LMFDB - Embedded newform 272.4.a.h.1.1

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			1.1
Root			$-3.58966$ 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)

$f(q)$	$=$	$q-8.47535 q^{3} +0.885690 q^{5} -3.81828 q^{7} +44.8316 q^{9} +O(q^{10})$	$q-8.47535 q^{3} +0.885690 q^{5} -3.81828 q^{7} +44.8316 q^{9} +52.3720 q^{11} -8.06025 q^{13} -7.50653 q^{15} -17.0000 q^{17} +66.5154 q^{19} +32.3613 q^{21} -180.226 q^{23} -124.216 q^{25} -151.129 q^{27} -41.2800 q^{29} +34.9114 q^{31} -443.871 q^{33} -3.38182 q^{35} +130.368 q^{37} +68.3134 q^{39} -17.9081 q^{41} -277.620 q^{43} +39.7069 q^{45} -463.789 q^{47} -328.421 q^{49} +144.081 q^{51} -329.944 q^{53} +46.3853 q^{55} -563.741 q^{57} -678.656 q^{59} +340.280 q^{61} -171.180 q^{63} -7.13888 q^{65} -15.3925 q^{67} +1527.48 q^{69} +670.203 q^{71} +193.480 q^{73} +1052.77 q^{75} -199.971 q^{77} -1080.15 q^{79} +70.4207 q^{81} +865.668 q^{83} -15.0567 q^{85} +349.863 q^{87} +1129.46 q^{89} +30.7763 q^{91} -295.886 q^{93} +58.9120 q^{95} -379.412 q^{97} +2347.92 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−8.47535	−1.63108	−0.815541	−	0.578699i	$-0.803561\pi$
$3$	−8.47535	−1.63108	−0.815541	+	0.578699i	$0.803561\pi$
$4$	0	0
$5$	0.885690	0.0792185	0.0396092	−	0.999215i	$-0.487389\pi$
$5$	0.885690	0.0792185	0.0396092	+	0.999215i	$0.487389\pi$
$6$	0	0
$7$	−3.81828	−0.206168	−0.103084	−	0.994673i	$-0.532871\pi$
$7$	−3.81828	−0.206168	−0.103084	+	0.994673i	$0.532871\pi$
$8$	0	0
$9$	44.8316	1.66043
$10$	0	0
$11$	52.3720	1.43552	0.717761	−	0.696289i	$-0.245167\pi$
$11$	52.3720	1.43552	0.717761	+	0.696289i	$0.245167\pi$
$12$	0	0
$13$	−8.06025	−0.171962	−0.0859811	−	0.996297i	$-0.527402\pi$
$13$	−8.06025	−0.171962	−0.0859811	+	0.996297i	$0.527402\pi$
$14$	0	0
$15$	−7.50653	−0.129212
$16$	0	0
$17$	−17.0000	−0.242536
$17$	−17.0000	−0.242536
$18$	0	0
$19$	66.5154	0.803141	0.401570	−	0.915828i	$-0.368465\pi$
$19$	66.5154	0.803141	0.401570	+	0.915828i	$0.368465\pi$
$20$	0	0
$21$	32.3613	0.336277
$22$	0	0
$23$	−180.226	−1.63390	−0.816948	−	0.576711i	$-0.804336\pi$
$23$	−180.226	−1.63390	−0.816948	+	0.576711i	$0.804336\pi$
$24$	0	0
$25$	−124.216	−0.993724
$26$	0	0
$27$	−151.129	−1.07722
$28$	0	0
$29$	−41.2800	−0.264328	−0.132164	−	0.991228i	$-0.542193\pi$
$29$	−41.2800	−0.264328	−0.132164	+	0.991228i	$0.542193\pi$
$30$	0	0
$31$	34.9114	0.202267	0.101133	−	0.994873i	$-0.467753\pi$
$31$	34.9114	0.202267	0.101133	+	0.994873i	$0.467753\pi$
$32$	0	0
$33$	−443.871	−2.34146
$34$	0	0
$35$	−3.38182	−0.0163323
$36$	0	0
$37$	130.368	0.579255	0.289627	−	0.957139i	$-0.406469\pi$
$37$	130.368	0.579255	0.289627	+	0.957139i	$0.406469\pi$
$38$	0	0
$39$	68.3134	0.280485
$40$	0	0
$41$	−17.9081	−0.0682142	−0.0341071	−	0.999418i	$-0.510859\pi$
$41$	−17.9081	−0.0682142	−0.0341071	+	0.999418i	$0.510859\pi$
$42$	0	0
$43$	−277.620	−0.984573	−0.492287	−	0.870433i	$-0.663839\pi$
$43$	−277.620	−0.984573	−0.492287	+	0.870433i	$0.663839\pi$
$44$	0	0
$45$	39.7069	0.131537
$46$	0	0
$47$	−463.789	−1.43937	−0.719687	−	0.694299i	$-0.755715\pi$
$47$	−463.789	−1.43937	−0.719687	+	0.694299i	$0.755715\pi$
$48$	0	0
$49$	−328.421	−0.957495
$50$	0	0
$51$	144.081	0.395596
$52$	0	0
$53$	−329.944	−0.855118	−0.427559	−	0.903987i	$-0.640626\pi$
$53$	−329.944	−0.855118	−0.427559	+	0.903987i	$0.640626\pi$
$54$	0	0
$55$	46.3853	0.113720
$56$	0	0
$57$	−563.741	−1.30999
$58$	0	0
$59$	−678.656	−1.49752	−0.748759	−	0.662843i	$-0.769350\pi$
$59$	−678.656	−1.49752	−0.748759	+	0.662843i	$0.769350\pi$
$60$	0	0
$61$	340.280	0.714237	0.357118	−	0.934059i	$-0.383759\pi$
$61$	340.280	0.714237	0.357118	+	0.934059i	$0.383759\pi$
$62$	0	0
$63$	−171.180	−0.342328
$64$	0	0
$65$	−7.13888	−0.0136226
$66$	0	0
$67$	−15.3925	−0.0280671	−0.0140336	−	0.999902i	$-0.504467\pi$
$67$	−15.3925	−0.0280671	−0.0140336	+	0.999902i	$0.504467\pi$
$68$	0	0
$69$	1527.48	2.66502
$70$	0	0
$71$	670.203	1.12026	0.560130	−	0.828405i	$-0.310751\pi$
$71$	670.203	1.12026	0.560130	+	0.828405i	$0.310751\pi$
$72$	0	0
$73$	193.480	0.310207	0.155103	−	0.987898i	$-0.450429\pi$
$73$	193.480	0.310207	0.155103	+	0.987898i	$0.450429\pi$
$74$	0	0
$75$	1052.77	1.62085
$76$	0	0
$77$	−199.971	−0.295959
$78$	0	0
$79$	−1080.15	−1.53831	−0.769156	−	0.639061i	$-0.779323\pi$
$79$	−1080.15	−1.53831	−0.769156	+	0.639061i	$0.779323\pi$
$80$	0	0
$81$	70.4207	0.0965990
$82$	0	0
$83$	865.668	1.14481	0.572406	−	0.819970i	$-0.306010\pi$
$83$	865.668	1.14481	0.572406	+	0.819970i	$0.306010\pi$
$84$	0	0
$85$	−15.0567	−0.0192133
$86$	0	0
$87$	349.863	0.431141
$88$	0	0
$89$	1129.46	1.34520	0.672599	−	0.740008i	$-0.265178\pi$
$89$	1129.46	1.34520	0.672599	+	0.740008i	$0.265178\pi$
$90$	0	0
$91$	30.7763	0.0354531
$92$	0	0
$93$	−295.886	−0.329914
$94$	0	0
$95$	58.9120	0.0636236
$96$	0	0
$97$	−379.412	−0.397149	−0.198574	−	0.980086i	$-0.563631\pi$
$97$	−379.412	−0.397149	−0.198574	+	0.980086i	$0.563631\pi$
$98$	0	0
$99$	2347.92	2.38359
$100$	0	0
$101$	131.732	0.129780	0.0648902	−	0.997892i	$-0.479330\pi$
$101$	131.732	0.129780	0.0648902	+	0.997892i	$0.479330\pi$
$102$	0	0
$103$	−195.988	−0.187488	−0.0937442	−	0.995596i	$-0.529884\pi$
$103$	−195.988	−0.187488	−0.0937442	+	0.995596i	$0.529884\pi$
$104$	0	0
$105$	28.6621	0.0266394
$106$	0	0
$107$	485.147	0.438326	0.219163	−	0.975688i	$-0.429667\pi$
$107$	485.147	0.438326	0.219163	+	0.975688i	$0.429667\pi$
$108$	0	0
$109$	−1255.12	−1.10292	−0.551460	−	0.834201i	$-0.685929\pi$
$109$	−1255.12	−1.10292	−0.551460	+	0.834201i	$0.685929\pi$
$110$	0	0
$111$	−1104.92	−0.944812
$112$	0	0
$113$	−1013.35	−0.843612	−0.421806	−	0.906686i	$-0.638604\pi$
$113$	−1013.35	−0.843612	−0.421806	+	0.906686i	$0.638604\pi$
$114$	0	0
$115$	−159.624	−0.129435
$116$	0	0
$117$	−361.354	−0.285531
$118$	0	0
$119$	64.9108	0.0500031
$120$	0	0
$121$	1411.83	1.06073
$122$	0	0
$123$	151.778	0.111263
$124$	0	0
$125$	−220.728	−0.157940
$126$	0	0
$127$	−1927.72	−1.34691	−0.673456	−	0.739227i	$-0.735191\pi$
$127$	−1927.72	−1.34691	−0.673456	+	0.739227i	$0.735191\pi$
$128$	0	0
$129$	2352.93	1.60592
$130$	0	0
$131$	406.738	0.271274	0.135637	−	0.990759i	$-0.456692\pi$
$131$	406.738	0.271274	0.135637	+	0.990759i	$0.456692\pi$
$132$	0	0
$133$	−253.975	−0.165582
$134$	0	0
$135$	−133.854	−0.0853355
$136$	0	0
$137$	−130.552	−0.0814149	−0.0407074	−	0.999171i	$-0.512961\pi$
$137$	−130.552	−0.0814149	−0.0407074	+	0.999171i	$0.512961\pi$
$138$	0	0
$139$	−2073.54	−1.26529	−0.632644	−	0.774443i	$-0.718030\pi$
$139$	−2073.54	−1.26529	−0.632644	+	0.774443i	$0.718030\pi$
$140$	0	0
$141$	3930.78	2.34774
$142$	0	0
$143$	−422.131	−0.246856
$144$	0	0
$145$	−36.5613	−0.0209397
$146$	0	0
$147$	2783.48	1.56175
$148$	0	0
$149$	−1852.73	−1.01867	−0.509334	−	0.860569i	$-0.670108\pi$
$149$	−1852.73	−1.01867	−0.509334	+	0.860569i	$0.670108\pi$
$150$	0	0
$151$	−2050.86	−1.10527	−0.552637	−	0.833422i	$-0.686378\pi$
$151$	−2050.86	−1.10527	−0.552637	+	0.833422i	$0.686378\pi$
$152$	0	0
$153$	−762.138	−0.402714
$154$	0	0
$155$	30.9207	0.0160233
$156$	0	0
$157$	−262.991	−0.133688	−0.0668438	−	0.997763i	$-0.521293\pi$
$157$	−262.991	−0.133688	−0.0668438	+	0.997763i	$0.521293\pi$
$158$	0	0
$159$	2796.39	1.39477
$160$	0	0
$161$	688.152	0.336857
$162$	0	0
$163$	1444.98	0.694354	0.347177	−	0.937800i	$-0.387140\pi$
$163$	1444.98	0.694354	0.347177	+	0.937800i	$0.387140\pi$
$164$	0	0
$165$	−393.132	−0.185487
$166$	0	0
$167$	501.565	0.232409	0.116204	−	0.993225i	$-0.462927\pi$
$167$	501.565	0.232409	0.116204	+	0.993225i	$0.462927\pi$
$168$	0	0
$169$	−2132.03	−0.970429
$170$	0	0
$171$	2981.99	1.33356
$172$	0	0
$173$	−2590.14	−1.13829	−0.569146	−	0.822237i	$-0.692726\pi$
$173$	−2590.14	−1.13829	−0.569146	+	0.822237i	$0.692726\pi$
$174$	0	0
$175$	474.290	0.204874
$176$	0	0
$177$	5751.85	2.44257
$178$	0	0
$179$	−2165.65	−0.904294	−0.452147	−	0.891943i	$-0.649342\pi$
$179$	−2165.65	−0.904294	−0.452147	+	0.891943i	$0.649342\pi$
$180$	0	0
$181$	−1925.56	−0.790750	−0.395375	−	0.918520i	$-0.629385\pi$
$181$	−1925.56	−0.790750	−0.395375	+	0.918520i	$0.629385\pi$
$182$	0	0
$183$	−2884.00	−1.16498
$184$	0	0
$185$	115.466	0.0458877
$186$	0	0
$187$	−890.324	−0.348165
$188$	0	0
$189$	577.055	0.222088
$190$	0	0
$191$	2783.52	1.05449	0.527247	−	0.849712i	$-0.323224\pi$
$191$	2783.52	1.05449	0.527247	+	0.849712i	$0.323224\pi$
$192$	0	0
$193$	2258.27	0.842246	0.421123	−	0.907004i	$-0.361636\pi$
$193$	2258.27	0.842246	0.421123	+	0.907004i	$0.361636\pi$
$194$	0	0
$195$	60.5045	0.0222196
$196$	0	0
$197$	−1270.70	−0.459560	−0.229780	−	0.973243i	$-0.573801\pi$
$197$	−1270.70	−0.459560	−0.229780	+	0.973243i	$0.573801\pi$
$198$	0	0
$199$	4794.36	1.70786	0.853928	−	0.520392i	$-0.174214\pi$
$199$	4794.36	1.70786	0.853928	+	0.520392i	$0.174214\pi$
$200$	0	0
$201$	130.457	0.0457798
$202$	0	0
$203$	157.619	0.0544960
$204$	0	0
$205$	−15.8611	−0.00540383
$206$	0	0
$207$	−8079.80	−2.71297
$208$	0	0
$209$	3483.54	1.15293
$210$	0	0
$211$	2807.00	0.915837	0.457918	−	0.888994i	$-0.348595\pi$
$211$	2807.00	0.915837	0.457918	+	0.888994i	$0.348595\pi$
$212$	0	0
$213$	−5680.21	−1.82724
$214$	0	0
$215$	−245.885	−0.0779964
$216$	0	0
$217$	−133.302	−0.0417009
$218$	0	0
$219$	−1639.81	−0.505973
$220$	0	0
$221$	137.024	0.0417070
$222$	0	0
$223$	−4684.30	−1.40665	−0.703327	−	0.710866i	$-0.748303\pi$
$223$	−4684.30	−1.40665	−0.703327	+	0.710866i	$0.748303\pi$
$224$	0	0
$225$	−5568.79	−1.65001
$226$	0	0
$227$	1395.72	0.408095	0.204047	−	0.978961i	$-0.434590\pi$
$227$	1395.72	0.408095	0.204047	+	0.978961i	$0.434590\pi$
$228$	0	0
$229$	894.638	0.258163	0.129082	−	0.991634i	$-0.458797\pi$
$229$	894.638	0.258163	0.129082	+	0.991634i	$0.458797\pi$
$230$	0	0
$231$	1694.83	0.482733
$232$	0	0
$233$	1196.13	0.336313	0.168156	−	0.985760i	$-0.446219\pi$
$233$	1196.13	0.336313	0.168156	+	0.985760i	$0.446219\pi$
$234$	0	0
$235$	−410.773	−0.114025
$236$	0	0
$237$	9154.67	2.50911
$238$	0	0
$239$	−4948.82	−1.33938	−0.669691	−	0.742639i	$-0.733574\pi$
$239$	−4948.82	−1.33938	−0.669691	+	0.742639i	$0.733574\pi$
$240$	0	0
$241$	−6702.73	−1.79154	−0.895770	−	0.444518i	$-0.853375\pi$
$241$	−6702.73	−1.79154	−0.895770	+	0.444518i	$0.853375\pi$
$242$	0	0
$243$	3483.65	0.919656
$244$	0	0
$245$	−290.879	−0.0758513
$246$	0	0
$247$	−536.130	−0.138110
$248$	0	0
$249$	−7336.85	−1.86728
$250$	0	0
$251$	4756.08	1.19602	0.598010	−	0.801489i	$-0.295958\pi$
$251$	4756.08	1.19602	0.598010	+	0.801489i	$0.295958\pi$
$252$	0	0
$253$	−9438.77	−2.34550
$254$	0	0
$255$	127.611	0.0313385
$256$	0	0
$257$	2892.84	0.702143	0.351071	−	0.936349i	$-0.385817\pi$
$257$	2892.84	0.702143	0.351071	+	0.936349i	$0.385817\pi$
$258$	0	0
$259$	−497.784	−0.119424
$260$	0	0
$261$	−1850.65	−0.438898
$262$	0	0
$263$	−5415.48	−1.26971	−0.634853	−	0.772633i	$-0.718939\pi$
$263$	−5415.48	−1.26971	−0.634853	+	0.772633i	$0.718939\pi$
$264$	0	0
$265$	−292.228	−0.0677412
$266$	0	0
$267$	−9572.58	−2.19413
$268$	0	0
$269$	5787.00	1.31167	0.655835	−	0.754904i	$-0.272317\pi$
$269$	5787.00	1.31167	0.655835	+	0.754904i	$0.272317\pi$
$270$	0	0
$271$	−5465.13	−1.22503	−0.612515	−	0.790459i	$-0.709842\pi$
$271$	−5465.13	−1.22503	−0.612515	+	0.790459i	$0.709842\pi$
$272$	0	0
$273$	−260.840	−0.0578270
$274$	0	0
$275$	−6505.42	−1.42651
$276$	0	0
$277$	−1207.65	−0.261952	−0.130976	−	0.991386i	$-0.541811\pi$
$277$	−1207.65	−0.261952	−0.130976	+	0.991386i	$0.541811\pi$
$278$	0	0
$279$	1565.13	0.335850
$280$	0	0
$281$	−1197.18	−0.254155	−0.127077	−	0.991893i	$-0.540560\pi$
$281$	−1197.18	−0.254155	−0.127077	+	0.991893i	$0.540560\pi$
$282$	0	0
$283$	−3164.73	−0.664748	−0.332374	−	0.943148i	$-0.607850\pi$
$283$	−3164.73	−0.664748	−0.332374	+	0.943148i	$0.607850\pi$
$284$	0	0
$285$	−499.300	−0.103775
$286$	0	0
$287$	68.3784	0.0140636
$288$	0	0
$289$	289.000	0.0588235
$290$	0	0
$291$	3215.65	0.647782
$292$	0	0
$293$	7456.21	1.48668	0.743339	−	0.668915i	$-0.233241\pi$
$293$	7456.21	1.48668	0.743339	+	0.668915i	$0.233241\pi$
$294$	0	0
$295$	−601.079	−0.118631
$296$	0	0
$297$	−7914.94	−1.54637
$298$	0	0
$299$	1452.66	0.280969
$300$	0	0
$301$	1060.03	0.202988
$302$	0	0
$303$	−1116.47	−0.211683
$304$	0	0
$305$	301.383	0.0565808
$306$	0	0
$307$	6535.48	1.21498	0.607491	−	0.794327i	$-0.292176\pi$
$307$	6535.48	1.21498	0.607491	+	0.794327i	$0.292176\pi$
$308$	0	0
$309$	1661.07	0.305809
$310$	0	0
$311$	8935.89	1.62928	0.814642	−	0.579963i	$-0.196933\pi$
$311$	8935.89	1.62928	0.814642	+	0.579963i	$0.196933\pi$
$312$	0	0
$313$	−2628.71	−0.474707	−0.237353	−	0.971423i	$-0.576280\pi$
$313$	−2628.71	−0.474707	−0.237353	+	0.971423i	$0.576280\pi$
$314$	0	0
$315$	−151.612	−0.0271187
$316$	0	0
$317$	4268.54	0.756293	0.378147	−	0.925746i	$-0.376562\pi$
$317$	4268.54	0.756293	0.378147	+	0.925746i	$0.376562\pi$
$318$	0	0
$319$	−2161.92	−0.379449
$320$	0	0
$321$	−4111.79	−0.714946
$322$	0	0
$323$	−1130.76	−0.194790
$324$	0	0
$325$	1001.21	0.170883
$326$	0	0
$327$	10637.5	1.79895
$328$	0	0
$329$	1770.88	0.296753
$330$	0	0
$331$	−992.298	−0.164778	−0.0823892	−	0.996600i	$-0.526255\pi$
$331$	−992.298	−0.164778	−0.0823892	+	0.996600i	$0.526255\pi$
$332$	0	0
$333$	5844.63	0.961812
$334$	0	0
$335$	−13.6330	−0.00222344
$336$	0	0
$337$	8042.26	1.29997	0.649985	−	0.759947i	$-0.274775\pi$
$337$	8042.26	1.29997	0.649985	+	0.759947i	$0.274775\pi$
$338$	0	0
$339$	8588.52	1.37600
$340$	0	0
$341$	1828.38	0.290359
$342$	0	0
$343$	2563.68	0.403573
$344$	0	0
$345$	1352.87	0.211119
$346$	0	0
$347$	7414.16	1.14701	0.573506	−	0.819202i	$-0.305583\pi$
$347$	7414.16	1.14701	0.573506	+	0.819202i	$0.305583\pi$
$348$	0	0
$349$	−859.194	−0.131781	−0.0658905	−	0.997827i	$-0.520989\pi$
$349$	−859.194	−0.131781	−0.0658905	+	0.997827i	$0.520989\pi$
$350$	0	0
$351$	1218.14	0.185241
$352$	0	0
$353$	569.084	0.0858053	0.0429027	−	0.999079i	$-0.486339\pi$
$353$	569.084	0.0858053	0.0429027	+	0.999079i	$0.486339\pi$
$354$	0	0
$355$	593.592	0.0887453
$356$	0	0
$357$	−550.142	−0.0815592
$358$	0	0
$359$	5005.21	0.735835	0.367918	−	0.929858i	$-0.380071\pi$
$359$	5005.21	0.735835	0.367918	+	0.929858i	$0.380071\pi$
$360$	0	0
$361$	−2434.71	−0.354965
$362$	0	0
$363$	−11965.7	−1.73013
$364$	0	0
$365$	171.363	0.0245741
$366$	0	0
$367$	10975.3	1.56105	0.780523	−	0.625127i	$-0.214953\pi$
$367$	10975.3	1.56105	0.780523	+	0.625127i	$0.214953\pi$
$368$	0	0
$369$	−802.851	−0.113265
$370$	0	0
$371$	1259.82	0.176298
$372$	0	0
$373$	−3211.72	−0.445835	−0.222918	−	0.974837i	$-0.571558\pi$
$373$	−3211.72	−0.445835	−0.222918	+	0.974837i	$0.571558\pi$
$374$	0	0
$375$	1870.74	0.257613
$376$	0	0
$377$	332.727	0.0454544
$378$	0	0
$379$	−8051.48	−1.09123	−0.545616	−	0.838035i	$-0.683704\pi$
$379$	−8051.48	−1.09123	−0.545616	+	0.838035i	$0.683704\pi$
$380$	0	0
$381$	16338.1	2.19692
$382$	0	0
$383$	2584.16	0.344763	0.172382	−	0.985030i	$-0.444854\pi$
$383$	2584.16	0.344763	0.172382	+	0.985030i	$0.444854\pi$
$384$	0	0
$385$	−177.112	−0.0234454
$386$	0	0
$387$	−12446.2	−1.63482
$388$	0	0
$389$	−5174.31	−0.674417	−0.337208	−	0.941430i	$-0.609483\pi$
$389$	−5174.31	−0.674417	−0.337208	+	0.941430i	$0.609483\pi$
$390$	0	0
$391$	3063.83	0.396278
$392$	0	0
$393$	−3447.25	−0.442470
$394$	0	0
$395$	−956.680	−0.121863
$396$	0	0
$397$	−5149.36	−0.650980	−0.325490	−	0.945545i	$-0.605529\pi$
$397$	−5149.36	−0.650980	−0.325490	+	0.945545i	$0.605529\pi$
$398$	0	0
$399$	2152.53	0.270078
$400$	0	0
$401$	8700.49	1.08350	0.541748	−	0.840541i	$-0.317763\pi$
$401$	8700.49	1.08350	0.541748	+	0.840541i	$0.317763\pi$
$402$	0	0
$403$	−281.394	−0.0347823
$404$	0	0
$405$	62.3709	0.00765243
$406$	0	0
$407$	6827.65	0.831533
$408$	0	0
$409$	12346.0	1.49260	0.746299	−	0.665611i	$-0.231829\pi$
$409$	12346.0	1.49260	0.746299	+	0.665611i	$0.231829\pi$
$410$	0	0
$411$	1106.48	0.132794
$412$	0	0
$413$	2591.30	0.308740
$414$	0	0
$415$	766.713	0.0906903
$416$	0	0
$417$	17574.0	2.06379
$418$	0	0
$419$	5763.33	0.671974	0.335987	−	0.941867i	$-0.390930\pi$
$419$	5763.33	0.671974	0.335987	+	0.941867i	$0.390930\pi$
$420$	0	0
$421$	−1876.12	−0.217188	−0.108594	−	0.994086i	$-0.534635\pi$
$421$	−1876.12	−0.217188	−0.108594	+	0.994086i	$0.534635\pi$
$422$	0	0
$423$	−20792.4	−2.38998
$424$	0	0
$425$	2111.66	0.241014
$426$	0	0
$427$	−1299.29	−0.147253
$428$	0	0
$429$	3577.71	0.402642
$430$	0	0
$431$	−83.9299	−0.00937996	−0.00468998	−	0.999989i	$-0.501493\pi$
$431$	−83.9299	−0.00937996	−0.00468998	+	0.999989i	$0.501493\pi$
$432$	0	0
$433$	−15345.0	−1.70308	−0.851539	−	0.524291i	$-0.824331\pi$
$433$	−15345.0	−1.70308	−0.851539	+	0.524291i	$0.824331\pi$
$434$	0	0
$435$	309.870	0.0341543
$436$	0	0
$437$	−11987.8	−1.31225
$438$	0	0
$439$	−3064.74	−0.333194	−0.166597	−	0.986025i	$-0.553278\pi$
$439$	−3064.74	−0.333194	−0.166597	+	0.986025i	$0.553278\pi$
$440$	0	0
$441$	−14723.6	−1.58985
$442$	0	0
$443$	1792.97	0.192295	0.0961474	−	0.995367i	$-0.469348\pi$
$443$	1792.97	0.192295	0.0961474	+	0.995367i	$0.469348\pi$
$444$	0	0
$445$	1000.35	0.106564
$446$	0	0
$447$	15702.6	1.66153
$448$	0	0
$449$	2499.19	0.262681	0.131341	−	0.991337i	$-0.458072\pi$
$449$	2499.19	0.262681	0.131341	+	0.991337i	$0.458072\pi$
$450$	0	0
$451$	−937.885	−0.0979231
$452$	0	0
$453$	17381.7	1.80279
$454$	0	0
$455$	27.2583	0.00280854
$456$	0	0
$457$	14784.4	1.51331	0.756656	−	0.653813i	$-0.226832\pi$
$457$	14784.4	1.51331	0.756656	+	0.653813i	$0.226832\pi$
$458$	0	0
$459$	2569.20	0.261263
$460$	0	0
$461$	−17746.9	−1.79297	−0.896483	−	0.443078i	$-0.853887\pi$
$461$	−17746.9	−1.79297	−0.896483	+	0.443078i	$0.853887\pi$
$462$	0	0
$463$	−18486.4	−1.85559	−0.927793	−	0.373096i	$-0.878296\pi$
$463$	−18486.4	−1.85559	−0.927793	+	0.373096i	$0.878296\pi$
$464$	0	0
$465$	−262.064	−0.0261353
$466$	0	0
$467$	−7406.57	−0.733908	−0.366954	−	0.930239i	$-0.619599\pi$
$467$	−7406.57	−0.733908	−0.366954	+	0.930239i	$0.619599\pi$
$468$	0	0
$469$	58.7731	0.00578655
$470$	0	0
$471$	2228.94	0.218055
$472$	0	0
$473$	−14539.5	−1.41338
$474$	0	0
$475$	−8262.24	−0.798101
$476$	0	0
$477$	−14791.9	−1.41986
$478$	0	0
$479$	18550.9	1.76955	0.884775	−	0.466019i	$-0.154312\pi$
$479$	18550.9	1.76955	0.884775	+	0.466019i	$0.154312\pi$
$480$	0	0
$481$	−1050.80	−0.0996100
$482$	0	0
$483$	−5832.34	−0.549442
$484$	0	0
$485$	−336.041	−0.0314615
$486$	0	0
$487$	−10203.4	−0.949406	−0.474703	−	0.880146i	$-0.657444\pi$
$487$	−10203.4	−0.949406	−0.474703	+	0.880146i	$0.657444\pi$
$488$	0	0
$489$	−12246.7	−1.13255
$490$	0	0
$491$	1247.46	0.114658	0.0573290	−	0.998355i	$-0.481742\pi$
$491$	1247.46	0.114658	0.0573290	+	0.998355i	$0.481742\pi$
$492$	0	0
$493$	701.760	0.0641089
$494$	0	0
$495$	2079.53	0.188824
$496$	0	0
$497$	−2559.03	−0.230962
$498$	0	0
$499$	−70.0303	−0.00628254	−0.00314127	−	0.999995i	$-0.501000\pi$
$499$	−70.0303	−0.00628254	−0.00314127	+	0.999995i	$0.501000\pi$
$500$	0	0
$501$	−4250.94	−0.379078
$502$	0	0
$503$	−1444.29	−0.128028	−0.0640138	−	0.997949i	$-0.520390\pi$
$503$	−1444.29	−0.128028	−0.0640138	+	0.997949i	$0.520390\pi$
$504$	0	0
$505$	116.674	0.0102810
$506$	0	0
$507$	18069.7	1.58285
$508$	0	0
$509$	14272.8	1.24289	0.621445	−	0.783458i	$-0.286546\pi$
$509$	14272.8	1.24289	0.621445	+	0.783458i	$0.286546\pi$
$510$	0	0
$511$	−738.761	−0.0639547
$512$	0	0
$513$	−10052.4	−0.865157
$514$	0	0
$515$	−173.585	−0.0148525
$516$	0	0
$517$	−24289.6	−2.06625
$518$	0	0
$519$	21952.3	1.85665
$520$	0	0
$521$	14874.0	1.25075	0.625376	−	0.780324i	$-0.284946\pi$
$521$	14874.0	1.25075	0.625376	+	0.780324i	$0.284946\pi$
$522$	0	0
$523$	8142.90	0.680811	0.340406	−	0.940279i	$-0.389436\pi$
$523$	8142.90	0.680811	0.340406	+	0.940279i	$0.389436\pi$
$524$	0	0
$525$	−4019.78	−0.334167
$526$	0	0
$527$	−593.494	−0.0490569
$528$	0	0
$529$	20314.2	1.66962
$530$	0	0
$531$	−30425.3	−2.48652
$532$	0	0
$533$	144.344	0.0117303
$534$	0	0
$535$	429.689	0.0347235
$536$	0	0
$537$	18354.7	1.47498
$538$	0	0
$539$	−17200.0	−1.37451
$540$	0	0
$541$	3179.67	0.252689	0.126344	−	0.991986i	$-0.459676\pi$
$541$	3179.67	0.252689	0.126344	+	0.991986i	$0.459676\pi$
$542$	0	0
$543$	16319.8	1.28978
$544$	0	0
$545$	−1111.64	−0.0873716
$546$	0	0
$547$	−2107.07	−0.164702	−0.0823509	−	0.996603i	$-0.526243\pi$
$547$	−2107.07	−0.164702	−0.0823509	+	0.996603i	$0.526243\pi$
$548$	0	0
$549$	15255.3	1.18594
$550$	0	0
$551$	−2745.76	−0.212292
$552$	0	0
$553$	4124.33	0.317151
$554$	0	0
$555$	−978.614	−0.0748466
$556$	0	0
$557$	467.382	0.0355540	0.0177770	−	0.999842i	$-0.494341\pi$
$557$	467.382	0.0355540	0.0177770	+	0.999842i	$0.494341\pi$
$558$	0	0
$559$	2237.69	0.169309
$560$	0	0
$561$	7545.81	0.567887
$562$	0	0
$563$	−14612.6	−1.09387	−0.546935	−	0.837175i	$-0.684206\pi$
$563$	−14612.6	−1.09387	−0.546935	+	0.837175i	$0.684206\pi$
$564$	0	0
$565$	−897.515	−0.0668297
$566$	0	0
$567$	−268.886	−0.0199156
$568$	0	0
$569$	11602.3	0.854821	0.427410	−	0.904058i	$-0.359426\pi$
$569$	11602.3	0.854821	0.427410	+	0.904058i	$0.359426\pi$
$570$	0	0
$571$	10534.9	0.772104	0.386052	−	0.922477i	$-0.373839\pi$
$571$	10534.9	0.772104	0.386052	+	0.922477i	$0.373839\pi$
$572$	0	0
$573$	−23591.3	−1.71997
$574$	0	0
$575$	22386.8	1.62364
$576$	0	0
$577$	14404.7	1.03930	0.519650	−	0.854379i	$-0.326062\pi$
$577$	14404.7	1.03930	0.519650	+	0.854379i	$0.326062\pi$
$578$	0	0
$579$	−19139.6	−1.37377
$580$	0	0
$581$	−3305.37	−0.236024
$582$	0	0
$583$	−17279.8	−1.22754
$584$	0	0
$585$	−320.047	−0.0226194
$586$	0	0
$587$	11004.9	0.773799	0.386900	−	0.922122i	$-0.373546\pi$
$587$	11004.9	0.773799	0.386900	+	0.922122i	$0.373546\pi$
$588$	0	0
$589$	2322.14	0.162449
$590$	0	0
$591$	10769.6	0.749581
$592$	0	0
$593$	1853.59	0.128361	0.0641804	−	0.997938i	$-0.479557\pi$
$593$	1853.59	0.128361	0.0641804	+	0.997938i	$0.479557\pi$
$594$	0	0
$595$	57.4909	0.00396117
$596$	0	0
$597$	−40633.9	−2.78565
$598$	0	0
$599$	−19074.7	−1.30112	−0.650559	−	0.759456i	$-0.725465\pi$
$599$	−19074.7	−1.30112	−0.650559	+	0.759456i	$0.725465\pi$
$600$	0	0
$601$	−27776.0	−1.88520	−0.942600	−	0.333923i	$-0.891627\pi$
$601$	−27776.0	−1.88520	−0.942600	+	0.333923i	$0.891627\pi$
$602$	0	0
$603$	−690.073	−0.0466035
$604$	0	0
$605$	1250.44	0.0840291
$606$	0	0
$607$	−18728.3	−1.25232	−0.626159	−	0.779695i	$-0.715374\pi$
$607$	−18728.3	−1.25232	−0.626159	+	0.779695i	$0.715374\pi$
$608$	0	0
$609$	−1335.88	−0.0888874
$610$	0	0
$611$	3738.25	0.247518
$612$	0	0
$613$	−24405.3	−1.60802	−0.804012	−	0.594613i	$-0.797305\pi$
$613$	−24405.3	−1.60802	−0.804012	+	0.594613i	$0.797305\pi$
$614$	0	0
$615$	134.428	0.00881409
$616$	0	0
$617$	−22516.4	−1.46917	−0.734584	−	0.678518i	$-0.762623\pi$
$617$	−22516.4	−1.46917	−0.734584	+	0.678518i	$0.762623\pi$
$618$	0	0
$619$	5146.53	0.334179	0.167089	−	0.985942i	$-0.446563\pi$
$619$	5146.53	0.334179	0.167089	+	0.985942i	$0.446563\pi$
$620$	0	0
$621$	27237.4	1.76006
$622$	0	0
$623$	−4312.60	−0.277337
$624$	0	0
$625$	15331.4	0.981213
$626$	0	0
$627$	−29524.3	−1.88052
$628$	0	0
$629$	−2216.26	−0.140490
$630$	0	0
$631$	3858.77	0.243447	0.121724	−	0.992564i	$-0.461158\pi$
$631$	3858.77	0.243447	0.121724	+	0.992564i	$0.461158\pi$
$632$	0	0
$633$	−23790.3	−1.49381
$634$	0	0
$635$	−1707.36	−0.106700
$636$	0	0
$637$	2647.15	0.164653
$638$	0	0
$639$	30046.3	1.86011
$640$	0	0
$641$	18689.3	1.15161	0.575805	−	0.817587i	$-0.304689\pi$
$641$	18689.3	1.15161	0.575805	+	0.817587i	$0.304689\pi$
$642$	0	0
$643$	−26473.5	−1.62366	−0.811831	−	0.583893i	$-0.801529\pi$
$643$	−26473.5	−1.62366	−0.811831	+	0.583893i	$0.801529\pi$
$644$	0	0
$645$	2083.96	0.127219
$646$	0	0
$647$	−14397.7	−0.874855	−0.437427	−	0.899254i	$-0.644110\pi$
$647$	−14397.7	−0.874855	−0.437427	+	0.899254i	$0.644110\pi$
$648$	0	0
$649$	−35542.6	−2.14972
$650$	0	0
$651$	1129.78	0.0680177
$652$	0	0
$653$	20939.5	1.25486	0.627431	−	0.778672i	$-0.284107\pi$
$653$	20939.5	1.25486	0.627431	+	0.778672i	$0.284107\pi$
$654$	0	0
$655$	360.244	0.0214899
$656$	0	0
$657$	8674.02	0.515077
$658$	0	0
$659$	−4031.76	−0.238323	−0.119162	−	0.992875i	$-0.538021\pi$
$659$	−4031.76	−0.238323	−0.119162	+	0.992875i	$0.538021\pi$
$660$	0	0
$661$	6691.52	0.393752	0.196876	−	0.980428i	$-0.436920\pi$
$661$	6691.52	0.393752	0.196876	+	0.980428i	$0.436920\pi$
$662$	0	0
$663$	−1161.33	−0.0680275
$664$	0	0
$665$	−224.943	−0.0131171
$666$	0	0
$667$	7439.71	0.431884
$668$	0	0
$669$	39701.1	2.29437
$670$	0	0
$671$	17821.2	1.02530
$672$	0	0
$673$	10319.2	0.591048	0.295524	−	0.955335i	$-0.404506\pi$
$673$	10319.2	0.591048	0.295524	+	0.955335i	$0.404506\pi$
$674$	0	0
$675$	18772.6	1.07046
$676$	0	0
$677$	−19813.3	−1.12480	−0.562398	−	0.826866i	$-0.690121\pi$
$677$	−19813.3	−1.12480	−0.562398	+	0.826866i	$0.690121\pi$
$678$	0	0
$679$	1448.70	0.0818793
$680$	0	0
$681$	−11829.3	−0.665636
$682$	0	0
$683$	−5924.61	−0.331916	−0.165958	−	0.986133i	$-0.553072\pi$
$683$	−5924.61	−0.331916	−0.165958	+	0.986133i	$0.553072\pi$
$684$	0	0
$685$	−115.629	−0.00644957
$686$	0	0
$687$	−7582.38	−0.421085
$688$	0	0
$689$	2659.43	0.147048
$690$	0	0
$691$	−1973.16	−0.108629	−0.0543143	−	0.998524i	$-0.517297\pi$
$691$	−1973.16	−0.108629	−0.0543143	+	0.998524i	$0.517297\pi$
$692$	0	0
$693$	−8965.03	−0.491419
$694$	0	0
$695$	−1836.51	−0.100234
$696$	0	0
$697$	304.439	0.0165444
$698$	0	0
$699$	−10137.6	−0.548554
$700$	0	0
$701$	−12840.1	−0.691815	−0.345907	−	0.938269i	$-0.612429\pi$
$701$	−12840.1	−0.691815	−0.345907	+	0.938269i	$0.612429\pi$
$702$	0	0
$703$	8671.50	0.465223
$704$	0	0
$705$	3481.45	0.185984
$706$	0	0
$707$	−502.990	−0.0267566
$708$	0	0
$709$	−27749.7	−1.46990	−0.734952	−	0.678119i	$-0.762796\pi$
$709$	−27749.7	−1.46990	−0.734952	+	0.678119i	$0.762796\pi$
$710$	0	0
$711$	−48425.0	−2.55426
$712$	0	0
$713$	−6291.92	−0.330483
$714$	0	0
$715$	−373.877	−0.0195555
$716$	0	0
$717$	41943.0	2.18464
$718$	0	0
$719$	−16888.3	−0.875979	−0.437989	−	0.898980i	$-0.644309\pi$
$719$	−16888.3	−0.875979	−0.437989	+	0.898980i	$0.644309\pi$
$720$	0	0
$721$	748.339	0.0386541
$722$	0	0
$723$	56808.0	2.92215
$724$	0	0
$725$	5127.62	0.262669
$726$	0	0
$727$	−2135.25	−0.108930	−0.0544649	−	0.998516i	$-0.517345\pi$
$727$	−2135.25	−0.108930	−0.0544649	+	0.998516i	$0.517345\pi$
$728$	0	0
$729$	−31426.5	−1.59663
$730$	0	0
$731$	4719.54	0.238794
$732$	0	0
$733$	4795.27	0.241633	0.120817	−	0.992675i	$-0.461449\pi$
$733$	4795.27	0.241633	0.120817	+	0.992675i	$0.461449\pi$
$734$	0	0
$735$	2465.30	0.123720
$736$	0	0
$737$	−806.138	−0.0402910
$738$	0	0
$739$	32747.6	1.63010	0.815048	−	0.579393i	$-0.196710\pi$
$739$	32747.6	1.63010	0.815048	+	0.579393i	$0.196710\pi$
$740$	0	0
$741$	4543.89	0.225269
$742$	0	0
$743$	−12299.4	−0.607298	−0.303649	−	0.952784i	$-0.598205\pi$
$743$	−12299.4	−0.607298	−0.303649	+	0.952784i	$0.598205\pi$
$744$	0	0
$745$	−1640.94	−0.0806974
$746$	0	0
$747$	38809.3	1.90088
$748$	0	0
$749$	−1852.43	−0.0903688
$750$	0	0
$751$	−30102.6	−1.46266	−0.731332	−	0.682021i	$-0.761101\pi$
$751$	−30102.6	−1.46266	−0.731332	+	0.682021i	$0.761101\pi$
$752$	0	0
$753$	−40309.4	−1.95081
$754$	0	0
$755$	−1816.42	−0.0875581
$756$	0	0
$757$	38826.3	1.86416	0.932078	−	0.362257i	$-0.117994\pi$
$757$	38826.3	1.86416	0.932078	+	0.362257i	$0.117994\pi$
$758$	0	0
$759$	79996.9	3.82570
$760$	0	0
$761$	19981.6	0.951815	0.475907	−	0.879495i	$-0.342120\pi$
$761$	19981.6	0.951815	0.475907	+	0.879495i	$0.342120\pi$
$762$	0	0
$763$	4792.39	0.227387
$764$	0	0
$765$	−675.017	−0.0319024
$766$	0	0
$767$	5470.14	0.257517
$768$	0	0
$769$	−22407.7	−1.05077	−0.525384	−	0.850865i	$-0.676078\pi$
$769$	−22407.7	−1.05077	−0.525384	+	0.850865i	$0.676078\pi$
$770$	0	0
$771$	−24517.9	−1.14525
$772$	0	0
$773$	−6902.77	−0.321184	−0.160592	−	0.987021i	$-0.551340\pi$
$773$	−6902.77	−0.321184	−0.160592	+	0.987021i	$0.551340\pi$
$774$	0	0
$775$	−4336.54	−0.200997
$776$	0	0
$777$	4218.89	0.194790
$778$	0	0
$779$	−1191.17	−0.0547856
$780$	0	0
$781$	35099.9	1.60816
$782$	0	0
$783$	6238.62	0.284738
$784$	0	0
$785$	−232.928	−0.0105905
$786$	0	0
$787$	22185.9	1.00488	0.502442	−	0.864611i	$-0.332435\pi$
$787$	22185.9	1.00488	0.502442	+	0.864611i	$0.332435\pi$
$788$	0	0
$789$	45898.1	2.07099
$790$	0	0
$791$	3869.27	0.173926
$792$	0	0
$793$	−2742.74	−0.122822
$794$	0	0
$795$	2476.73	0.110491
$796$	0	0
$797$	−16291.1	−0.724040	−0.362020	−	0.932170i	$-0.617913\pi$
$797$	−16291.1	−0.724040	−0.362020	+	0.932170i	$0.617913\pi$
$798$	0	0
$799$	7884.41	0.349100
$800$	0	0
$801$	50635.5	2.23361
$802$	0	0
$803$	10132.9	0.445309
$804$	0	0
$805$	609.489	0.0266853
$806$	0	0
$807$	−49046.8	−2.13944
$808$	0	0
$809$	17696.8	0.769082	0.384541	−	0.923108i	$-0.374360\pi$
$809$	17696.8	0.769082	0.384541	+	0.923108i	$0.374360\pi$
$810$	0	0
$811$	3095.34	0.134022	0.0670111	−	0.997752i	$-0.478654\pi$
$811$	3095.34	0.134022	0.0670111	+	0.997752i	$0.478654\pi$
$812$	0	0
$813$	46318.9	1.99812
$814$	0	0
$815$	1279.81	0.0550057
$816$	0	0
$817$	−18466.0	−0.790751
$818$	0	0
$819$	1379.75	0.0588674
$820$	0	0
$821$	12323.5	0.523864	0.261932	−	0.965086i	$-0.415640\pi$
$821$	12323.5	0.523864	0.261932	+	0.965086i	$0.415640\pi$
$822$	0	0
$823$	34436.5	1.45854	0.729271	−	0.684225i	$-0.239860\pi$
$823$	34436.5	1.45854	0.729271	+	0.684225i	$0.239860\pi$
$824$	0	0
$825$	55135.7	2.32676
$826$	0	0
$827$	−18761.6	−0.788880	−0.394440	−	0.918922i	$-0.629061\pi$
$827$	−18761.6	−0.788880	−0.394440	+	0.918922i	$0.629061\pi$
$828$	0	0
$829$	22423.8	0.939457	0.469728	−	0.882811i	$-0.344352\pi$
$829$	22423.8	0.939457	0.469728	+	0.882811i	$0.344352\pi$
$830$	0	0
$831$	10235.3	0.427265
$832$	0	0
$833$	5583.15	0.232227
$834$	0	0
$835$	444.231	0.0184111
$836$	0	0
$837$	−5276.13	−0.217885
$838$	0	0
$839$	−9128.63	−0.375632	−0.187816	−	0.982204i	$-0.560141\pi$
$839$	−9128.63	−0.375632	−0.187816	+	0.982204i	$0.560141\pi$
$840$	0	0
$841$	−22685.0	−0.930131
$842$	0	0
$843$	10146.5	0.414547
$844$	0	0
$845$	−1888.32	−0.0768759
$846$	0	0
$847$	−5390.75	−0.218688
$848$	0	0
$849$	26822.2	1.08426
$850$	0	0
$851$	−23495.7	−0.946442
$852$	0	0
$853$	27204.8	1.09200	0.545999	−	0.837786i	$-0.316150\pi$
$853$	27204.8	1.09200	0.545999	+	0.837786i	$0.316150\pi$
$854$	0	0
$855$	2641.12	0.105643
$856$	0	0
$857$	−38060.0	−1.51704	−0.758520	−	0.651649i	$-0.774077\pi$
$857$	−38060.0	−1.51704	−0.758520	+	0.651649i	$0.774077\pi$
$858$	0	0
$859$	33326.2	1.32372	0.661860	−	0.749627i	$-0.269767\pi$
$859$	33326.2	1.32372	0.661860	+	0.749627i	$0.269767\pi$
$860$	0	0
$861$	−579.531	−0.0229389
$862$	0	0
$863$	41724.2	1.64578	0.822890	−	0.568201i	$-0.192360\pi$
$863$	41724.2	1.64578	0.822890	+	0.568201i	$0.192360\pi$
$864$	0	0
$865$	−2294.06	−0.0901737
$866$	0	0
$867$	−2449.38	−0.0959460
$868$	0	0
$869$	−56569.7	−2.20828
$870$	0	0
$871$	124.068	0.00482649
$872$	0	0
$873$	−17009.6	−0.659438
$874$	0	0
$875$	842.801	0.0325621
$876$	0	0
$877$	−49337.3	−1.89966	−0.949830	−	0.312767i	$-0.898744\pi$
$877$	−49337.3	−1.89966	−0.949830	+	0.312767i	$0.898744\pi$
$878$	0	0
$879$	−63194.0	−2.42489
$880$	0	0
$881$	8845.46	0.338265	0.169132	−	0.985593i	$-0.445903\pi$
$881$	8845.46	0.338265	0.169132	+	0.985593i	$0.445903\pi$
$882$	0	0
$883$	−14724.2	−0.561165	−0.280582	−	0.959830i	$-0.590528\pi$
$883$	−14724.2	−0.561165	−0.280582	+	0.959830i	$0.590528\pi$
$884$	0	0
$885$	5094.36	0.193497
$886$	0	0
$887$	−3864.38	−0.146283	−0.0731415	−	0.997322i	$-0.523302\pi$
$887$	−3864.38	−0.146283	−0.0731415	+	0.997322i	$0.523302\pi$
$888$	0	0
$889$	7360.60	0.277690
$890$	0	0
$891$	3688.07	0.138670
$892$	0	0
$893$	−30849.1	−1.15602
$894$	0	0
$895$	−1918.10	−0.0716368
$896$	0	0
$897$	−12311.8	−0.458283
$898$	0	0
$899$	−1441.14	−0.0534648
$900$	0	0
$901$	5609.04	0.207397
$902$	0	0
$903$	−8984.15	−0.331089
$904$	0	0
$905$	−1705.45	−0.0626421
$906$	0	0
$907$	743.409	0.0272155	0.0136078	−	0.999907i	$-0.495668\pi$
$907$	743.409	0.0272155	0.0136078	+	0.999907i	$0.495668\pi$
$908$	0	0
$909$	5905.76	0.215491
$910$	0	0
$911$	−16291.0	−0.592475	−0.296238	−	0.955114i	$-0.595732\pi$
$911$	−16291.0	−0.592475	−0.296238	+	0.955114i	$0.595732\pi$
$912$	0	0
$913$	45336.8	1.64340
$914$	0	0
$915$	−2554.33	−0.0922879
$916$	0	0
$917$	−1553.04	−0.0559280
$918$	0	0
$919$	6188.99	0.222150	0.111075	−	0.993812i	$-0.464571\pi$
$919$	6188.99	0.222150	0.111075	+	0.993812i	$0.464571\pi$
$920$	0	0
$921$	−55390.5	−1.98174
$922$	0	0
$923$	−5402.00	−0.192643
$924$	0	0
$925$	−16193.8	−0.575620
$926$	0	0
$927$	−8786.47	−0.311311
$928$	0	0
$929$	−31661.7	−1.11818	−0.559089	−	0.829108i	$-0.688849\pi$
$929$	−31661.7	−1.11818	−0.559089	+	0.829108i	$0.688849\pi$
$930$	0	0
$931$	−21845.0	−0.769003
$932$	0	0
$933$	−75734.8	−2.65750
$934$	0	0
$935$	−788.551	−0.0275811
$936$	0	0
$937$	35010.5	1.22064	0.610322	−	0.792153i	$-0.291040\pi$
$937$	35010.5	1.22064	0.610322	+	0.792153i	$0.291040\pi$
$938$	0	0
$939$	22279.2	0.774286
$940$	0	0
$941$	−45625.8	−1.58061	−0.790307	−	0.612711i	$-0.790079\pi$
$941$	−45625.8	−1.58061	−0.790307	+	0.612711i	$0.790079\pi$
$942$	0	0
$943$	3227.51	0.111455
$944$	0	0
$945$	511.092	0.0175934
$946$	0	0
$947$	21508.4	0.738044	0.369022	−	0.929421i	$-0.379693\pi$
$947$	21508.4	0.738044	0.369022	+	0.929421i	$0.379693\pi$
$948$	0	0
$949$	−1559.50	−0.0533439
$950$	0	0
$951$	−36177.4	−1.23358
$952$	0	0
$953$	35686.7	1.21302	0.606509	−	0.795076i	$-0.292569\pi$
$953$	35686.7	1.21302	0.606509	+	0.795076i	$0.292569\pi$
$954$	0	0
$955$	2465.34	0.0835355
$956$	0	0
$957$	18323.0	0.618912
$958$	0	0
$959$	498.486	0.0167851
$960$	0	0
$961$	−28572.2	−0.959088
$962$	0	0
$963$	21749.9	0.727810
$964$	0	0
$965$	2000.12	0.0667215
$966$	0	0
$967$	3731.33	0.124086	0.0620432	−	0.998073i	$-0.480238\pi$
$967$	3731.33	0.124086	0.0620432	+	0.998073i	$0.480238\pi$
$968$	0	0
$969$	9583.60	0.317719
$970$	0	0
$971$	−17645.1	−0.583171	−0.291585	−	0.956545i	$-0.594183\pi$
$971$	−17645.1	−0.583171	−0.291585	+	0.956545i	$0.594183\pi$
$972$	0	0
$973$	7917.35	0.260862
$974$	0	0
$975$	−8485.59	−0.278725
$976$	0	0
$977$	24941.2	0.816723	0.408362	−	0.912820i	$-0.366100\pi$
$977$	24941.2	0.816723	0.408362	+	0.912820i	$0.366100\pi$
$978$	0	0
$979$	59152.1	1.93106
$980$	0	0
$981$	−56268.9	−1.83132
$982$	0	0
$983$	22506.2	0.730252	0.365126	−	0.930958i	$-0.381026\pi$
$983$	22506.2	0.730252	0.365126	+	0.930958i	$0.381026\pi$
$984$	0	0
$985$	−1125.44	−0.0364057
$986$	0	0
$987$	−15008.8	−0.484029
$988$	0	0
$989$	50034.2	1.60869
$990$	0	0
$991$	−32694.1	−1.04799	−0.523997	−	0.851720i	$-0.675560\pi$
$991$	−32694.1	−1.04799	−0.523997	+	0.851720i	$0.675560\pi$
$992$	0	0
$993$	8410.08	0.268767
$994$	0	0
$995$	4246.32	0.135294
$996$	0	0
$997$	18248.8	0.579686	0.289843	−	0.957074i	$-0.406397\pi$
$997$	18248.8	0.579686	0.289843	+	0.957074i	$0.406397\pi$
$998$	0	0
$999$	−19702.5	−0.623983

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	272.4.a.h.1.1		3
3.2	odd	2		2448.4.a.bi.1.2		3
4.3	odd	2		17.4.a.b.1.1	✓	3
8.3	odd	2		1088.4.a.v.1.1		3
8.5	even	2		1088.4.a.x.1.3		3
12.11	even	2		153.4.a.g.1.3		3
20.3	even	4		425.4.b.f.324.6		6
20.7	even	4		425.4.b.f.324.1		6
20.19	odd	2		425.4.a.g.1.3		3
28.27	even	2		833.4.a.d.1.1		3
44.43	even	2		2057.4.a.e.1.3		3
68.47	odd	4		289.4.b.b.288.6		6
68.55	odd	4		289.4.b.b.288.5		6
68.67	odd	2		289.4.a.b.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.4.a.b.1.1	✓	3	4.3	odd	2
153.4.a.g.1.3		3	12.11	even	2
272.4.a.h.1.1		3	1.1	even	1	trivial
289.4.a.b.1.1		3	68.67	odd	2
289.4.b.b.288.5		6	68.55	odd	4
289.4.b.b.288.6		6	68.47	odd	4
425.4.a.g.1.3		3	20.19	odd	2
425.4.b.f.324.1		6	20.7	even	4
425.4.b.f.324.6		6	20.3	even	4
833.4.a.d.1.1		3	28.27	even	2
1088.4.a.v.1.1		3	8.3	odd	2
1088.4.a.x.1.3		3	8.5	even	2
2057.4.a.e.1.3		3	44.43	even	2
2448.4.a.bi.1.2		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)

\(f(q)\)	\(=\)	\(q-8.47535 q^{3} +0.885690 q^{5} -3.81828 q^{7} +44.8316 q^{9} +O(q^{10})\)	\(q-8.47535 q^{3} +0.885690 q^{5} -3.81828 q^{7} +44.8316 q^{9} +52.3720 q^{11} -8.06025 q^{13} -7.50653 q^{15} -17.0000 q^{17} +66.5154 q^{19} +32.3613 q^{21} -180.226 q^{23} -124.216 q^{25} -151.129 q^{27} -41.2800 q^{29} +34.9114 q^{31} -443.871 q^{33} -3.38182 q^{35} +130.368 q^{37} +68.3134 q^{39} -17.9081 q^{41} -277.620 q^{43} +39.7069 q^{45} -463.789 q^{47} -328.421 q^{49} +144.081 q^{51} -329.944 q^{53} +46.3853 q^{55} -563.741 q^{57} -678.656 q^{59} +340.280 q^{61} -171.180 q^{63} -7.13888 q^{65} -15.3925 q^{67} +1527.48 q^{69} +670.203 q^{71} +193.480 q^{73} +1052.77 q^{75} -199.971 q^{77} -1080.15 q^{79} +70.4207 q^{81} +865.668 q^{83} -15.0567 q^{85} +349.863 q^{87} +1129.46 q^{89} +30.7763 q^{91} -295.886 q^{93} +58.9120 q^{95} -379.412 q^{97} +2347.92 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 4 q^{3} - 8 q^{5} - 22 q^{7} + 59 q^{9}+O(q^{10}) \) 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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