LMFDB - Embedded newform 196.4.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [196,4,Mod(1,196)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(196, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("196.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 196 = 2^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	196.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:	$11.5643743611$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{37}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 9 $ x^2 - x - 9
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 28)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.54138$ of defining polynomial
Character	$\chi$	$=$	196.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6.08276 q^{3} +19.1655 q^{5} +10.0000 q^{9} +O(q^{10})$	$q-6.08276 q^{3} +19.1655 q^{5} +10.0000 q^{9} -26.5793 q^{11} -10.3311 q^{13} -116.579 q^{15} +101.331 q^{17} +93.7517 q^{19} -8.57934 q^{23} +242.317 q^{25} +103.407 q^{27} +52.3174 q^{29} +55.4207 q^{31} +161.676 q^{33} +428.476 q^{37} +62.8413 q^{39} +137.007 q^{41} -172.000 q^{43} +191.655 q^{45} +49.2414 q^{47} -616.373 q^{51} -474.476 q^{53} -509.407 q^{55} -570.269 q^{57} -197.062 q^{59} +401.124 q^{61} -198.000 q^{65} +125.421 q^{67} +52.1861 q^{69} +788.635 q^{71} -604.655 q^{73} -1473.96 q^{75} +783.007 q^{79} -899.000 q^{81} +339.283 q^{83} +1942.06 q^{85} -318.234 q^{87} -511.966 q^{89} -337.111 q^{93} +1796.80 q^{95} +672.290 q^{97} -265.793 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 14 q^{5} + 20 q^{9}+O(q^{10}) $ 2 * q + 14 * q^5 + 20 * q^9	$ 2 q + 14 q^{5} + 20 q^{9} + 32 q^{11} + 28 q^{13} - 148 q^{15} + 154 q^{17} + 224 q^{19} + 68 q^{23} + 144 q^{25} - 236 q^{29} + 196 q^{31} + 518 q^{33} + 346 q^{37} + 296 q^{39} + 420 q^{41} - 344 q^{43} + 140 q^{45} - 84 q^{47} - 296 q^{51} - 438 q^{53} - 812 q^{55} + 222 q^{57} + 56 q^{59} - 98 q^{61} - 396 q^{65} + 336 q^{67} + 518 q^{69} + 896 q^{71} - 966 q^{73} - 2072 q^{75} - 52 q^{79} - 1798 q^{81} - 392 q^{83} + 1670 q^{85} - 2072 q^{87} - 294 q^{89} + 518 q^{93} + 1124 q^{95} + 420 q^{97} + 320 q^{99}+O(q^{100}) $ 2 * q + 14 * q^5 + 20 * q^9 + 32 * q^11 + 28 * q^13 - 148 * q^15 + 154 * q^17 + 224 * q^19 + 68 * q^23 + 144 * q^25 - 236 * q^29 + 196 * q^31 + 518 * q^33 + 346 * q^37 + 296 * q^39 + 420 * q^41 - 344 * q^43 + 140 * q^45 - 84 * q^47 - 296 * q^51 - 438 * q^53 - 812 * q^55 + 222 * q^57 + 56 * q^59 - 98 * q^61 - 396 * q^65 + 336 * q^67 + 518 * q^69 + 896 * q^71 - 966 * q^73 - 2072 * q^75 - 52 * q^79 - 1798 * q^81 - 392 * q^83 + 1670 * q^85 - 2072 * q^87 - 294 * q^89 + 518 * q^93 + 1124 * q^95 + 420 * q^97 + 320 * 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
$3$	−6.08276	−1.17063	−0.585314	−	0.810807i	$-0.699029\pi$
$3$	−6.08276	−1.17063	−0.585314	+	0.810807i	$0.699029\pi$
$4$	0	0
$5$	19.1655	1.71422	0.857108	−	0.515136i	$-0.172259\pi$
$5$	19.1655	1.71422	0.857108	+	0.515136i	$0.172259\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	10.0000	0.370370
$10$	0	0
$11$	−26.5793	−0.728543	−0.364271	−	0.931293i	$-0.618682\pi$
$11$	−26.5793	−0.728543	−0.364271	+	0.931293i	$0.618682\pi$
$12$	0	0
$13$	−10.3311	−0.220409	−0.110205	−	0.993909i	$-0.535151\pi$
$13$	−10.3311	−0.220409	−0.110205	+	0.993909i	$0.535151\pi$
$14$	0	0
$15$	−116.579	−2.00671
$16$	0	0
$17$	101.331	1.44567	0.722835	−	0.691021i	$-0.242839\pi$
$17$	101.331	1.44567	0.722835	+	0.691021i	$0.242839\pi$
$18$	0	0
$19$	93.7517	1.13201	0.566003	−	0.824403i	$-0.308489\pi$
$19$	93.7517	1.13201	0.566003	+	0.824403i	$0.308489\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−8.57934	−0.0777789	−0.0388895	−	0.999244i	$-0.512382\pi$
$23$	−8.57934	−0.0777789	−0.0388895	+	0.999244i	$0.512382\pi$
$24$	0	0
$25$	242.317	1.93854
$26$	0	0
$27$	103.407	0.737062
$28$	0	0
$29$	52.3174	0.335003	0.167502	−	0.985872i	$-0.446430\pi$
$29$	52.3174	0.335003	0.167502	+	0.985872i	$0.446430\pi$
$30$	0	0
$31$	55.4207	0.321092	0.160546	−	0.987028i	$-0.448675\pi$
$31$	55.4207	0.321092	0.160546	+	0.987028i	$0.448675\pi$
$32$	0	0
$33$	161.676	0.852853
$34$	0	0
$35$	0	0
$36$	0	0
$37$	428.476	1.90381	0.951906	−	0.306391i	$-0.0991215\pi$
$37$	428.476	1.90381	0.951906	+	0.306391i	$0.0991215\pi$
$38$	0	0
$39$	62.8413	0.258017
$40$	0	0
$41$	137.007	0.521875	0.260938	−	0.965356i	$-0.415968\pi$
$41$	137.007	0.521875	0.260938	+	0.965356i	$0.415968\pi$
$42$	0	0
$43$	−172.000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−172.000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	191.655	0.634895
$46$	0	0
$47$	49.2414	0.152821	0.0764107	−	0.997076i	$-0.475654\pi$
$47$	49.2414	0.152821	0.0764107	+	0.997076i	$0.475654\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−616.373	−1.69234
$52$	0	0
$53$	−474.476	−1.22970	−0.614852	−	0.788643i	$-0.710784\pi$
$53$	−474.476	−1.22970	−0.614852	+	0.788643i	$0.710784\pi$
$54$	0	0
$55$	−509.407	−1.24888
$56$	0	0
$57$	−570.269	−1.32516
$58$	0	0
$59$	−197.062	−0.434836	−0.217418	−	0.976079i	$-0.569763\pi$
$59$	−197.062	−0.434836	−0.217418	+	0.976079i	$0.569763\pi$
$60$	0	0
$61$	401.124	0.841946	0.420973	−	0.907073i	$-0.361689\pi$
$61$	401.124	0.841946	0.420973	+	0.907073i	$0.361689\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−198.000	−0.377829
$66$	0	0
$67$	125.421	0.228695	0.114348	−	0.993441i	$-0.463522\pi$
$67$	125.421	0.228695	0.114348	+	0.993441i	$0.463522\pi$
$68$	0	0
$69$	52.1861	0.0910502
$70$	0	0
$71$	788.635	1.31822	0.659111	−	0.752046i	$-0.270933\pi$
$71$	788.635	1.31822	0.659111	+	0.752046i	$0.270933\pi$
$72$	0	0
$73$	−604.655	−0.969446	−0.484723	−	0.874668i	$-0.661080\pi$
$73$	−604.655	−0.969446	−0.484723	+	0.874668i	$0.661080\pi$
$74$	0	0
$75$	−1473.96	−2.26931
$76$	0	0
$77$	0	0
$78$	0	0
$79$	783.007	1.11513	0.557565	−	0.830134i	$-0.311736\pi$
$79$	783.007	1.11513	0.557565	+	0.830134i	$0.311736\pi$
$80$	0	0
$81$	−899.000	−1.23320
$82$	0	0
$83$	339.283	0.448689	0.224344	−	0.974510i	$-0.427976\pi$
$83$	339.283	0.448689	0.224344	+	0.974510i	$0.427976\pi$
$84$	0	0
$85$	1942.06	2.47819
$86$	0	0
$87$	−318.234	−0.392164
$88$	0	0
$89$	−511.966	−0.609756	−0.304878	−	0.952391i	$-0.598616\pi$
$89$	−511.966	−0.609756	−0.304878	+	0.952391i	$0.598616\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−337.111	−0.375879
$94$	0	0
$95$	1796.80	1.94050
$96$	0	0
$97$	672.290	0.703719	0.351859	−	0.936053i	$-0.385550\pi$
$97$	672.290	0.703719	0.351859	+	0.936053i	$0.385550\pi$
$98$	0	0
$99$	−265.793	−0.269831
$100$	0	0
$101$	−133.773	−0.131791	−0.0658955	−	0.997827i	$-0.520990\pi$
$101$	−133.773	−0.131791	−0.0658955	+	0.997827i	$0.520990\pi$
$102$	0	0
$103$	−1160.74	−1.11040	−0.555202	−	0.831716i	$-0.687359\pi$
$103$	−1160.74	−1.11040	−0.555202	+	0.831716i	$0.687359\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1308.79	−1.18248	−0.591239	−	0.806497i	$-0.701361\pi$
$107$	−1308.79	−1.18248	−0.591239	+	0.806497i	$0.701361\pi$
$108$	0	0
$109$	−1045.84	−0.919022	−0.459511	−	0.888172i	$-0.651975\pi$
$109$	−1045.84	−0.919022	−0.459511	+	0.888172i	$0.651975\pi$
$110$	0	0
$111$	−2606.32	−2.22866
$112$	0	0
$113$	−117.587	−0.0978905	−0.0489453	−	0.998801i	$-0.515586\pi$
$113$	−117.587	−0.0978905	−0.0489453	+	0.998801i	$0.515586\pi$
$114$	0	0
$115$	−164.428	−0.133330
$116$	0	0
$117$	−103.311	−0.0816330
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−624.539	−0.469225
$122$	0	0
$123$	−833.380	−0.610922
$124$	0	0
$125$	2248.45	1.60886
$126$	0	0
$127$	−1788.63	−1.24973	−0.624865	−	0.780733i	$-0.714846\pi$
$127$	−1788.63	−1.24973	−0.624865	+	0.780733i	$0.714846\pi$
$128$	0	0
$129$	1046.24	0.714077
$130$	0	0
$131$	1949.28	1.30007	0.650037	−	0.759903i	$-0.274753\pi$
$131$	1949.28	1.30007	0.650037	+	0.759903i	$0.274753\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	1981.85	1.26348
$136$	0	0
$137$	−571.539	−0.356422	−0.178211	−	0.983992i	$-0.557031\pi$
$137$	−571.539	−0.356422	−0.178211	+	0.983992i	$0.557031\pi$
$138$	0	0
$139$	−1176.58	−0.717958	−0.358979	−	0.933346i	$-0.616875\pi$
$139$	−1176.58	−0.717958	−0.358979	+	0.933346i	$0.616875\pi$
$140$	0	0
$141$	−299.524	−0.178897
$142$	0	0
$143$	274.592	0.160577
$144$	0	0
$145$	1002.69	0.574268
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1773.33	0.975014	0.487507	−	0.873119i	$-0.337906\pi$
$149$	1773.33	0.975014	0.487507	+	0.873119i	$0.337906\pi$
$150$	0	0
$151$	−503.007	−0.271087	−0.135544	−	0.990771i	$-0.543278\pi$
$151$	−503.007	−0.271087	−0.135544	+	0.990771i	$0.543278\pi$
$152$	0	0
$153$	1013.31	0.535433
$154$	0	0
$155$	1062.17	0.550421
$156$	0	0
$157$	−2336.31	−1.18763	−0.593815	−	0.804601i	$-0.702379\pi$
$157$	−2336.31	−1.18763	−0.593815	+	0.804601i	$0.702379\pi$
$158$	0	0
$159$	2886.12	1.43953
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−1736.59	−0.834482	−0.417241	−	0.908796i	$-0.637003\pi$
$163$	−1736.59	−0.834482	−0.417241	+	0.908796i	$0.637003\pi$
$164$	0	0
$165$	3098.60	1.46197
$166$	0	0
$167$	−668.331	−0.309683	−0.154841	−	0.987939i	$-0.549487\pi$
$167$	−668.331	−0.309683	−0.154841	+	0.987939i	$0.549487\pi$
$168$	0	0
$169$	−2090.27	−0.951420
$170$	0	0
$171$	937.517	0.419262
$172$	0	0
$173$	2145.23	0.942769	0.471385	−	0.881928i	$-0.343754\pi$
$173$	2145.23	0.942769	0.471385	+	0.881928i	$0.343754\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1198.68	0.509031
$178$	0	0
$179$	3767.55	1.57318	0.786591	−	0.617474i	$-0.211844\pi$
$179$	3767.55	1.57318	0.786591	+	0.617474i	$0.211844\pi$
$180$	0	0
$181$	1752.51	0.719686	0.359843	−	0.933013i	$-0.382830\pi$
$181$	1752.51	0.719686	0.359843	+	0.933013i	$0.382830\pi$
$182$	0	0
$183$	−2439.94	−0.985606
$184$	0	0
$185$	8211.97	3.26355
$186$	0	0
$187$	−2693.31	−1.05323
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−2011.12	−0.761882	−0.380941	−	0.924599i	$-0.624400\pi$
$191$	−2011.12	−0.761882	−0.380941	+	0.924599i	$0.624400\pi$
$192$	0	0
$193$	−1579.95	−0.589261	−0.294631	−	0.955611i	$-0.595197\pi$
$193$	−1579.95	−0.589261	−0.294631	+	0.955611i	$0.595197\pi$
$194$	0	0
$195$	1204.39	0.442297
$196$	0	0
$197$	−26.4132	−0.00955262	−0.00477631	−	0.999989i	$-0.501520\pi$
$197$	−26.4132	−0.00955262	−0.00477631	+	0.999989i	$0.501520\pi$
$198$	0	0
$199$	57.5462	0.0204992	0.0102496	−	0.999947i	$-0.496737\pi$
$199$	57.5462	0.0204992	0.0102496	+	0.999947i	$0.496737\pi$
$200$	0	0
$201$	−762.904	−0.267717
$202$	0	0
$203$	0	0
$204$	0	0
$205$	2625.81	0.894607
$206$	0	0
$207$	−85.7934	−0.0288070
$208$	0	0
$209$	−2491.86	−0.824715
$210$	0	0
$211$	−1193.90	−0.389534	−0.194767	−	0.980850i	$-0.562395\pi$
$211$	−1193.90	−0.389534	−0.194767	+	0.980850i	$0.562395\pi$
$212$	0	0
$213$	−4797.08	−1.54315
$214$	0	0
$215$	−3296.47	−1.04566
$216$	0	0
$217$	0	0
$218$	0	0
$219$	3677.97	1.13486
$220$	0	0
$221$	−1046.86	−0.318639
$222$	0	0
$223$	2921.66	0.877348	0.438674	−	0.898646i	$-0.355448\pi$
$223$	2921.66	0.877348	0.438674	+	0.898646i	$0.355448\pi$
$224$	0	0
$225$	2423.17	0.717977
$226$	0	0
$227$	3112.34	0.910016	0.455008	−	0.890487i	$-0.349636\pi$
$227$	3112.34	0.910016	0.455008	+	0.890487i	$0.349636\pi$
$228$	0	0
$229$	5335.31	1.53959	0.769797	−	0.638289i	$-0.220357\pi$
$229$	5335.31	1.53959	0.769797	+	0.638289i	$0.220357\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−5238.08	−1.47278	−0.736390	−	0.676557i	$-0.763471\pi$
$233$	−5238.08	−1.47278	−0.736390	+	0.676557i	$0.763471\pi$
$234$	0	0
$235$	943.738	0.261969
$236$	0	0
$237$	−4762.85	−1.30540
$238$	0	0
$239$	4181.05	1.13159	0.565794	−	0.824547i	$-0.308570\pi$
$239$	4181.05	1.13159	0.565794	+	0.824547i	$0.308570\pi$
$240$	0	0
$241$	−5464.88	−1.46068	−0.730340	−	0.683084i	$-0.760638\pi$
$241$	−5464.88	−1.46068	−0.730340	+	0.683084i	$0.760638\pi$
$242$	0	0
$243$	2676.42	0.706552
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−968.554	−0.249504
$248$	0	0
$249$	−2063.78	−0.525248
$250$	0	0
$251$	5718.95	1.43816	0.719078	−	0.694930i	$-0.244564\pi$
$251$	5718.95	1.43816	0.719078	+	0.694930i	$0.244564\pi$
$252$	0	0
$253$	228.033	0.0566653
$254$	0	0
$255$	−11813.1	−2.90104
$256$	0	0
$257$	−1018.28	−0.247154	−0.123577	−	0.992335i	$-0.539437\pi$
$257$	−1018.28	−0.247154	−0.123577	+	0.992335i	$0.539437\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	523.174	0.124075
$262$	0	0
$263$	−8224.31	−1.92826	−0.964130	−	0.265430i	$-0.914486\pi$
$263$	−8224.31	−1.92826	−0.964130	+	0.265430i	$0.914486\pi$
$264$	0	0
$265$	−9093.58	−2.10798
$266$	0	0
$267$	3114.17	0.713797
$268$	0	0
$269$	5867.88	1.33000	0.665002	−	0.746841i	$-0.268431\pi$
$269$	5867.88	1.33000	0.665002	+	0.746841i	$0.268431\pi$
$270$	0	0
$271$	−6216.77	−1.39351	−0.696757	−	0.717307i	$-0.745374\pi$
$271$	−6216.77	−1.39351	−0.696757	+	0.717307i	$0.745374\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−6440.63	−1.41231
$276$	0	0
$277$	1477.52	0.320490	0.160245	−	0.987077i	$-0.448772\pi$
$277$	1477.52	0.320490	0.160245	+	0.987077i	$0.448772\pi$
$278$	0	0
$279$	554.207	0.118923
$280$	0	0
$281$	−3611.93	−0.766797	−0.383398	−	0.923583i	$-0.625246\pi$
$281$	−3611.93	−0.766797	−0.383398	+	0.923583i	$0.625246\pi$
$282$	0	0
$283$	7628.07	1.60227	0.801134	−	0.598485i	$-0.204230\pi$
$283$	7628.07	1.60227	0.801134	+	0.598485i	$0.204230\pi$
$284$	0	0
$285$	−10929.5	−2.27161
$286$	0	0
$287$	0	0
$288$	0	0
$289$	5354.98	1.08996
$290$	0	0
$291$	−4089.38	−0.823793
$292$	0	0
$293$	5393.67	1.07543	0.537716	−	0.843126i	$-0.319287\pi$
$293$	5393.67	1.07543	0.537716	+	0.843126i	$0.319287\pi$
$294$	0	0
$295$	−3776.80	−0.745403
$296$	0	0
$297$	−2748.49	−0.536981
$298$	0	0
$299$	88.6336	0.0171432
$300$	0	0
$301$	0	0
$302$	0	0
$303$	813.708	0.154278
$304$	0	0
$305$	7687.76	1.44328
$306$	0	0
$307$	−8054.16	−1.49731	−0.748656	−	0.662958i	$-0.769301\pi$
$307$	−8054.16	−1.49731	−0.748656	+	0.662958i	$0.769301\pi$
$308$	0	0
$309$	7060.54	1.29987
$310$	0	0
$311$	−2284.61	−0.416554	−0.208277	−	0.978070i	$-0.566785\pi$
$311$	−2284.61	−0.416554	−0.208277	+	0.978070i	$0.566785\pi$
$312$	0	0
$313$	−8336.13	−1.50539	−0.752694	−	0.658371i	$-0.771246\pi$
$313$	−8336.13	−1.50539	−0.752694	+	0.658371i	$0.771246\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−10416.2	−1.84552	−0.922762	−	0.385369i	$-0.874074\pi$
$317$	−10416.2	−1.84552	−0.922762	+	0.385369i	$0.874074\pi$
$318$	0	0
$319$	−1390.56	−0.244064
$320$	0	0
$321$	7961.03	1.38424
$322$	0	0
$323$	9499.96	1.63651
$324$	0	0
$325$	−2503.39	−0.427272
$326$	0	0
$327$	6361.60	1.07583
$328$	0	0
$329$	0	0
$330$	0	0
$331$	6943.88	1.15308	0.576541	−	0.817068i	$-0.304402\pi$
$331$	6943.88	1.15308	0.576541	+	0.817068i	$0.304402\pi$
$332$	0	0
$333$	4284.76	0.705115
$334$	0	0
$335$	2403.75	0.392033
$336$	0	0
$337$	4951.93	0.800442	0.400221	−	0.916419i	$-0.368933\pi$
$337$	4951.93	0.800442	0.400221	+	0.916419i	$0.368933\pi$
$338$	0	0
$339$	715.252	0.114593
$340$	0	0
$341$	−1473.04	−0.233929
$342$	0	0
$343$	0	0
$344$	0	0
$345$	1000.17	0.156080
$346$	0	0
$347$	7275.35	1.12554	0.562769	−	0.826614i	$-0.309736\pi$
$347$	7275.35	1.12554	0.562769	+	0.826614i	$0.309736\pi$
$348$	0	0
$349$	−3433.29	−0.526589	−0.263294	−	0.964716i	$-0.584809\pi$
$349$	−3433.29	−0.526589	−0.263294	+	0.964716i	$0.584809\pi$
$350$	0	0
$351$	−1068.30	−0.162455
$352$	0	0
$353$	−6159.09	−0.928655	−0.464328	−	0.885664i	$-0.653704\pi$
$353$	−6159.09	−0.928655	−0.464328	+	0.885664i	$0.653704\pi$
$354$	0	0
$355$	15114.6	2.25972
$356$	0	0
$357$	0	0
$358$	0	0
$359$	8006.62	1.17708	0.588542	−	0.808466i	$-0.299702\pi$
$359$	8006.62	1.17708	0.588542	+	0.808466i	$0.299702\pi$
$360$	0	0
$361$	1930.38	0.281438
$362$	0	0
$363$	3798.92	0.549288
$364$	0	0
$365$	−11588.5	−1.66184
$366$	0	0
$367$	−1184.49	−0.168474	−0.0842372	−	0.996446i	$-0.526845\pi$
$367$	−1184.49	−0.168474	−0.0842372	+	0.996446i	$0.526845\pi$
$368$	0	0
$369$	1370.07	0.193287
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−944.218	−0.131072	−0.0655359	−	0.997850i	$-0.520876\pi$
$373$	−944.218	−0.131072	−0.0655359	+	0.997850i	$0.520876\pi$
$374$	0	0
$375$	−13676.8	−1.88338
$376$	0	0
$377$	−540.493	−0.0738377
$378$	0	0
$379$	−2906.32	−0.393898	−0.196949	−	0.980414i	$-0.563103\pi$
$379$	−2906.32	−0.393898	−0.196949	+	0.980414i	$0.563103\pi$
$380$	0	0
$381$	10879.8	1.46297
$382$	0	0
$383$	1146.84	0.153005	0.0765023	−	0.997069i	$-0.475625\pi$
$383$	1146.84	0.153005	0.0765023	+	0.997069i	$0.475625\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1720.00	−0.225924
$388$	0	0
$389$	−5589.49	−0.728530	−0.364265	−	0.931295i	$-0.618680\pi$
$389$	−5589.49	−0.728530	−0.364265	+	0.931295i	$0.618680\pi$
$390$	0	0
$391$	−869.353	−0.112443
$392$	0	0
$393$	−11857.0	−1.52190
$394$	0	0
$395$	15006.7	1.91157
$396$	0	0
$397$	−3154.10	−0.398740	−0.199370	−	0.979924i	$-0.563890\pi$
$397$	−3154.10	−0.398740	−0.199370	+	0.979924i	$0.563890\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−6391.76	−0.795983	−0.397992	−	0.917389i	$-0.630293\pi$
$401$	−6391.76	−0.795983	−0.397992	+	0.917389i	$0.630293\pi$
$402$	0	0
$403$	−572.554	−0.0707715
$404$	0	0
$405$	−17229.8	−2.11397
$406$	0	0
$407$	−11388.6	−1.38701
$408$	0	0
$409$	−196.630	−0.0237720	−0.0118860	−	0.999929i	$-0.503784\pi$
$409$	−196.630	−0.0237720	−0.0118860	+	0.999929i	$0.503784\pi$
$410$	0	0
$411$	3476.53	0.417238
$412$	0	0
$413$	0	0
$414$	0	0
$415$	6502.54	0.769150
$416$	0	0
$417$	7156.86	0.840462
$418$	0	0
$419$	15699.9	1.83052	0.915262	−	0.402858i	$-0.131983\pi$
$419$	15699.9	1.83052	0.915262	+	0.402858i	$0.131983\pi$
$420$	0	0
$421$	−8097.20	−0.937372	−0.468686	−	0.883365i	$-0.655272\pi$
$421$	−8097.20	−0.937372	−0.468686	+	0.883365i	$0.655272\pi$
$422$	0	0
$423$	492.414	0.0566005
$424$	0	0
$425$	24554.3	2.80249
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−1670.28	−0.187976
$430$	0	0
$431$	11578.4	1.29399	0.646995	−	0.762494i	$-0.276026\pi$
$431$	11578.4	1.29399	0.646995	+	0.762494i	$0.276026\pi$
$432$	0	0
$433$	7270.25	0.806896	0.403448	−	0.915003i	$-0.367812\pi$
$433$	7270.25	0.806896	0.403448	+	0.915003i	$0.367812\pi$
$434$	0	0
$435$	−6099.12	−0.672254
$436$	0	0
$437$	−804.328	−0.0880462
$438$	0	0
$439$	4127.39	0.448723	0.224362	−	0.974506i	$-0.427970\pi$
$439$	4127.39	0.448723	0.224362	+	0.974506i	$0.427970\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−190.439	−0.0204244	−0.0102122	−	0.999948i	$-0.503251\pi$
$443$	−190.439	−0.0204244	−0.0102122	+	0.999948i	$0.503251\pi$
$444$	0	0
$445$	−9812.09	−1.04525
$446$	0	0
$447$	−10786.8	−1.14138
$448$	0	0
$449$	17175.2	1.80523	0.902613	−	0.430454i	$-0.141646\pi$
$449$	17175.2	1.80523	0.902613	+	0.430454i	$0.141646\pi$
$450$	0	0
$451$	−3641.55	−0.380208
$452$	0	0
$453$	3059.67	0.317342
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−3505.80	−0.358850	−0.179425	−	0.983772i	$-0.557424\pi$
$457$	−3505.80	−0.358850	−0.179425	+	0.983772i	$0.557424\pi$
$458$	0	0
$459$	10478.3	1.06555
$460$	0	0
$461$	−5983.80	−0.604541	−0.302270	−	0.953222i	$-0.597745\pi$
$461$	−5983.80	−0.604541	−0.302270	+	0.953222i	$0.597745\pi$
$462$	0	0
$463$	14057.4	1.41102	0.705510	−	0.708700i	$-0.250718\pi$
$463$	14057.4	1.41102	0.705510	+	0.708700i	$0.250718\pi$
$464$	0	0
$465$	−6460.90	−0.644338
$466$	0	0
$467$	−18773.8	−1.86027	−0.930137	−	0.367211i	$-0.880313\pi$
$467$	−18773.8	−1.86027	−0.930137	+	0.367211i	$0.880313\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	14211.2	1.39027
$472$	0	0
$473$	4571.65	0.444407
$474$	0	0
$475$	22717.7	2.19444
$476$	0	0
$477$	−4744.76	−0.455446
$478$	0	0
$479$	−263.773	−0.0251610	−0.0125805	−	0.999921i	$-0.504005\pi$
$479$	−263.773	−0.0251610	−0.0125805	+	0.999921i	$0.504005\pi$
$480$	0	0
$481$	−4426.61	−0.419617
$482$	0	0
$483$	0	0
$484$	0	0
$485$	12884.8	1.20633
$486$	0	0
$487$	12654.9	1.17751	0.588757	−	0.808310i	$-0.299617\pi$
$487$	12654.9	1.17751	0.588757	+	0.808310i	$0.299617\pi$
$488$	0	0
$489$	10563.3	0.976868
$490$	0	0
$491$	19041.6	1.75017	0.875087	−	0.483965i	$-0.160804\pi$
$491$	19041.6	1.75017	0.875087	+	0.483965i	$0.160804\pi$
$492$	0	0
$493$	5301.37	0.484304
$494$	0	0
$495$	−5094.07	−0.462548
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−17999.4	−1.61476	−0.807378	−	0.590035i	$-0.799114\pi$
$499$	−17999.4	−1.61476	−0.807378	+	0.590035i	$0.799114\pi$
$500$	0	0
$501$	4065.30	0.362523
$502$	0	0
$503$	−15245.1	−1.35139	−0.675693	−	0.737183i	$-0.736155\pi$
$503$	−15245.1	−1.35139	−0.675693	+	0.737183i	$0.736155\pi$
$504$	0	0
$505$	−2563.83	−0.225918
$506$	0	0
$507$	12714.6	1.11376
$508$	0	0
$509$	−4807.85	−0.418672	−0.209336	−	0.977844i	$-0.567130\pi$
$509$	−4807.85	−0.418672	−0.209336	+	0.977844i	$0.567130\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	9694.58	0.834359
$514$	0	0
$515$	−22246.3	−1.90347
$516$	0	0
$517$	−1308.80	−0.111337
$518$	0	0
$519$	−13048.9	−1.10363
$520$	0	0
$521$	300.226	0.0252460	0.0126230	−	0.999920i	$-0.495982\pi$
$521$	300.226	0.0252460	0.0126230	+	0.999920i	$0.495982\pi$
$522$	0	0
$523$	−795.488	−0.0665091	−0.0332546	−	0.999447i	$-0.510587\pi$
$523$	−795.488	−0.0665091	−0.0332546	+	0.999447i	$0.510587\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5615.83	0.464193
$528$	0	0
$529$	−12093.4	−0.993950
$530$	0	0
$531$	−1970.62	−0.161050
$532$	0	0
$533$	−1415.42	−0.115026
$534$	0	0
$535$	−25083.6	−2.02702
$536$	0	0
$537$	−22917.1	−1.84161
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−18669.7	−1.48368	−0.741841	−	0.670576i	$-0.766047\pi$
$541$	−18669.7	−1.48368	−0.741841	+	0.670576i	$0.766047\pi$
$542$	0	0
$543$	−10660.1	−0.842485
$544$	0	0
$545$	−20044.1	−1.57540
$546$	0	0
$547$	−3264.72	−0.255191	−0.127596	−	0.991826i	$-0.540726\pi$
$547$	−3264.72	−0.255191	−0.127596	+	0.991826i	$0.540726\pi$
$548$	0	0
$549$	4011.24	0.311832
$550$	0	0
$551$	4904.84	0.379226
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−49951.5	−3.82040
$556$	0	0
$557$	13404.7	1.01970	0.509852	−	0.860262i	$-0.329700\pi$
$557$	13404.7	1.01970	0.509852	+	0.860262i	$0.329700\pi$
$558$	0	0
$559$	1776.94	0.134448
$560$	0	0
$561$	16382.8	1.23294
$562$	0	0
$563$	−9299.00	−0.696103	−0.348051	−	0.937475i	$-0.613157\pi$
$563$	−9299.00	−0.696103	−0.348051	+	0.937475i	$0.613157\pi$
$564$	0	0
$565$	−2253.61	−0.167806
$566$	0	0
$567$	0	0
$568$	0	0
$569$	20197.8	1.48811	0.744055	−	0.668119i	$-0.232900\pi$
$569$	20197.8	1.48811	0.744055	+	0.668119i	$0.232900\pi$
$570$	0	0
$571$	−9413.56	−0.689922	−0.344961	−	0.938617i	$-0.612108\pi$
$571$	−9413.56	−0.689922	−0.344961	+	0.938617i	$0.612108\pi$
$572$	0	0
$573$	12233.2	0.891880
$574$	0	0
$575$	−2078.92	−0.150777
$576$	0	0
$577$	−7737.70	−0.558275	−0.279138	−	0.960251i	$-0.590049\pi$
$577$	−7737.70	−0.558275	−0.279138	+	0.960251i	$0.590049\pi$
$578$	0	0
$579$	9610.47	0.689806
$580$	0	0
$581$	0	0
$582$	0	0
$583$	12611.3	0.895892
$584$	0	0
$585$	−1980.00	−0.139937
$586$	0	0
$587$	−16755.0	−1.17811	−0.589057	−	0.808091i	$-0.700501\pi$
$587$	−16755.0	−1.17811	−0.589057	+	0.808091i	$0.700501\pi$
$588$	0	0
$589$	5195.78	0.363478
$590$	0	0
$591$	160.666	0.0111826
$592$	0	0
$593$	11772.9	0.815270	0.407635	−	0.913145i	$-0.366354\pi$
$593$	11772.9	0.815270	0.407635	+	0.913145i	$0.366354\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−350.040	−0.0239970
$598$	0	0
$599$	−25259.3	−1.72299	−0.861493	−	0.507769i	$-0.830470\pi$
$599$	−25259.3	−1.72299	−0.861493	+	0.507769i	$0.830470\pi$
$600$	0	0
$601$	−187.801	−0.0127464	−0.00637319	−	0.999980i	$-0.502029\pi$
$601$	−187.801	−0.0127464	−0.00637319	+	0.999980i	$0.502029\pi$
$602$	0	0
$603$	1254.21	0.0847019
$604$	0	0
$605$	−11969.6	−0.804354
$606$	0	0
$607$	−17655.3	−1.18057	−0.590284	−	0.807196i	$-0.700984\pi$
$607$	−17655.3	−1.18057	−0.590284	+	0.807196i	$0.700984\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−508.716	−0.0336832
$612$	0	0
$613$	−1004.93	−0.0662130	−0.0331065	−	0.999452i	$-0.510540\pi$
$613$	−1004.93	−0.0662130	−0.0331065	+	0.999452i	$0.510540\pi$
$614$	0	0
$615$	−15972.2	−1.04725
$616$	0	0
$617$	8416.05	0.549137	0.274568	−	0.961568i	$-0.411465\pi$
$617$	8416.05	0.549137	0.274568	+	0.961568i	$0.411465\pi$
$618$	0	0
$619$	−6285.24	−0.408118	−0.204059	−	0.978959i	$-0.565413\pi$
$619$	−6285.24	−0.408118	−0.204059	+	0.978959i	$0.565413\pi$
$620$	0	0
$621$	−887.163	−0.0573279
$622$	0	0
$623$	0	0
$624$	0	0
$625$	12803.0	0.819394
$626$	0	0
$627$	15157.4	0.965435
$628$	0	0
$629$	43417.9	2.75228
$630$	0	0
$631$	15928.9	1.00495	0.502473	−	0.864593i	$-0.332424\pi$
$631$	15928.9	1.00495	0.502473	+	0.864593i	$0.332424\pi$
$632$	0	0
$633$	7262.24	0.456000
$634$	0	0
$635$	−34280.1	−2.14231
$636$	0	0
$637$	0	0
$638$	0	0
$639$	7886.35	0.488230
$640$	0	0
$641$	−12683.0	−0.781513	−0.390756	−	0.920494i	$-0.627787\pi$
$641$	−12683.0	−0.781513	−0.390756	+	0.920494i	$0.627787\pi$
$642$	0	0
$643$	1461.21	0.0896179	0.0448090	−	0.998996i	$-0.485732\pi$
$643$	1461.21	0.0896179	0.0448090	+	0.998996i	$0.485732\pi$
$644$	0	0
$645$	20051.6	1.22408
$646$	0	0
$647$	−24001.4	−1.45841	−0.729206	−	0.684294i	$-0.760110\pi$
$647$	−24001.4	−1.45841	−0.729206	+	0.684294i	$0.760110\pi$
$648$	0	0
$649$	5237.78	0.316797
$650$	0	0
$651$	0	0
$652$	0	0
$653$	9097.14	0.545174	0.272587	−	0.962131i	$-0.412121\pi$
$653$	9097.14	0.545174	0.272587	+	0.962131i	$0.412121\pi$
$654$	0	0
$655$	37359.0	2.22861
$656$	0	0
$657$	−6046.55	−0.359054
$658$	0	0
$659$	−15018.0	−0.887738	−0.443869	−	0.896092i	$-0.646394\pi$
$659$	−15018.0	−0.887738	−0.443869	+	0.896092i	$0.646394\pi$
$660$	0	0
$661$	24497.2	1.44150	0.720748	−	0.693197i	$-0.243798\pi$
$661$	24497.2	1.44150	0.720748	+	0.693197i	$0.243798\pi$
$662$	0	0
$663$	6367.78	0.373008
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−448.848	−0.0260562
$668$	0	0
$669$	−17771.7	−1.02705
$670$	0	0
$671$	−10661.6	−0.613394
$672$	0	0
$673$	1378.86	0.0789762	0.0394881	−	0.999220i	$-0.487427\pi$
$673$	1378.86	0.0789762	0.0394881	+	0.999220i	$0.487427\pi$
$674$	0	0
$675$	25057.3	1.42882
$676$	0	0
$677$	12535.0	0.711607	0.355804	−	0.934561i	$-0.384207\pi$
$677$	12535.0	0.711607	0.355804	+	0.934561i	$0.384207\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−18931.7	−1.06529
$682$	0	0
$683$	27054.1	1.51566	0.757830	−	0.652452i	$-0.226259\pi$
$683$	27054.1	1.51566	0.757830	+	0.652452i	$0.226259\pi$
$684$	0	0
$685$	−10953.8	−0.610985
$686$	0	0
$687$	−32453.4	−1.80229
$688$	0	0
$689$	4901.84	0.271038
$690$	0	0
$691$	25843.5	1.42277	0.711385	−	0.702803i	$-0.248068\pi$
$691$	25843.5	1.42277	0.711385	+	0.702803i	$0.248068\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−22549.8	−1.23074
$696$	0	0
$697$	13883.0	0.754459
$698$	0	0
$699$	31862.0	1.72408
$700$	0	0
$701$	−29351.6	−1.58145	−0.790723	−	0.612174i	$-0.790295\pi$
$701$	−29351.6	−1.58145	−0.790723	+	0.612174i	$0.790295\pi$
$702$	0	0
$703$	40170.4	2.15513
$704$	0	0
$705$	−5740.53	−0.306668
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−17013.4	−0.901202	−0.450601	−	0.892725i	$-0.648790\pi$
$709$	−17013.4	−0.901202	−0.450601	+	0.892725i	$0.648790\pi$
$710$	0	0
$711$	7830.07	0.413011
$712$	0	0
$713$	−475.473	−0.0249742
$714$	0	0
$715$	5262.71	0.275265
$716$	0	0
$717$	−25432.3	−1.32467
$718$	0	0
$719$	−14546.5	−0.754510	−0.377255	−	0.926109i	$-0.623132\pi$
$719$	−14546.5	−0.754510	−0.377255	+	0.926109i	$0.623132\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	33241.5	1.70991
$724$	0	0
$725$	12677.4	0.649416
$726$	0	0
$727$	−25023.9	−1.27660	−0.638298	−	0.769790i	$-0.720361\pi$
$727$	−25023.9	−1.27660	−0.638298	+	0.769790i	$0.720361\pi$
$728$	0	0
$729$	7993.00	0.406086
$730$	0	0
$731$	−17428.9	−0.881850
$732$	0	0
$733$	12899.2	0.649988	0.324994	−	0.945716i	$-0.394638\pi$
$733$	12899.2	0.649988	0.324994	+	0.945716i	$0.394638\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−3333.60	−0.166614
$738$	0	0
$739$	−15253.3	−0.759270	−0.379635	−	0.925136i	$-0.623950\pi$
$739$	−15253.3	−0.759270	−0.379635	+	0.925136i	$0.623950\pi$
$740$	0	0
$741$	5891.48	0.292077
$742$	0	0
$743$	−7984.56	−0.394247	−0.197123	−	0.980379i	$-0.563160\pi$
$743$	−7984.56	−0.394247	−0.197123	+	0.980379i	$0.563160\pi$
$744$	0	0
$745$	33986.8	1.67138
$746$	0	0
$747$	3392.83	0.166181
$748$	0	0
$749$	0	0
$750$	0	0
$751$	32781.6	1.59283	0.796416	−	0.604750i	$-0.206727\pi$
$751$	32781.6	1.59283	0.796416	+	0.604750i	$0.206727\pi$
$752$	0	0
$753$	−34787.0	−1.68355
$754$	0	0
$755$	−9640.40	−0.464702
$756$	0	0
$757$	−16431.5	−0.788920	−0.394460	−	0.918913i	$-0.629068\pi$
$757$	−16431.5	−0.788920	−0.394460	+	0.918913i	$0.629068\pi$
$758$	0	0
$759$	−1387.07	−0.0663340
$760$	0	0
$761$	33858.3	1.61283	0.806415	−	0.591350i	$-0.201405\pi$
$761$	33858.3	1.61283	0.806415	+	0.591350i	$0.201405\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	19420.6	0.917849
$766$	0	0
$767$	2035.86	0.0958418
$768$	0	0
$769$	36173.6	1.69630	0.848149	−	0.529758i	$-0.177717\pi$
$769$	36173.6	1.69630	0.848149	+	0.529758i	$0.177717\pi$
$770$	0	0
$771$	6193.96	0.289326
$772$	0	0
$773$	−20812.8	−0.968416	−0.484208	−	0.874953i	$-0.660892\pi$
$773$	−20812.8	−0.968416	−0.484208	+	0.874953i	$0.660892\pi$
$774$	0	0
$775$	13429.4	0.622449
$776$	0	0
$777$	0	0
$778$	0	0
$779$	12844.6	0.590766
$780$	0	0
$781$	−20961.4	−0.960381
$782$	0	0
$783$	5409.98	0.246918
$784$	0	0
$785$	−44776.6	−2.03586
$786$	0	0
$787$	29722.2	1.34623	0.673114	−	0.739538i	$-0.264956\pi$
$787$	29722.2	1.34623	0.673114	+	0.739538i	$0.264956\pi$
$788$	0	0
$789$	50026.5	2.25728
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−4144.04	−0.185573
$794$	0	0
$795$	55314.1	2.46766
$796$	0	0
$797$	−6475.82	−0.287811	−0.143905	−	0.989591i	$-0.545966\pi$
$797$	−6475.82	−0.287811	−0.143905	+	0.989591i	$0.545966\pi$
$798$	0	0
$799$	4989.69	0.220929
$800$	0	0
$801$	−5119.66	−0.225835
$802$	0	0
$803$	16071.3	0.706283
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−35692.9	−1.55694
$808$	0	0
$809$	−39418.9	−1.71309	−0.856547	−	0.516069i	$-0.827395\pi$
$809$	−39418.9	−1.71309	−0.856547	+	0.516069i	$0.827395\pi$
$810$	0	0
$811$	−38728.6	−1.67688	−0.838438	−	0.544997i	$-0.816531\pi$
$811$	−38728.6	−1.67688	−0.838438	+	0.544997i	$0.816531\pi$
$812$	0	0
$813$	37815.2	1.63129
$814$	0	0
$815$	−33282.7	−1.43048
$816$	0	0
$817$	−16125.3	−0.690517
$818$	0	0
$819$	0	0
$820$	0	0
$821$	32010.0	1.36073	0.680363	−	0.732876i	$-0.261822\pi$
$821$	32010.0	1.36073	0.680363	+	0.732876i	$0.261822\pi$
$822$	0	0
$823$	10137.8	0.429384	0.214692	−	0.976682i	$-0.431125\pi$
$823$	10137.8	0.429384	0.214692	+	0.976682i	$0.431125\pi$
$824$	0	0
$825$	39176.9	1.65329
$826$	0	0
$827$	9064.96	0.381160	0.190580	−	0.981672i	$-0.438963\pi$
$827$	9064.96	0.381160	0.190580	+	0.981672i	$0.438963\pi$
$828$	0	0
$829$	−19310.3	−0.809017	−0.404508	−	0.914534i	$-0.632557\pi$
$829$	−19310.3	−0.809017	−0.404508	+	0.914534i	$0.632557\pi$
$830$	0	0
$831$	−8987.43	−0.375175
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−12808.9	−0.530863
$836$	0	0
$837$	5730.88	0.236665
$838$	0	0
$839$	−11458.4	−0.471498	−0.235749	−	0.971814i	$-0.575754\pi$
$839$	−11458.4	−0.471498	−0.235749	+	0.971814i	$0.575754\pi$
$840$	0	0
$841$	−21651.9	−0.887773
$842$	0	0
$843$	21970.5	0.897634
$844$	0	0
$845$	−40061.1	−1.63094
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−46399.8	−1.87566
$850$	0	0
$851$	−3676.04	−0.148076
$852$	0	0
$853$	−6203.94	−0.249026	−0.124513	−	0.992218i	$-0.539737\pi$
$853$	−6203.94	−0.249026	−0.124513	+	0.992218i	$0.539737\pi$
$854$	0	0
$855$	17968.0	0.718705
$856$	0	0
$857$	15577.8	0.620920	0.310460	−	0.950587i	$-0.399517\pi$
$857$	15577.8	0.620920	0.310460	+	0.950587i	$0.399517\pi$
$858$	0	0
$859$	11893.7	0.472420	0.236210	−	0.971702i	$-0.424095\pi$
$859$	11893.7	0.472420	0.236210	+	0.971702i	$0.424095\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−11673.6	−0.460455	−0.230227	−	0.973137i	$-0.573947\pi$
$863$	−11673.6	−0.460455	−0.230227	+	0.973137i	$0.573947\pi$
$864$	0	0
$865$	41114.5	1.61611
$866$	0	0
$867$	−32573.1	−1.27594
$868$	0	0
$869$	−20811.8	−0.812420
$870$	0	0
$871$	−1295.73	−0.0504065
$872$	0	0
$873$	6722.90	0.260637
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−3063.66	−0.117962	−0.0589808	−	0.998259i	$-0.518785\pi$
$877$	−3063.66	−0.117962	−0.0589808	+	0.998259i	$0.518785\pi$
$878$	0	0
$879$	−32808.4	−1.25893
$880$	0	0
$881$	−19236.9	−0.735649	−0.367825	−	0.929895i	$-0.619897\pi$
$881$	−19236.9	−0.735649	−0.367825	+	0.929895i	$0.619897\pi$
$882$	0	0
$883$	−35556.9	−1.35514	−0.677569	−	0.735459i	$-0.736966\pi$
$883$	−35556.9	−1.35514	−0.677569	+	0.735459i	$0.736966\pi$
$884$	0	0
$885$	22973.4	0.872590
$886$	0	0
$887$	−23218.8	−0.878928	−0.439464	−	0.898260i	$-0.644832\pi$
$887$	−23218.8	−0.878928	−0.439464	+	0.898260i	$0.644832\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	23894.8	0.898436
$892$	0	0
$893$	4616.47	0.172995
$894$	0	0
$895$	72207.0	2.69678
$896$	0	0
$897$	−539.137	−0.0200683
$898$	0	0
$899$	2899.46	0.107567
$900$	0	0
$901$	−48079.2	−1.77775
$902$	0	0
$903$	0	0
$904$	0	0
$905$	33587.8	1.23370
$906$	0	0
$907$	−20516.7	−0.751098	−0.375549	−	0.926803i	$-0.622546\pi$
$907$	−20516.7	−0.751098	−0.375549	+	0.926803i	$0.622546\pi$
$908$	0	0
$909$	−1337.73	−0.0488115
$910$	0	0
$911$	18247.6	0.663634	0.331817	−	0.943344i	$-0.392338\pi$
$911$	18247.6	0.663634	0.331817	+	0.943344i	$0.392338\pi$
$912$	0	0
$913$	−9017.92	−0.326889
$914$	0	0
$915$	−46762.8	−1.68954
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−27362.2	−0.982149	−0.491074	−	0.871118i	$-0.663396\pi$
$919$	−27362.2	−0.982149	−0.491074	+	0.871118i	$0.663396\pi$
$920$	0	0
$921$	48991.5	1.75280
$922$	0	0
$923$	−8147.42	−0.290548
$924$	0	0
$925$	103827.	3.69061
$926$	0	0
$927$	−11607.4	−0.411261
$928$	0	0
$929$	−43542.6	−1.53777	−0.768884	−	0.639389i	$-0.779187\pi$
$929$	−43542.6	−1.53777	−0.768884	+	0.639389i	$0.779187\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	13896.7	0.487629
$934$	0	0
$935$	−51618.7	−1.80547
$936$	0	0
$937$	−32990.1	−1.15020	−0.575101	−	0.818083i	$-0.695037\pi$
$937$	−32990.1	−1.15020	−0.575101	+	0.818083i	$0.695037\pi$
$938$	0	0
$939$	50706.7	1.76225
$940$	0	0
$941$	17311.6	0.599726	0.299863	−	0.953982i	$-0.403059\pi$
$941$	17311.6	0.599726	0.299863	+	0.953982i	$0.403059\pi$
$942$	0	0
$943$	−1175.43	−0.0405909
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1335.71	−0.0458339	−0.0229169	−	0.999737i	$-0.507295\pi$
$947$	−1335.71	−0.0458339	−0.0229169	+	0.999737i	$0.507295\pi$
$948$	0	0
$949$	6246.72	0.213675
$950$	0	0
$951$	63359.2	2.16042
$952$	0	0
$953$	18877.6	0.641663	0.320831	−	0.947136i	$-0.396038\pi$
$953$	18877.6	0.641663	0.320831	+	0.947136i	$0.396038\pi$
$954$	0	0
$955$	−38544.1	−1.30603
$956$	0	0
$957$	8458.45	0.285708
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−26719.6	−0.896900
$962$	0	0
$963$	−13087.9	−0.437955
$964$	0	0
$965$	−30280.6	−1.01012
$966$	0	0
$967$	−39814.6	−1.32405	−0.662023	−	0.749483i	$-0.730302\pi$
$967$	−39814.6	−1.32405	−0.662023	+	0.749483i	$0.730302\pi$
$968$	0	0
$969$	−57786.0	−1.91574
$970$	0	0
$971$	22844.8	0.755021	0.377511	−	0.926005i	$-0.376780\pi$
$971$	22844.8	0.755021	0.377511	+	0.926005i	$0.376780\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	15227.5	0.500176
$976$	0	0
$977$	24396.6	0.798891	0.399446	−	0.916757i	$-0.369203\pi$
$977$	24396.6	0.798891	0.399446	+	0.916757i	$0.369203\pi$
$978$	0	0
$979$	13607.7	0.444233
$980$	0	0
$981$	−10458.4	−0.340379
$982$	0	0
$983$	−32975.5	−1.06994	−0.534972	−	0.844870i	$-0.679678\pi$
$983$	−32975.5	−1.06994	−0.534972	+	0.844870i	$0.679678\pi$
$984$	0	0
$985$	−506.224	−0.0163753
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1475.65	0.0474447
$990$	0	0
$991$	−11877.1	−0.380717	−0.190358	−	0.981715i	$-0.560965\pi$
$991$	−11877.1	−0.380717	−0.190358	+	0.981715i	$0.560965\pi$
$992$	0	0
$993$	−42238.0	−1.34983
$994$	0	0
$995$	1102.90	0.0351401
$996$	0	0
$997$	−16386.3	−0.520522	−0.260261	−	0.965538i	$-0.583809\pi$
$997$	−16386.3	−0.520522	−0.260261	+	0.965538i	$0.583809\pi$
$998$	0	0
$999$	44307.4	1.40323

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	196.4.a.g.1.1		2
3.2	odd	2		1764.4.a.n.1.1		2
4.3	odd	2		784.4.a.ba.1.2		2
7.2	even	3		196.4.e.g.165.2		4
7.3	odd	6		28.4.e.a.9.1	✓	4
7.4	even	3		196.4.e.g.177.2		4
7.5	odd	6		28.4.e.a.25.1	yes	4
7.6	odd	2		196.4.a.e.1.2		2
21.2	odd	6		1764.4.k.ba.361.2		4
21.5	even	6		252.4.k.c.109.1		4
21.11	odd	6		1764.4.k.ba.1549.2		4
21.17	even	6		252.4.k.c.37.1		4
21.20	even	2		1764.4.a.z.1.2		2
28.3	even	6		112.4.i.d.65.2		4
28.19	even	6		112.4.i.d.81.2		4
28.27	even	2		784.4.a.u.1.1		2
35.3	even	12		700.4.r.d.149.4		8
35.12	even	12		700.4.r.d.249.4		8
35.17	even	12		700.4.r.d.149.1		8
35.19	odd	6		700.4.i.g.501.2		4
35.24	odd	6		700.4.i.g.401.2		4
35.33	even	12		700.4.r.d.249.1		8
56.3	even	6		448.4.i.g.65.1		4
56.5	odd	6		448.4.i.h.193.2		4
56.19	even	6		448.4.i.g.193.1		4
56.45	odd	6		448.4.i.h.65.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.4.e.a.9.1	✓	4	7.3	odd	6
28.4.e.a.25.1	yes	4	7.5	odd	6
112.4.i.d.65.2		4	28.3	even	6
112.4.i.d.81.2		4	28.19	even	6
196.4.a.e.1.2		2	7.6	odd	2
196.4.a.g.1.1		2	1.1	even	1	trivial
196.4.e.g.165.2		4	7.2	even	3
196.4.e.g.177.2		4	7.4	even	3
252.4.k.c.37.1		4	21.17	even	6
252.4.k.c.109.1		4	21.5	even	6
448.4.i.g.65.1		4	56.3	even	6
448.4.i.g.193.1		4	56.19	even	6
448.4.i.h.65.2		4	56.45	odd	6
448.4.i.h.193.2		4	56.5	odd	6
700.4.i.g.401.2		4	35.24	odd	6
700.4.i.g.501.2		4	35.19	odd	6
700.4.r.d.149.1		8	35.17	even	12
700.4.r.d.149.4		8	35.3	even	12
700.4.r.d.249.1		8	35.33	even	12
700.4.r.d.249.4		8	35.12	even	12
784.4.a.u.1.1		2	28.27	even	2
784.4.a.ba.1.2		2	4.3	odd	2
1764.4.a.n.1.1		2	3.2	odd	2
1764.4.a.z.1.2		2	21.20	even	2
1764.4.k.ba.361.2		4	21.2	odd	6
1764.4.k.ba.1549.2		4	21.11	odd	6

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:	\( N \)	\(=\)	\( 196 = 2^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	196.a (trivial)

\(f(q)\)	\(=\)	\(q-6.08276 q^{3} +19.1655 q^{5} +10.0000 q^{9} +O(q^{10})\)	\(q-6.08276 q^{3} +19.1655 q^{5} +10.0000 q^{9} -26.5793 q^{11} -10.3311 q^{13} -116.579 q^{15} +101.331 q^{17} +93.7517 q^{19} -8.57934 q^{23} +242.317 q^{25} +103.407 q^{27} +52.3174 q^{29} +55.4207 q^{31} +161.676 q^{33} +428.476 q^{37} +62.8413 q^{39} +137.007 q^{41} -172.000 q^{43} +191.655 q^{45} +49.2414 q^{47} -616.373 q^{51} -474.476 q^{53} -509.407 q^{55} -570.269 q^{57} -197.062 q^{59} +401.124 q^{61} -198.000 q^{65} +125.421 q^{67} +52.1861 q^{69} +788.635 q^{71} -604.655 q^{73} -1473.96 q^{75} +783.007 q^{79} -899.000 q^{81} +339.283 q^{83} +1942.06 q^{85} -318.234 q^{87} -511.966 q^{89} -337.111 q^{93} +1796.80 q^{95} +672.290 q^{97} -265.793 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 14 q^{5} + 20 q^{9}+O(q^{10}) \) 2 * q + 14 * q^5 + 20 * q^9	\( 2 q + 14 q^{5} + 20 q^{9} + 32 q^{11} + 28 q^{13} - 148 q^{15} + 154 q^{17} + 224 q^{19} + 68 q^{23} + 144 q^{25} - 236 q^{29} + 196 q^{31} + 518 q^{33} + 346 q^{37} + 296 q^{39} + 420 q^{41} - 344 q^{43} + 140 q^{45} - 84 q^{47} - 296 q^{51} - 438 q^{53} - 812 q^{55} + 222 q^{57} + 56 q^{59} - 98 q^{61} - 396 q^{65} + 336 q^{67} + 518 q^{69} + 896 q^{71} - 966 q^{73} - 2072 q^{75} - 52 q^{79} - 1798 q^{81} - 392 q^{83} + 1670 q^{85} - 2072 q^{87} - 294 q^{89} + 518 q^{93} + 1124 q^{95} + 420 q^{97} + 320 q^{99}+O(q^{100}) \) 2 * q + 14 * q^5 + 20 * q^9 + 32 * q^11 + 28 * q^13 - 148 * q^15 + 154 * q^17 + 224 * q^19 + 68 * q^23 + 144 * q^25 - 236 * q^29 + 196 * q^31 + 518 * q^33 + 346 * q^37 + 296 * q^39 + 420 * q^41 - 344 * q^43 + 140 * q^45 - 84 * q^47 - 296 * q^51 - 438 * q^53 - 812 * q^55 + 222 * q^57 + 56 * q^59 - 98 * q^61 - 396 * q^65 + 336 * q^67 + 518 * q^69 + 896 * q^71 - 966 * q^73 - 2072 * q^75 - 52 * q^79 - 1798 * q^81 - 392 * q^83 + 1670 * q^85 - 2072 * q^87 - 294 * q^89 + 518 * q^93 + 1124 * q^95 + 420 * q^97 + 320 * q^99

Introduction

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