LMFDB - Embedded newform 272.4.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [272,4,Mod(1,272)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(272, 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("272.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 272 = 2^{4} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	272.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:	$16.0485195216$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.2636.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 14x - 4 $ x^3 - 14*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 17)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.87707$ of defining polynomial
Character	$\chi$	$=$	272.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.15463 q^{3} +3.03171 q^{5} +7.94049 q^{7} -17.0483 q^{9} +O(q^{10})$	$q-3.15463 q^{3} +3.03171 q^{5} +7.94049 q^{7} -17.0483 q^{9} -27.6161 q^{11} +58.1117 q^{13} -9.56391 q^{15} -17.0000 q^{17} -89.1688 q^{19} -25.0493 q^{21} +115.269 q^{23} -115.809 q^{25} +138.956 q^{27} -128.558 q^{29} -273.460 q^{31} +87.1187 q^{33} +24.0732 q^{35} -132.351 q^{37} -183.321 q^{39} -470.559 q^{41} -352.642 q^{43} -51.6854 q^{45} -152.598 q^{47} -279.949 q^{49} +53.6287 q^{51} +527.614 q^{53} -83.7239 q^{55} +281.295 q^{57} +292.020 q^{59} -53.8962 q^{61} -135.372 q^{63} +176.178 q^{65} -52.9572 q^{67} -363.632 q^{69} -788.400 q^{71} +295.780 q^{73} +365.334 q^{75} -219.285 q^{77} +720.325 q^{79} +21.9487 q^{81} +116.051 q^{83} -51.5390 q^{85} +405.552 q^{87} -813.329 q^{89} +461.435 q^{91} +862.664 q^{93} -270.334 q^{95} +794.693 q^{97} +470.808 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 4 q^{3} - 8 q^{5} - 22 q^{7} + 59 q^{9}+O(q^{10}) $ 3 * q - 4 * q^3 - 8 * q^5 - 22 * q^7 + 59 * q^9	$ 3 q - 4 q^{3} - 8 q^{5} - 22 q^{7} + 59 q^{9} + 28 q^{11} + 30 q^{13} - 108 q^{15} - 51 q^{17} - 80 q^{19} - 192 q^{21} - 142 q^{23} - 223 q^{25} + 20 q^{27} - 456 q^{29} - 230 q^{31} - 332 q^{33} + 332 q^{35} + 356 q^{37} - 268 q^{39} - 294 q^{41} - 556 q^{43} - 384 q^{45} - 640 q^{47} - 269 q^{49} + 68 q^{51} + 302 q^{53} - 76 q^{55} - 720 q^{57} - 636 q^{59} - 84 q^{61} - 1122 q^{63} + 408 q^{65} - 1008 q^{67} + 576 q^{69} + 402 q^{71} + 838 q^{73} + 1548 q^{75} - 504 q^{77} + 594 q^{79} - 505 q^{81} + 2396 q^{83} + 136 q^{85} - 1428 q^{87} - 170 q^{89} + 1016 q^{91} + 632 q^{93} + 472 q^{95} - 270 q^{97} + 2920 q^{99}+O(q^{100}) $ 3 * q - 4 * q^3 - 8 * q^5 - 22 * q^7 + 59 * q^9 + 28 * q^11 + 30 * q^13 - 108 * q^15 - 51 * q^17 - 80 * q^19 - 192 * q^21 - 142 * q^23 - 223 * q^25 + 20 * q^27 - 456 * q^29 - 230 * q^31 - 332 * q^33 + 332 * q^35 + 356 * q^37 - 268 * q^39 - 294 * q^41 - 556 * q^43 - 384 * q^45 - 640 * q^47 - 269 * q^49 + 68 * q^51 + 302 * q^53 - 76 * q^55 - 720 * q^57 - 636 * q^59 - 84 * q^61 - 1122 * q^63 + 408 * q^65 - 1008 * q^67 + 576 * q^69 + 402 * q^71 + 838 * q^73 + 1548 * q^75 - 504 * q^77 + 594 * q^79 - 505 * q^81 + 2396 * q^83 + 136 * q^85 - 1428 * q^87 - 170 * q^89 + 1016 * q^91 + 632 * q^93 + 472 * q^95 - 270 * q^97 + 2920 * 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
$3$	−3.15463	−0.607109	−0.303555	−	0.952814i	$-0.598173\pi$
$3$	−3.15463	−0.607109	−0.303555	+	0.952814i	$0.598173\pi$
$4$	0	0
$5$	3.03171	0.271164	0.135582	−	0.990766i	$-0.456710\pi$
$5$	3.03171	0.271164	0.135582	+	0.990766i	$0.456710\pi$
$6$	0	0
$7$	7.94049	0.428746	0.214373	−	0.976752i	$-0.431229\pi$
$7$	7.94049	0.428746	0.214373	+	0.976752i	$0.431229\pi$
$8$	0	0
$9$	−17.0483	−0.631419
$10$	0	0
$11$	−27.6161	−0.756961	−0.378481	−	0.925609i	$-0.623553\pi$
$11$	−27.6161	−0.756961	−0.378481	+	0.925609i	$0.623553\pi$
$12$	0	0
$13$	58.1117	1.23979	0.619896	−	0.784684i	$-0.287175\pi$
$13$	58.1117	1.23979	0.619896	+	0.784684i	$0.287175\pi$
$14$	0	0
$15$	−9.56391	−0.164626
$16$	0	0
$17$	−17.0000	−0.242536
$17$	−17.0000	−0.242536
$18$	0	0
$19$	−89.1688	−1.07667	−0.538335	−	0.842731i	$-0.680946\pi$
$19$	−89.1688	−1.07667	−0.538335	+	0.842731i	$0.680946\pi$
$20$	0	0
$21$	−25.0493	−0.260296
$22$	0	0
$23$	115.269	1.04501	0.522507	−	0.852635i	$-0.324997\pi$
$23$	115.269	1.04501	0.522507	+	0.852635i	$0.324997\pi$
$24$	0	0
$25$	−115.809	−0.926470
$26$	0	0
$27$	138.956	0.990449
$28$	0	0
$29$	−128.558	−0.823191	−0.411596	−	0.911367i	$-0.635028\pi$
$29$	−128.558	−0.823191	−0.411596	+	0.911367i	$0.635028\pi$
$30$	0	0
$31$	−273.460	−1.58435	−0.792174	−	0.610295i	$-0.791051\pi$
$31$	−273.460	−1.58435	−0.792174	+	0.610295i	$0.791051\pi$
$32$	0	0
$33$	87.1187	0.459558
$34$	0	0
$35$	24.0732	0.116260
$36$	0	0
$37$	−132.351	−0.588063	−0.294031	−	0.955796i	$-0.594997\pi$
$37$	−132.351	−0.588063	−0.294031	+	0.955796i	$0.594997\pi$
$38$	0	0
$39$	−183.321	−0.752689
$40$	0	0
$41$	−470.559	−1.79241	−0.896207	−	0.443636i	$-0.853688\pi$
$41$	−470.559	−1.79241	−0.896207	+	0.443636i	$0.853688\pi$
$42$	0	0
$43$	−352.642	−1.25064	−0.625318	−	0.780370i	$-0.715031\pi$
$43$	−352.642	−1.25064	−0.625318	+	0.780370i	$0.715031\pi$
$44$	0	0
$45$	−51.6854	−0.171218
$46$	0	0
$47$	−152.598	−0.473589	−0.236795	−	0.971560i	$-0.576097\pi$
$47$	−152.598	−0.473589	−0.236795	+	0.971560i	$0.576097\pi$
$48$	0	0
$49$	−279.949	−0.816177
$50$	0	0
$51$	53.6287	0.147246
$52$	0	0
$53$	527.614	1.36742	0.683711	−	0.729753i	$-0.260365\pi$
$53$	527.614	1.36742	0.683711	+	0.729753i	$0.260365\pi$
$54$	0	0
$55$	−83.7239	−0.205261
$56$	0	0
$57$	281.295	0.653656
$58$	0	0
$59$	292.020	0.644368	0.322184	−	0.946677i	$-0.395583\pi$
$59$	292.020	0.644368	0.322184	+	0.946677i	$0.395583\pi$
$60$	0	0
$61$	−53.8962	−0.113126	−0.0565632	−	0.998399i	$-0.518014\pi$
$61$	−53.8962	−0.113126	−0.0565632	+	0.998399i	$0.518014\pi$
$62$	0	0
$63$	−135.372	−0.270718
$64$	0	0
$65$	176.178	0.336187
$66$	0	0
$67$	−52.9572	−0.0965635	−0.0482817	−	0.998834i	$-0.515375\pi$
$67$	−52.9572	−0.0965635	−0.0482817	+	0.998834i	$0.515375\pi$
$68$	0	0
$69$	−363.632	−0.634437
$70$	0	0
$71$	−788.400	−1.31783	−0.658915	−	0.752218i	$-0.728984\pi$
$71$	−788.400	−1.31783	−0.658915	+	0.752218i	$0.728984\pi$
$72$	0	0
$73$	295.780	0.474224	0.237112	−	0.971482i	$-0.423799\pi$
$73$	295.780	0.474224	0.237112	+	0.971482i	$0.423799\pi$
$74$	0	0
$75$	365.334	0.562468
$76$	0	0
$77$	−219.285	−0.324544
$78$	0	0
$79$	720.325	1.02586	0.512930	−	0.858430i	$-0.328560\pi$
$79$	720.325	1.02586	0.512930	+	0.858430i	$0.328560\pi$
$80$	0	0
$81$	21.9487	0.0301079
$82$	0	0
$83$	116.051	0.153473	0.0767363	−	0.997051i	$-0.475550\pi$
$83$	116.051	0.153473	0.0767363	+	0.997051i	$0.475550\pi$
$84$	0	0
$85$	−51.5390	−0.0657669
$86$	0	0
$87$	405.552	0.499767
$88$	0	0
$89$	−813.329	−0.968682	−0.484341	−	0.874879i	$-0.660941\pi$
$89$	−813.329	−0.968682	−0.484341	+	0.874879i	$0.660941\pi$
$90$	0	0
$91$	461.435	0.531556
$92$	0	0
$93$	862.664	0.961872
$94$	0	0
$95$	−270.334	−0.291954
$96$	0	0
$97$	794.693	0.831844	0.415922	−	0.909400i	$-0.363459\pi$
$97$	794.693	0.831844	0.415922	+	0.909400i	$0.363459\pi$
$98$	0	0
$99$	470.808	0.477959
$100$	0	0
$101$	265.513	0.261579	0.130790	−	0.991410i	$-0.458249\pi$
$101$	265.513	0.261579	0.130790	+	0.991410i	$0.458249\pi$
$102$	0	0
$103$	−523.107	−0.500420	−0.250210	−	0.968192i	$-0.580500\pi$
$103$	−523.107	−0.500420	−0.250210	+	0.968192i	$0.580500\pi$
$104$	0	0
$105$	−75.9421	−0.0705828
$106$	0	0
$107$	986.039	0.890878	0.445439	−	0.895312i	$-0.353048\pi$
$107$	986.039	0.890878	0.445439	+	0.895312i	$0.353048\pi$
$108$	0	0
$109$	1814.39	1.59438	0.797188	−	0.603732i	$-0.206320\pi$
$109$	1814.39	1.59438	0.797188	+	0.603732i	$0.206320\pi$
$110$	0	0
$111$	417.518	0.357018
$112$	0	0
$113$	−707.339	−0.588857	−0.294429	−	0.955673i	$-0.595129\pi$
$113$	−707.339	−0.588857	−0.294429	+	0.955673i	$0.595129\pi$
$114$	0	0
$115$	349.463	0.283370
$116$	0	0
$117$	−990.706	−0.782827
$118$	0	0
$119$	−134.988	−0.103986
$120$	0	0
$121$	−568.350	−0.427010
$122$	0	0
$123$	1484.44	1.08819
$124$	0	0
$125$	−730.061	−0.522389
$126$	0	0
$127$	−2648.18	−1.85030	−0.925151	−	0.379600i	$-0.876062\pi$
$127$	−2648.18	−1.85030	−0.925151	+	0.379600i	$0.876062\pi$
$128$	0	0
$129$	1112.46	0.759273
$130$	0	0
$131$	1979.08	1.31995	0.659974	−	0.751289i	$-0.270567\pi$
$131$	1979.08	1.31995	0.659974	+	0.751289i	$0.270567\pi$
$132$	0	0
$133$	−708.044	−0.461618
$134$	0	0
$135$	421.274	0.268574
$136$	0	0
$137$	3141.92	1.95936	0.979679	−	0.200570i	$-0.0642794\pi$
$137$	3141.92	1.95936	0.979679	+	0.200570i	$0.0642794\pi$
$138$	0	0
$139$	−1468.07	−0.895830	−0.447915	−	0.894076i	$-0.647833\pi$
$139$	−1468.07	−0.895830	−0.447915	+	0.894076i	$0.647833\pi$
$140$	0	0
$141$	481.390	0.287520
$142$	0	0
$143$	−1604.82	−0.938474
$144$	0	0
$145$	−389.749	−0.223220
$146$	0	0
$147$	883.135	0.495508
$148$	0	0
$149$	−286.027	−0.157263	−0.0786316	−	0.996904i	$-0.525055\pi$
$149$	−286.027	−0.157263	−0.0786316	+	0.996904i	$0.525055\pi$
$150$	0	0
$151$	669.626	0.360883	0.180442	−	0.983586i	$-0.442247\pi$
$151$	669.626	0.360883	0.180442	+	0.983586i	$0.442247\pi$
$152$	0	0
$153$	289.821	0.153141
$154$	0	0
$155$	−829.049	−0.429618
$156$	0	0
$157$	720.809	0.366413	0.183206	−	0.983074i	$-0.441352\pi$
$157$	720.809	0.366413	0.183206	+	0.983074i	$0.441352\pi$
$158$	0	0
$159$	−1664.43	−0.830174
$160$	0	0
$161$	915.294	0.448045
$162$	0	0
$163$	676.599	0.325125	0.162562	−	0.986698i	$-0.448024\pi$
$163$	676.599	0.325125	0.162562	+	0.986698i	$0.448024\pi$
$164$	0	0
$165$	264.118	0.124616
$166$	0	0
$167$	2835.67	1.31396	0.656979	−	0.753909i	$-0.271834\pi$
$167$	2835.67	1.31396	0.656979	+	0.753909i	$0.271834\pi$
$168$	0	0
$169$	1179.97	0.537083
$170$	0	0
$171$	1520.18	0.679829
$172$	0	0
$173$	−177.314	−0.0779243	−0.0389621	−	0.999241i	$-0.512405\pi$
$173$	−177.314	−0.0779243	−0.0389621	+	0.999241i	$0.512405\pi$
$174$	0	0
$175$	−919.578	−0.397220
$176$	0	0
$177$	−921.214	−0.391202
$178$	0	0
$179$	1023.76	0.427483	0.213741	−	0.976890i	$-0.431435\pi$
$179$	1023.76	0.427483	0.213741	+	0.976890i	$0.431435\pi$
$180$	0	0
$181$	−3450.21	−1.41686	−0.708432	−	0.705779i	$-0.750597\pi$
$181$	−3450.21	−1.41686	−0.708432	+	0.705779i	$0.750597\pi$
$182$	0	0
$183$	170.023	0.0686800
$184$	0	0
$185$	−401.248	−0.159461
$186$	0	0
$187$	469.474	0.183590
$188$	0	0
$189$	1103.38	0.424651
$190$	0	0
$191$	490.894	0.185968	0.0929839	−	0.995668i	$-0.470360\pi$
$191$	490.894	0.185968	0.0929839	+	0.995668i	$0.470360\pi$
$192$	0	0
$193$	−3548.80	−1.32357	−0.661783	−	0.749696i	$-0.730200\pi$
$193$	−3548.80	−1.32357	−0.661783	+	0.749696i	$0.730200\pi$
$194$	0	0
$195$	−555.775	−0.204102
$196$	0	0
$197$	1363.15	0.492996	0.246498	−	0.969143i	$-0.420720\pi$
$197$	1363.15	0.492996	0.246498	+	0.969143i	$0.420720\pi$
$198$	0	0
$199$	−3737.46	−1.33137	−0.665683	−	0.746235i	$-0.731860\pi$
$199$	−3737.46	−1.33137	−0.665683	+	0.746235i	$0.731860\pi$
$200$	0	0
$201$	167.060	0.0586246
$202$	0	0
$203$	−1020.81	−0.352940
$204$	0	0
$205$	−1426.60	−0.486038
$206$	0	0
$207$	−1965.15	−0.659841
$208$	0	0
$209$	2462.50	0.814997
$210$	0	0
$211$	5266.12	1.71817	0.859087	−	0.511829i	$-0.171032\pi$
$211$	5266.12	1.71817	0.859087	+	0.511829i	$0.171032\pi$
$212$	0	0
$213$	2487.11	0.800066
$214$	0	0
$215$	−1069.11	−0.339128
$216$	0	0
$217$	−2171.40	−0.679283
$218$	0	0
$219$	−933.075	−0.287906
$220$	0	0
$221$	−987.899	−0.300694
$222$	0	0
$223$	−704.546	−0.211569	−0.105785	−	0.994389i	$-0.533735\pi$
$223$	−704.546	−0.211569	−0.105785	+	0.994389i	$0.533735\pi$
$224$	0	0
$225$	1974.34	0.584990
$226$	0	0
$227$	2151.26	0.629006	0.314503	−	0.949256i	$-0.398162\pi$
$227$	2151.26	0.629006	0.314503	+	0.949256i	$0.398162\pi$
$228$	0	0
$229$	−3916.94	−1.13030	−0.565149	−	0.824989i	$-0.691181\pi$
$229$	−3916.94	−1.13030	−0.565149	+	0.824989i	$0.691181\pi$
$230$	0	0
$231$	691.764	0.197034
$232$	0	0
$233$	−5192.74	−1.46003	−0.730017	−	0.683429i	$-0.760488\pi$
$233$	−5192.74	−1.46003	−0.730017	+	0.683429i	$0.760488\pi$
$234$	0	0
$235$	−462.632	−0.128420
$236$	0	0
$237$	−2272.36	−0.622809
$238$	0	0
$239$	−334.305	−0.0904786	−0.0452393	−	0.998976i	$-0.514405\pi$
$239$	−334.305	−0.0904786	−0.0452393	+	0.998976i	$0.514405\pi$
$240$	0	0
$241$	−1918.45	−0.512773	−0.256386	−	0.966574i	$-0.582532\pi$
$241$	−1918.45	−0.512773	−0.256386	+	0.966574i	$0.582532\pi$
$242$	0	0
$243$	−3821.06	−1.00873
$244$	0	0
$245$	−848.722	−0.221318
$246$	0	0
$247$	−5181.75	−1.33485
$248$	0	0
$249$	−366.097	−0.0931746
$250$	0	0
$251$	−7695.71	−1.93525	−0.967627	−	0.252385i	$-0.918785\pi$
$251$	−7695.71	−1.93525	−0.967627	+	0.252385i	$0.918785\pi$
$252$	0	0
$253$	−3183.29	−0.791035
$254$	0	0
$255$	162.587	0.0399277
$256$	0	0
$257$	5335.10	1.29492	0.647460	−	0.762099i	$-0.275831\pi$
$257$	5335.10	1.29492	0.647460	+	0.762099i	$0.275831\pi$
$258$	0	0
$259$	−1050.93	−0.252130
$260$	0	0
$261$	2191.69	0.519778
$262$	0	0
$263$	−3934.15	−0.922396	−0.461198	−	0.887297i	$-0.652580\pi$
$263$	−3934.15	−0.922396	−0.461198	+	0.887297i	$0.652580\pi$
$264$	0	0
$265$	1599.57	0.370795
$266$	0	0
$267$	2565.75	0.588095
$268$	0	0
$269$	3424.04	0.776088	0.388044	−	0.921641i	$-0.373151\pi$
$269$	3424.04	0.776088	0.388044	+	0.921641i	$0.373151\pi$
$270$	0	0
$271$	−549.034	−0.123068	−0.0615340	−	0.998105i	$-0.519599\pi$
$271$	−549.034	−0.123068	−0.0615340	+	0.998105i	$0.519599\pi$
$272$	0	0
$273$	−1455.66	−0.322712
$274$	0	0
$275$	3198.19	0.701302
$276$	0	0
$277$	5203.65	1.12873	0.564363	−	0.825527i	$-0.309122\pi$
$277$	5203.65	1.12873	0.564363	+	0.825527i	$0.309122\pi$
$278$	0	0
$279$	4662.02	1.00039
$280$	0	0
$281$	−1986.73	−0.421774	−0.210887	−	0.977510i	$-0.567635\pi$
$281$	−1986.73	−0.421774	−0.210887	+	0.977510i	$0.567635\pi$
$282$	0	0
$283$	−753.696	−0.158313	−0.0791565	−	0.996862i	$-0.525223\pi$
$283$	−753.696	−0.158313	−0.0791565	+	0.996862i	$0.525223\pi$
$284$	0	0
$285$	852.803	0.177248
$286$	0	0
$287$	−3736.47	−0.768490
$288$	0	0
$289$	289.000	0.0588235
$290$	0	0
$291$	−2506.96	−0.505020
$292$	0	0
$293$	−7202.22	−1.43603	−0.718017	−	0.696025i	$-0.754950\pi$
$293$	−7202.22	−1.43603	−0.718017	+	0.696025i	$0.754950\pi$
$294$	0	0
$295$	885.318	0.174729
$296$	0	0
$297$	−3837.43	−0.749731
$298$	0	0
$299$	6698.50	1.29560
$300$	0	0
$301$	−2800.15	−0.536205
$302$	0	0
$303$	−837.595	−0.158807
$304$	0	0
$305$	−163.398	−0.0306758
$306$	0	0
$307$	−2425.71	−0.450953	−0.225477	−	0.974249i	$-0.572394\pi$
$307$	−2425.71	−0.450953	−0.225477	+	0.974249i	$0.572394\pi$
$308$	0	0
$309$	1650.21	0.303809
$310$	0	0
$311$	9544.94	1.74033	0.870167	−	0.492757i	$-0.164011\pi$
$311$	9544.94	1.74033	0.870167	+	0.492757i	$0.164011\pi$
$312$	0	0
$313$	588.379	0.106253	0.0531264	−	0.998588i	$-0.483081\pi$
$313$	588.379	0.106253	0.0531264	+	0.998588i	$0.483081\pi$
$314$	0	0
$315$	−410.407	−0.0734090
$316$	0	0
$317$	7653.31	1.35600	0.678001	−	0.735061i	$-0.262846\pi$
$317$	7653.31	1.35600	0.678001	+	0.735061i	$0.262846\pi$
$318$	0	0
$319$	3550.26	0.623124
$320$	0	0
$321$	−3110.59	−0.540860
$322$	0	0
$323$	1515.87	0.261131
$324$	0	0
$325$	−6729.85	−1.14863
$326$	0	0
$327$	−5723.73	−0.967960
$328$	0	0
$329$	−1211.70	−0.203050
$330$	0	0
$331$	−752.266	−0.124919	−0.0624597	−	0.998047i	$-0.519894\pi$
$331$	−752.266	−0.124919	−0.0624597	+	0.998047i	$0.519894\pi$
$332$	0	0
$333$	2256.36	0.371314
$334$	0	0
$335$	−160.551	−0.0261845
$336$	0	0
$337$	−1968.57	−0.318204	−0.159102	−	0.987262i	$-0.550860\pi$
$337$	−1968.57	−0.318204	−0.159102	+	0.987262i	$0.550860\pi$
$338$	0	0
$339$	2231.39	0.357501
$340$	0	0
$341$	7551.89	1.19929
$342$	0	0
$343$	−4946.52	−0.778678
$344$	0	0
$345$	−1102.43	−0.172037
$346$	0	0
$347$	−3983.10	−0.616207	−0.308104	−	0.951353i	$-0.599694\pi$
$347$	−3983.10	−0.616207	−0.308104	+	0.951353i	$0.599694\pi$
$348$	0	0
$349$	1495.61	0.229393	0.114697	−	0.993401i	$-0.463410\pi$
$349$	1495.61	0.229393	0.114697	+	0.993401i	$0.463410\pi$
$350$	0	0
$351$	8074.98	1.22795
$352$	0	0
$353$	6482.49	0.977417	0.488708	−	0.872447i	$-0.337468\pi$
$353$	6482.49	0.977417	0.488708	+	0.872447i	$0.337468\pi$
$354$	0	0
$355$	−2390.20	−0.357348
$356$	0	0
$357$	425.838	0.0631309
$358$	0	0
$359$	4943.42	0.726751	0.363376	−	0.931643i	$-0.381624\pi$
$359$	4943.42	0.726751	0.363376	+	0.931643i	$0.381624\pi$
$360$	0	0
$361$	1092.08	0.159218
$362$	0	0
$363$	1792.94	0.259242
$364$	0	0
$365$	896.717	0.128593
$366$	0	0
$367$	14.8871	0.00211743	0.00105872	−	0.999999i	$-0.499663\pi$
$367$	14.8871	0.00211743	0.00105872	+	0.999999i	$0.499663\pi$
$368$	0	0
$369$	8022.23	1.13176
$370$	0	0
$371$	4189.51	0.586276
$372$	0	0
$373$	1923.18	0.266966	0.133483	−	0.991051i	$-0.457384\pi$
$373$	1923.18	0.266966	0.133483	+	0.991051i	$0.457384\pi$
$374$	0	0
$375$	2303.07	0.317147
$376$	0	0
$377$	−7470.70	−1.02059
$378$	0	0
$379$	9592.87	1.30014	0.650069	−	0.759875i	$-0.274740\pi$
$379$	9592.87	1.30014	0.650069	+	0.759875i	$0.274740\pi$
$380$	0	0
$381$	8354.04	1.12333
$382$	0	0
$383$	9083.77	1.21190	0.605951	−	0.795502i	$-0.292793\pi$
$383$	9083.77	1.21190	0.605951	+	0.795502i	$0.292793\pi$
$384$	0	0
$385$	−664.809	−0.0880046
$386$	0	0
$387$	6011.95	0.789675
$388$	0	0
$389$	−1143.78	−0.149079	−0.0745396	−	0.997218i	$-0.523749\pi$
$389$	−1143.78	−0.149079	−0.0745396	+	0.997218i	$0.523749\pi$
$390$	0	0
$391$	−1959.58	−0.253453
$392$	0	0
$393$	−6243.27	−0.801352
$394$	0	0
$395$	2183.81	0.278176
$396$	0	0
$397$	10604.5	1.34061	0.670307	−	0.742084i	$-0.266162\pi$
$397$	10604.5	1.34061	0.670307	+	0.742084i	$0.266162\pi$
$398$	0	0
$399$	2233.62	0.280252
$400$	0	0
$401$	13785.4	1.71674	0.858368	−	0.513035i	$-0.171479\pi$
$401$	13785.4	1.71674	0.858368	+	0.513035i	$0.171479\pi$
$402$	0	0
$403$	−15891.2	−1.96426
$404$	0	0
$405$	66.5420	0.00816419
$406$	0	0
$407$	3655.01	0.445141
$408$	0	0
$409$	−9505.94	−1.14924	−0.574619	−	0.818421i	$-0.694850\pi$
$409$	−9505.94	−1.14924	−0.574619	+	0.818421i	$0.694850\pi$
$410$	0	0
$411$	−9911.59	−1.18954
$412$	0	0
$413$	2318.78	0.276270
$414$	0	0
$415$	351.832	0.0416162
$416$	0	0
$417$	4631.23	0.543866
$418$	0	0
$419$	−9680.86	−1.12874	−0.564369	−	0.825523i	$-0.690880\pi$
$419$	−9680.86	−1.12874	−0.564369	+	0.825523i	$0.690880\pi$
$420$	0	0
$421$	−12360.3	−1.43089	−0.715444	−	0.698671i	$-0.753775\pi$
$421$	−12360.3	−1.43089	−0.715444	+	0.698671i	$0.753775\pi$
$422$	0	0
$423$	2601.54	0.299033
$424$	0	0
$425$	1968.75	0.224702
$426$	0	0
$427$	−427.962	−0.0485025
$428$	0	0
$429$	5062.61	0.569756
$430$	0	0
$431$	−2970.58	−0.331990	−0.165995	−	0.986127i	$-0.553084\pi$
$431$	−2970.58	−0.331990	−0.165995	+	0.986127i	$0.553084\pi$
$432$	0	0
$433$	6131.50	0.680510	0.340255	−	0.940333i	$-0.389487\pi$
$433$	6131.50	0.680510	0.340255	+	0.940333i	$0.389487\pi$
$434$	0	0
$435$	1229.51	0.135519
$436$	0	0
$437$	−10278.4	−1.12513
$438$	0	0
$439$	2544.91	0.276679	0.138339	−	0.990385i	$-0.455824\pi$
$439$	2544.91	0.276679	0.138339	+	0.990385i	$0.455824\pi$
$440$	0	0
$441$	4772.65	0.515349
$442$	0	0
$443$	−8529.82	−0.914817	−0.457408	−	0.889257i	$-0.651222\pi$
$443$	−8529.82	−0.914817	−0.457408	+	0.889257i	$0.651222\pi$
$444$	0	0
$445$	−2465.77	−0.262672
$446$	0	0
$447$	902.308	0.0954759
$448$	0	0
$449$	8855.74	0.930798	0.465399	−	0.885101i	$-0.345911\pi$
$449$	8855.74	0.930798	0.465399	+	0.885101i	$0.345911\pi$
$450$	0	0
$451$	12995.0	1.35679
$452$	0	0
$453$	−2112.42	−0.219095
$454$	0	0
$455$	1398.94	0.144139
$456$	0	0
$457$	−7154.78	−0.732356	−0.366178	−	0.930545i	$-0.619334\pi$
$457$	−7154.78	−0.732356	−0.366178	+	0.930545i	$0.619334\pi$
$458$	0	0
$459$	−2362.25	−0.240219
$460$	0	0
$461$	−7263.06	−0.733784	−0.366892	−	0.930264i	$-0.619578\pi$
$461$	−7263.06	−0.733784	−0.366892	+	0.930264i	$0.619578\pi$
$462$	0	0
$463$	−352.898	−0.0354224	−0.0177112	−	0.999843i	$-0.505638\pi$
$463$	−352.898	−0.0354224	−0.0177112	+	0.999843i	$0.505638\pi$
$464$	0	0
$465$	2615.34	0.260825
$466$	0	0
$467$	−1483.02	−0.146951	−0.0734753	−	0.997297i	$-0.523409\pi$
$467$	−1483.02	−0.146951	−0.0734753	+	0.997297i	$0.523409\pi$
$468$	0	0
$469$	−420.506	−0.0414012
$470$	0	0
$471$	−2273.89	−0.222452
$472$	0	0
$473$	9738.60	0.946683
$474$	0	0
$475$	10326.5	0.997502
$476$	0	0
$477$	−8994.92	−0.863415
$478$	0	0
$479$	9990.10	0.952942	0.476471	−	0.879190i	$-0.341916\pi$
$479$	9990.10	0.952942	0.476471	+	0.879190i	$0.341916\pi$
$480$	0	0
$481$	−7691.13	−0.729075
$482$	0	0
$483$	−2887.42	−0.272012
$484$	0	0
$485$	2409.27	0.225566
$486$	0	0
$487$	1129.88	0.105133	0.0525663	−	0.998617i	$-0.483260\pi$
$487$	1129.88	0.105133	0.0525663	+	0.998617i	$0.483260\pi$
$488$	0	0
$489$	−2134.42	−0.197386
$490$	0	0
$491$	−18774.9	−1.72566	−0.862832	−	0.505491i	$-0.831311\pi$
$491$	−18774.9	−1.72566	−0.862832	+	0.505491i	$0.831311\pi$
$492$	0	0
$493$	2185.48	0.199653
$494$	0	0
$495$	1427.35	0.129605
$496$	0	0
$497$	−6260.28	−0.565014
$498$	0	0
$499$	−17329.1	−1.55462	−0.777310	−	0.629118i	$-0.783416\pi$
$499$	−17329.1	−1.55462	−0.777310	+	0.629118i	$0.783416\pi$
$500$	0	0
$501$	−8945.50	−0.797716
$502$	0	0
$503$	20837.0	1.84707	0.923533	−	0.383518i	$-0.125288\pi$
$503$	20837.0	1.84707	0.923533	+	0.383518i	$0.125288\pi$
$504$	0	0
$505$	804.957	0.0709309
$506$	0	0
$507$	−3722.37	−0.326068
$508$	0	0
$509$	11835.0	1.03060	0.515301	−	0.857009i	$-0.327680\pi$
$509$	11835.0	1.03060	0.515301	+	0.857009i	$0.327680\pi$
$510$	0	0
$511$	2348.63	0.203322
$512$	0	0
$513$	−12390.6	−1.06639
$514$	0	0
$515$	−1585.91	−0.135696
$516$	0	0
$517$	4214.16	0.358489
$518$	0	0
$519$	559.359	0.0473086
$520$	0	0
$521$	7686.37	0.646346	0.323173	−	0.946340i	$-0.395250\pi$
$521$	7686.37	0.646346	0.323173	+	0.946340i	$0.395250\pi$
$522$	0	0
$523$	−11476.4	−0.959518	−0.479759	−	0.877400i	$-0.659276\pi$
$523$	−11476.4	−0.959518	−0.479759	+	0.877400i	$0.659276\pi$
$524$	0	0
$525$	2900.93	0.241156
$526$	0	0
$527$	4648.81	0.384261
$528$	0	0
$529$	1120.01	0.0920535
$530$	0	0
$531$	−4978.44	−0.406866
$532$	0	0
$533$	−27345.0	−2.22222
$534$	0	0
$535$	2989.38	0.241574
$536$	0	0
$537$	−3229.59	−0.259529
$538$	0	0
$539$	7731.09	0.617814
$540$	0	0
$541$	−546.481	−0.0434289	−0.0217145	−	0.999764i	$-0.506912\pi$
$541$	−546.481	−0.0434289	−0.0217145	+	0.999764i	$0.506912\pi$
$542$	0	0
$543$	10884.1	0.860191
$544$	0	0
$545$	5500.69	0.432337
$546$	0	0
$547$	−8397.33	−0.656388	−0.328194	−	0.944610i	$-0.606440\pi$
$547$	−8397.33	−0.656388	−0.328194	+	0.944610i	$0.606440\pi$
$548$	0	0
$549$	918.839	0.0714301
$550$	0	0
$551$	11463.3	0.886305
$552$	0	0
$553$	5719.73	0.439833
$554$	0	0
$555$	1265.79	0.0968105
$556$	0	0
$557$	−4881.65	−0.371350	−0.185675	−	0.982611i	$-0.559447\pi$
$557$	−4881.65	−0.371350	−0.185675	+	0.982611i	$0.559447\pi$
$558$	0	0
$559$	−20492.6	−1.55053
$560$	0	0
$561$	−1481.02	−0.111459
$562$	0	0
$563$	−7198.57	−0.538870	−0.269435	−	0.963019i	$-0.586837\pi$
$563$	−7198.57	−0.538870	−0.269435	+	0.963019i	$0.586837\pi$
$564$	0	0
$565$	−2144.44	−0.159677
$566$	0	0
$567$	174.283	0.0129087
$568$	0	0
$569$	−23946.9	−1.76433	−0.882167	−	0.470937i	$-0.843916\pi$
$569$	−23946.9	−1.76433	−0.882167	+	0.470937i	$0.843916\pi$
$570$	0	0
$571$	−1593.15	−0.116763	−0.0583813	−	0.998294i	$-0.518594\pi$
$571$	−1593.15	−0.116763	−0.0583813	+	0.998294i	$0.518594\pi$
$572$	0	0
$573$	−1548.59	−0.112903
$574$	0	0
$575$	−13349.2	−0.968174
$576$	0	0
$577$	12937.4	0.933435	0.466717	−	0.884406i	$-0.345436\pi$
$577$	12937.4	0.933435	0.466717	+	0.884406i	$0.345436\pi$
$578$	0	0
$579$	11195.2	0.803549
$580$	0	0
$581$	921.499	0.0658007
$582$	0	0
$583$	−14570.6	−1.03508
$584$	0	0
$585$	−3003.53	−0.212275
$586$	0	0
$587$	12899.2	0.906998	0.453499	−	0.891257i	$-0.350176\pi$
$587$	12899.2	0.906998	0.453499	+	0.891257i	$0.350176\pi$
$588$	0	0
$589$	24384.1	1.70582
$590$	0	0
$591$	−4300.23	−0.299302
$592$	0	0
$593$	4357.13	0.301730	0.150865	−	0.988554i	$-0.451794\pi$
$593$	4357.13	0.301730	0.150865	+	0.988554i	$0.451794\pi$
$594$	0	0
$595$	−409.245	−0.0281973
$596$	0	0
$597$	11790.3	0.808284
$598$	0	0
$599$	−13726.8	−0.936328	−0.468164	−	0.883642i	$-0.655084\pi$
$599$	−13726.8	−0.936328	−0.468164	+	0.883642i	$0.655084\pi$
$600$	0	0
$601$	2531.41	0.171811	0.0859056	−	0.996303i	$-0.472622\pi$
$601$	2531.41	0.171811	0.0859056	+	0.996303i	$0.472622\pi$
$602$	0	0
$603$	902.830	0.0609720
$604$	0	0
$605$	−1723.07	−0.115790
$606$	0	0
$607$	−185.004	−0.0123708	−0.00618540	−	0.999981i	$-0.501969\pi$
$607$	−185.004	−0.0123708	−0.00618540	+	0.999981i	$0.501969\pi$
$608$	0	0
$609$	3220.28	0.214273
$610$	0	0
$611$	−8867.73	−0.587152
$612$	0	0
$613$	−17706.9	−1.16668	−0.583339	−	0.812228i	$-0.698254\pi$
$613$	−17706.9	−1.16668	−0.583339	+	0.812228i	$0.698254\pi$
$614$	0	0
$615$	4500.39	0.295078
$616$	0	0
$617$	−6183.89	−0.403491	−0.201746	−	0.979438i	$-0.564661\pi$
$617$	−6183.89	−0.403491	−0.201746	+	0.979438i	$0.564661\pi$
$618$	0	0
$619$	1247.51	0.0810046	0.0405023	−	0.999179i	$-0.487104\pi$
$619$	1247.51	0.0810046	0.0405023	+	0.999179i	$0.487104\pi$
$620$	0	0
$621$	16017.4	1.03503
$622$	0	0
$623$	−6458.23	−0.415318
$624$	0	0
$625$	12262.8	0.784817
$626$	0	0
$627$	−7768.27	−0.494792
$628$	0	0
$629$	2249.96	0.142626
$630$	0	0
$631$	−24053.3	−1.51750	−0.758752	−	0.651379i	$-0.774191\pi$
$631$	−24053.3	−1.51750	−0.758752	+	0.651379i	$0.774191\pi$
$632$	0	0
$633$	−16612.7	−1.04312
$634$	0	0
$635$	−8028.51	−0.501735
$636$	0	0
$637$	−16268.3	−1.01189
$638$	0	0
$639$	13440.9	0.832102
$640$	0	0
$641$	−21286.8	−1.31167	−0.655834	−	0.754905i	$-0.727683\pi$
$641$	−21286.8	−1.31167	−0.655834	+	0.754905i	$0.727683\pi$
$642$	0	0
$643$	1789.41	0.109747	0.0548736	−	0.998493i	$-0.482524\pi$
$643$	1789.41	0.109747	0.0548736	+	0.998493i	$0.482524\pi$
$644$	0	0
$645$	3372.64	0.205888
$646$	0	0
$647$	4378.61	0.266060	0.133030	−	0.991112i	$-0.457529\pi$
$647$	4378.61	0.266060	0.133030	+	0.991112i	$0.457529\pi$
$648$	0	0
$649$	−8064.45	−0.487762
$650$	0	0
$651$	6849.97	0.412399
$652$	0	0
$653$	−7665.15	−0.459358	−0.229679	−	0.973266i	$-0.573768\pi$
$653$	−7665.15	−0.459358	−0.229679	+	0.973266i	$0.573768\pi$
$654$	0	0
$655$	5999.99	0.357922
$656$	0	0
$657$	−5042.54	−0.299434
$658$	0	0
$659$	4710.22	0.278428	0.139214	−	0.990262i	$-0.455542\pi$
$659$	4710.22	0.278428	0.139214	+	0.990262i	$0.455542\pi$
$660$	0	0
$661$	−31266.6	−1.83983	−0.919916	−	0.392116i	$-0.871743\pi$
$661$	−31266.6	−1.83983	−0.919916	+	0.392116i	$0.871743\pi$
$662$	0	0
$663$	3116.46	0.182554
$664$	0	0
$665$	−2146.58	−0.125174
$666$	0	0
$667$	−14818.7	−0.860246
$668$	0	0
$669$	2222.58	0.128446
$670$	0	0
$671$	1488.40	0.0856322
$672$	0	0
$673$	11723.0	0.671454	0.335727	−	0.941959i	$-0.391018\pi$
$673$	11723.0	0.671454	0.335727	+	0.941959i	$0.391018\pi$
$674$	0	0
$675$	−16092.3	−0.917621
$676$	0	0
$677$	−289.531	−0.0164366	−0.00821829	−	0.999966i	$-0.502616\pi$
$677$	−289.531	−0.0164366	−0.00821829	+	0.999966i	$0.502616\pi$
$678$	0	0
$679$	6310.25	0.356650
$680$	0	0
$681$	−6786.45	−0.381875
$682$	0	0
$683$	−1720.10	−0.0963660	−0.0481830	−	0.998839i	$-0.515343\pi$
$683$	−1720.10	−0.0963660	−0.0481830	+	0.998839i	$0.515343\pi$
$684$	0	0
$685$	9525.37	0.531308
$686$	0	0
$687$	12356.5	0.686215
$688$	0	0
$689$	30660.5	1.69532
$690$	0	0
$691$	16777.7	0.923665	0.461832	−	0.886967i	$-0.347192\pi$
$691$	16777.7	0.923665	0.461832	+	0.886967i	$0.347192\pi$
$692$	0	0
$693$	3738.44	0.204923
$694$	0	0
$695$	−4450.77	−0.242917
$696$	0	0
$697$	7999.50	0.434724
$698$	0	0
$699$	16381.2	0.886400
$700$	0	0
$701$	23981.1	1.29209	0.646043	−	0.763301i	$-0.276423\pi$
$701$	23981.1	1.29209	0.646043	+	0.763301i	$0.276423\pi$
$702$	0	0
$703$	11801.6	0.633150
$704$	0	0
$705$	1459.43	0.0779652
$706$	0	0
$707$	2108.30	0.112151
$708$	0	0
$709$	−7709.28	−0.408361	−0.204181	−	0.978933i	$-0.565453\pi$
$709$	−7709.28	−0.408361	−0.204181	+	0.978933i	$0.565453\pi$
$710$	0	0
$711$	−12280.3	−0.647747
$712$	0	0
$713$	−31521.5	−1.65567
$714$	0	0
$715$	−4865.34	−0.254480
$716$	0	0
$717$	1054.61	0.0549304
$718$	0	0
$719$	11976.5	0.621209	0.310605	−	0.950539i	$-0.399468\pi$
$719$	11976.5	0.621209	0.310605	+	0.950539i	$0.399468\pi$
$720$	0	0
$721$	−4153.72	−0.214553
$722$	0	0
$723$	6052.00	0.311309
$724$	0	0
$725$	14888.1	0.762662
$726$	0	0
$727$	18597.3	0.948745	0.474372	−	0.880324i	$-0.342675\pi$
$727$	18597.3	0.948745	0.474372	+	0.880324i	$0.342675\pi$
$728$	0	0
$729$	11461.4	0.582300
$730$	0	0
$731$	5994.91	0.303324
$732$	0	0
$733$	−23569.5	−1.18767	−0.593833	−	0.804588i	$-0.702386\pi$
$733$	−23569.5	−1.18767	−0.593833	+	0.804588i	$0.702386\pi$
$734$	0	0
$735$	2677.41	0.134364
$736$	0	0
$737$	1462.47	0.0730948
$738$	0	0
$739$	−10149.1	−0.505199	−0.252599	−	0.967571i	$-0.581285\pi$
$739$	−10149.1	−0.505199	−0.252599	+	0.967571i	$0.581285\pi$
$740$	0	0
$741$	16346.5	0.810397
$742$	0	0
$743$	−27758.0	−1.37058	−0.685291	−	0.728269i	$-0.740325\pi$
$743$	−27758.0	−1.37058	−0.685291	+	0.728269i	$0.740325\pi$
$744$	0	0
$745$	−867.148	−0.0426441
$746$	0	0
$747$	−1978.47	−0.0969054
$748$	0	0
$749$	7829.63	0.381960
$750$	0	0
$751$	815.225	0.0396112	0.0198056	−	0.999804i	$-0.493695\pi$
$751$	815.225	0.0396112	0.0198056	+	0.999804i	$0.493695\pi$
$752$	0	0
$753$	24277.1	1.17491
$754$	0	0
$755$	2030.11	0.0978585
$756$	0	0
$757$	−13239.4	−0.635659	−0.317829	−	0.948148i	$-0.602954\pi$
$757$	−13239.4	−0.635659	−0.317829	+	0.948148i	$0.602954\pi$
$758$	0	0
$759$	10042.1	0.480244
$760$	0	0
$761$	−11028.2	−0.525324	−0.262662	−	0.964888i	$-0.584600\pi$
$761$	−11028.2	−0.525324	−0.262662	+	0.964888i	$0.584600\pi$
$762$	0	0
$763$	14407.1	0.683582
$764$	0	0
$765$	878.652	0.0415265
$766$	0	0
$767$	16969.8	0.798882
$768$	0	0
$769$	−18921.2	−0.887277	−0.443639	−	0.896206i	$-0.646313\pi$
$769$	−18921.2	−0.887277	−0.443639	+	0.896206i	$0.646313\pi$
$770$	0	0
$771$	−16830.3	−0.786158
$772$	0	0
$773$	38728.6	1.80203	0.901016	−	0.433786i	$-0.142823\pi$
$773$	38728.6	1.80203	0.901016	+	0.433786i	$0.142823\pi$
$774$	0	0
$775$	31669.0	1.46785
$776$	0	0
$777$	3315.29	0.153070
$778$	0	0
$779$	41959.2	1.92984
$780$	0	0
$781$	21772.5	0.997545
$782$	0	0
$783$	−17863.9	−0.815329
$784$	0	0
$785$	2185.28	0.0993579
$786$	0	0
$787$	20587.3	0.932477	0.466239	−	0.884659i	$-0.345609\pi$
$787$	20587.3	0.932477	0.466239	+	0.884659i	$0.345609\pi$
$788$	0	0
$789$	12410.8	0.559995
$790$	0	0
$791$	−5616.62	−0.252470
$792$	0	0
$793$	−3132.00	−0.140253
$794$	0	0
$795$	−5046.05	−0.225113
$796$	0	0
$797$	−15871.4	−0.705385	−0.352693	−	0.935739i	$-0.614734\pi$
$797$	−15871.4	−0.705385	−0.352693	+	0.935739i	$0.614734\pi$
$798$	0	0
$799$	2594.17	0.114862
$800$	0	0
$801$	13865.9	0.611644
$802$	0	0
$803$	−8168.28	−0.358969
$804$	0	0
$805$	2774.90	0.121494
$806$	0	0
$807$	−10801.6	−0.471170
$808$	0	0
$809$	39667.1	1.72388	0.861942	−	0.507007i	$-0.169248\pi$
$809$	39667.1	1.72388	0.861942	+	0.507007i	$0.169248\pi$
$810$	0	0
$811$	−8003.87	−0.346552	−0.173276	−	0.984873i	$-0.555435\pi$
$811$	−8003.87	−0.346552	−0.173276	+	0.984873i	$0.555435\pi$
$812$	0	0
$813$	1732.00	0.0747157
$814$	0	0
$815$	2051.25	0.0881621
$816$	0	0
$817$	31444.7	1.34652
$818$	0	0
$819$	−7866.69	−0.335634
$820$	0	0
$821$	−13279.1	−0.564489	−0.282244	−	0.959343i	$-0.591079\pi$
$821$	−13279.1	−0.564489	−0.282244	+	0.959343i	$0.591079\pi$
$822$	0	0
$823$	−28934.0	−1.22549	−0.612745	−	0.790281i	$-0.709935\pi$
$823$	−28934.0	−1.22549	−0.612745	+	0.790281i	$0.709935\pi$
$824$	0	0
$825$	−10089.1	−0.425767
$826$	0	0
$827$	13679.6	0.575193	0.287597	−	0.957752i	$-0.407144\pi$
$827$	13679.6	0.575193	0.287597	+	0.957752i	$0.407144\pi$
$828$	0	0
$829$	16514.5	0.691886	0.345943	−	0.938256i	$-0.387559\pi$
$829$	16514.5	0.691886	0.345943	+	0.938256i	$0.387559\pi$
$830$	0	0
$831$	−16415.6	−0.685260
$832$	0	0
$833$	4759.13	0.197952
$834$	0	0
$835$	8596.93	0.356298
$836$	0	0
$837$	−37998.9	−1.56922
$838$	0	0
$839$	87.9839	0.00362043	0.00181022	−	0.999998i	$-0.499424\pi$
$839$	87.9839	0.00362043	0.00181022	+	0.999998i	$0.499424\pi$
$840$	0	0
$841$	−7861.94	−0.322356
$842$	0	0
$843$	6267.41	0.256063
$844$	0	0
$845$	3577.33	0.145638
$846$	0	0
$847$	−4512.98	−0.183079
$848$	0	0
$849$	2377.63	0.0961133
$850$	0	0
$851$	−15256.0	−0.614534
$852$	0	0
$853$	8162.96	0.327660	0.163830	−	0.986489i	$-0.447615\pi$
$853$	8162.96	0.327660	0.163830	+	0.986489i	$0.447615\pi$
$854$	0	0
$855$	4608.73	0.184345
$856$	0	0
$857$	18724.9	0.746361	0.373181	−	0.927759i	$-0.378267\pi$
$857$	18724.9	0.746361	0.373181	+	0.927759i	$0.378267\pi$
$858$	0	0
$859$	46422.5	1.84391	0.921953	−	0.387301i	$-0.126593\pi$
$859$	46422.5	1.84391	0.921953	+	0.387301i	$0.126593\pi$
$860$	0	0
$861$	11787.2	0.466557
$862$	0	0
$863$	−29112.3	−1.14831	−0.574157	−	0.818746i	$-0.694670\pi$
$863$	−29112.3	−1.14831	−0.574157	+	0.818746i	$0.694670\pi$
$864$	0	0
$865$	−537.563	−0.0211303
$866$	0	0
$867$	−911.688	−0.0357123
$868$	0	0
$869$	−19892.6	−0.776536
$870$	0	0
$871$	−3077.43	−0.119719
$872$	0	0
$873$	−13548.2	−0.525241
$874$	0	0
$875$	−5797.04	−0.223972
$876$	0	0
$877$	39163.0	1.50791	0.753957	−	0.656924i	$-0.228143\pi$
$877$	39163.0	1.50791	0.753957	+	0.656924i	$0.228143\pi$
$878$	0	0
$879$	22720.3	0.871830
$880$	0	0
$881$	−35073.2	−1.34125	−0.670627	−	0.741795i	$-0.733975\pi$
$881$	−35073.2	−1.34125	−0.670627	+	0.741795i	$0.733975\pi$
$882$	0	0
$883$	48775.7	1.85893	0.929463	−	0.368915i	$-0.120271\pi$
$883$	48775.7	1.85893	0.929463	+	0.368915i	$0.120271\pi$
$884$	0	0
$885$	−2792.85	−0.106080
$886$	0	0
$887$	−13296.0	−0.503309	−0.251654	−	0.967817i	$-0.580975\pi$
$887$	−13296.0	−0.503309	−0.251654	+	0.967817i	$0.580975\pi$
$888$	0	0
$889$	−21027.9	−0.793309
$890$	0	0
$891$	−606.137	−0.0227905
$892$	0	0
$893$	13607.0	0.509899
$894$	0	0
$895$	3103.74	0.115918
$896$	0	0
$897$	−21131.3	−0.786570
$898$	0	0
$899$	35155.3	1.30422
$900$	0	0
$901$	−8969.43	−0.331648
$902$	0	0
$903$	8833.44	0.325535
$904$	0	0
$905$	−10460.0	−0.384202
$906$	0	0
$907$	11675.0	0.427410	0.213705	−	0.976898i	$-0.431447\pi$
$907$	11675.0	0.427410	0.213705	+	0.976898i	$0.431447\pi$
$908$	0	0
$909$	−4526.54	−0.165166
$910$	0	0
$911$	−18552.9	−0.674738	−0.337369	−	0.941372i	$-0.609537\pi$
$911$	−18552.9	−0.674738	−0.337369	+	0.941372i	$0.609537\pi$
$912$	0	0
$913$	−3204.87	−0.116173
$914$	0	0
$915$	515.459	0.0186235
$916$	0	0
$917$	15714.9	0.565922
$918$	0	0
$919$	−33956.8	−1.21886	−0.609429	−	0.792841i	$-0.708601\pi$
$919$	−33956.8	−1.21886	−0.609429	+	0.792841i	$0.708601\pi$
$920$	0	0
$921$	7652.23	0.273778
$922$	0	0
$923$	−45815.3	−1.63383
$924$	0	0
$925$	15327.4	0.544823
$926$	0	0
$927$	8918.08	0.315974
$928$	0	0
$929$	−23695.3	−0.836832	−0.418416	−	0.908256i	$-0.637415\pi$
$929$	−23695.3	−0.836832	−0.418416	+	0.908256i	$0.637415\pi$
$930$	0	0
$931$	24962.7	0.878753
$932$	0	0
$933$	−30110.8	−1.05657
$934$	0	0
$935$	1423.31	0.0497830
$936$	0	0
$937$	7990.62	0.278593	0.139297	−	0.990251i	$-0.455516\pi$
$937$	7990.62	0.278593	0.139297	+	0.990251i	$0.455516\pi$
$938$	0	0
$939$	−1856.12	−0.0645071
$940$	0	0
$941$	24385.9	0.844799	0.422400	−	0.906410i	$-0.361188\pi$
$941$	24385.9	0.844799	0.422400	+	0.906410i	$0.361188\pi$
$942$	0	0
$943$	−54241.0	−1.87310
$944$	0	0
$945$	3345.12	0.115150
$946$	0	0
$947$	1174.62	0.0403064	0.0201532	−	0.999797i	$-0.493585\pi$
$947$	1174.62	0.0403064	0.0201532	+	0.999797i	$0.493585\pi$
$948$	0	0
$949$	17188.3	0.587939
$950$	0	0
$951$	−24143.4	−0.823241
$952$	0	0
$953$	−33546.9	−1.14029	−0.570143	−	0.821546i	$-0.693112\pi$
$953$	−33546.9	−1.14029	−0.570143	+	0.821546i	$0.693112\pi$
$954$	0	0
$955$	1488.25	0.0504277
$956$	0	0
$957$	−11199.8	−0.378304
$958$	0	0
$959$	24948.4	0.840067
$960$	0	0
$961$	44989.1	1.51016
$962$	0	0
$963$	−16810.3	−0.562517
$964$	0	0
$965$	−10758.9	−0.358903
$966$	0	0
$967$	−24766.8	−0.823625	−0.411813	−	0.911269i	$-0.635104\pi$
$967$	−24766.8	−0.823625	−0.411813	+	0.911269i	$0.635104\pi$
$968$	0	0
$969$	−4782.01	−0.158535
$970$	0	0
$971$	−42324.3	−1.39882	−0.699409	−	0.714721i	$-0.746553\pi$
$971$	−42324.3	−1.39882	−0.699409	+	0.714721i	$0.746553\pi$
$972$	0	0
$973$	−11657.2	−0.384083
$974$	0	0
$975$	21230.2	0.697344
$976$	0	0
$977$	−11320.4	−0.370698	−0.185349	−	0.982673i	$-0.559342\pi$
$977$	−11320.4	−0.370698	−0.185349	+	0.982673i	$0.559342\pi$
$978$	0	0
$979$	22461.0	0.733254
$980$	0	0
$981$	−30932.2	−1.00672
$982$	0	0
$983$	11311.9	0.367032	0.183516	−	0.983017i	$-0.441252\pi$
$983$	11311.9	0.367032	0.183516	+	0.983017i	$0.441252\pi$
$984$	0	0
$985$	4132.66	0.133683
$986$	0	0
$987$	3822.47	0.123273
$988$	0	0
$989$	−40648.8	−1.30693
$990$	0	0
$991$	29405.5	0.942580	0.471290	−	0.881978i	$-0.343788\pi$
$991$	29405.5	0.942580	0.471290	+	0.881978i	$0.343788\pi$
$992$	0	0
$993$	2373.12	0.0758397
$994$	0	0
$995$	−11330.9	−0.361018
$996$	0	0
$997$	−54905.9	−1.74412	−0.872060	−	0.489398i	$-0.837216\pi$
$997$	−54905.9	−1.74412	−0.872060	+	0.489398i	$0.837216\pi$
$998$	0	0
$999$	−18390.9	−0.582446

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	272.4.a.h.1.2		3
3.2	odd	2		2448.4.a.bi.1.1		3
4.3	odd	2		17.4.a.b.1.2	✓	3
8.3	odd	2		1088.4.a.v.1.2		3
8.5	even	2		1088.4.a.x.1.2		3
12.11	even	2		153.4.a.g.1.2		3
20.3	even	4		425.4.b.f.324.3		6
20.7	even	4		425.4.b.f.324.4		6
20.19	odd	2		425.4.a.g.1.2		3
28.27	even	2		833.4.a.d.1.2		3
44.43	even	2		2057.4.a.e.1.2		3
68.47	odd	4		289.4.b.b.288.4		6
68.55	odd	4		289.4.b.b.288.3		6
68.67	odd	2		289.4.a.b.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.4.a.b.1.2	✓	3	4.3	odd	2
153.4.a.g.1.2		3	12.11	even	2
272.4.a.h.1.2		3	1.1	even	1	trivial
289.4.a.b.1.2		3	68.67	odd	2
289.4.b.b.288.3		6	68.55	odd	4
289.4.b.b.288.4		6	68.47	odd	4
425.4.a.g.1.2		3	20.19	odd	2
425.4.b.f.324.3		6	20.3	even	4
425.4.b.f.324.4		6	20.7	even	4
833.4.a.d.1.2		3	28.27	even	2
1088.4.a.v.1.2		3	8.3	odd	2
1088.4.a.x.1.2		3	8.5	even	2
2057.4.a.e.1.2		3	44.43	even	2
2448.4.a.bi.1.1		3	3.2	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 272 = 2^{4} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	272.a (trivial)

\(f(q)\)	\(=\)	\(q-3.15463 q^{3} +3.03171 q^{5} +7.94049 q^{7} -17.0483 q^{9} +O(q^{10})\)	\(q-3.15463 q^{3} +3.03171 q^{5} +7.94049 q^{7} -17.0483 q^{9} -27.6161 q^{11} +58.1117 q^{13} -9.56391 q^{15} -17.0000 q^{17} -89.1688 q^{19} -25.0493 q^{21} +115.269 q^{23} -115.809 q^{25} +138.956 q^{27} -128.558 q^{29} -273.460 q^{31} +87.1187 q^{33} +24.0732 q^{35} -132.351 q^{37} -183.321 q^{39} -470.559 q^{41} -352.642 q^{43} -51.6854 q^{45} -152.598 q^{47} -279.949 q^{49} +53.6287 q^{51} +527.614 q^{53} -83.7239 q^{55} +281.295 q^{57} +292.020 q^{59} -53.8962 q^{61} -135.372 q^{63} +176.178 q^{65} -52.9572 q^{67} -363.632 q^{69} -788.400 q^{71} +295.780 q^{73} +365.334 q^{75} -219.285 q^{77} +720.325 q^{79} +21.9487 q^{81} +116.051 q^{83} -51.5390 q^{85} +405.552 q^{87} -813.329 q^{89} +461.435 q^{91} +862.664 q^{93} -270.334 q^{95} +794.693 q^{97} +470.808 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 4 q^{3} - 8 q^{5} - 22 q^{7} + 59 q^{9}+O(q^{10}) \) 3 * q - 4 * q^3 - 8 * q^5 - 22 * q^7 + 59 * q^9	\( 3 q - 4 q^{3} - 8 q^{5} - 22 q^{7} + 59 q^{9} + 28 q^{11} + 30 q^{13} - 108 q^{15} - 51 q^{17} - 80 q^{19} - 192 q^{21} - 142 q^{23} - 223 q^{25} + 20 q^{27} - 456 q^{29} - 230 q^{31} - 332 q^{33} + 332 q^{35} + 356 q^{37} - 268 q^{39} - 294 q^{41} - 556 q^{43} - 384 q^{45} - 640 q^{47} - 269 q^{49} + 68 q^{51} + 302 q^{53} - 76 q^{55} - 720 q^{57} - 636 q^{59} - 84 q^{61} - 1122 q^{63} + 408 q^{65} - 1008 q^{67} + 576 q^{69} + 402 q^{71} + 838 q^{73} + 1548 q^{75} - 504 q^{77} + 594 q^{79} - 505 q^{81} + 2396 q^{83} + 136 q^{85} - 1428 q^{87} - 170 q^{89} + 1016 q^{91} + 632 q^{93} + 472 q^{95} - 270 q^{97} + 2920 q^{99}+O(q^{100}) \) 3 * q - 4 * q^3 - 8 * q^5 - 22 * q^7 + 59 * q^9 + 28 * q^11 + 30 * q^13 - 108 * q^15 - 51 * q^17 - 80 * q^19 - 192 * q^21 - 142 * q^23 - 223 * q^25 + 20 * q^27 - 456 * q^29 - 230 * q^31 - 332 * q^33 + 332 * q^35 + 356 * q^37 - 268 * q^39 - 294 * q^41 - 556 * q^43 - 384 * q^45 - 640 * q^47 - 269 * q^49 + 68 * q^51 + 302 * q^53 - 76 * q^55 - 720 * q^57 - 636 * q^59 - 84 * q^61 - 1122 * q^63 + 408 * q^65 - 1008 * q^67 + 576 * q^69 + 402 * q^71 + 838 * q^73 + 1548 * q^75 - 504 * q^77 + 594 * q^79 - 505 * q^81 + 2396 * q^83 + 136 * q^85 - 1428 * q^87 - 170 * q^89 + 1016 * q^91 + 632 * q^93 + 472 * q^95 - 270 * q^97 + 2920 * q^99

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