LMFDB - Embedded newform 1216.4.a.bc.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} - x^{4} - 20x^{3} + 24x^{2} + 2x - 4 $ x^5 - x^4 - 20x^3 + 24x^2 + 2*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 608)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.14402$ 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)

$f(q)$	$=$	$q-9.59869 q^{3} -0.0570048 q^{5} -24.6071 q^{7} +65.1348 q^{9} +O(q^{10})$	$q-9.59869 q^{3} -0.0570048 q^{5} -24.6071 q^{7} +65.1348 q^{9} -43.3012 q^{11} -36.6926 q^{13} +0.547171 q^{15} +65.8370 q^{17} +19.0000 q^{19} +236.196 q^{21} +110.168 q^{23} -124.997 q^{25} -366.044 q^{27} +91.1764 q^{29} -148.095 q^{31} +415.634 q^{33} +1.40272 q^{35} +89.4653 q^{37} +352.201 q^{39} +414.689 q^{41} -104.245 q^{43} -3.71299 q^{45} +602.823 q^{47} +262.508 q^{49} -631.948 q^{51} -187.011 q^{53} +2.46837 q^{55} -182.375 q^{57} +283.291 q^{59} -539.169 q^{61} -1602.78 q^{63} +2.09165 q^{65} +730.548 q^{67} -1057.46 q^{69} +707.857 q^{71} +989.094 q^{73} +1199.80 q^{75} +1065.51 q^{77} -897.609 q^{79} +1754.90 q^{81} +388.174 q^{83} -3.75302 q^{85} -875.174 q^{87} +17.3670 q^{89} +902.898 q^{91} +1421.52 q^{93} -1.08309 q^{95} -721.171 q^{97} -2820.41 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 3 q^{3} - 27 q^{5} + 20 q^{7} + 154 q^{9}+O(q^{10}) $ 5 * q + 3 * q^3 - 27 * q^5 + 20 * q^7 + 154 * q^9	$ 5 q + 3 q^{3} - 27 q^{5} + 20 q^{7} + 154 q^{9} - 67 q^{11} - 191 q^{13} - 36 q^{15} - 52 q^{17} + 95 q^{19} + 29 q^{21} + 35 q^{23} - 2 q^{25} - 27 q^{27} + 33 q^{29} - 694 q^{31} + 298 q^{33} - 387 q^{35} - 108 q^{37} - 31 q^{39} + 54 q^{41} + 427 q^{43} - 1699 q^{45} + 79 q^{47} + 1033 q^{49} - 363 q^{51} - 1025 q^{53} + 1431 q^{55} + 57 q^{57} + 649 q^{59} - 1413 q^{61} - 1449 q^{63} + 548 q^{65} - 915 q^{67} - 407 q^{69} + 1444 q^{71} + 2682 q^{73} + 465 q^{75} - 169 q^{77} - 836 q^{79} + 2925 q^{81} + 502 q^{83} - 373 q^{85} - 3491 q^{87} + 996 q^{89} - 2919 q^{91} - 2492 q^{93} - 513 q^{95} + 424 q^{97} - 7249 q^{99}+O(q^{100}) $ 5 * q + 3 * q^3 - 27 * q^5 + 20 * q^7 + 154 * q^9 - 67 * q^11 - 191 * q^13 - 36 * q^15 - 52 * q^17 + 95 * q^19 + 29 * q^21 + 35 * q^23 - 2 * q^25 - 27 * q^27 + 33 * q^29 - 694 * q^31 + 298 * q^33 - 387 * q^35 - 108 * q^37 - 31 * q^39 + 54 * q^41 + 427 * q^43 - 1699 * q^45 + 79 * q^47 + 1033 * q^49 - 363 * q^51 - 1025 * q^53 + 1431 * q^55 + 57 * q^57 + 649 * q^59 - 1413 * q^61 - 1449 * q^63 + 548 * q^65 - 915 * q^67 - 407 * q^69 + 1444 * q^71 + 2682 * q^73 + 465 * q^75 - 169 * q^77 - 836 * q^79 + 2925 * q^81 + 502 * q^83 - 373 * q^85 - 3491 * q^87 + 996 * q^89 - 2919 * q^91 - 2492 * q^93 - 513 * q^95 + 424 * q^97 - 7249 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−9.59869	−1.84727	−0.923634	−	0.383276i	$-0.874796\pi$
$3$	−9.59869	−1.84727	−0.923634	+	0.383276i	$0.874796\pi$
$4$	0	0
$5$	−0.0570048	−0.00509866	−0.00254933	−	0.999997i	$-0.500811\pi$
$5$	−0.0570048	−0.00509866	−0.00254933	+	0.999997i	$0.500811\pi$
$6$	0	0
$7$	−24.6071	−1.32866	−0.664329	−	0.747441i	$-0.731282\pi$
$7$	−24.6071	−1.32866	−0.664329	+	0.747441i	$0.731282\pi$
$8$	0	0
$9$	65.1348	2.41240
$10$	0	0
$11$	−43.3012	−1.18689	−0.593445	−	0.804875i	$-0.702233\pi$
$11$	−43.3012	−1.18689	−0.593445	+	0.804875i	$0.702233\pi$
$12$	0	0
$13$	−36.6926	−0.782823	−0.391411	−	0.920216i	$-0.628013\pi$
$13$	−36.6926	−0.782823	−0.391411	+	0.920216i	$0.628013\pi$
$14$	0	0
$15$	0.547171	0.00941859
$16$	0	0
$17$	65.8370	0.939283	0.469641	−	0.882857i	$-0.344383\pi$
$17$	65.8370	0.939283	0.469641	+	0.882857i	$0.344383\pi$
$18$	0	0
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	0	0
$21$	236.196	2.45439
$22$	0	0
$23$	110.168	0.998762	0.499381	−	0.866382i	$-0.333561\pi$
$23$	110.168	0.998762	0.499381	+	0.866382i	$0.333561\pi$
$24$	0	0
$25$	−124.997	−0.999974
$26$	0	0
$27$	−366.044	−2.60908
$28$	0	0
$29$	91.1764	0.583829	0.291914	−	0.956444i	$-0.405708\pi$
$29$	91.1764	0.583829	0.291914	+	0.956444i	$0.405708\pi$
$30$	0	0
$31$	−148.095	−0.858021	−0.429010	−	0.903300i	$-0.641138\pi$
$31$	−148.095	−0.858021	−0.429010	+	0.903300i	$0.641138\pi$
$32$	0	0
$33$	415.634	2.19250
$34$	0	0
$35$	1.40272	0.00677437
$36$	0	0
$37$	89.4653	0.397514	0.198757	−	0.980049i	$-0.436310\pi$
$37$	89.4653	0.397514	0.198757	+	0.980049i	$0.436310\pi$
$38$	0	0
$39$	352.201	1.44608
$40$	0	0
$41$	414.689	1.57960	0.789799	−	0.613366i	$-0.210185\pi$
$41$	414.689	1.57960	0.789799	+	0.613366i	$0.210185\pi$
$42$	0	0
$43$	−104.245	−0.369704	−0.184852	−	0.982766i	$-0.559181\pi$
$43$	−104.245	−0.369704	−0.184852	+	0.982766i	$0.559181\pi$
$44$	0	0
$45$	−3.71299	−0.0123000
$46$	0	0
$47$	602.823	1.87087	0.935435	−	0.353500i	$-0.115009\pi$
$47$	602.823	1.87087	0.935435	+	0.353500i	$0.115009\pi$
$48$	0	0
$49$	262.508	0.765330
$50$	0	0
$51$	−631.948	−1.73511
$52$	0	0
$53$	−187.011	−0.484678	−0.242339	−	0.970192i	$-0.577915\pi$
$53$	−187.011	−0.484678	−0.242339	+	0.970192i	$0.577915\pi$
$54$	0	0
$55$	2.46837	0.00605155
$56$	0	0
$57$	−182.375	−0.423792
$58$	0	0
$59$	283.291	0.625107	0.312553	−	0.949900i	$-0.398816\pi$
$59$	283.291	0.625107	0.312553	+	0.949900i	$0.398816\pi$
$60$	0	0
$61$	−539.169	−1.13170	−0.565848	−	0.824509i	$-0.691451\pi$
$61$	−539.169	−1.13170	−0.565848	+	0.824509i	$0.691451\pi$
$62$	0	0
$63$	−1602.78	−3.20525
$64$	0	0
$65$	2.09165	0.00399135
$66$	0	0
$67$	730.548	1.33210	0.666050	−	0.745907i	$-0.267984\pi$
$67$	730.548	1.33210	0.666050	+	0.745907i	$0.267984\pi$
$68$	0	0
$69$	−1057.46	−1.84498
$70$	0	0
$71$	707.857	1.18320	0.591599	−	0.806232i	$-0.298497\pi$
$71$	707.857	1.18320	0.591599	+	0.806232i	$0.298497\pi$
$72$	0	0
$73$	989.094	1.58582	0.792909	−	0.609340i	$-0.208566\pi$
$73$	989.094	1.58582	0.792909	+	0.609340i	$0.208566\pi$
$74$	0	0
$75$	1199.80	1.84722
$76$	0	0
$77$	1065.51	1.57697
$78$	0	0
$79$	−897.609	−1.27834	−0.639170	−	0.769065i	$-0.720722\pi$
$79$	−897.609	−1.27834	−0.639170	+	0.769065i	$0.720722\pi$
$80$	0	0
$81$	1754.90	2.40727
$82$	0	0
$83$	388.174	0.513345	0.256673	−	0.966498i	$-0.417374\pi$
$83$	388.174	0.513345	0.256673	+	0.966498i	$0.417374\pi$
$84$	0	0
$85$	−3.75302	−0.00478908
$86$	0	0
$87$	−875.174	−1.07849
$88$	0	0
$89$	17.3670	0.0206842	0.0103421	−	0.999947i	$-0.496708\pi$
$89$	17.3670	0.0206842	0.0103421	+	0.999947i	$0.496708\pi$
$90$	0	0
$91$	902.898	1.04010
$92$	0	0
$93$	1421.52	1.58499
$94$	0	0
$95$	−1.08309	−0.00116971
$96$	0	0
$97$	−721.171	−0.754885	−0.377442	−	0.926033i	$-0.623196\pi$
$97$	−721.171	−0.754885	−0.377442	+	0.926033i	$0.623196\pi$
$98$	0	0
$99$	−2820.41	−2.86325
$100$	0	0
$101$	−1280.69	−1.26172	−0.630859	−	0.775897i	$-0.717297\pi$
$101$	−1280.69	−1.26172	−0.630859	+	0.775897i	$0.717297\pi$
$102$	0	0
$103$	−220.460	−0.210899	−0.105449	−	0.994425i	$-0.533628\pi$
$103$	−220.460	−0.210899	−0.105449	+	0.994425i	$0.533628\pi$
$104$	0	0
$105$	−13.4643	−0.0125141
$106$	0	0
$107$	1771.18	1.60024	0.800122	−	0.599838i	$-0.204768\pi$
$107$	1771.18	1.60024	0.800122	+	0.599838i	$0.204768\pi$
$108$	0	0
$109$	1129.40	0.992446	0.496223	−	0.868195i	$-0.334720\pi$
$109$	1129.40	0.992446	0.496223	+	0.868195i	$0.334720\pi$
$110$	0	0
$111$	−858.749	−0.734314
$112$	0	0
$113$	−116.302	−0.0968213	−0.0484106	−	0.998828i	$-0.515416\pi$
$113$	−116.302	−0.0968213	−0.0484106	+	0.998828i	$0.515416\pi$
$114$	0	0
$115$	−6.28008	−0.00509235
$116$	0	0
$117$	−2389.96	−1.88848
$118$	0	0
$119$	−1620.06	−1.24799
$120$	0	0
$121$	543.990	0.408708
$122$	0	0
$123$	−3980.47	−2.91794
$124$	0	0
$125$	14.2510	0.0101972
$126$	0	0
$127$	−2032.51	−1.42013	−0.710064	−	0.704137i	$-0.751334\pi$
$127$	−2032.51	−1.42013	−0.710064	+	0.704137i	$0.751334\pi$
$128$	0	0
$129$	1000.62	0.682942
$130$	0	0
$131$	−2860.19	−1.90760	−0.953802	−	0.300436i	$-0.902868\pi$
$131$	−2860.19	−1.90760	−0.953802	+	0.300436i	$0.902868\pi$
$132$	0	0
$133$	−467.534	−0.304815
$134$	0	0
$135$	20.8662	0.0133028
$136$	0	0
$137$	−823.336	−0.513448	−0.256724	−	0.966485i	$-0.582643\pi$
$137$	−823.336	−0.513448	−0.256724	+	0.966485i	$0.582643\pi$
$138$	0	0
$139$	−1750.55	−1.06820	−0.534101	−	0.845421i	$-0.679350\pi$
$139$	−1750.55	−1.06820	−0.534101	+	0.845421i	$0.679350\pi$
$140$	0	0
$141$	−5786.31	−3.45600
$142$	0	0
$143$	1588.83	0.929125
$144$	0	0
$145$	−5.19749	−0.00297675
$146$	0	0
$147$	−2519.73	−1.41377
$148$	0	0
$149$	1794.34	0.986562	0.493281	−	0.869870i	$-0.335797\pi$
$149$	1794.34	0.986562	0.493281	+	0.869870i	$0.335797\pi$
$150$	0	0
$151$	−3102.87	−1.67224	−0.836120	−	0.548546i	$-0.815182\pi$
$151$	−3102.87	−1.67224	−0.836120	+	0.548546i	$0.815182\pi$
$152$	0	0
$153$	4288.28	2.26593
$154$	0	0
$155$	8.44212	0.00437476
$156$	0	0
$157$	899.606	0.457302	0.228651	−	0.973508i	$-0.426569\pi$
$157$	899.606	0.457302	0.228651	+	0.973508i	$0.426569\pi$
$158$	0	0
$159$	1795.06	0.895331
$160$	0	0
$161$	−2710.90	−1.32701
$162$	0	0
$163$	3572.38	1.71663	0.858314	−	0.513125i	$-0.171512\pi$
$163$	3572.38	1.71663	0.858314	+	0.513125i	$0.171512\pi$
$164$	0	0
$165$	−23.6931	−0.0111788
$166$	0	0
$167$	−1128.82	−0.523060	−0.261530	−	0.965195i	$-0.584227\pi$
$167$	−1128.82	−0.523060	−0.261530	+	0.965195i	$0.584227\pi$
$168$	0	0
$169$	−850.653	−0.387188
$170$	0	0
$171$	1237.56	0.553442
$172$	0	0
$173$	702.227	0.308609	0.154304	−	0.988023i	$-0.450686\pi$
$173$	702.227	0.308609	0.154304	+	0.988023i	$0.450686\pi$
$174$	0	0
$175$	3075.80	1.32862
$176$	0	0
$177$	−2719.22	−1.15474
$178$	0	0
$179$	1429.46	0.596888	0.298444	−	0.954427i	$-0.403532\pi$
$179$	1429.46	0.596888	0.298444	+	0.954427i	$0.403532\pi$
$180$	0	0
$181$	−4023.61	−1.65234	−0.826168	−	0.563424i	$-0.809484\pi$
$181$	−4023.61	−1.65234	−0.826168	+	0.563424i	$0.809484\pi$
$182$	0	0
$183$	5175.31	2.09055
$184$	0	0
$185$	−5.09995	−0.00202679
$186$	0	0
$187$	−2850.82	−1.11483
$188$	0	0
$189$	9007.26	3.46657
$190$	0	0
$191$	−4063.94	−1.53956	−0.769781	−	0.638308i	$-0.779635\pi$
$191$	−4063.94	−1.53956	−0.769781	+	0.638308i	$0.779635\pi$
$192$	0	0
$193$	−4267.76	−1.59171	−0.795856	−	0.605487i	$-0.792979\pi$
$193$	−4267.76	−1.59171	−0.795856	+	0.605487i	$0.792979\pi$
$194$	0	0
$195$	−20.0771	−0.00737309
$196$	0	0
$197$	−3220.31	−1.16466	−0.582329	−	0.812953i	$-0.697859\pi$
$197$	−3220.31	−1.16466	−0.582329	+	0.812953i	$0.697859\pi$
$198$	0	0
$199$	602.161	0.214503	0.107251	−	0.994232i	$-0.465795\pi$
$199$	602.161	0.214503	0.107251	+	0.994232i	$0.465795\pi$
$200$	0	0
$201$	−7012.30	−2.46074
$202$	0	0
$203$	−2243.59	−0.775709
$204$	0	0
$205$	−23.6392	−0.00805383
$206$	0	0
$207$	7175.74	2.40941
$208$	0	0
$209$	−822.722	−0.272291
$210$	0	0
$211$	4258.47	1.38941	0.694705	−	0.719295i	$-0.255535\pi$
$211$	4258.47	1.38941	0.694705	+	0.719295i	$0.255535\pi$
$212$	0	0
$213$	−6794.49	−2.18569
$214$	0	0
$215$	5.94248	0.00188499
$216$	0	0
$217$	3644.18	1.14002
$218$	0	0
$219$	−9494.00	−2.92943
$220$	0	0
$221$	−2415.73	−0.735292
$222$	0	0
$223$	−1785.06	−0.536039	−0.268020	−	0.963413i	$-0.586369\pi$
$223$	−1785.06	−0.536039	−0.268020	+	0.963413i	$0.586369\pi$
$224$	0	0
$225$	−8141.63	−2.41234
$226$	0	0
$227$	2581.96	0.754937	0.377468	−	0.926022i	$-0.376795\pi$
$227$	2581.96	0.754937	0.377468	+	0.926022i	$0.376795\pi$
$228$	0	0
$229$	1778.50	0.513217	0.256608	−	0.966515i	$-0.417395\pi$
$229$	1778.50	0.513217	0.256608	+	0.966515i	$0.417395\pi$
$230$	0	0
$231$	−10227.5	−2.91309
$232$	0	0
$233$	6415.20	1.80375	0.901874	−	0.431998i	$-0.142191\pi$
$233$	6415.20	1.80375	0.901874	+	0.431998i	$0.142191\pi$
$234$	0	0
$235$	−34.3638	−0.00953893
$236$	0	0
$237$	8615.87	2.36144
$238$	0	0
$239$	−2560.36	−0.692953	−0.346476	−	0.938059i	$-0.612622\pi$
$239$	−2560.36	−0.692953	−0.346476	+	0.938059i	$0.612622\pi$
$240$	0	0
$241$	1141.41	0.305081	0.152541	−	0.988297i	$-0.451255\pi$
$241$	1141.41	0.305081	0.152541	+	0.988297i	$0.451255\pi$
$242$	0	0
$243$	−6961.55	−1.83779
$244$	0	0
$245$	−14.9642	−0.00390216
$246$	0	0
$247$	−697.159	−0.179592
$248$	0	0
$249$	−3725.96	−0.948286
$250$	0	0
$251$	80.1561	0.0201570	0.0100785	−	0.999949i	$-0.496792\pi$
$251$	80.1561	0.0201570	0.0100785	+	0.999949i	$0.496792\pi$
$252$	0	0
$253$	−4770.38	−1.18542
$254$	0	0
$255$	36.0241	0.00884672
$256$	0	0
$257$	−1674.71	−0.406481	−0.203240	−	0.979129i	$-0.565147\pi$
$257$	−1674.71	−0.406481	−0.203240	+	0.979129i	$0.565147\pi$
$258$	0	0
$259$	−2201.48	−0.528159
$260$	0	0
$261$	5938.76	1.40843
$262$	0	0
$263$	909.698	0.213287	0.106643	−	0.994297i	$-0.465990\pi$
$263$	909.698	0.213287	0.106643	+	0.994297i	$0.465990\pi$
$264$	0	0
$265$	10.6605	0.00247121
$266$	0	0
$267$	−166.700	−0.0382093
$268$	0	0
$269$	3338.47	0.756691	0.378346	−	0.925664i	$-0.376493\pi$
$269$	3338.47	0.756691	0.378346	+	0.925664i	$0.376493\pi$
$270$	0	0
$271$	−1156.39	−0.259209	−0.129605	−	0.991566i	$-0.541371\pi$
$271$	−1156.39	−0.259209	−0.129605	+	0.991566i	$0.541371\pi$
$272$	0	0
$273$	−8666.63	−1.92135
$274$	0	0
$275$	5412.50	1.18686
$276$	0	0
$277$	−5717.67	−1.24022	−0.620111	−	0.784514i	$-0.712912\pi$
$277$	−5717.67	−1.24022	−0.620111	+	0.784514i	$0.712912\pi$
$278$	0	0
$279$	−9646.13	−2.06989
$280$	0	0
$281$	7501.41	1.59251	0.796257	−	0.604958i	$-0.206810\pi$
$281$	7501.41	1.59251	0.796257	+	0.604958i	$0.206810\pi$
$282$	0	0
$283$	1206.82	0.253492	0.126746	−	0.991935i	$-0.459547\pi$
$283$	1206.82	0.253492	0.126746	+	0.991935i	$0.459547\pi$
$284$	0	0
$285$	10.3962	0.00216077
$286$	0	0
$287$	−10204.3	−2.09874
$288$	0	0
$289$	−578.494	−0.117748
$290$	0	0
$291$	6922.29	1.39447
$292$	0	0
$293$	3365.46	0.671032	0.335516	−	0.942035i	$-0.391089\pi$
$293$	3365.46	0.671032	0.335516	+	0.942035i	$0.391089\pi$
$294$	0	0
$295$	−16.1489	−0.00318721
$296$	0	0
$297$	15850.1	3.09669
$298$	0	0
$299$	−4042.34	−0.781854
$300$	0	0
$301$	2565.17	0.491210
$302$	0	0
$303$	12293.0	2.33073
$304$	0	0
$305$	30.7352	0.00577014
$306$	0	0
$307$	4699.93	0.873743	0.436871	−	0.899524i	$-0.356086\pi$
$307$	4699.93	0.873743	0.436871	+	0.899524i	$0.356086\pi$
$308$	0	0
$309$	2116.13	0.389586
$310$	0	0
$311$	−8300.02	−1.51335	−0.756673	−	0.653793i	$-0.773177\pi$
$311$	−8300.02	−1.51335	−0.756673	+	0.653793i	$0.773177\pi$
$312$	0	0
$313$	−2609.06	−0.471160	−0.235580	−	0.971855i	$-0.575699\pi$
$313$	−2609.06	−0.471160	−0.235580	+	0.971855i	$0.575699\pi$
$314$	0	0
$315$	91.3659	0.0163425
$316$	0	0
$317$	1392.56	0.246731	0.123366	−	0.992361i	$-0.460631\pi$
$317$	1392.56	0.246731	0.123366	+	0.992361i	$0.460631\pi$
$318$	0	0
$319$	−3948.05	−0.692941
$320$	0	0
$321$	−17001.0	−2.95608
$322$	0	0
$323$	1250.90	0.215486
$324$	0	0
$325$	4586.46	0.782802
$326$	0	0
$327$	−10840.7	−1.83331
$328$	0	0
$329$	−14833.7	−2.48574
$330$	0	0
$331$	−8795.26	−1.46052	−0.730258	−	0.683171i	$-0.760600\pi$
$331$	−8795.26	−1.46052	−0.730258	+	0.683171i	$0.760600\pi$
$332$	0	0
$333$	5827.30	0.958961
$334$	0	0
$335$	−41.6447	−0.00679192
$336$	0	0
$337$	639.206	0.103323	0.0516614	−	0.998665i	$-0.483548\pi$
$337$	639.206	0.103323	0.0516614	+	0.998665i	$0.483548\pi$
$338$	0	0
$339$	1116.35	0.178855
$340$	0	0
$341$	6412.68	1.01838
$342$	0	0
$343$	1980.67	0.311796
$344$	0	0
$345$	60.2805	0.00940693
$346$	0	0
$347$	7973.41	1.23353	0.616765	−	0.787147i	$-0.288443\pi$
$347$	7973.41	1.23353	0.616765	+	0.787147i	$0.288443\pi$
$348$	0	0
$349$	2886.35	0.442702	0.221351	−	0.975194i	$-0.428953\pi$
$349$	2886.35	0.442702	0.221351	+	0.975194i	$0.428953\pi$
$350$	0	0
$351$	13431.1	2.04245
$352$	0	0
$353$	8955.16	1.35024	0.675120	−	0.737707i	$-0.264092\pi$
$353$	8955.16	1.35024	0.675120	+	0.737707i	$0.264092\pi$
$354$	0	0
$355$	−40.3512	−0.00603273
$356$	0	0
$357$	15550.4	2.30536
$358$	0	0
$359$	2841.36	0.417720	0.208860	−	0.977946i	$-0.433025\pi$
$359$	2841.36	0.417720	0.208860	+	0.977946i	$0.433025\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	−5221.59	−0.754993
$364$	0	0
$365$	−56.3831	−0.00808555
$366$	0	0
$367$	6955.82	0.989348	0.494674	−	0.869079i	$-0.335287\pi$
$367$	6955.82	0.989348	0.494674	+	0.869079i	$0.335287\pi$
$368$	0	0
$369$	27010.7	3.81062
$370$	0	0
$371$	4601.80	0.643971
$372$	0	0
$373$	−14076.4	−1.95402	−0.977009	−	0.213199i	$-0.931612\pi$
$373$	−14076.4	−1.95402	−0.977009	+	0.213199i	$0.931612\pi$
$374$	0	0
$375$	−136.791	−0.0188369
$376$	0	0
$377$	−3345.50	−0.457035
$378$	0	0
$379$	9658.59	1.30905	0.654523	−	0.756042i	$-0.272869\pi$
$379$	9658.59	1.30905	0.654523	+	0.756042i	$0.272869\pi$
$380$	0	0
$381$	19509.4	2.62336
$382$	0	0
$383$	2037.30	0.271805	0.135903	−	0.990722i	$-0.456607\pi$
$383$	2037.30	0.271805	0.135903	+	0.990722i	$0.456607\pi$
$384$	0	0
$385$	−60.7394	−0.00804043
$386$	0	0
$387$	−6790.00	−0.891873
$388$	0	0
$389$	9276.35	1.20907	0.604537	−	0.796577i	$-0.293358\pi$
$389$	9276.35	1.20907	0.604537	+	0.796577i	$0.293358\pi$
$390$	0	0
$391$	7253.10	0.938120
$392$	0	0
$393$	27454.1	3.52386
$394$	0	0
$395$	51.1680	0.00651783
$396$	0	0
$397$	−8770.65	−1.10878	−0.554391	−	0.832257i	$-0.687049\pi$
$397$	−8770.65	−1.10878	−0.554391	+	0.832257i	$0.687049\pi$
$398$	0	0
$399$	4487.72	0.563075
$400$	0	0
$401$	−10514.4	−1.30939	−0.654693	−	0.755895i	$-0.727202\pi$
$401$	−10514.4	−1.30939	−0.654693	+	0.755895i	$0.727202\pi$
$402$	0	0
$403$	5433.99	0.671678
$404$	0	0
$405$	−100.038	−0.0122738
$406$	0	0
$407$	−3873.95	−0.471805
$408$	0	0
$409$	3645.19	0.440692	0.220346	−	0.975422i	$-0.429281\pi$
$409$	3645.19	0.440692	0.220346	+	0.975422i	$0.429281\pi$
$410$	0	0
$411$	7902.95	0.948476
$412$	0	0
$413$	−6970.95	−0.830553
$414$	0	0
$415$	−22.1278	−0.00261737
$416$	0	0
$417$	16803.0	1.97325
$418$	0	0
$419$	−9725.93	−1.13399	−0.566996	−	0.823721i	$-0.691894\pi$
$419$	−9725.93	−1.13399	−0.566996	+	0.823721i	$0.691894\pi$
$420$	0	0
$421$	−14391.3	−1.66601	−0.833004	−	0.553266i	$-0.813381\pi$
$421$	−14391.3	−1.66601	−0.833004	+	0.553266i	$0.813381\pi$
$422$	0	0
$423$	39264.8	4.51328
$424$	0	0
$425$	−8229.41	−0.939259
$426$	0	0
$427$	13267.4	1.50364
$428$	0	0
$429$	−15250.7	−1.71634
$430$	0	0
$431$	14981.5	1.67433	0.837163	−	0.546953i	$-0.184212\pi$
$431$	14981.5	1.67433	0.837163	+	0.546953i	$0.184212\pi$
$432$	0	0
$433$	2866.64	0.318157	0.159079	−	0.987266i	$-0.449148\pi$
$433$	2866.64	0.318157	0.159079	+	0.987266i	$0.449148\pi$
$434$	0	0
$435$	49.8891	0.00549885
$436$	0	0
$437$	2093.18	0.229132
$438$	0	0
$439$	17012.1	1.84953	0.924763	−	0.380543i	$-0.124263\pi$
$439$	17012.1	1.84953	0.924763	+	0.380543i	$0.124263\pi$
$440$	0	0
$441$	17098.4	1.84628
$442$	0	0
$443$	−3724.18	−0.399416	−0.199708	−	0.979855i	$-0.563999\pi$
$443$	−3724.18	−0.399416	−0.199708	+	0.979855i	$0.563999\pi$
$444$	0	0
$445$	−0.990001	−0.000105462 0
$446$	0	0
$447$	−17223.3	−1.82244
$448$	0	0
$449$	−1558.94	−0.163855	−0.0819275	−	0.996638i	$-0.526108\pi$
$449$	−1558.94	−0.163855	−0.0819275	+	0.996638i	$0.526108\pi$
$450$	0	0
$451$	−17956.5	−1.87481
$452$	0	0
$453$	29783.5	3.08908
$454$	0	0
$455$	−51.4695	−0.00530313
$456$	0	0
$457$	−17435.7	−1.78470	−0.892352	−	0.451341i	$-0.850946\pi$
$457$	−17435.7	−1.78470	−0.892352	+	0.451341i	$0.850946\pi$
$458$	0	0
$459$	−24099.2	−2.45066
$460$	0	0
$461$	−5830.59	−0.589062	−0.294531	−	0.955642i	$-0.595163\pi$
$461$	−5830.59	−0.589062	−0.294531	+	0.955642i	$0.595163\pi$
$462$	0	0
$463$	−11241.8	−1.12841	−0.564203	−	0.825636i	$-0.690816\pi$
$463$	−11241.8	−1.12841	−0.564203	+	0.825636i	$0.690816\pi$
$464$	0	0
$465$	−81.0332	−0.00808135
$466$	0	0
$467$	−6039.72	−0.598469	−0.299234	−	0.954180i	$-0.596731\pi$
$467$	−6039.72	−0.598469	−0.299234	+	0.954180i	$0.596731\pi$
$468$	0	0
$469$	−17976.7	−1.76990
$470$	0	0
$471$	−8635.04	−0.844759
$472$	0	0
$473$	4513.94	0.438798
$474$	0	0
$475$	−2374.94	−0.229410
$476$	0	0
$477$	−12180.9	−1.16924
$478$	0	0
$479$	5861.23	0.559095	0.279548	−	0.960132i	$-0.409816\pi$
$479$	5861.23	0.559095	0.279548	+	0.960132i	$0.409816\pi$
$480$	0	0
$481$	−3282.71	−0.311183
$482$	0	0
$483$	26021.1	2.45135
$484$	0	0
$485$	41.1102	0.00384890
$486$	0	0
$487$	2002.93	0.186368	0.0931842	−	0.995649i	$-0.470295\pi$
$487$	2002.93	0.186368	0.0931842	+	0.995649i	$0.470295\pi$
$488$	0	0
$489$	−34290.1	−3.17107
$490$	0	0
$491$	−5771.71	−0.530496	−0.265248	−	0.964180i	$-0.585454\pi$
$491$	−5771.71	−0.530496	−0.265248	+	0.964180i	$0.585454\pi$
$492$	0	0
$493$	6002.78	0.548381
$494$	0	0
$495$	160.777	0.0145987
$496$	0	0
$497$	−17418.3	−1.57207
$498$	0	0
$499$	−2258.01	−0.202570	−0.101285	−	0.994857i	$-0.532295\pi$
$499$	−2258.01	−0.202570	−0.101285	+	0.994857i	$0.532295\pi$
$500$	0	0
$501$	10835.2	0.966233
$502$	0	0
$503$	−1733.98	−0.153706	−0.0768532	−	0.997042i	$-0.524487\pi$
$503$	−1733.98	−0.153706	−0.0768532	+	0.997042i	$0.524487\pi$
$504$	0	0
$505$	73.0055	0.00643307
$506$	0	0
$507$	8165.15	0.715241
$508$	0	0
$509$	354.549	0.0308745	0.0154372	−	0.999881i	$-0.495086\pi$
$509$	354.549	0.0308745	0.0154372	+	0.999881i	$0.495086\pi$
$510$	0	0
$511$	−24338.7	−2.10701
$512$	0	0
$513$	−6954.83	−0.598564
$514$	0	0
$515$	12.5673	0.00107530
$516$	0	0
$517$	−26103.0	−2.22052
$518$	0	0
$519$	−6740.46	−0.570083
$520$	0	0
$521$	20496.1	1.72352	0.861758	−	0.507320i	$-0.169364\pi$
$521$	20496.1	1.72352	0.861758	+	0.507320i	$0.169364\pi$
$522$	0	0
$523$	12180.8	1.01841	0.509206	−	0.860645i	$-0.329939\pi$
$523$	12180.8	1.01841	0.509206	+	0.860645i	$0.329939\pi$
$524$	0	0
$525$	−29523.7	−2.45432
$526$	0	0
$527$	−9750.12	−0.805924
$528$	0	0
$529$	−30.1038	−0.00247422
$530$	0	0
$531$	18452.1	1.50801
$532$	0	0
$533$	−15216.0	−1.23654
$534$	0	0
$535$	−100.965	−0.00815910
$536$	0	0
$537$	−13721.0	−1.10261
$538$	0	0
$539$	−11366.9	−0.908362
$540$	0	0
$541$	8926.76	0.709411	0.354706	−	0.934978i	$-0.384581\pi$
$541$	8926.76	0.709411	0.354706	+	0.934978i	$0.384581\pi$
$542$	0	0
$543$	38621.4	3.05231
$544$	0	0
$545$	−64.3810	−0.00506014
$546$	0	0
$547$	6752.62	0.527827	0.263913	−	0.964546i	$-0.414987\pi$
$547$	6752.62	0.527827	0.263913	+	0.964546i	$0.414987\pi$
$548$	0	0
$549$	−35118.6	−2.73010
$550$	0	0
$551$	1732.35	0.133940
$552$	0	0
$553$	22087.5	1.69848
$554$	0	0
$555$	48.9528	0.00374402
$556$	0	0
$557$	−15003.6	−1.14134	−0.570668	−	0.821181i	$-0.693315\pi$
$557$	−15003.6	−1.14134	−0.570668	+	0.821181i	$0.693315\pi$
$558$	0	0
$559$	3825.03	0.289413
$560$	0	0
$561$	27364.1	2.05938
$562$	0	0
$563$	15587.1	1.16682	0.583410	−	0.812178i	$-0.301718\pi$
$563$	15587.1	1.16682	0.583410	+	0.812178i	$0.301718\pi$
$564$	0	0
$565$	6.62979	0.000493659 0
$566$	0	0
$567$	−43182.9	−3.19844
$568$	0	0
$569$	−4603.20	−0.339150	−0.169575	−	0.985517i	$-0.554239\pi$
$569$	−4603.20	−0.339150	−0.169575	+	0.985517i	$0.554239\pi$
$570$	0	0
$571$	−17145.3	−1.25658	−0.628291	−	0.777978i	$-0.716245\pi$
$571$	−17145.3	−1.25658	−0.628291	+	0.777978i	$0.716245\pi$
$572$	0	0
$573$	39008.5	2.84398
$574$	0	0
$575$	−13770.6	−0.998736
$576$	0	0
$577$	−2960.62	−0.213609	−0.106804	−	0.994280i	$-0.534062\pi$
$577$	−2960.62	−0.213609	−0.106804	+	0.994280i	$0.534062\pi$
$578$	0	0
$579$	40964.9	2.94032
$580$	0	0
$581$	−9551.83	−0.682060
$582$	0	0
$583$	8097.80	0.575260
$584$	0	0
$585$	136.239	0.00962872
$586$	0	0
$587$	21030.3	1.47873	0.739365	−	0.673304i	$-0.235126\pi$
$587$	21030.3	1.47873	0.739365	+	0.673304i	$0.235126\pi$
$588$	0	0
$589$	−2813.80	−0.196843
$590$	0	0
$591$	30910.8	2.15144
$592$	0	0
$593$	−11951.6	−0.827642	−0.413821	−	0.910358i	$-0.635806\pi$
$593$	−11951.6	−0.827642	−0.413821	+	0.910358i	$0.635806\pi$
$594$	0	0
$595$	92.3509	0.00636305
$596$	0	0
$597$	−5779.96	−0.396244
$598$	0	0
$599$	−16250.4	−1.10847	−0.554233	−	0.832361i	$-0.686988\pi$
$599$	−16250.4	−1.10847	−0.554233	+	0.832361i	$0.686988\pi$
$600$	0	0
$601$	−3085.90	−0.209445	−0.104722	−	0.994501i	$-0.533395\pi$
$601$	−3085.90	−0.209445	−0.104722	+	0.994501i	$0.533395\pi$
$602$	0	0
$603$	47584.1	3.21356
$604$	0	0
$605$	−31.0100	−0.00208386
$606$	0	0
$607$	11987.4	0.801572	0.400786	−	0.916172i	$-0.368737\pi$
$607$	11987.4	0.801572	0.400786	+	0.916172i	$0.368737\pi$
$608$	0	0
$609$	21535.5	1.43294
$610$	0	0
$611$	−22119.2	−1.46456
$612$	0	0
$613$	−24008.8	−1.58190	−0.790952	−	0.611878i	$-0.790414\pi$
$613$	−24008.8	−1.58190	−0.790952	+	0.611878i	$0.790414\pi$
$614$	0	0
$615$	226.906	0.0148776
$616$	0	0
$617$	−18159.1	−1.18486	−0.592429	−	0.805623i	$-0.701831\pi$
$617$	−18159.1	−1.18486	−0.592429	+	0.805623i	$0.701831\pi$
$618$	0	0
$619$	14599.6	0.947995	0.473997	−	0.880526i	$-0.342811\pi$
$619$	14599.6	0.947995	0.473997	+	0.880526i	$0.342811\pi$
$620$	0	0
$621$	−40326.1	−2.60585
$622$	0	0
$623$	−427.351	−0.0274823
$624$	0	0
$625$	15623.8	0.999922
$626$	0	0
$627$	7897.05	0.502995
$628$	0	0
$629$	5890.12	0.373378
$630$	0	0
$631$	−24728.3	−1.56009	−0.780045	−	0.625724i	$-0.784804\pi$
$631$	−24728.3	−1.56009	−0.780045	+	0.625724i	$0.784804\pi$
$632$	0	0
$633$	−40875.7	−2.56661
$634$	0	0
$635$	115.863	0.00724075
$636$	0	0
$637$	−9632.11	−0.599118
$638$	0	0
$639$	46106.1	2.85435
$640$	0	0
$641$	−22562.2	−1.39026	−0.695128	−	0.718886i	$-0.744652\pi$
$641$	−22562.2	−1.39026	−0.695128	+	0.718886i	$0.744652\pi$
$642$	0	0
$643$	−4168.62	−0.255668	−0.127834	−	0.991796i	$-0.540802\pi$
$643$	−4168.62	−0.255668	−0.127834	+	0.991796i	$0.540802\pi$
$644$	0	0
$645$	−57.0400	−0.00348209
$646$	0	0
$647$	13452.2	0.817402	0.408701	−	0.912668i	$-0.365982\pi$
$647$	13452.2	0.817402	0.408701	+	0.912668i	$0.365982\pi$
$648$	0	0
$649$	−12266.8	−0.741933
$650$	0	0
$651$	−34979.4	−2.10591
$652$	0	0
$653$	−11996.7	−0.718939	−0.359469	−	0.933157i	$-0.617042\pi$
$653$	−11996.7	−0.718939	−0.359469	+	0.933157i	$0.617042\pi$
$654$	0	0
$655$	163.045	0.00972622
$656$	0	0
$657$	64424.4	3.82562
$658$	0	0
$659$	−28779.1	−1.70117	−0.850586	−	0.525835i	$-0.823753\pi$
$659$	−28779.1	−1.70117	−0.850586	+	0.525835i	$0.823753\pi$
$660$	0	0
$661$	20145.1	1.18540	0.592702	−	0.805422i	$-0.298061\pi$
$661$	20145.1	1.18540	0.592702	+	0.805422i	$0.298061\pi$
$662$	0	0
$663$	23187.8	1.35828
$664$	0	0
$665$	26.6517	0.00155415
$666$	0	0
$667$	10044.7	0.583106
$668$	0	0
$669$	17134.3	0.990208
$670$	0	0
$671$	23346.6	1.34320
$672$	0	0
$673$	13609.8	0.779523	0.389761	−	0.920916i	$-0.372557\pi$
$673$	13609.8	0.779523	0.389761	+	0.920916i	$0.372557\pi$
$674$	0	0
$675$	45754.3	2.60901
$676$	0	0
$677$	16098.6	0.913913	0.456957	−	0.889489i	$-0.348939\pi$
$677$	16098.6	0.913913	0.456957	+	0.889489i	$0.348939\pi$
$678$	0	0
$679$	17745.9	1.00298
$680$	0	0
$681$	−24783.4	−1.39457
$682$	0	0
$683$	−11107.3	−0.622266	−0.311133	−	0.950366i	$-0.600708\pi$
$683$	−11107.3	−0.622266	−0.311133	+	0.950366i	$0.600708\pi$
$684$	0	0
$685$	46.9341	0.00261790
$686$	0	0
$687$	−17071.3	−0.948049
$688$	0	0
$689$	6861.92	0.379417
$690$	0	0
$691$	9405.37	0.517796	0.258898	−	0.965905i	$-0.416641\pi$
$691$	9405.37	0.517796	0.258898	+	0.965905i	$0.416641\pi$
$692$	0	0
$693$	69402.1	3.80428
$694$	0	0
$695$	99.7899	0.00544640
$696$	0	0
$697$	27301.8	1.48369
$698$	0	0
$699$	−61577.5	−3.33201
$700$	0	0
$701$	20205.4	1.08865	0.544327	−	0.838873i	$-0.316785\pi$
$701$	20205.4	1.08865	0.544327	+	0.838873i	$0.316785\pi$
$702$	0	0
$703$	1699.84	0.0911959
$704$	0	0
$705$	329.847	0.0176210
$706$	0	0
$707$	31514.1	1.67639
$708$	0	0
$709$	25892.5	1.37153	0.685763	−	0.727825i	$-0.259469\pi$
$709$	25892.5	1.37153	0.685763	+	0.727825i	$0.259469\pi$
$710$	0	0
$711$	−58465.6	−3.08387
$712$	0	0
$713$	−16315.3	−0.856959
$714$	0	0
$715$	−90.5710	−0.00473729
$716$	0	0
$717$	24576.1	1.28007
$718$	0	0
$719$	26267.6	1.36247	0.681235	−	0.732064i	$-0.261443\pi$
$719$	26267.6	1.36247	0.681235	+	0.732064i	$0.261443\pi$
$720$	0	0
$721$	5424.87	0.280212
$722$	0	0
$723$	−10956.0	−0.563567
$724$	0	0
$725$	−11396.8	−0.583814
$726$	0	0
$727$	−29489.6	−1.50442	−0.752208	−	0.658926i	$-0.771011\pi$
$727$	−29489.6	−1.50442	−0.752208	+	0.658926i	$0.771011\pi$
$728$	0	0
$729$	19439.4	0.987626
$730$	0	0
$731$	−6863.20	−0.347257
$732$	0	0
$733$	−1452.30	−0.0731812	−0.0365906	−	0.999330i	$-0.511650\pi$
$733$	−1452.30	−0.0731812	−0.0365906	+	0.999330i	$0.511650\pi$
$734$	0	0
$735$	143.637	0.00720833
$736$	0	0
$737$	−31633.6	−1.58106
$738$	0	0
$739$	−654.985	−0.0326035	−0.0163018	−	0.999867i	$-0.505189\pi$
$739$	−654.985	−0.0326035	−0.0163018	+	0.999867i	$0.505189\pi$
$740$	0	0
$741$	6691.81	0.331754
$742$	0	0
$743$	−10410.0	−0.514008	−0.257004	−	0.966410i	$-0.582735\pi$
$743$	−10410.0	−0.514008	−0.257004	+	0.966410i	$0.582735\pi$
$744$	0	0
$745$	−102.286	−0.00503015
$746$	0	0
$747$	25283.6	1.23839
$748$	0	0
$749$	−43583.5	−2.12617
$750$	0	0
$751$	21744.2	1.05653	0.528267	−	0.849079i	$-0.322842\pi$
$751$	21744.2	1.05653	0.528267	+	0.849079i	$0.322842\pi$
$752$	0	0
$753$	−769.393	−0.0372354
$754$	0	0
$755$	176.879	0.00852619
$756$	0	0
$757$	−2073.45	−0.0995518	−0.0497759	−	0.998760i	$-0.515851\pi$
$757$	−2073.45	−0.0995518	−0.0497759	+	0.998760i	$0.515851\pi$
$758$	0	0
$759$	45789.4	2.18979
$760$	0	0
$761$	−35323.0	−1.68260	−0.841299	−	0.540571i	$-0.818208\pi$
$761$	−35323.0	−1.68260	−0.841299	+	0.540571i	$0.818208\pi$
$762$	0	0
$763$	−27791.2	−1.31862
$764$	0	0
$765$	−244.452	−0.0115532
$766$	0	0
$767$	−10394.7	−0.489348
$768$	0	0
$769$	−1210.78	−0.0567775	−0.0283887	−	0.999597i	$-0.509038\pi$
$769$	−1210.78	−0.0567775	−0.0283887	+	0.999597i	$0.509038\pi$
$770$	0	0
$771$	16075.0	0.750879
$772$	0	0
$773$	12359.3	0.575074	0.287537	−	0.957769i	$-0.407164\pi$
$773$	12359.3	0.575074	0.287537	+	0.957769i	$0.407164\pi$
$774$	0	0
$775$	18511.4	0.857998
$776$	0	0
$777$	21131.3	0.975652
$778$	0	0
$779$	7879.09	0.362385
$780$	0	0
$781$	−30651.0	−1.40433
$782$	0	0
$783$	−33374.6	−1.52326
$784$	0	0
$785$	−51.2818	−0.00233163
$786$	0	0
$787$	−14821.7	−0.671332	−0.335666	−	0.941981i	$-0.608961\pi$
$787$	−14821.7	−0.671332	−0.335666	+	0.941981i	$0.608961\pi$
$788$	0	0
$789$	−8731.91	−0.393998
$790$	0	0
$791$	2861.86	0.128642
$792$	0	0
$793$	19783.5	0.885918
$794$	0	0
$795$	−102.327	−0.00456499
$796$	0	0
$797$	8213.76	0.365052	0.182526	−	0.983201i	$-0.441573\pi$
$797$	8213.76	0.365052	0.182526	+	0.983201i	$0.441573\pi$
$798$	0	0
$799$	39688.1	1.75728
$800$	0	0
$801$	1131.20	0.0498987
$802$	0	0
$803$	−42828.9	−1.88219
$804$	0	0
$805$	154.534	0.00676599
$806$	0	0
$807$	−32044.9	−1.39781
$808$	0	0
$809$	17199.4	0.747463	0.373731	−	0.927537i	$-0.378078\pi$
$809$	17199.4	0.747463	0.373731	+	0.927537i	$0.378078\pi$
$810$	0	0
$811$	−535.506	−0.0231864	−0.0115932	−	0.999933i	$-0.503690\pi$
$811$	−535.506	−0.0231864	−0.0115932	+	0.999933i	$0.503690\pi$
$812$	0	0
$813$	11099.8	0.478829
$814$	0	0
$815$	−203.643	−0.00875250
$816$	0	0
$817$	−1980.66	−0.0848159
$818$	0	0
$819$	58810.0	2.50914
$820$	0	0
$821$	−40618.3	−1.72666	−0.863330	−	0.504640i	$-0.831625\pi$
$821$	−40618.3	−1.72666	−0.863330	+	0.504640i	$0.831625\pi$
$822$	0	0
$823$	24003.4	1.01665	0.508327	−	0.861164i	$-0.330264\pi$
$823$	24003.4	1.01665	0.508327	+	0.861164i	$0.330264\pi$
$824$	0	0
$825$	−51952.9	−2.19245
$826$	0	0
$827$	−43837.2	−1.84325	−0.921625	−	0.388081i	$-0.873138\pi$
$827$	−43837.2	−1.84325	−0.921625	+	0.388081i	$0.873138\pi$
$828$	0	0
$829$	−20706.3	−0.867503	−0.433752	−	0.901033i	$-0.642810\pi$
$829$	−20706.3	−0.867503	−0.433752	+	0.901033i	$0.642810\pi$
$830$	0	0
$831$	54882.1	2.29102
$832$	0	0
$833$	17282.7	0.718861
$834$	0	0
$835$	64.3484	0.00266691
$836$	0	0
$837$	54209.2	2.23864
$838$	0	0
$839$	−41630.9	−1.71306	−0.856530	−	0.516098i	$-0.827384\pi$
$839$	−41630.9	−1.71306	−0.856530	+	0.516098i	$0.827384\pi$
$840$	0	0
$841$	−16075.9	−0.659144
$842$	0	0
$843$	−72003.7	−2.94180
$844$	0	0
$845$	48.4913	0.00197414
$846$	0	0
$847$	−13386.0	−0.543033
$848$	0	0
$849$	−11583.9	−0.468267
$850$	0	0
$851$	9856.18	0.397022
$852$	0	0
$853$	6032.28	0.242135	0.121068	−	0.992644i	$-0.461368\pi$
$853$	6032.28	0.242135	0.121068	+	0.992644i	$0.461368\pi$
$854$	0	0
$855$	−70.5468	−0.00282181
$856$	0	0
$857$	7238.25	0.288511	0.144255	−	0.989540i	$-0.453921\pi$
$857$	7238.25	0.288511	0.144255	+	0.989540i	$0.453921\pi$
$858$	0	0
$859$	−14061.0	−0.558503	−0.279251	−	0.960218i	$-0.590086\pi$
$859$	−14061.0	−0.558503	−0.279251	+	0.960218i	$0.590086\pi$
$860$	0	0
$861$	97947.6	3.87694
$862$	0	0
$863$	−20525.0	−0.809593	−0.404797	−	0.914407i	$-0.632658\pi$
$863$	−20525.0	−0.809593	−0.404797	+	0.914407i	$0.632658\pi$
$864$	0	0
$865$	−40.0303	−0.00157349
$866$	0	0
$867$	5552.78	0.217511
$868$	0	0
$869$	38867.5	1.51725
$870$	0	0
$871$	−26805.7	−1.04280
$872$	0	0
$873$	−46973.3	−1.82108
$874$	0	0
$875$	−350.676	−0.0135486
$876$	0	0
$877$	−2364.44	−0.0910394	−0.0455197	−	0.998963i	$-0.514494\pi$
$877$	−2364.44	−0.0910394	−0.0455197	+	0.998963i	$0.514494\pi$
$878$	0	0
$879$	−32304.0	−1.23958
$880$	0	0
$881$	−10705.0	−0.409377	−0.204688	−	0.978827i	$-0.565618\pi$
$881$	−10705.0	−0.409377	−0.204688	+	0.978827i	$0.565618\pi$
$882$	0	0
$883$	13097.7	0.499177	0.249588	−	0.968352i	$-0.419705\pi$
$883$	13097.7	0.499177	0.249588	+	0.968352i	$0.419705\pi$
$884$	0	0
$885$	155.008	0.00588763
$886$	0	0
$887$	16986.7	0.643018	0.321509	−	0.946907i	$-0.395810\pi$
$887$	16986.7	0.643018	0.321509	+	0.946907i	$0.395810\pi$
$888$	0	0
$889$	50014.2	1.88686
$890$	0	0
$891$	−75989.2	−2.85716
$892$	0	0
$893$	11453.6	0.429207
$894$	0	0
$895$	−81.4862	−0.00304333
$896$	0	0
$897$	38801.1	1.44429
$898$	0	0
$899$	−13502.8	−0.500937
$900$	0	0
$901$	−12312.2	−0.455250
$902$	0	0
$903$	−24622.3	−0.907396
$904$	0	0
$905$	229.365	0.00842470
$906$	0	0
$907$	−17790.9	−0.651309	−0.325654	−	0.945489i	$-0.605585\pi$
$907$	−17790.9	−0.651309	−0.325654	+	0.945489i	$0.605585\pi$
$908$	0	0
$909$	−83417.5	−3.04377
$910$	0	0
$911$	−15308.1	−0.556730	−0.278365	−	0.960475i	$-0.589793\pi$
$911$	−15308.1	−0.556730	−0.278365	+	0.960475i	$0.589793\pi$
$912$	0	0
$913$	−16808.4	−0.609284
$914$	0	0
$915$	−295.017	−0.0106590
$916$	0	0
$917$	70380.9	2.53455
$918$	0	0
$919$	−41114.6	−1.47578	−0.737892	−	0.674919i	$-0.764179\pi$
$919$	−41114.6	−1.47578	−0.737892	+	0.674919i	$0.764179\pi$
$920$	0	0
$921$	−45113.1	−1.61404
$922$	0	0
$923$	−25973.1	−0.926235
$924$	0	0
$925$	−11182.9	−0.397503
$926$	0	0
$927$	−14359.6	−0.508772
$928$	0	0
$929$	−27352.7	−0.965998	−0.482999	−	0.875621i	$-0.660453\pi$
$929$	−27352.7	−0.965998	−0.482999	+	0.875621i	$0.660453\pi$
$930$	0	0
$931$	4987.66	0.175579
$932$	0	0
$933$	79669.2	2.79556
$934$	0	0
$935$	162.510	0.00568412
$936$	0	0
$937$	35879.0	1.25092	0.625462	−	0.780254i	$-0.284910\pi$
$937$	35879.0	1.25092	0.625462	+	0.780254i	$0.284910\pi$
$938$	0	0
$939$	25043.6	0.870358
$940$	0	0
$941$	−4435.13	−0.153646	−0.0768232	−	0.997045i	$-0.524478\pi$
$941$	−4435.13	−0.153646	−0.0768232	+	0.997045i	$0.524478\pi$
$942$	0	0
$943$	45685.3	1.57764
$944$	0	0
$945$	−513.457	−0.0176749
$946$	0	0
$947$	−22077.9	−0.757589	−0.378794	−	0.925481i	$-0.623661\pi$
$947$	−22077.9	−0.757589	−0.378794	+	0.925481i	$0.623661\pi$
$948$	0	0
$949$	−36292.4	−1.24141
$950$	0	0
$951$	−13366.7	−0.455779
$952$	0	0
$953$	33060.2	1.12374	0.561870	−	0.827226i	$-0.310082\pi$
$953$	33060.2	1.12374	0.561870	+	0.827226i	$0.310082\pi$
$954$	0	0
$955$	231.664	0.00784971
$956$	0	0
$957$	37896.0	1.28005
$958$	0	0
$959$	20259.9	0.682196
$960$	0	0
$961$	−7858.88	−0.263801
$962$	0	0
$963$	115365.	3.86042
$964$	0	0
$965$	243.283	0.00811559
$966$	0	0
$967$	26626.6	0.885474	0.442737	−	0.896651i	$-0.354007\pi$
$967$	26626.6	0.885474	0.442737	+	0.896651i	$0.354007\pi$
$968$	0	0
$969$	−12007.0	−0.398061
$970$	0	0
$971$	−3039.26	−0.100447	−0.0502237	−	0.998738i	$-0.515993\pi$
$971$	−3039.26	−0.100447	−0.0502237	+	0.998738i	$0.515993\pi$
$972$	0	0
$973$	43076.0	1.41927
$974$	0	0
$975$	−44023.9	−1.44605
$976$	0	0
$977$	2405.10	0.0787574	0.0393787	−	0.999224i	$-0.487462\pi$
$977$	2405.10	0.0787574	0.0393787	+	0.999224i	$0.487462\pi$
$978$	0	0
$979$	−752.011	−0.0245499
$980$	0	0
$981$	73563.0	2.39418
$982$	0	0
$983$	24676.7	0.800675	0.400337	−	0.916368i	$-0.368893\pi$
$983$	24676.7	0.800675	0.400337	+	0.916368i	$0.368893\pi$
$984$	0	0
$985$	183.573	0.00593820
$986$	0	0
$987$	142384.	4.59183
$988$	0	0
$989$	−11484.5	−0.369246
$990$	0	0
$991$	−24590.8	−0.788246	−0.394123	−	0.919058i	$-0.628952\pi$
$991$	−24590.8	−0.788246	−0.394123	+	0.919058i	$0.628952\pi$
$992$	0	0
$993$	84422.9	2.69797
$994$	0	0
$995$	−34.3261	−0.00109368
$996$	0	0
$997$	−42352.3	−1.34535	−0.672674	−	0.739939i	$-0.734854\pi$
$997$	−42352.3	−1.34535	−0.672674	+	0.739939i	$0.734854\pi$
$998$	0	0
$999$	−32748.2	−1.03714

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.4.a.bc.1.1		5
4.3	odd	2		1216.4.a.z.1.5		5
8.3	odd	2		608.4.a.g.1.1	yes	5
8.5	even	2		608.4.a.f.1.5	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
608.4.a.f.1.5	✓	5	8.5	even	2
608.4.a.g.1.1	yes	5	8.3	odd	2
1216.4.a.z.1.5		5	4.3	odd	2
1216.4.a.bc.1.1		5	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1216.a (trivial)

\(f(q)\)	\(=\)	\(q-9.59869 q^{3} -0.0570048 q^{5} -24.6071 q^{7} +65.1348 q^{9} +O(q^{10})\)	\(q-9.59869 q^{3} -0.0570048 q^{5} -24.6071 q^{7} +65.1348 q^{9} -43.3012 q^{11} -36.6926 q^{13} +0.547171 q^{15} +65.8370 q^{17} +19.0000 q^{19} +236.196 q^{21} +110.168 q^{23} -124.997 q^{25} -366.044 q^{27} +91.1764 q^{29} -148.095 q^{31} +415.634 q^{33} +1.40272 q^{35} +89.4653 q^{37} +352.201 q^{39} +414.689 q^{41} -104.245 q^{43} -3.71299 q^{45} +602.823 q^{47} +262.508 q^{49} -631.948 q^{51} -187.011 q^{53} +2.46837 q^{55} -182.375 q^{57} +283.291 q^{59} -539.169 q^{61} -1602.78 q^{63} +2.09165 q^{65} +730.548 q^{67} -1057.46 q^{69} +707.857 q^{71} +989.094 q^{73} +1199.80 q^{75} +1065.51 q^{77} -897.609 q^{79} +1754.90 q^{81} +388.174 q^{83} -3.75302 q^{85} -875.174 q^{87} +17.3670 q^{89} +902.898 q^{91} +1421.52 q^{93} -1.08309 q^{95} -721.171 q^{97} -2820.41 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 3 q^{3} - 27 q^{5} + 20 q^{7} + 154 q^{9}+O(q^{10}) \) 5 * q + 3 * q^3 - 27 * q^5 + 20 * q^7 + 154 * q^9	\( 5 q + 3 q^{3} - 27 q^{5} + 20 q^{7} + 154 q^{9} - 67 q^{11} - 191 q^{13} - 36 q^{15} - 52 q^{17} + 95 q^{19} + 29 q^{21} + 35 q^{23} - 2 q^{25} - 27 q^{27} + 33 q^{29} - 694 q^{31} + 298 q^{33} - 387 q^{35} - 108 q^{37} - 31 q^{39} + 54 q^{41} + 427 q^{43} - 1699 q^{45} + 79 q^{47} + 1033 q^{49} - 363 q^{51} - 1025 q^{53} + 1431 q^{55} + 57 q^{57} + 649 q^{59} - 1413 q^{61} - 1449 q^{63} + 548 q^{65} - 915 q^{67} - 407 q^{69} + 1444 q^{71} + 2682 q^{73} + 465 q^{75} - 169 q^{77} - 836 q^{79} + 2925 q^{81} + 502 q^{83} - 373 q^{85} - 3491 q^{87} + 996 q^{89} - 2919 q^{91} - 2492 q^{93} - 513 q^{95} + 424 q^{97} - 7249 q^{99}+O(q^{100}) \) 5 * q + 3 * q^3 - 27 * q^5 + 20 * q^7 + 154 * q^9 - 67 * q^11 - 191 * q^13 - 36 * q^15 - 52 * q^17 + 95 * q^19 + 29 * q^21 + 35 * q^23 - 2 * q^25 - 27 * q^27 + 33 * q^29 - 694 * q^31 + 298 * q^33 - 387 * q^35 - 108 * q^37 - 31 * q^39 + 54 * q^41 + 427 * q^43 - 1699 * q^45 + 79 * q^47 + 1033 * q^49 - 363 * q^51 - 1025 * q^53 + 1431 * q^55 + 57 * q^57 + 649 * q^59 - 1413 * q^61 - 1449 * q^63 + 548 * q^65 - 915 * q^67 - 407 * q^69 + 1444 * q^71 + 2682 * q^73 + 465 * q^75 - 169 * q^77 - 836 * q^79 + 2925 * q^81 + 502 * q^83 - 373 * q^85 - 3491 * q^87 + 996 * q^89 - 2919 * q^91 - 2492 * q^93 - 513 * q^95 + 424 * q^97 - 7249 * q^99

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