$197$	−3947.92	−1.42781	−0.713904	−	0.700244i	$-0.753075\pi$
$197$	−3947.92	−1.42781	−0.713904	+	0.700244i	$0.753075\pi$
$198$	0	0
$199$	−5073.60	−1.80733	−0.903663	−	0.428245i	$-0.859132\pi$
$199$	−5073.60	−1.80733	−0.903663	+	0.428245i	$0.859132\pi$
$200$	0	0
$201$	−677.725	−0.237826
$202$	0	0
$203$	281.399	0.0972924
$204$	0	0
$205$	−643.656	−0.219292
$206$	0	0
$207$	−3722.11	−1.24978
$208$	0	0
$209$	8.66790	0.00286876
$210$	0	0
$211$	−1782.72	−0.581646	−0.290823	−	0.956777i	$-0.593929\pi$
$211$	−1782.72	−0.581646	−0.290823	+	0.956777i	$0.593929\pi$
$212$	0	0
$213$	−3307.71	−1.06404
$214$	0	0
$215$	522.561	0.165760
$216$	0	0
$217$	578.118	0.180854
$218$	0	0
$219$	−505.986	−0.156125
$220$	0	0
$221$	7.57366	0.00230525
$222$	0	0
$223$	−602.235	−0.180846	−0.0904230	−	0.995903i	$-0.528822\pi$
$223$	−602.235	−0.180846	−0.0904230	+	0.995903i	$0.528822\pi$
$224$	0	0
$225$	3461.77	1.02571
$226$	0	0
$227$	−3549.26	−1.03776	−0.518882	−	0.854846i	$-0.673652\pi$
$227$	−3549.26	−1.03776	−0.518882	+	0.854846i	$0.673652\pi$
$228$	0	0
$229$	−664.711	−0.191814	−0.0959069	−	0.995390i	$-0.530575\pi$
$229$	−664.711	−0.191814	−0.0959069	+	0.995390i	$0.530575\pi$
$230$	0	0
$231$	9.79779	0.00279068
$232$	0	0
$233$	597.040	0.167869	0.0839344	−	0.996471i	$-0.473251\pi$
$233$	597.040	0.167869	0.0839344	+	0.996471i	$0.473251\pi$
$234$	0	0
$235$	−6007.96	−1.66773
$236$	0	0
$237$	−4061.61	−1.11321
$238$	0	0
$239$	6306.26	1.70677	0.853385	−	0.521281i	$-0.174546\pi$
$239$	6306.26	1.70677	0.853385	+	0.521281i	$0.174546\pi$
$240$	0	0
$241$	−5991.75	−1.60150	−0.800752	−	0.598995i	$-0.795567\pi$
$241$	−5991.75	−1.60150	−0.800752	+	0.598995i	$0.795567\pi$
$242$	0	0
$243$	5405.36	1.42697
$244$	0	0
$245$	−5115.91	−1.33405
$246$	0	0
$247$	11.7518	0.00302732
$248$	0	0
$249$	4555.75	1.15947
$250$	0	0
$251$	−6010.68	−1.51152	−0.755759	−	0.654850i	$-0.772732\pi$
$251$	−6010.68	−1.51152	−0.755759	+	0.654850i	$0.772732\pi$
$252$	0	0
$253$	−52.9503	−0.0131579
$254$	0	0
$255$	−1436.36	−0.352739
$256$	0	0
$257$	−1646.63	−0.399664	−0.199832	−	0.979830i	$-0.564040\pi$
$257$	−1646.63	−0.399664	−0.199832	+	0.979830i	$0.564040\pi$
$258$	0	0
$259$	−594.607	−0.142653
$260$	0	0
$261$	−3229.35	−0.765869
$262$	0	0
$263$	−1824.24	−0.427708	−0.213854	−	0.976866i	$-0.568602\pi$
$263$	−1824.24	−0.427708	−0.213854	+	0.976866i	$0.568602\pi$
$264$	0	0
$265$	7592.96	1.76012
$266$	0	0
$267$	−8852.97	−2.02919
$268$	0	0
$269$	−4156.89	−0.942193	−0.471097	−	0.882082i	$-0.656142\pi$
$269$	−4156.89	−0.942193	−0.471097	+	0.882082i	$0.656142\pi$
$270$	0	0
$271$	790.092	0.177102	0.0885510	−	0.996072i	$-0.471776\pi$
$271$	790.092	0.177102	0.0885510	+	0.996072i	$0.471776\pi$
$272$	0	0
$273$	13.2837	0.00294492
$274$	0	0
$275$	49.2467	0.0107989
$276$	0	0
$277$	2474.69	0.536785	0.268393	−	0.963310i	$-0.413508\pi$
$277$	2474.69	0.536785	0.268393	+	0.963310i	$0.413508\pi$
$278$	0	0
$279$	−6634.51	−1.42365
$280$	0	0
$281$	5244.82	1.11345	0.556726	−	0.830696i	$-0.312057\pi$
$281$	5244.82	1.11345	0.556726	+	0.830696i	$0.312057\pi$
$282$	0	0
$283$	4045.21	0.849691	0.424846	−	0.905266i	$-0.360328\pi$
$283$	4045.21	0.849691	0.424846	+	0.905266i	$0.360328\pi$
$284$	0	0
$285$	−2228.75	−0.463228
$286$	0	0
$287$	117.845	0.0242376
$288$	0	0
$289$	−4763.06	−0.969481
$290$	0	0
$291$	397.476	0.0800702
$292$	0	0
$293$	6654.39	1.32680	0.663402	−	0.748264i	$-0.269112\pi$
$293$	6654.39	1.32680	0.663402	+	0.748264i	$0.269112\pi$
$294$	0	0
$295$	1265.57	0.249778
$296$	0	0
$297$	−17.7719	−0.00347216
$298$	0	0
$299$	−71.7891	−0.0138852
$300$	0	0
$301$	−95.6745	−0.0183209
$302$	0	0
$303$	12732.2	2.41402
$304$	0	0
$305$	10007.5	1.87878
$306$	0	0
$307$	−5788.70	−1.07615	−0.538075	−	0.842897i	$-0.680848\pi$
$307$	−5788.70	−1.07615	−0.538075	+	0.842897i	$0.680848\pi$
$308$	0	0
$309$	−1209.68	−0.222706
$310$	0	0
$311$	−7503.90	−1.36819	−0.684095	−	0.729393i	$-0.739803\pi$
$311$	−7503.90	−1.36819	−0.684095	+	0.729393i	$0.739803\pi$
$312$	0	0
$313$	10133.5	1.82997	0.914983	−	0.403492i	$-0.132204\pi$
$313$	10133.5	1.82997	0.914983	+	0.403492i	$0.132204\pi$
$314$	0	0
$315$	−1367.73	−0.244644
$316$	0	0
$317$	8752.21	1.55070	0.775352	−	0.631529i	$-0.217572\pi$
$317$	8752.21	1.55070	0.775352	+	0.631529i	$0.217572\pi$
$318$	0	0
$319$	−45.9404	−0.00806322
$320$	0	0
$321$	7251.41	1.26086
$322$	0	0
$323$	232.653	0.0400779
$324$	0	0
$325$	66.7678	0.0113957
$326$	0	0
$327$	7520.97	1.27190
$328$	0	0
$329$	1099.98	0.184328
$330$	0	0
$331$	−6268.49	−1.04093	−0.520464	−	0.853883i	$-0.674241\pi$
$331$	−6268.49	−1.04093	−0.520464	+	0.853883i	$0.674241\pi$
$332$	0	0
$333$	6823.74	1.12294
$334$	0	0
$335$	1345.87	0.219501
$336$	0	0
$337$	−8373.12	−1.35345	−0.676725	−	0.736236i	$-0.736602\pi$
$337$	−8373.12	−1.35345	−0.676725	+	0.736236i	$0.736602\pi$
$338$	0	0
$339$	7528.85	1.20623
$340$	0	0
$341$	−94.3818	−0.0149885
$342$	0	0
$343$	1895.14	0.298332
$344$	0	0
$345$	13615.0	2.12465
$346$	0	0
$347$	3172.02	0.490729	0.245365	−	0.969431i	$-0.421092\pi$
$347$	3172.02	0.490729	0.245365	+	0.969431i	$0.421092\pi$
$348$	0	0
$349$	−7706.05	−1.18194	−0.590968	−	0.806695i	$-0.701254\pi$
$349$	−7706.05	−1.18194	−0.590968	+	0.806695i	$0.701254\pi$
$350$	0	0
$351$	−24.0948	−0.00366407
$352$	0	0
$353$	−9170.03	−1.38264	−0.691319	−	0.722550i	$-0.742970\pi$
$353$	−9170.03	−1.38264	−0.691319	+	0.722550i	$0.742970\pi$
$354$	0	0
$355$	6568.70	0.982057
$356$	0	0
$357$	262.980	0.0389871
$358$	0	0
$359$	3545.04	0.521171	0.260585	−	0.965451i	$-0.416084\pi$
$359$	3545.04	0.521171	0.260585	+	0.965451i	$0.416084\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	10228.0	1.47887
$364$	0	0
$365$	1004.82	0.144096
$366$	0	0
$367$	10096.4	1.43604	0.718021	−	0.696021i	$-0.245048\pi$
$367$	10096.4	1.43604	0.718021	+	0.696021i	$0.245048\pi$
$368$	0	0
$369$	−1352.40	−0.190794
$370$	0	0
$371$	−1390.18	−0.194540
$372$	0	0
$373$	8445.91	1.17242	0.586210	−	0.810159i	$-0.300619\pi$
$373$	8445.91	1.17242	0.586210	+	0.810159i	$0.300619\pi$
$374$	0	0
$375$	2000.19	0.275438
$376$	0	0
$377$	−62.2852	−0.00850888
$378$	0	0
$379$	4057.60	0.549933	0.274967	−	0.961454i	$-0.411333\pi$
$379$	4057.60	0.549933	0.274967	+	0.961454i	$0.411333\pi$
$380$	0	0
$381$	10202.0	1.37182
$382$	0	0
$383$	1348.64	0.179927	0.0899637	−	0.995945i	$-0.471325\pi$
$383$	1348.64	0.179927	0.0899637	+	0.995945i	$0.471325\pi$
$384$	0	0
$385$	−19.4572	−0.00257566
$386$	0	0
$387$	1097.96	0.144219
$388$	0	0
$389$	−1764.98	−0.230047	−0.115024	−	0.993363i	$-0.536694\pi$
$389$	−1764.98	−0.230047	−0.115024	+	0.993363i	$0.536694\pi$
$390$	0	0
$391$	−1421.23	−0.183823
$392$	0	0
$393$	−15634.9	−2.00680
$394$	0	0
$395$	8065.85	1.02743
$396$	0	0
$397$	−1813.78	−0.229297	−0.114648	−	0.993406i	$-0.536574\pi$
$397$	−1813.78	−0.229297	−0.114648	+	0.993406i	$0.536574\pi$
$398$	0	0
$399$	408.057	0.0511990
$400$	0	0
$401$	1987.91	0.247560	0.123780	−	0.992310i	$-0.460498\pi$
$401$	1987.91	0.247560	0.123780	+	0.992310i	$0.460498\pi$
$402$	0	0
$403$	−127.961	−0.0158169
$404$	0	0
$405$	−8645.59	−1.06075
$406$	0	0
$407$	97.0737	0.0118225
$408$	0	0
$409$	−9369.41	−1.13273	−0.566366	−	0.824154i	$-0.691651\pi$
$409$	−9369.41	−1.13273	−0.566366	+	0.824154i	$0.691651\pi$
$410$	0	0
$411$	13186.9	1.58264
$412$	0	0
$413$	−231.711	−0.0276071
$414$	0	0
$415$	−9047.14	−1.07014
$416$	0	0
$417$	13544.3	1.59057
$418$	0	0
$419$	−3469.26	−0.404497	−0.202249	−	0.979334i	$-0.564825\pi$
$419$	−3469.26	−0.404497	−0.202249	+	0.979334i	$0.564825\pi$
$420$	0	0
$421$	6978.44	0.807858	0.403929	−	0.914790i	$-0.367644\pi$
$421$	6978.44	0.807858	0.403929	+	0.914790i	$0.367644\pi$
$422$	0	0
$423$	−12623.4	−1.45100
$424$	0	0
$425$	1321.82	0.150865
$426$	0	0
$427$	−1832.25	−0.207655
$428$	0	0
$429$	−2.16865	−0.000244064 0
$430$	0	0
$431$	−141.266	−0.0157878	−0.00789391	−	0.999969i	$-0.502513\pi$
$431$	−141.266	−0.0157878	−0.00789391	+	0.999969i	$0.502513\pi$
$432$	0	0
$433$	4320.19	0.479480	0.239740	−	0.970837i	$-0.422938\pi$
$433$	4320.19	0.479480	0.239740	+	0.970837i	$0.422938\pi$
$434$	0	0
$435$	11812.5	1.30199
$436$	0	0
$437$	−2205.27	−0.241401
$438$	0	0
$439$	−14353.3	−1.56047	−0.780234	−	0.625488i	$-0.784900\pi$
$439$	−14353.3	−1.56047	−0.780234	+	0.625488i	$0.784900\pi$
$440$	0	0
$441$	−10749.1	−1.16069
$442$	0	0
$443$	10320.9	1.10691	0.553453	−	0.832881i	$-0.313310\pi$
$443$	10320.9	1.10691	0.553453	+	0.832881i	$0.313310\pi$
$444$	0	0
$445$	17580.9	1.87284
$446$	0	0
$447$	15875.8	1.67987
$448$	0	0
$449$	−15853.2	−1.66628	−0.833140	−	0.553062i	$-0.813459\pi$
$449$	−15853.2	−1.66628	−0.833140	+	0.553062i	$0.813459\pi$
$450$	0	0
$451$	−19.2391	−0.00200872
$452$	0	0
$453$	−9046.65	−0.938298
$454$	0	0
$455$	−26.3797	−0.00271802
$456$	0	0
$457$	6963.89	0.712816	0.356408	−	0.934330i	$-0.384001\pi$
$457$	6963.89	0.712816	0.356408	+	0.934330i	$0.384001\pi$
$458$	0	0
$459$	−477.012	−0.0485077
$460$	0	0
$461$	−9994.12	−1.00970	−0.504851	−	0.863207i	$-0.668453\pi$
$461$	−9994.12	−1.00970	−0.504851	+	0.863207i	$0.668453\pi$
$462$	0	0
$463$	3474.18	0.348723	0.174362	−	0.984682i	$-0.444214\pi$
$463$	3474.18	0.348723	0.174362	+	0.984682i	$0.444214\pi$
$464$	0	0
$465$	24268.1	2.42023
$466$	0	0
$467$	−6248.92	−0.619199	−0.309599	−	0.950867i	$-0.600195\pi$
$467$	−6248.92	−0.619199	−0.309599	+	0.950867i	$0.600195\pi$
$468$	0	0
$469$	−246.413	−0.0242607
$470$	0	0
$471$	−13853.4	−1.35527
$472$	0	0
$473$	15.6195	0.00151836
$474$	0	0
$475$	2051.02	0.198121
$476$	0	0
$477$	15953.7	1.53139
$478$	0	0
$479$	6277.23	0.598777	0.299388	−	0.954131i	$-0.403217\pi$
$479$	6277.23	0.598777	0.299388	+	0.954131i	$0.403217\pi$
$480$	0	0
$481$	131.611	0.0124760
$482$	0	0
$483$	−2492.73	−0.234831
$484$	0	0
$485$	−789.336	−0.0739008
$486$	0	0
$487$	7311.96	0.680363	0.340181	−	0.940360i	$-0.389512\pi$
$487$	7311.96	0.680363	0.340181	+	0.940360i	$0.389512\pi$
$488$	0	0
$489$	16023.0	1.48177
$490$	0	0
$491$	−17240.0	−1.58458	−0.792289	−	0.610145i	$-0.791111\pi$
$491$	−17240.0	−1.58458	−0.792289	+	0.610145i	$0.791111\pi$
$492$	0	0
$493$	−1233.08	−0.112647
$494$	0	0
$495$	223.291	0.0202751
$496$	0	0
$497$	−1202.65	−0.108543
$498$	0	0
$499$	−843.857	−0.0757039	−0.0378519	−	0.999283i	$-0.512052\pi$
$499$	−843.857	−0.0757039	−0.0378519	+	0.999283i	$0.512052\pi$
$500$	0	0
$501$	−7220.18	−0.643860
$502$	0	0
$503$	18235.7	1.61648	0.808242	−	0.588850i	$-0.200419\pi$
$503$	18235.7	1.61648	0.808242	+	0.588850i	$0.200419\pi$
$504$	0	0
$505$	−25284.6	−2.22802
$506$	0	0
$507$	16882.4	1.47884
$508$	0	0
$509$	4978.87	0.433565	0.216783	−	0.976220i	$-0.430444\pi$
$509$	4978.87	0.433565	0.216783	+	0.976220i	$0.430444\pi$
$510$	0	0
$511$	−183.971	−0.0159264
$512$	0	0
$513$	−740.163	−0.0637018
$514$	0	0
$515$	2402.26	0.205546
$516$	0	0
$517$	−179.580	−0.0152764
$518$	0	0
$519$	−1646.81	−0.139281
$520$	0	0
$521$	−9413.57	−0.791585	−0.395793	−	0.918340i	$-0.629530\pi$
$521$	−9413.57	−0.791585	−0.395793	+	0.918340i	$0.629530\pi$
$522$	0	0
$523$	−18142.6	−1.51687	−0.758434	−	0.651750i	$-0.774035\pi$
$523$	−18142.6	−1.51687	−0.758434	+	0.651750i	$0.774035\pi$
$524$	0	0
$525$	2318.38	0.192728
$526$	0	0
$527$	−2533.28	−0.209396
$528$	0	0
$529$	1304.52	0.107218
$530$	0	0
$531$	2659.12	0.217319
$532$	0	0
$533$	−26.0840	−0.00211974
$534$	0	0
$535$	−14400.4	−1.16371
$536$	0	0
$537$	−20322.5	−1.63311
$538$	0	0
$539$	−152.916	−0.0122200
$540$	0	0
$541$	−7901.85	−0.627961	−0.313981	−	0.949429i	$-0.601663\pi$
$541$	−7901.85	−0.627961	−0.313981	+	0.949429i	$0.601663\pi$
$542$	0	0
$543$	10803.9	0.853850
$544$	0	0
$545$	−14935.7	−1.17390
$546$	0	0
$547$	−12821.6	−1.00222	−0.501109	−	0.865384i	$-0.667074\pi$
$547$	−12821.6	−1.00222	−0.501109	+	0.865384i	$0.667074\pi$
$548$	0	0
$549$	21026.9	1.63462
$550$	0	0
$551$	−1913.32	−0.147931
$552$	0	0
$553$	−1476.76	−0.113559
$554$	0	0
$555$	−24960.3	−1.90902
$556$	0	0
$557$	−14141.8	−1.07577	−0.537886	−	0.843017i	$-0.680777\pi$
$557$	−14141.8	−1.07577	−0.537886	+	0.843017i	$0.680777\pi$
$558$	0	0
$559$	21.1767	0.00160229
$560$	0	0
$561$	−42.9334	−0.00323110
$562$	0	0
$563$	20912.4	1.56546	0.782729	−	0.622363i	$-0.213827\pi$
$563$	20912.4	1.56546	0.782729	+	0.622363i	$0.213827\pi$
$564$	0	0
$565$	−14951.3	−1.11329
$566$	0	0
$567$	1582.90	0.117241
$568$	0	0
$569$	−20364.1	−1.50036	−0.750181	−	0.661232i	$-0.770034\pi$
$569$	−20364.1	−1.50036	−0.750181	+	0.661232i	$0.770034\pi$
$570$	0	0
$571$	14151.3	1.03715	0.518575	−	0.855032i	$-0.326463\pi$
$571$	14151.3	1.03715	0.518575	+	0.855032i	$0.326463\pi$
$572$	0	0
$573$	−1227.99	−0.0895285
$574$	0	0
$575$	−12529.2	−0.908706
$576$	0	0
$577$	−22908.6	−1.65285	−0.826427	−	0.563044i	$-0.809630\pi$
$577$	−22908.6	−1.65285	−0.826427	+	0.563044i	$0.809630\pi$
$578$	0	0
$579$	15309.7	1.09887
$580$	0	0
$581$	1656.42	0.118278
$582$	0	0
$583$	226.956	0.0161227
$584$	0	0
$585$	302.734	0.0213958
$586$	0	0
$587$	−22949.0	−1.61364	−0.806819	−	0.590798i	$-0.798813\pi$
$587$	−22949.0	−1.61364	−0.806819	+	0.590798i	$0.798813\pi$
$588$	0	0
$589$	−3930.80	−0.274985
$590$	0	0
$591$	30342.2	2.11187
$592$	0	0
$593$	−17528.1	−1.21381	−0.606907	−	0.794773i	$-0.707590\pi$
$593$	−17528.1	−1.21381	−0.606907	+	0.794773i	$0.707590\pi$
$594$	0	0
$595$	−522.245	−0.0359831
$596$	0	0
$597$	38993.7	2.67321
$598$	0	0
$599$	8232.99	0.561587	0.280794	−	0.959768i	$-0.409402\pi$
$599$	8232.99	0.561587	0.280794	+	0.959768i	$0.409402\pi$
$600$	0	0
$601$	−25516.6	−1.73185	−0.865927	−	0.500170i	$-0.833271\pi$
$601$	−25516.6	−1.73185	−0.865927	+	0.500170i	$0.833271\pi$
$602$	0	0
$603$	2827.85	0.190976
$604$	0	0
$605$	−20311.4	−1.36492
$606$	0	0
$607$	28319.1	1.89363	0.946816	−	0.321775i	$-0.104279\pi$
$607$	28319.1	1.89363	0.946816	+	0.321775i	$0.104279\pi$
$608$	0	0
$609$	−2162.73	−0.143905
$610$	0	0
$611$	−243.471	−0.0161207
$612$	0	0
$613$	22313.5	1.47020	0.735100	−	0.677959i	$-0.237135\pi$
$613$	22313.5	1.47020	0.735100	+	0.677959i	$0.237135\pi$
$614$	0	0
$615$	4946.89	0.324354
$616$	0	0
$617$	4432.54	0.289218	0.144609	−	0.989489i	$-0.453808\pi$
$617$	4432.54	0.289218	0.144609	+	0.989489i	$0.453808\pi$
$618$	0	0
$619$	24084.8	1.56390	0.781948	−	0.623344i	$-0.214226\pi$
$619$	24084.8	1.56390	0.781948	+	0.623344i	$0.214226\pi$
$620$	0	0
$621$	4521.50	0.292176
$622$	0	0
$623$	−3218.84	−0.206998
$624$	0	0
$625$	−17465.7	−1.11780
$626$	0	0
$627$	−66.6182	−0.00424318
$628$	0	0
$629$	2605.53	0.165166
$630$	0	0
$631$	16357.1	1.03196	0.515979	−	0.856601i	$-0.327428\pi$
$631$	16357.1	1.03196	0.515979	+	0.856601i	$0.327428\pi$
$632$	0	0
$633$	13701.3	0.860311
$634$	0	0
$635$	−20259.9	−1.26612
$636$	0	0
$637$	−207.321	−0.0128954
$638$	0	0
$639$	13801.6	0.854436
$640$	0	0
$641$	9044.36	0.557302	0.278651	−	0.960392i	$-0.410113\pi$
$641$	9044.36	0.557302	0.278651	+	0.960392i	$0.410113\pi$
$642$	0	0
$643$	16.0941	0.000987073 0	0.000493537	−	1.00000i	$-0.499843\pi$
$643$	16.0941	0.000987073 0	0.000493537		1.00000i	$0.499843\pi$
$644$	0	0
$645$	−4016.21	−0.245175
$646$	0	0
$647$	−5673.50	−0.344743	−0.172371	−	0.985032i	$-0.555143\pi$
$647$	−5673.50	−0.344743	−0.172371	+	0.985032i	$0.555143\pi$
$648$	0	0
$649$	37.8284	0.00228797
$650$	0	0
$651$	−4443.19	−0.267500
$652$	0	0
$653$	−24349.4	−1.45921	−0.729605	−	0.683869i	$-0.760296\pi$
$653$	−24349.4	−1.45921	−0.729605	+	0.683869i	$0.760296\pi$
$654$	0	0
$655$	31048.8	1.85218
$656$	0	0
$657$	2111.26	0.125370
$658$	0	0
$659$	19990.2	1.18165	0.590825	−	0.806800i	$-0.298802\pi$
$659$	19990.2	1.18165	0.590825	+	0.806800i	$0.298802\pi$
$660$	0	0
$661$	−7138.48	−0.420052	−0.210026	−	0.977696i	$-0.567355\pi$
$661$	−7138.48	−0.420052	−0.210026	+	0.977696i	$0.567355\pi$
$662$	0	0
$663$	−58.2083	−0.00340969
$664$	0	0
$665$	−810.349	−0.0472541
$666$	0	0
$667$	11688.1	0.678506
$668$	0	0
$669$	4628.55	0.267489
$670$	0	0
$671$	299.127	0.0172096
$672$	0	0
$673$	331.701	0.0189987	0.00949936	−	0.999955i	$-0.496976\pi$
$673$	331.701	0.0189987	0.00949936	+	0.999955i	$0.496976\pi$
$674$	0	0
$675$	−4205.24	−0.239792
$676$	0	0
$677$	23542.7	1.33651	0.668257	−	0.743930i	$-0.267041\pi$
$677$	23542.7	1.33651	0.668257	+	0.743930i	$0.267041\pi$
$678$	0	0
$679$	144.518	0.00816801
$680$	0	0
$681$	27278.2	1.53495
$682$	0	0
$683$	20417.1	1.14383	0.571917	−	0.820312i	$-0.306200\pi$
$683$	20417.1	1.14383	0.571917	+	0.820312i	$0.306200\pi$
$684$	0	0
$685$	−26187.6	−1.46069
$686$	0	0
$687$	5108.71	0.283711
$688$	0	0
$689$	307.703	0.0170138
$690$	0	0
$691$	15846.0	0.872372	0.436186	−	0.899857i	$-0.356329\pi$
$691$	15846.0	0.872372	0.436186	+	0.899857i	$0.356329\pi$
$692$	0	0
$693$	−40.8818	−0.00224094
$694$	0	0
$695$	−26897.3	−1.46802
$696$	0	0
$697$	−516.392	−0.0280627
$698$	0	0
$699$	−4588.62	−0.248294
$700$	0	0
$701$	16511.6	0.889636	0.444818	−	0.895621i	$-0.353268\pi$
$701$	16511.6	0.889636	0.444818	+	0.895621i	$0.353268\pi$
$702$	0	0
$703$	4042.92	0.216901
$704$	0	0
$705$	46174.8	2.46673
$706$	0	0
$707$	4629.29	0.246255
$708$	0	0
$709$	−5585.31	−0.295854	−0.147927	−	0.988998i	$-0.547260\pi$
$709$	−5585.31	−0.295854	−0.147927	+	0.988998i	$0.547260\pi$
$710$	0	0
$711$	16947.3	0.893916
$712$	0	0
$713$	24012.4	1.26125
$714$	0	0
$715$	4.30666	0.000225259 0
$716$	0	0
$717$	−48467.5	−2.52448
$718$	0	0
$719$	−19604.0	−1.01684	−0.508418	−	0.861110i	$-0.669770\pi$
$719$	−19604.0	−1.01684	−0.508418	+	0.861110i	$0.669770\pi$
$720$	0	0
$721$	−439.824	−0.0227183
$722$	0	0
$723$	46050.3	2.36878
$724$	0	0
$725$	−10870.5	−0.556857
$726$	0	0
$727$	−19691.4	−1.00456	−0.502279	−	0.864706i	$-0.667505\pi$
$727$	−19691.4	−1.00456	−0.502279	+	0.864706i	$0.667505\pi$
$728$	0	0
$729$	−26249.2	−1.33360
$730$	0	0
$731$	419.240	0.0212123
$732$	0	0
$733$	20622.2	1.03915	0.519576	−	0.854424i	$-0.326090\pi$
$733$	20622.2	1.03915	0.519576	+	0.854424i	$0.326090\pi$
$734$	0	0
$735$	39318.9	1.97320
$736$	0	0
$737$	40.2286	0.00201064
$738$	0	0
$739$	29541.2	1.47049	0.735243	−	0.677804i	$-0.237068\pi$
$739$	29541.2	1.47049	0.735243	+	0.677804i	$0.237068\pi$
$740$	0	0
$741$	−90.3197	−0.00447770
$742$	0	0
$743$	−10131.7	−0.500263	−0.250132	−	0.968212i	$-0.580474\pi$
$743$	−10131.7	−0.500263	−0.250132	+	0.968212i	$0.580474\pi$
$744$	0	0
$745$	−31527.4	−1.55043
$746$	0	0
$747$	−19009.1	−0.931068
$748$	0	0
$749$	2636.53	0.128620
$750$	0	0
$751$	−31366.0	−1.52405	−0.762025	−	0.647547i	$-0.775795\pi$
$751$	−31366.0	−1.52405	−0.762025	+	0.647547i	$0.775795\pi$
$752$	0	0
$753$	46195.8	2.23568
$754$	0	0
$755$	17965.5	0.866002
$756$	0	0
$757$	2836.37	0.136182	0.0680909	−	0.997679i	$-0.478309\pi$
$757$	2836.37	0.136182	0.0680909	+	0.997679i	$0.478309\pi$
$758$	0	0
$759$	406.956	0.0194619
$760$	0	0
$761$	33195.4	1.58125	0.790626	−	0.612300i	$-0.209755\pi$
$761$	33195.4	1.58125	0.790626	+	0.612300i	$0.209755\pi$
$762$	0	0
$763$	2734.54	0.129747
$764$	0	0
$765$	5993.31	0.283253
$766$	0	0
$767$	51.2870	0.00241443
$768$	0	0
$769$	−378.413	−0.0177450	−0.00887252	−	0.999961i	$-0.502824\pi$
$769$	−378.413	−0.0177450	−0.00887252	+	0.999961i	$0.502824\pi$
$770$	0	0
$771$	12655.3	0.591142
$772$	0	0
$773$	−9764.06	−0.454319	−0.227160	−	0.973858i	$-0.572944\pi$
$773$	−9764.06	−0.454319	−0.227160	+	0.973858i	$0.572944\pi$
$774$	0	0
$775$	−22332.9	−1.03512
$776$	0	0
$777$	4569.92	0.210997
$778$	0	0
$779$	−801.267	−0.0368528
$780$	0	0
$781$	196.341	0.00899567
$782$	0	0
$783$	3922.91	0.179046
$784$	0	0
$785$	27511.1	1.25084
$786$	0	0
$787$	−11958.2	−0.541631	−0.270816	−	0.962631i	$-0.587293\pi$
$787$	−11958.2	−0.541631	−0.270816	+	0.962631i	$0.587293\pi$
$788$	0	0
$789$	14020.4	0.632622
$790$	0	0
$791$	2737.40	0.123048
$792$	0	0
$793$	405.551	0.0181608
$794$	0	0
$795$	−58356.6	−2.60339
$796$	0	0
$797$	−14502.1	−0.644529	−0.322264	−	0.946650i	$-0.604444\pi$
$797$	−14502.1	−0.644529	−0.322264	+	0.946650i	$0.604444\pi$
$798$	0	0
$799$	−4820.06	−0.213419
$800$	0	0
$801$	36939.5	1.62946
$802$	0	0
$803$	30.0345	0.00131992
$804$	0	0
$805$	4950.25	0.216737
$806$	0	0
$807$	31948.3	1.39360
$808$	0	0
$809$	−23743.3	−1.03185	−0.515926	−	0.856633i	$-0.672552\pi$
$809$	−23743.3	−1.03185	−0.515926	+	0.856633i	$0.672552\pi$
$810$	0	0
$811$	1788.53	0.0774398	0.0387199	−	0.999250i	$-0.487672\pi$
$811$	1788.53	0.0774398	0.0387199	+	0.999250i	$0.487672\pi$
$812$	0	0
$813$	−6072.34	−0.261951
$814$	0	0
$815$	−31819.7	−1.36760
$816$	0	0
$817$	650.520	0.0278566
$818$	0	0
$819$	−55.4269	−0.00236480
$820$	0	0
$821$	−16960.6	−0.720987	−0.360493	−	0.932762i	$-0.617392\pi$
$821$	−16960.6	−0.720987	−0.360493	+	0.932762i	$0.617392\pi$
$822$	0	0
$823$	−9066.84	−0.384022	−0.192011	−	0.981393i	$-0.561501\pi$
$823$	−9066.84	−0.384022	−0.192011	+	0.981393i	$0.561501\pi$
$824$	0	0
$825$	−378.491	−0.0159726
$826$	0	0
$827$	−31217.7	−1.31263	−0.656315	−	0.754487i	$-0.727886\pi$
$827$	−31217.7	−1.31263	−0.656315	+	0.754487i	$0.727886\pi$
$828$	0	0
$829$	33513.6	1.40407	0.702035	−	0.712142i	$-0.252275\pi$
$829$	33513.6	1.40407	0.702035	+	0.712142i	$0.252275\pi$
$830$	0	0
$831$	−19019.5	−0.793958
$832$	0	0
$833$	−4104.39	−0.170719
$834$	0	0
$835$	14338.3	0.594250
$836$	0	0
$837$	8059.38	0.332823
$838$	0	0
$839$	−46369.9	−1.90807	−0.954033	−	0.299703i	$-0.903112\pi$
$839$	−46369.9	−1.90807	−0.954033	+	0.299703i	$0.903112\pi$
$840$	0	0
$841$	−14248.3	−0.584210
$842$	0	0
$843$	−40309.7	−1.64690
$844$	0	0
$845$	−33526.2	−1.36490
$846$	0	0
$847$	3718.77	0.150860
$848$	0	0
$849$	−31089.9	−1.25678
$850$	0	0
$851$	−24697.3	−0.994844
$852$	0	0
$853$	217.304	0.00872257	0.00436129	−	0.999990i	$-0.498612\pi$
$853$	217.304	0.00872257	0.00436129	+	0.999990i	$0.498612\pi$
$854$	0	0
$855$	9299.61	0.371977
$856$	0	0
$857$	27096.2	1.08003	0.540016	−	0.841654i	$-0.318418\pi$
$857$	27096.2	1.08003	0.540016	+	0.841654i	$0.318418\pi$
$858$	0	0
$859$	11922.6	0.473565	0.236783	−	0.971563i	$-0.423907\pi$
$859$	11922.6	0.473565	0.236783	+	0.971563i	$0.423907\pi$
$860$	0	0
$861$	−905.714	−0.0358498
$862$	0	0
$863$	−7376.00	−0.290941	−0.145471	−	0.989363i	$-0.546470\pi$
$863$	−7376.00	−0.290941	−0.145471	+	0.989363i	$0.546470\pi$
$864$	0	0
$865$	3270.35	0.128549
$866$	0	0
$867$	36607.1	1.43396
$868$	0	0
$869$	241.091	0.00941132
$870$	0	0
$871$	54.5412	0.00212177
$872$	0	0
$873$	−1658.49	−0.0642971
$874$	0	0
$875$	727.245	0.0280976
$876$	0	0
$877$	50755.4	1.95426	0.977131	−	0.212637i	$-0.0682052\pi$
$877$	50755.4	1.95426	0.977131	+	0.212637i	$0.0682052\pi$
$878$	0	0
$879$	−51143.0	−1.96247
$880$	0	0
$881$	28803.5	1.10149	0.550746	−	0.834673i	$-0.314343\pi$
$881$	28803.5	1.10149	0.550746	+	0.834673i	$0.314343\pi$
$882$	0	0
$883$	51864.4	1.97664	0.988322	−	0.152381i	$-0.0486941\pi$
$883$	51864.4	1.97664	0.988322	+	0.152381i	$0.0486941\pi$
$884$	0	0
$885$	−9726.71	−0.369446
$886$	0	0
$887$	14211.4	0.537962	0.268981	−	0.963145i	$-0.413313\pi$
$887$	14211.4	0.537962	0.268981	+	0.963145i	$0.413313\pi$
$888$	0	0
$889$	3709.33	0.139940
$890$	0	0
$891$	−258.419	−0.00971647
$892$	0	0
$893$	−7479.12	−0.280268
$894$	0	0
$895$	40357.9	1.50728
$896$	0	0
$897$	551.743	0.0205375
$898$	0	0
$899$	20833.5	0.772899
$900$	0	0
$901$	6091.68	0.225242
$902$	0	0
$903$	735.317	0.0270984
$904$	0	0
$905$	−21455.2	−0.788061
$906$	0	0
$907$	−2766.42	−0.101276	−0.0506381	−	0.998717i	$-0.516125\pi$
$907$	−2766.42	−0.101276	−0.0506381	+	0.998717i	$0.516125\pi$
$908$	0	0
$909$	−53126.0	−1.93848
$910$	0	0
$911$	−1282.52	−0.0466430	−0.0233215	−	0.999728i	$-0.507424\pi$
$911$	−1282.52	−0.0466430	−0.0233215	+	0.999728i	$0.507424\pi$
$912$	0	0
$913$	−270.422	−0.00980247
$914$	0	0
$915$	−76913.7	−2.77889
$916$	0	0
$917$	−5684.65	−0.204715
$918$	0	0
$919$	−31183.4	−1.11931	−0.559655	−	0.828725i	$-0.689066\pi$
$919$	−31183.4	−1.11931	−0.559655	+	0.828725i	$0.689066\pi$
$920$	0	0
$921$	44489.7	1.59173
$922$	0	0
$923$	266.195	0.00949286
$924$	0	0
$925$	22969.8	0.816479
$926$	0	0
$927$	5047.44	0.178835
$928$	0	0
$929$	35037.3	1.23739	0.618696	−	0.785630i	$-0.287661\pi$
$929$	35037.3	1.23739	0.618696	+	0.785630i	$0.287661\pi$
$930$	0	0
$931$	−6368.64	−0.224193
$932$	0	0
$933$	57672.1	2.02369
$934$	0	0
$935$	85.2602	0.00298214
$936$	0	0
$937$	12969.3	0.452175	0.226087	−	0.974107i	$-0.427407\pi$
$937$	12969.3	0.452175	0.226087	+	0.974107i	$0.427407\pi$
$938$	0	0
$939$	−77882.2	−2.70670
$940$	0	0
$941$	−11958.5	−0.414279	−0.207140	−	0.978311i	$-0.566415\pi$
$941$	−11958.5	−0.414279	−0.207140	+	0.978311i	$0.566415\pi$
$942$	0	0
$943$	4894.76	0.169030
$944$	0	0
$945$	1661.47	0.0571933
$946$	0	0
$947$	−39481.6	−1.35478	−0.677392	−	0.735622i	$-0.736890\pi$
$947$	−39481.6	−1.35478	−0.677392	+	0.735622i	$0.736890\pi$
$948$	0	0
$949$	40.7202	0.00139287
$950$	0	0
$951$	−67266.1	−2.29364
$952$	0	0
$953$	31129.5	1.05811	0.529057	−	0.848586i	$-0.322546\pi$
$953$	31129.5	1.05811	0.529057	+	0.848586i	$0.322546\pi$
$954$	0	0
$955$	2438.62	0.0826303
$956$	0	0
$957$	353.080	0.0119263
$958$	0	0
$959$	4794.62	0.161446
$960$	0	0
$961$	13010.2	0.436715
$962$	0	0
$963$	−30257.0	−1.01248
$964$	0	0
$965$	−30403.1	−1.01421
$966$	0	0
$967$	25627.5	0.852250	0.426125	−	0.904664i	$-0.359878\pi$
$967$	25627.5	0.852250	0.426125	+	0.904664i	$0.359878\pi$
$968$	0	0
$969$	−1788.08	−0.0592792
$970$	0	0
$971$	−18600.8	−0.614756	−0.307378	−	0.951587i	$-0.599452\pi$
$971$	−18600.8	−0.614756	−0.307378	+	0.951587i	$0.599452\pi$
$972$	0	0
$973$	4924.56	0.162255
$974$	0	0
$975$	−513.151	−0.0168554
$976$	0	0
$977$	14081.2	0.461103	0.230552	−	0.973060i	$-0.425947\pi$
$977$	14081.2	0.461103	0.230552	+	0.973060i	$0.425947\pi$
$978$	0	0
$979$	525.498	0.0171552
$980$	0	0
$981$	−31381.7	−1.02135
$982$	0	0
$983$	−13359.2	−0.433462	−0.216731	−	0.976231i	$-0.569540\pi$
$983$	−13359.2	−0.433462	−0.216731	+	0.976231i	$0.569540\pi$
$984$	0	0
$985$	−60255.8	−1.94915
$986$	0	0
$987$	−8454.04	−0.272639
$988$	0	0
$989$	−3973.89	−0.127768
$990$	0	0
$991$	52748.2	1.69082	0.845409	−	0.534119i	$-0.179357\pi$
$991$	52748.2	1.69082	0.845409	+	0.534119i	$0.179357\pi$
$992$	0	0
$993$	48177.2	1.53963
$994$	0	0
$995$	−77436.5	−2.46724
$996$	0	0
$997$	6334.73	0.201226	0.100613	−	0.994926i	$-0.467920\pi$
$997$	6334.73	0.201226	0.100613	+	0.994926i	$0.467920\pi$
$998$	0	0
$999$	−8289.24	−0.262523

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.4.a.y.1.1		5
4.3	odd	2		1216.4.a.bd.1.5		5
8.3	odd	2		608.4.a.e.1.1	✓	5
8.5	even	2		608.4.a.h.1.5	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
608.4.a.e.1.1	✓	5	8.3	odd	2
608.4.a.h.1.5	yes	5	8.5	even	2
1216.4.a.y.1.1		5	1.1	even	1	trivial
1216.4.a.bd.1.5		5	4.3	odd	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:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1216.a (trivial)

\(f(q)\)	\(=\)	\(q-7.68562 q^{3} +15.2627 q^{5} -2.79440 q^{7} +32.0687 q^{9} +0.456205 q^{11} +0.618515 q^{13} -117.303 q^{15} +12.2449 q^{17} +19.0000 q^{19} +21.4767 q^{21} -116.067 q^{23} +107.949 q^{25} -38.9560 q^{27} -100.701 q^{29} -206.884 q^{31} -3.50622 q^{33} -42.6500 q^{35} +212.785 q^{37} -4.75367 q^{39} -42.1719 q^{41} +34.2379 q^{43} +489.453 q^{45} -393.638 q^{47} -335.191 q^{49} -94.1097 q^{51} +497.486 q^{53} +6.96291 q^{55} -146.027 q^{57} +82.9196 q^{59} +655.685 q^{61} -89.6128 q^{63} +9.44018 q^{65} +88.1809 q^{67} +892.045 q^{69} +430.377 q^{71} +65.8355 q^{73} -829.651 q^{75} -1.27482 q^{77} +528.469 q^{79} -566.454 q^{81} -592.763 q^{83} +186.890 q^{85} +773.950 q^{87} +1151.89 q^{89} -1.72838 q^{91} +1590.03 q^{93} +289.990 q^{95} -51.7168 q^{97} +14.6299 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 6 q^{3} - 5 q^{5} - 7 q^{7} - 5 q^{9} - 13 q^{11} + 72 q^{13} - 72 q^{15} - 59 q^{17} + 95 q^{19} + 224 q^{21} - 52 q^{23} - 86 q^{25} - 54 q^{27} + 128 q^{29} + 110 q^{31} - 68 q^{33} + 45 q^{35}+ \cdots - 601 q^{99}+O(q^{100}) \) 5 * q - 6 * q^3 - 5 * q^5 - 7 * q^7 - 5 * q^9 - 13 * q^11 + 72 * q^13 - 72 * q^15 - 59 * q^17 + 95 * q^19 + 224 * q^21 - 52 * q^23 - 86 * q^25 - 54 * q^27 + 128 * q^29 + 110 * q^31 - 68 * q^33 + 45 * q^35 + 436 * q^37 - 356 * q^39 - 804 * q^41 + 143 * q^43 + 579 * q^45 - 661 * q^47 - 406 * q^49 - 570 * q^51 + 898 * q^53 - 17 * q^55 - 114 * q^57 - 196 * q^59 + 1079 * q^61 - 143 * q^63 - 1632 * q^65 - 832 * q^67 + 1644 * q^69 - 834 * q^71 - 1375 * q^73 - 1054 * q^75 + 1473 * q^77 - 154 * q^79 - 1859 * q^81 - 2224 * q^83 + 1807 * q^85 - 2004 * q^87 - 542 * q^89 - 1890 * q^91 + 2788 * q^93 - 95 * q^95 - 1528 * q^97 - 601 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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