LMFDB - Embedded newform 4900.2.e.s.2549.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4900,2,Mod(2549,4900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4900, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4900.2549");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4900.e (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$39.1266969904$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.4227136.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 9x^{4} + 22x^{2} + 9 $ x^6 + 9x^4 + 22x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 700)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2549.1
Root			$-0.713538i$ of defining polynomial
Character	$\chi$	$=$	4900.2549
Dual form			4900.2.e.s.2549.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.20440i q^{3} -7.26819 q^{9} +O(q^{10})$	$q-3.20440i q^{3} -7.26819 q^{9} +4.20440 q^{11} +0.204402i q^{13} +5.06379i q^{17} +1.06379 q^{19} -2.14061i q^{23} +13.6770i q^{27} +7.47259 q^{29} +8.47259 q^{31} -13.4726i q^{33} -10.6132i q^{37} +0.654985 q^{39} -10.5494 q^{41} -8.26819i q^{43} -3.26819i q^{47} +16.2264 q^{51} -5.67699i q^{53} -3.40880i q^{57} +1.20440 q^{59} -1.65498 q^{61} -12.4088i q^{67} -6.85939 q^{69} -0.591197 q^{71} -4.00000i q^{73} -6.54942 q^{79} +22.0220 q^{81} +3.88139i q^{83} -23.9452i q^{87} -9.26819 q^{89} -27.1496i q^{93} -1.33198i q^{97} -30.5584 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 16 q^{9}+O(q^{10}) $ 6 * q - 16 * q^9	$ 6 q - 16 q^{9} + 8 q^{11} - 4 q^{19} + 6 q^{31} + 28 q^{39} - 22 q^{41} - 6 q^{51} - 10 q^{59} - 34 q^{61} - 48 q^{69} - 38 q^{71} + 2 q^{79} + 46 q^{81} - 28 q^{89} - 42 q^{99}+O(q^{100}) $ 6 * q - 16 * q^9 + 8 * q^11 - 4 * q^19 + 6 * q^31 + 28 * q^39 - 22 * q^41 - 6 * q^51 - 10 * q^59 - 34 * q^61 - 48 * q^69 - 38 * q^71 + 2 * q^79 + 46 * q^81 - 28 * q^89 - 42 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4900\mathbb{Z}\right)^\times$.

$n$	$101$	$1177$	$2451$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	− 3.20440i	− 1.85006i	−0.379892	−	0.925031i	$-0.624039\pi$
$3$	− 3.20440i	− 1.85006i	0.379892	−	0.925031i	$-0.375961\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−7.26819	−2.42273
$10$	0	0
$11$	4.20440	1.26767	0.633837	−	0.773466i	$-0.281479\pi$
$11$	4.20440	1.26767	0.633837	+	0.773466i	$0.281479\pi$
$12$	0	0
$13$	0.204402i	0.0566908i	0.999598	+	0.0283454i	$0.00902383\pi$
$13$	0.204402i	0.0566908i	−0.999598	+	0.0283454i	$0.990976\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.06379i	1.22815i	0.789248	+	0.614074i	$0.210471\pi$
$17$	5.06379i	1.22815i	−0.789248	+	0.614074i	$0.789529\pi$
$18$	0	0
$19$	1.06379	0.244050	0.122025	−	0.992527i	$-0.461061\pi$
$19$	1.06379	0.244050	0.122025	+	0.992527i	$0.461061\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 2.14061i	− 0.446349i	−0.974779	−	0.223174i	$-0.928358\pi$
$23$	− 2.14061i	− 0.446349i	0.974779	−	0.223174i	$-0.0716419\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	13.6770i	2.63214i
$28$	0	0
$29$	7.47259	1.38763	0.693813	−	0.720156i	$-0.255930\pi$
$29$	7.47259	1.38763	0.693813	+	0.720156i	$0.255930\pi$
$30$	0	0
$31$	8.47259	1.52172	0.760861	−	0.648915i	$-0.224777\pi$
$31$	8.47259	1.52172	0.760861	+	0.648915i	$0.224777\pi$
$32$	0	0
$33$	− 13.4726i	− 2.34528i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 10.6132i	− 1.74480i	−0.488793	−	0.872400i	$-0.662563\pi$
$37$	− 10.6132i	− 1.74480i	0.488793	−	0.872400i	$-0.337437\pi$
$38$	0	0
$39$	0.654985	0.104881
$40$	0	0
$41$	−10.5494	−1.64754	−0.823771	−	0.566923i	$-0.808134\pi$
$41$	−10.5494	−1.64754	−0.823771	+	0.566923i	$0.808134\pi$
$42$	0	0
$43$	− 8.26819i	− 1.26089i	−0.776235	−	0.630444i	$-0.782873\pi$
$43$	− 8.26819i	− 1.26089i	0.776235	−	0.630444i	$-0.217127\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 3.26819i	− 0.476714i	−0.971178	−	0.238357i	$-0.923391\pi$
$47$	− 3.26819i	− 0.476714i	0.971178	−	0.238357i	$-0.0766089\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	16.2264	2.27215
$52$	0	0
$53$	− 5.67699i	− 0.779795i	−0.920858	−	0.389897i	$-0.872510\pi$
$53$	− 5.67699i	− 0.779795i	0.920858	−	0.389897i	$-0.127490\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 3.40880i	− 0.451507i
$58$	0	0
$59$	1.20440	0.156800	0.0783999	−	0.996922i	$-0.475019\pi$
$59$	1.20440	0.156800	0.0783999	+	0.996922i	$0.475019\pi$
$60$	0	0
$61$	−1.65498	−0.211899	−0.105950	−	0.994372i	$-0.533788\pi$
$61$	−1.65498	−0.211899	−0.105950	+	0.994372i	$0.533788\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 12.4088i	− 1.51598i	−0.652268	−	0.757988i	$-0.726182\pi$
$67$	− 12.4088i	− 1.51598i	0.652268	−	0.757988i	$-0.273818\pi$
$68$	0	0
$69$	−6.85939	−0.825773
$70$	0	0
$71$	−0.591197	−0.0701622	−0.0350811	−	0.999384i	$-0.511169\pi$
$71$	−0.591197	−0.0701622	−0.0350811	+	0.999384i	$0.511169\pi$
$72$	0	0
$73$	− 4.00000i	− 0.468165i	−0.972217	−	0.234082i	$-0.924791\pi$
$73$	− 4.00000i	− 0.468165i	0.972217	−	0.234082i	$-0.0752085\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−6.54942	−0.736867	−0.368433	−	0.929654i	$-0.620106\pi$
$79$	−6.54942	−0.736867	−0.368433	+	0.929654i	$0.620106\pi$
$80$	0	0
$81$	22.0220	2.44689
$82$	0	0
$83$	3.88139i	0.426038i	0.977048	+	0.213019i	$0.0683297\pi$
$83$	3.88139i	0.426038i	−0.977048	+	0.213019i	$0.931670\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 23.9452i	− 2.56719i
$88$	0	0
$89$	−9.26819	−0.982426	−0.491213	−	0.871039i	$-0.663446\pi$
$89$	−9.26819	−0.982426	−0.491213	+	0.871039i	$0.663446\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	− 27.1496i	− 2.81528i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 1.33198i	− 0.135242i	−0.997711	−	0.0676209i	$-0.978459\pi$
$97$	− 1.33198i	− 0.135242i	0.997711	−	0.0676209i	$-0.0215408\pi$
$98$	0	0
$99$	−30.5584	−3.07123
$100$	0	0
$101$	−6.19136	−0.616064	−0.308032	−	0.951376i	$-0.599670\pi$
$101$	−6.19136	−0.616064	−0.308032	+	0.951376i	$0.599670\pi$
$102$	0	0
$103$	9.74078i	0.959788i	0.877327	+	0.479894i	$0.159325\pi$
$103$	9.74078i	0.959788i	−0.877327	+	0.479894i	$0.840675\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	9.85939i	0.953143i	0.879136	+	0.476571i	$0.158121\pi$
$107$	9.85939i	0.953143i	−0.879136	+	0.476571i	$0.841879\pi$
$108$	0	0
$109$	4.07683	0.390489	0.195245	−	0.980755i	$-0.437450\pi$
$109$	4.07683	0.390489	0.195245	+	0.980755i	$0.437450\pi$
$110$	0	0
$111$	−34.0090	−3.22799
$112$	0	0
$113$	− 2.21744i	− 0.208599i	−0.994546	−	0.104300i	$-0.966740\pi$
$113$	− 2.21744i	− 0.208599i	0.994546	−	0.104300i	$-0.0332601\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 1.48563i	− 0.137346i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	6.67699	0.606999
$122$	0	0
$123$	33.8046i	3.04806i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	6.54942i	0.581167i	0.956850	+	0.290583i	$0.0938494\pi$
$127$	6.54942i	0.581167i	−0.956850	+	0.290583i	$0.906151\pi$
$128$	0	0
$129$	−26.4946	−2.33272
$130$	0	0
$131$	17.0090	1.48608	0.743040	−	0.669247i	$-0.233383\pi$
$131$	17.0090	1.48608	0.743040	+	0.669247i	$0.233383\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 17.0858i	− 1.45974i	−0.683587	−	0.729869i	$-0.739581\pi$
$137$	− 17.0858i	− 1.45974i	0.683587	−	0.729869i	$-0.260419\pi$
$138$	0	0
$139$	0.740780	0.0628322	0.0314161	−	0.999506i	$-0.489998\pi$
$139$	0.740780	0.0628322	0.0314161	+	0.999506i	$0.489998\pi$
$140$	0	0
$141$	−10.4726	−0.881951
$142$	0	0
$143$	0.859386i	0.0718655i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	16.4178	1.34500	0.672498	−	0.740099i	$-0.265221\pi$
$149$	16.4178	1.34500	0.672498	+	0.740099i	$0.265221\pi$
$150$	0	0
$151$	0.527409	0.0429199	0.0214600	−	0.999770i	$-0.493169\pi$
$151$	0.527409	0.0429199	0.0214600	+	0.999770i	$0.493169\pi$
$152$	0	0
$153$	− 36.8046i	− 2.97547i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	9.48563i	0.757036i	0.925594	+	0.378518i	$0.123566\pi$
$157$	9.48563i	0.757036i	−0.925594	+	0.378518i	$0.876434\pi$
$158$	0	0
$159$	−18.1914	−1.44267
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 8.85939i	− 0.693921i	−0.937880	−	0.346960i	$-0.887214\pi$
$163$	− 8.85939i	− 0.693921i	0.937880	−	0.346960i	$-0.112786\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 8.33198i	− 0.644748i	−0.946612	−	0.322374i	$-0.895519\pi$
$167$	− 8.33198i	− 0.644748i	0.946612	−	0.322374i	$-0.104481\pi$
$168$	0	0
$169$	12.9582	0.996786
$170$	0	0
$171$	−7.73181	−0.591266
$172$	0	0
$173$	− 18.3450i	− 1.39475i	−0.716709	−	0.697373i	$-0.754352\pi$
$173$	− 18.3450i	− 1.39475i	0.716709	−	0.697373i	$-0.245648\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 3.85939i	− 0.290089i
$178$	0	0
$179$	4.28123	0.319994	0.159997	−	0.987118i	$-0.448852\pi$
$179$	4.28123	0.319994	0.159997	+	0.987118i	$0.448852\pi$
$180$	0	0
$181$	−2.93621	−0.218247	−0.109123	−	0.994028i	$-0.534804\pi$
$181$	−2.93621	−0.218247	−0.109123	+	0.994028i	$0.534804\pi$
$182$	0	0
$183$	5.30324i	0.392026i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	21.2902i	1.55689i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−11.9452	−0.864323	−0.432162	−	0.901796i	$-0.642249\pi$
$191$	−11.9452	−0.864323	−0.432162	+	0.901796i	$0.642249\pi$
$192$	0	0
$193$	− 7.08580i	− 0.510047i	−0.966935	−	0.255023i	$-0.917917\pi$
$193$	− 7.08580i	− 0.510047i	0.966935	−	0.255023i	$-0.0820832\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 5.48563i	− 0.390835i	−0.980720	−	0.195417i	$-0.937394\pi$
$197$	− 5.48563i	− 0.390835i	0.980720	−	0.195417i	$-0.0626062\pi$
$198$	0	0
$199$	16.0858	1.14029	0.570146	−	0.821543i	$-0.306887\pi$
$199$	16.0858	1.14029	0.570146	+	0.821543i	$0.306887\pi$
$200$	0	0
$201$	−39.7628	−2.80465
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	15.5584i	1.08138i
$208$	0	0
$209$	4.47259	0.309376
$210$	0	0
$211$	16.6002	1.14280	0.571401	−	0.820671i	$-0.306400\pi$
$211$	16.6002	1.14280	0.571401	+	0.820671i	$0.306400\pi$
$212$	0	0
$213$	1.89443i	0.129804i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−12.8176	−0.866134
$220$	0	0
$221$	−1.03505	−0.0696247
$222$	0	0
$223$	− 15.4856i	− 1.03699i	−0.855079	−	0.518497i	$-0.826492\pi$
$223$	− 15.4856i	− 1.03699i	0.855079	−	0.518497i	$-0.173508\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3.78256i	0.251057i	0.992090	+	0.125529i	$0.0400627\pi$
$227$	3.78256i	0.251057i	−0.992090	+	0.125529i	$0.959937\pi$
$228$	0	0
$229$	−11.2044	−0.740408	−0.370204	−	0.928951i	$-0.620712\pi$
$229$	−11.2044	−0.740408	−0.370204	+	0.928951i	$0.620712\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	11.9452i	0.782555i	0.920273	+	0.391277i	$0.127967\pi$
$233$	11.9452i	0.782555i	−0.920273	+	0.391277i	$0.872033\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	20.9870i	1.36325i
$238$	0	0
$239$	−15.3450	−0.992587	−0.496293	−	0.868155i	$-0.665306\pi$
$239$	−15.3450	−0.992587	−0.496293	+	0.868155i	$0.665306\pi$
$240$	0	0
$241$	14.3958	0.927313	0.463656	−	0.886015i	$-0.346537\pi$
$241$	14.3958	0.927313	0.463656	+	0.886015i	$0.346537\pi$
$242$	0	0
$243$	− 29.5364i	− 1.89476i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0.217440i	0.0138354i
$248$	0	0
$249$	12.4375	0.788197
$250$	0	0
$251$	19.4178	1.22564	0.612819	−	0.790223i	$-0.290035\pi$
$251$	19.4178	1.22564	0.612819	+	0.790223i	$0.290035\pi$
$252$	0	0
$253$	− 9.00000i	− 0.565825i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	13.8724i	0.865338i	0.901553	+	0.432669i	$0.142428\pi$
$257$	13.8724i	0.865338i	−0.901553	+	0.432669i	$0.857572\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−54.3122	−3.36184
$262$	0	0
$263$	− 3.45955i	− 0.213325i	−0.994295	−	0.106663i	$-0.965984\pi$
$263$	− 3.45955i	− 0.213325i	0.994295	−	0.106663i	$-0.0340165\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	29.6990i	1.81755i
$268$	0	0
$269$	25.6640	1.56476	0.782379	−	0.622802i	$-0.214006\pi$
$269$	25.6640	1.56476	0.782379	+	0.622802i	$0.214006\pi$
$270$	0	0
$271$	−19.6770	−1.19529	−0.597646	−	0.801760i	$-0.703897\pi$
$271$	−19.6770	−1.19529	−0.597646	+	0.801760i	$0.703897\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	18.1496i	1.09050i	0.838273	+	0.545251i	$0.183566\pi$
$277$	18.1496i	1.09050i	−0.838273	+	0.545251i	$0.816434\pi$
$278$	0	0
$279$	−61.5804	−3.68672
$280$	0	0
$281$	−18.5364	−1.10579	−0.552894	−	0.833252i	$-0.686476\pi$
$281$	−18.5364	−1.10579	−0.552894	+	0.833252i	$0.686476\pi$
$282$	0	0
$283$	− 28.5584i	− 1.69762i	−0.528698	−	0.848810i	$-0.677320\pi$
$283$	− 28.5584i	− 1.69762i	0.528698	−	0.848810i	$-0.322680\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−8.64195	−0.508350
$290$	0	0
$291$	−4.26819	−0.250206
$292$	0	0
$293$	− 25.5494i	− 1.49261i	−0.665603	−	0.746306i	$-0.731825\pi$
$293$	− 25.5494i	− 1.49261i	0.665603	−	0.746306i	$-0.268175\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	57.5036i	3.33670i
$298$	0	0
$299$	0.437545	0.0253039
$300$	0	0
$301$	0	0
$302$	0	0
$303$	19.8396i	1.13976i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	10.1145i	0.577267i	0.957440	+	0.288634i	$0.0932009\pi$
$307$	10.1145i	0.577267i	−0.957440	+	0.288634i	$0.906799\pi$
$308$	0	0
$309$	31.2134	1.77567
$310$	0	0
$311$	−34.1716	−1.93769	−0.968847	−	0.247662i	$-0.920338\pi$
$311$	−34.1716	−1.93769	−0.968847	+	0.247662i	$0.920338\pi$
$312$	0	0
$313$	8.61320i	0.486847i	0.969920	+	0.243424i	$0.0782705\pi$
$313$	8.61320i	0.486847i	−0.969920	+	0.243424i	$0.921730\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	7.81761i	0.439081i	0.975603	+	0.219540i	$0.0704557\pi$
$317$	7.81761i	0.439081i	−0.975603	+	0.219540i	$0.929544\pi$
$318$	0	0
$319$	31.4178	1.75906
$320$	0	0
$321$	31.5934	1.76337
$322$	0	0
$323$	5.38680i	0.299729i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 13.0638i	− 0.722429i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−22.0988	−1.21466	−0.607331	−	0.794449i	$-0.707760\pi$
$331$	−22.0988	−1.21466	−0.607331	+	0.794449i	$0.707760\pi$
$332$	0	0
$333$	77.1388i	4.22718i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 1.29020i	− 0.0702815i	−0.999382	−	0.0351408i	$-0.988812\pi$
$337$	− 1.29020i	− 0.0702815i	0.999382	−	0.0351408i	$-0.0111880\pi$
$338$	0	0
$339$	−7.10557	−0.385921
$340$	0	0
$341$	35.6222	1.92905
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 9.45955i	− 0.507815i	−0.967228	−	0.253908i	$-0.918284\pi$
$347$	− 9.45955i	− 0.507815i	0.967228	−	0.253908i	$-0.0817159\pi$
$348$	0	0
$349$	15.5494	0.832341	0.416171	−	0.909287i	$-0.363372\pi$
$349$	15.5494	0.832341	0.416171	+	0.909287i	$0.363372\pi$
$350$	0	0
$351$	−2.79560	−0.149218
$352$	0	0
$353$	15.1365i	0.805637i	0.915280	+	0.402818i	$0.131969\pi$
$353$	15.1365i	0.805637i	−0.915280	+	0.402818i	$0.868031\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−10.3189	−0.544613	−0.272306	−	0.962211i	$-0.587786\pi$
$359$	−10.3189	−0.544613	−0.272306	+	0.962211i	$0.587786\pi$
$360$	0	0
$361$	−17.8684	−0.940440
$362$	0	0
$363$	− 21.3958i	− 1.12299i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 17.9362i	− 0.936263i	−0.883659	−	0.468131i	$-0.844927\pi$
$367$	− 17.9362i	− 0.936263i	0.883659	−	0.468131i	$-0.155073\pi$
$368$	0	0
$369$	76.6752	3.99155
$370$	0	0
$371$	0	0
$372$	0	0
$373$	16.1914i	0.838357i	0.907904	+	0.419179i	$0.137682\pi$
$373$	16.1914i	0.838357i	−0.907904	+	0.419179i	$0.862318\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.52741i	0.0786656i
$378$	0	0
$379$	12.6002	0.647227	0.323614	−	0.946189i	$-0.395102\pi$
$379$	12.6002	0.647227	0.323614	+	0.946189i	$0.395102\pi$
$380$	0	0
$381$	20.9870	1.07519
$382$	0	0
$383$	15.1757i	0.775440i	0.921777	+	0.387720i	$0.126737\pi$
$383$	15.1757i	0.775440i	−0.921777	+	0.387720i	$0.873263\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	60.0948i	3.05479i
$388$	0	0
$389$	29.2394	1.48250	0.741249	−	0.671230i	$-0.234234\pi$
$389$	29.2394	1.48250	0.741249	+	0.671230i	$0.234234\pi$
$390$	0	0
$391$	10.8396	0.548183
$392$	0	0
$393$	− 54.5036i	− 2.74934i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 11.2682i	− 0.565534i	−0.959189	−	0.282767i	$-0.908748\pi$
$397$	− 11.2682i	− 0.565534i	0.959189	−	0.282767i	$-0.0912524\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−18.0220	−0.899976	−0.449988	−	0.893035i	$-0.648572\pi$
$401$	−18.0220	−0.899976	−0.449988	+	0.893035i	$0.648572\pi$
$402$	0	0
$403$	1.73181i	0.0862676i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 44.6222i	− 2.21184i
$408$	0	0
$409$	−15.7915	−0.780841	−0.390420	−	0.920637i	$-0.627670\pi$
$409$	−15.7915	−0.780841	−0.390420	+	0.920637i	$0.627670\pi$
$410$	0	0
$411$	−54.7497	−2.70061
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 2.37376i	− 0.116243i
$418$	0	0
$419$	−13.4726	−0.658179	−0.329090	−	0.944299i	$-0.606742\pi$
$419$	−13.4726	−0.658179	−0.329090	+	0.944299i	$0.606742\pi$
$420$	0	0
$421$	−26.8396	−1.30808	−0.654041	−	0.756459i	$-0.726928\pi$
$421$	−26.8396	−1.30808	−0.654041	+	0.756459i	$0.726928\pi$
$422$	0	0
$423$	23.7538i	1.15495i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	2.75382	0.132956
$430$	0	0
$431$	16.0507	0.773137	0.386569	−	0.922261i	$-0.373660\pi$
$431$	16.0507	0.773137	0.386569	+	0.922261i	$0.373660\pi$
$432$	0	0
$433$	0.910136i	0.0437383i	0.999761	+	0.0218692i	$0.00696173\pi$
$433$	0.910136i	0.0437383i	−0.999761	+	0.0218692i	$0.993038\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 2.27716i	− 0.108931i
$438$	0	0
$439$	−37.9034	−1.80903	−0.904515	−	0.426441i	$-0.859767\pi$
$439$	−37.9034	−1.80903	−0.904515	+	0.426441i	$0.859767\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	6.88139i	0.326945i	0.986548	+	0.163472i	$0.0522695\pi$
$443$	6.88139i	0.326945i	−0.986548	+	0.163472i	$0.947731\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 52.6091i	− 2.48833i
$448$	0	0
$449$	−6.43754	−0.303807	−0.151903	−	0.988395i	$-0.548540\pi$
$449$	−6.43754	−0.303807	−0.151903	+	0.988395i	$0.548540\pi$
$450$	0	0
$451$	−44.3540	−2.08855
$452$	0	0
$453$	− 1.69003i	− 0.0794046i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 9.29427i	− 0.434767i	−0.976086	−	0.217384i	$-0.930248\pi$
$457$	− 9.29427i	− 0.434767i	0.976086	−	0.217384i	$-0.0697523\pi$
$458$	0	0
$459$	−69.2574	−3.23266
$460$	0	0
$461$	−9.29020	−0.432688	−0.216344	−	0.976317i	$-0.569413\pi$
$461$	−9.29020	−0.432688	−0.216344	+	0.976317i	$0.569413\pi$
$462$	0	0
$463$	29.6091i	1.37605i	0.725685	+	0.688027i	$0.241523\pi$
$463$	29.6091i	1.37605i	−0.725685	+	0.688027i	$0.758477\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	26.0638i	1.20609i	0.797708	+	0.603044i	$0.206046\pi$
$467$	26.0638i	1.20609i	−0.797708	+	0.603044i	$0.793954\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	30.3958	1.40056
$472$	0	0
$473$	− 34.7628i	− 1.59839i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	41.2615i	1.88923i
$478$	0	0
$479$	−1.10557	−0.0505147	−0.0252573	−	0.999681i	$-0.508041\pi$
$479$	−1.10557	−0.0505147	−0.0252573	+	0.999681i	$0.508041\pi$
$480$	0	0
$481$	2.16936	0.0989141
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	5.19136i	0.235243i	0.993058	+	0.117622i	$0.0375270\pi$
$487$	5.19136i	0.235243i	−0.993058	+	0.117622i	$0.962473\pi$
$488$	0	0
$489$	−28.3890	−1.28380
$490$	0	0
$491$	−28.6640	−1.29359	−0.646793	−	0.762666i	$-0.723890\pi$
$491$	−28.6640	−1.29359	−0.646793	+	0.762666i	$0.723890\pi$
$492$	0	0
$493$	37.8396i	1.70421i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	26.0728	1.16718	0.583588	−	0.812050i	$-0.301648\pi$
$499$	26.0728	1.16718	0.583588	+	0.812050i	$0.301648\pi$
$500$	0	0
$501$	−26.6990	−1.19282
$502$	0	0
$503$	8.80864i	0.392758i	0.980528	+	0.196379i	$0.0629182\pi$
$503$	8.80864i	0.392758i	−0.980528	+	0.196379i	$0.937082\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 41.5233i	− 1.84412i
$508$	0	0
$509$	−22.0179	−0.975928	−0.487964	−	0.872864i	$-0.662260\pi$
$509$	−22.0179	−0.975928	−0.487964	+	0.872864i	$0.662260\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	14.5494i	0.642372i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 13.7408i	− 0.604319i
$518$	0	0
$519$	−58.7848	−2.58037
$520$	0	0
$521$	15.0988	0.661492	0.330746	−	0.943720i	$-0.392700\pi$
$521$	15.0988	0.661492	0.330746	+	0.943720i	$0.392700\pi$
$522$	0	0
$523$	− 4.59120i	− 0.200759i	−0.994949	−	0.100380i	$-0.967994\pi$
$523$	− 4.59120i	− 0.200759i	0.994949	−	0.100380i	$-0.0320057\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	42.9034i	1.86890i
$528$	0	0
$529$	18.4178	0.800773
$530$	0	0
$531$	−8.75382	−0.379883
$532$	0	0
$533$	− 2.15632i	− 0.0934005i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 13.7188i	− 0.592009i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	13.6770	0.588020	0.294010	−	0.955802i	$-0.405010\pi$
$541$	13.6770	0.588020	0.294010	+	0.955802i	$0.405010\pi$
$542$	0	0
$543$	9.40880i	0.403770i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 11.3581i	− 0.485635i	−0.970072	−	0.242818i	$-0.921928\pi$
$547$	− 11.3581i	− 0.485635i	0.970072	−	0.242818i	$-0.0780717\pi$
$548$	0	0
$549$	12.0287	0.513374
$550$	0	0
$551$	7.94925	0.338649
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	16.3958i	0.694711i	0.937734	+	0.347355i	$0.112920\pi$
$557$	16.3958i	0.694711i	−0.937734	+	0.347355i	$0.887080\pi$
$558$	0	0
$559$	1.69003	0.0714807
$560$	0	0
$561$	68.2223	2.88035
$562$	0	0
$563$	− 4.87242i	− 0.205348i	−0.994715	−	0.102674i	$-0.967260\pi$
$563$	− 4.87242i	− 0.205348i	0.994715	−	0.102674i	$-0.0327399\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	37.8553	1.58698	0.793489	−	0.608585i	$-0.208263\pi$
$569$	37.8553	1.58698	0.793489	+	0.608585i	$0.208263\pi$
$570$	0	0
$571$	−41.7408	−1.74680	−0.873399	−	0.487006i	$-0.838089\pi$
$571$	−41.7408	−1.74680	−0.873399	+	0.487006i	$0.838089\pi$
$572$	0	0
$573$	38.2772i	1.59905i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 43.0530i	− 1.79232i	−0.443732	−	0.896160i	$-0.646346\pi$
$577$	− 43.0530i	− 1.79232i	0.443732	−	0.896160i	$-0.353654\pi$
$578$	0	0
$579$	−22.7057	−0.943618
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 23.8684i	− 0.988526i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	5.67699i	0.234315i	0.993113	+	0.117157i	$0.0373782\pi$
$587$	5.67699i	0.234315i	−0.993113	+	0.117157i	$0.962622\pi$
$588$	0	0
$589$	9.01304	0.371376
$590$	0	0
$591$	−17.5782	−0.723069
$592$	0	0
$593$	8.94518i	0.367335i	0.982988	+	0.183667i	$0.0587969\pi$
$593$	8.94518i	0.367335i	−0.982988	+	0.183667i	$0.941203\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 51.5453i	− 2.10961i
$598$	0	0
$599$	25.2264	1.03072	0.515362	−	0.856973i	$-0.327658\pi$
$599$	25.2264	1.03072	0.515362	+	0.856973i	$0.327658\pi$
$600$	0	0
$601$	−15.3409	−0.625770	−0.312885	−	0.949791i	$-0.601295\pi$
$601$	−15.3409	−0.625770	−0.312885	+	0.949791i	$0.601295\pi$
$602$	0	0
$603$	90.1895i	3.67280i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 20.9362i	− 0.849775i	−0.905246	−	0.424887i	$-0.860314\pi$
$607$	− 20.9362i	− 0.849775i	0.905246	−	0.424887i	$-0.139686\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0.668023	0.0270253
$612$	0	0
$613$	5.31894i	0.214830i	0.994214	+	0.107415i	$0.0342573\pi$
$613$	5.31894i	0.214830i	−0.994214	+	0.107415i	$0.965743\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 28.6262i	− 1.15245i	−0.817291	−	0.576225i	$-0.804525\pi$
$617$	− 28.6262i	− 1.15245i	0.817291	−	0.576225i	$-0.195475\pi$
$618$	0	0
$619$	−21.5625	−0.866668	−0.433334	−	0.901233i	$-0.642663\pi$
$619$	−21.5625	−0.866668	−0.433334	+	0.901233i	$0.642663\pi$
$620$	0	0
$621$	29.2772	1.17485
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 14.3320i	− 0.572364i
$628$	0	0
$629$	53.7430	2.14287
$630$	0	0
$631$	−25.1365	−1.00067	−0.500335	−	0.865832i	$-0.666790\pi$
$631$	−25.1365	−1.00067	−0.500335	+	0.865832i	$0.666790\pi$
$632$	0	0
$633$	− 53.1936i	− 2.11426i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	4.29693	0.169984
$640$	0	0
$641$	44.1078	1.74215	0.871077	−	0.491147i	$-0.163422\pi$
$641$	44.1078	1.74215	0.871077	+	0.491147i	$0.163422\pi$
$642$	0	0
$643$	− 16.4816i	− 0.649969i	−0.945719	−	0.324985i	$-0.894641\pi$
$643$	− 16.4816i	− 0.649969i	0.945719	−	0.324985i	$-0.105359\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	31.4398i	1.23603i	0.786168	+	0.618013i	$0.212062\pi$
$647$	31.4398i	1.23603i	−0.786168	+	0.618013i	$0.787938\pi$
$648$	0	0
$649$	5.06379	0.198771
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 40.8773i	− 1.59965i	−0.600231	−	0.799827i	$-0.704925\pi$
$653$	− 40.8773i	− 1.59965i	0.600231	−	0.799827i	$-0.295075\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	29.0728i	1.13424i
$658$	0	0
$659$	46.6860	1.81863	0.909313	−	0.416112i	$-0.136608\pi$
$659$	46.6860	1.81863	0.909313	+	0.416112i	$0.136608\pi$
$660$	0	0
$661$	−0.127575	−0.00496211	−0.00248106	−	0.999997i	$-0.500790\pi$
$661$	−0.127575	−0.00496211	−0.00248106	+	0.999997i	$0.500790\pi$
$662$	0	0
$663$	3.31670i	0.128810i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 15.9959i	− 0.619365i
$668$	0	0
$669$	−49.6222	−1.91850
$670$	0	0
$671$	−6.95822	−0.268619
$672$	0	0
$673$	− 48.1936i	− 1.85773i	−0.370423	−	0.928863i	$-0.620787\pi$
$673$	− 48.1936i	− 1.85773i	0.370423	−	0.928863i	$-0.379213\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	46.4178i	1.78398i	0.452055	+	0.891990i	$0.350691\pi$
$677$	46.4178i	1.78398i	−0.452055	+	0.891990i	$0.649309\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	12.1208	0.464472
$682$	0	0
$683$	− 43.5494i	− 1.66637i	−0.552993	−	0.833186i	$-0.686514\pi$
$683$	− 43.5494i	− 1.66637i	0.552993	−	0.833186i	$-0.313486\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	35.9034i	1.36980i
$688$	0	0
$689$	1.16039	0.0442072
$690$	0	0
$691$	26.0948	0.992692	0.496346	−	0.868125i	$-0.334675\pi$
$691$	26.0948	0.992692	0.496346	+	0.868125i	$0.334675\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 53.4200i	− 2.02343i
$698$	0	0
$699$	38.2772	1.44778
$700$	0	0
$701$	21.7277	0.820645	0.410323	−	0.911940i	$-0.365416\pi$
$701$	21.7277	0.820645	0.410323	+	0.911940i	$0.365416\pi$
$702$	0	0
$703$	− 11.2902i	− 0.425818i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−17.0768	−0.641334	−0.320667	−	0.947192i	$-0.603907\pi$
$709$	−17.0768	−0.641334	−0.320667	+	0.947192i	$0.603907\pi$
$710$	0	0
$711$	47.6024	1.78523
$712$	0	0
$713$	− 18.1365i	− 0.679219i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	49.1716i	1.83635i
$718$	0	0
$719$	27.2462	1.01611	0.508056	−	0.861324i	$-0.330364\pi$
$719$	27.2462	1.01611	0.508056	+	0.861324i	$0.330364\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	− 46.1298i	− 1.71559i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 19.9232i	− 0.738910i	−0.929249	−	0.369455i	$-0.879544\pi$
$727$	− 19.9232i	− 0.738910i	0.929249	−	0.369455i	$-0.120456\pi$
$728$	0	0
$729$	−28.5804	−1.05853
$730$	0	0
$731$	41.8684	1.54856
$732$	0	0
$733$	31.1626i	1.15102i	0.817796	+	0.575509i	$0.195196\pi$
$733$	31.1626i	1.15102i	−0.817796	+	0.575509i	$0.804804\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 52.1716i	− 1.92177i
$738$	0	0
$739$	−1.90117	−0.0699355	−0.0349678	−	0.999388i	$-0.511133\pi$
$739$	−1.90117	−0.0699355	−0.0349678	+	0.999388i	$0.511133\pi$
$740$	0	0
$741$	0.696765	0.0255963
$742$	0	0
$743$	− 36.0660i	− 1.32313i	−0.749886	−	0.661567i	$-0.769892\pi$
$743$	− 36.0660i	− 1.32313i	0.749886	−	0.661567i	$-0.230108\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 28.2107i	− 1.03218i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−0.769522	−0.0280803	−0.0140401	−	0.999901i	$-0.504469\pi$
$751$	−0.769522	−0.0280803	−0.0140401	+	0.999901i	$0.504469\pi$
$752$	0	0
$753$	− 62.2223i	− 2.26751i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 14.9542i	− 0.543518i	−0.962365	−	0.271759i	$-0.912395\pi$
$757$	− 14.9542i	− 0.543518i	0.962365	−	0.271759i	$-0.0876053\pi$
$758$	0	0
$759$	−28.8396	−1.04681
$760$	0	0
$761$	−19.1716	−0.694970	−0.347485	−	0.937686i	$-0.612964\pi$
$761$	−19.1716	−0.694970	−0.347485	+	0.937686i	$0.612964\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0.246182i	0.00888910i
$768$	0	0
$769$	36.3189	1.30969	0.654847	−	0.755761i	$-0.272733\pi$
$769$	36.3189	1.30969	0.654847	+	0.755761i	$0.272733\pi$
$770$	0	0
$771$	44.4528	1.60093
$772$	0	0
$773$	− 47.4685i	− 1.70732i	−0.520827	−	0.853662i	$-0.674376\pi$
$773$	− 47.4685i	− 1.70732i	0.520827	−	0.853662i	$-0.325624\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−11.2223	−0.402082
$780$	0	0
$781$	−2.48563	−0.0889428
$782$	0	0
$783$	102.203i	3.65242i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 7.88546i	− 0.281086i	−0.990075	−	0.140543i	$-0.955115\pi$
$787$	− 7.88546i	− 0.281086i	0.990075	−	0.140543i	$-0.0448849\pi$
$788$	0	0
$789$	−11.0858	−0.394665
$790$	0	0
$791$	0	0
$792$	0	0
$793$	− 0.338281i	− 0.0120127i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	33.6132i	1.19064i	0.803488	+	0.595320i	$0.202975\pi$
$797$	33.6132i	1.19064i	−0.803488	+	0.595320i	$0.797025\pi$
$798$	0	0
$799$	16.5494	0.585476
$800$	0	0
$801$	67.3630	2.38015
$802$	0	0
$803$	− 16.8176i	− 0.593480i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 82.2376i	− 2.89490i
$808$	0	0
$809$	9.19136	0.323151	0.161576	−	0.986860i	$-0.448342\pi$
$809$	9.19136	0.323151	0.161576	+	0.986860i	$0.448342\pi$
$810$	0	0
$811$	−21.6483	−0.760173	−0.380086	−	0.924951i	$-0.624106\pi$
$811$	−21.6483	−0.760173	−0.380086	+	0.924951i	$0.624106\pi$
$812$	0	0
$813$	63.0530i	2.21136i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 8.79560i	− 0.307719i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	32.2615	1.12593	0.562966	−	0.826480i	$-0.309660\pi$
$821$	32.2615	1.12593	0.562966	+	0.826480i	$0.309660\pi$
$822$	0	0
$823$	− 11.4438i	− 0.398908i	−0.979907	−	0.199454i	$-0.936083\pi$
$823$	− 11.4438i	− 0.398908i	0.979907	−	0.199454i	$-0.0639168\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	9.82658i	0.341704i	0.985297	+	0.170852i	$0.0546519\pi$
$827$	9.82658i	0.341704i	−0.985297	+	0.170852i	$0.945348\pi$
$828$	0	0
$829$	38.2574	1.32873	0.664367	−	0.747407i	$-0.268701\pi$
$829$	38.2574	1.32873	0.664367	+	0.747407i	$0.268701\pi$
$830$	0	0
$831$	58.1586	2.01750
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	115.880i	4.00538i
$838$	0	0
$839$	30.6900	1.05954	0.529769	−	0.848142i	$-0.322279\pi$
$839$	30.6900	1.05954	0.529769	+	0.848142i	$0.322279\pi$
$840$	0	0
$841$	26.8396	0.925504
$842$	0	0
$843$	59.3980i	2.04578i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−91.5125	−3.14070
$850$	0	0
$851$	−22.7188	−0.778789
$852$	0	0
$853$	39.4569i	1.35098i	0.737370	+	0.675489i	$0.236067\pi$
$853$	39.4569i	1.35098i	−0.737370	+	0.675489i	$0.763933\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	39.8594i	1.36157i	0.732483	+	0.680785i	$0.238361\pi$
$857$	39.8594i	1.36157i	−0.732483	+	0.680785i	$0.761639\pi$
$858$	0	0
$859$	−19.5144	−0.665822	−0.332911	−	0.942958i	$-0.608031\pi$
$859$	−19.5144	−0.665822	−0.332911	+	0.942958i	$0.608031\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	37.9321i	1.29123i	0.763665	+	0.645613i	$0.223398\pi$
$863$	37.9321i	1.29123i	−0.763665	+	0.645613i	$0.776602\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	27.6923i	0.940479i
$868$	0	0
$869$	−27.5364	−0.934108
$870$	0	0
$871$	2.53638	0.0859419
$872$	0	0
$873$	9.68106i	0.327654i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	10.9452i	0.369593i	0.982777	+	0.184796i	$0.0591625\pi$
$877$	10.9452i	0.369593i	−0.982777	+	0.184796i	$0.940837\pi$
$878$	0	0
$879$	−81.8706	−2.76143
$880$	0	0
$881$	16.2592	0.547787	0.273894	−	0.961760i	$-0.411688\pi$
$881$	16.2592	0.547787	0.273894	+	0.961760i	$0.411688\pi$
$882$	0	0
$883$	− 23.2511i	− 0.782461i	−0.920293	−	0.391231i	$-0.872049\pi$
$883$	− 23.2511i	− 0.782461i	0.920293	−	0.391231i	$-0.127951\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	23.2394i	0.780304i	0.920750	+	0.390152i	$0.127578\pi$
$887$	23.2394i	0.780304i	−0.920750	+	0.390152i	$0.872422\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	92.5894	3.10186
$892$	0	0
$893$	− 3.47666i	− 0.116342i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 1.40207i	− 0.0468137i
$898$	0	0
$899$	63.3122	2.11158
$900$	0	0
$901$	28.7471	0.957704
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 44.2264i	− 1.46851i	−0.678872	−	0.734257i	$-0.737531\pi$
$907$	− 44.2264i	− 1.46851i	0.678872	−	0.734257i	$-0.262469\pi$
$908$	0	0
$909$	45.0000	1.49256
$910$	0	0
$911$	−27.4218	−0.908526	−0.454263	−	0.890868i	$-0.650097\pi$
$911$	−27.4218	−0.908526	−0.454263	+	0.890868i	$0.650097\pi$
$912$	0	0
$913$	16.3189i	0.540078i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	33.3163	1.09900	0.549501	−	0.835493i	$-0.314818\pi$
$919$	33.3163	1.09900	0.549501	+	0.835493i	$0.314818\pi$
$920$	0	0
$921$	32.4110	1.06798
$922$	0	0
$923$	− 0.120842i	− 0.00397755i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 70.7978i	− 2.32531i
$928$	0	0
$929$	−13.2484	−0.434666	−0.217333	−	0.976097i	$-0.569736\pi$
$929$	−13.2484	−0.434666	−0.217333	+	0.976097i	$0.569736\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	109.499i	3.58485i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	10.1365i	0.331146i	0.986197	+	0.165573i	$0.0529474\pi$
$937$	10.1365i	0.331146i	−0.986197	+	0.165573i	$0.947053\pi$
$938$	0	0
$939$	27.6002	0.900697
$940$	0	0
$941$	32.8684	1.07148	0.535739	−	0.844384i	$-0.320033\pi$
$941$	32.8684	1.07148	0.535739	+	0.844384i	$0.320033\pi$
$942$	0	0
$943$	22.5822i	0.735379i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 22.3252i	− 0.725473i	−0.931892	−	0.362736i	$-0.881843\pi$
$947$	− 22.3252i	− 0.725473i	0.931892	−	0.362736i	$-0.118157\pi$
$948$	0	0
$949$	0.817606	0.0265406
$950$	0	0
$951$	25.0507	0.812326
$952$	0	0
$953$	47.4685i	1.53766i	0.639455	+	0.768828i	$0.279160\pi$
$953$	47.4685i	1.53766i	−0.639455	+	0.768828i	$0.720840\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 100.675i	− 3.25437i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	40.7848	1.31564
$962$	0	0
$963$	− 71.6599i	− 2.30921i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	21.3189i	0.685571i	0.939414	+	0.342785i	$0.111370\pi$
$967$	21.3189i	0.685571i	−0.939414	+	0.342785i	$0.888630\pi$
$968$	0	0
$969$	17.2615	0.554518
$970$	0	0
$971$	29.5625	0.948704	0.474352	−	0.880335i	$-0.342682\pi$
$971$	29.5625	0.948704	0.474352	+	0.880335i	$0.342682\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 1.83961i	− 0.0588545i	−0.999567	−	0.0294272i	$-0.990632\pi$
$977$	− 1.83961i	− 0.0588545i	0.999567	−	0.0294272i	$-0.00936833\pi$
$978$	0	0
$979$	−38.9672	−1.24540
$980$	0	0
$981$	−29.6311	−0.946050
$982$	0	0
$983$	− 11.3760i	− 0.362838i	−0.983406	−	0.181419i	$-0.941931\pi$
$983$	− 11.3760i	− 0.362838i	0.983406	−	0.181419i	$-0.0580690\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−17.6990	−0.562795
$990$	0	0
$991$	7.79153	0.247506	0.123753	−	0.992313i	$-0.460507\pi$
$991$	7.79153	0.247506	0.123753	+	0.992313i	$0.460507\pi$
$992$	0	0
$993$	70.8135i	2.24720i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 11.4218i	− 0.361733i	−0.983508	−	0.180867i	$-0.942110\pi$
$997$	− 11.4218i	− 0.361733i	0.983508	−	0.180867i	$-0.0578903\pi$
$998$	0	0
$999$	145.157	4.59256

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4900.2.e.s.2549.1		6
5.2	odd	4		4900.2.a.bb.1.1		3
5.3	odd	4		4900.2.a.bd.1.3		3
5.4	even	2	inner	4900.2.e.s.2549.6		6
7.3	odd	6		700.2.r.d.149.6		12
7.5	odd	6		700.2.r.d.249.1		12
7.6	odd	2		4900.2.e.t.2549.6		6
35.3	even	12		700.2.i.e.401.3	yes	6
35.12	even	12		700.2.i.d.501.1	yes	6
35.13	even	4		4900.2.a.ba.1.1		3
35.17	even	12		700.2.i.d.401.1	✓	6
35.19	odd	6		700.2.r.d.249.6		12
35.24	odd	6		700.2.r.d.149.1		12
35.27	even	4		4900.2.a.bc.1.3		3
35.33	even	12		700.2.i.e.501.3	yes	6
35.34	odd	2		4900.2.e.t.2549.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
700.2.i.d.401.1	✓	6	35.17	even	12
700.2.i.d.501.1	yes	6	35.12	even	12
700.2.i.e.401.3	yes	6	35.3	even	12
700.2.i.e.501.3	yes	6	35.33	even	12
700.2.r.d.149.1		12	35.24	odd	6
700.2.r.d.149.6		12	7.3	odd	6
700.2.r.d.249.1		12	7.5	odd	6
700.2.r.d.249.6		12	35.19	odd	6
4900.2.a.ba.1.1		3	35.13	even	4
4900.2.a.bb.1.1		3	5.2	odd	4
4900.2.a.bc.1.3		3	35.27	even	4
4900.2.a.bd.1.3		3	5.3	odd	4
4900.2.e.s.2549.1		6	1.1	even	1	trivial
4900.2.e.s.2549.6		6	5.4	even	2	inner
4900.2.e.t.2549.1		6	35.34	odd	2
4900.2.e.t.2549.6		6	7.6	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:	\( N \)	\(=\)	\( 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4900.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.20440i q^{3} -7.26819 q^{9} +O(q^{10})\)	\(q-3.20440i q^{3} -7.26819 q^{9} +4.20440 q^{11} +0.204402i q^{13} +5.06379i q^{17} +1.06379 q^{19} -2.14061i q^{23} +13.6770i q^{27} +7.47259 q^{29} +8.47259 q^{31} -13.4726i q^{33} -10.6132i q^{37} +0.654985 q^{39} -10.5494 q^{41} -8.26819i q^{43} -3.26819i q^{47} +16.2264 q^{51} -5.67699i q^{53} -3.40880i q^{57} +1.20440 q^{59} -1.65498 q^{61} -12.4088i q^{67} -6.85939 q^{69} -0.591197 q^{71} -4.00000i q^{73} -6.54942 q^{79} +22.0220 q^{81} +3.88139i q^{83} -23.9452i q^{87} -9.26819 q^{89} -27.1496i q^{93} -1.33198i q^{97} -30.5584 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 16 q^{9}+O(q^{10}) \) 6 * q - 16 * q^9	\( 6 q - 16 q^{9} + 8 q^{11} - 4 q^{19} + 6 q^{31} + 28 q^{39} - 22 q^{41} - 6 q^{51} - 10 q^{59} - 34 q^{61} - 48 q^{69} - 38 q^{71} + 2 q^{79} + 46 q^{81} - 28 q^{89} - 42 q^{99}+O(q^{100}) \) 6 * q - 16 * q^9 + 8 * q^11 - 4 * q^19 + 6 * q^31 + 28 * q^39 - 22 * q^41 - 6 * q^51 - 10 * q^59 - 34 * q^61 - 48 * q^69 - 38 * q^71 + 2 * q^79 + 46 * q^81 - 28 * q^89 - 42 * q^99

Introduction

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