LMFDB - Embedded newform 6800.2.a.bq.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6800,2,Mod(1,6800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6800, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6800 = 2^{4} \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6800.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:	$54.2982733745$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.564.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 3 $ x^3 - x^2 - 5*x + 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3400)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.571993$ of defining polynomial
Character	$\chi$	$=$	6800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.571993 q^{3} -3.24482 q^{7} -2.67282 q^{9} +O(q^{10})$	$q+0.571993 q^{3} -3.24482 q^{7} -2.67282 q^{9} +2.67282 q^{11} +1.67282 q^{13} -1.00000 q^{17} -2.00000 q^{19} -1.85601 q^{21} +6.67282 q^{23} -3.24482 q^{27} +0.856013 q^{29} +1.42801 q^{31} +1.52884 q^{33} -0.856013 q^{37} +0.956844 q^{39} -2.85601 q^{41} +8.48963 q^{43} -1.14399 q^{47} +3.52884 q^{49} -0.571993 q^{51} -3.00000 q^{53} -1.14399 q^{57} -3.14399 q^{59} -3.71203 q^{61} +8.67282 q^{63} +7.05767 q^{67} +3.81681 q^{69} -4.95684 q^{71} -8.48963 q^{73} -8.67282 q^{77} -14.5905 q^{79} +6.16246 q^{81} +4.00000 q^{83} +0.489634 q^{87} -9.54731 q^{89} -5.42801 q^{91} +0.816810 q^{93} -2.28797 q^{97} -7.14399 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{3} + q^{7} + 2 q^{9}+O(q^{10}) $ 3 * q + q^3 + q^7 + 2 * q^9	$ 3 q + q^{3} + q^{7} + 2 q^{9} - 2 q^{11} - 5 q^{13} - 3 q^{17} - 6 q^{19} - 7 q^{21} + 10 q^{23} + q^{27} + 4 q^{29} + 5 q^{31} - 4 q^{33} - 4 q^{37} - 5 q^{39} - 10 q^{41} + 4 q^{43} - 2 q^{47} + 2 q^{49} - q^{51} - 9 q^{53} - 2 q^{57} - 8 q^{59} - 14 q^{61} + 16 q^{63} + 4 q^{67} - 7 q^{71} - 4 q^{73} - 16 q^{77} - 13 q^{79} - 13 q^{81} + 12 q^{83} - 20 q^{87} + 10 q^{89} - 17 q^{91} - 9 q^{93} - 4 q^{97} - 20 q^{99}+O(q^{100}) $ 3 * q + q^3 + q^7 + 2 * q^9 - 2 * q^11 - 5 * q^13 - 3 * q^17 - 6 * q^19 - 7 * q^21 + 10 * q^23 + q^27 + 4 * q^29 + 5 * q^31 - 4 * q^33 - 4 * q^37 - 5 * q^39 - 10 * q^41 + 4 * q^43 - 2 * q^47 + 2 * q^49 - q^51 - 9 * q^53 - 2 * q^57 - 8 * q^59 - 14 * q^61 + 16 * q^63 + 4 * q^67 - 7 * q^71 - 4 * q^73 - 16 * q^77 - 13 * q^79 - 13 * q^81 + 12 * q^83 - 20 * q^87 + 10 * q^89 - 17 * q^91 - 9 * q^93 - 4 * q^97 - 20 * 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
$3$	0.571993	0.330240	0.165120	−	0.986273i	$-0.447199\pi$
$3$	0.571993	0.330240	0.165120	+	0.986273i	$0.447199\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.24482	−1.22643	−0.613213	−	0.789918i	$-0.710123\pi$
$7$	−3.24482	−1.22643	−0.613213	+	0.789918i	$0.710123\pi$
$8$	0	0
$9$	−2.67282	−0.890941
$10$	0	0
$11$	2.67282	0.805887	0.402943	−	0.915225i	$-0.367987\pi$
$11$	2.67282	0.805887	0.402943	+	0.915225i	$0.367987\pi$
$12$	0	0
$13$	1.67282	0.463958	0.231979	−	0.972721i	$-0.425480\pi$
$13$	1.67282	0.463958	0.231979	+	0.972721i	$0.425480\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	−1.85601	−0.405015
$22$	0	0
$23$	6.67282	1.39138	0.695690	−	0.718342i	$-0.255099\pi$
$23$	6.67282	1.39138	0.695690	+	0.718342i	$0.255099\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−3.24482	−0.624465
$28$	0	0
$29$	0.856013	0.158958	0.0794789	−	0.996837i	$-0.474674\pi$
$29$	0.856013	0.158958	0.0794789	+	0.996837i	$0.474674\pi$
$30$	0	0
$31$	1.42801	0.256478	0.128239	−	0.991743i	$-0.459068\pi$
$31$	1.42801	0.256478	0.128239	+	0.991743i	$0.459068\pi$
$32$	0	0
$33$	1.52884	0.266136
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−0.856013	−0.140728	−0.0703639	−	0.997521i	$-0.522416\pi$
$37$	−0.856013	−0.140728	−0.0703639	+	0.997521i	$0.522416\pi$
$38$	0	0
$39$	0.956844	0.153218
$40$	0	0
$41$	−2.85601	−0.446034	−0.223017	−	0.974815i	$-0.571591\pi$
$41$	−2.85601	−0.446034	−0.223017	+	0.974815i	$0.571591\pi$
$42$	0	0
$43$	8.48963	1.29466	0.647329	−	0.762211i	$-0.275886\pi$
$43$	8.48963	1.29466	0.647329	+	0.762211i	$0.275886\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1.14399	−0.166868	−0.0834338	−	0.996513i	$-0.526589\pi$
$47$	−1.14399	−0.166868	−0.0834338	+	0.996513i	$0.526589\pi$
$48$	0	0
$49$	3.52884	0.504120
$50$	0	0
$51$	−0.571993	−0.0800951
$52$	0	0
$53$	−3.00000	−0.412082	−0.206041	−	0.978543i	$-0.566058\pi$
$53$	−3.00000	−0.412082	−0.206041	+	0.978543i	$0.566058\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−1.14399	−0.151525
$58$	0	0
$59$	−3.14399	−0.409312	−0.204656	−	0.978834i	$-0.565608\pi$
$59$	−3.14399	−0.409312	−0.204656	+	0.978834i	$0.565608\pi$
$60$	0	0
$61$	−3.71203	−0.475276	−0.237638	−	0.971354i	$-0.576373\pi$
$61$	−3.71203	−0.475276	−0.237638	+	0.971354i	$0.576373\pi$
$62$	0	0
$63$	8.67282	1.09267
$64$	0	0
$65$	0	0
$66$	0	0
$67$	7.05767	0.862232	0.431116	−	0.902296i	$-0.358120\pi$
$67$	7.05767	0.862232	0.431116	+	0.902296i	$0.358120\pi$
$68$	0	0
$69$	3.81681	0.459490
$70$	0	0
$71$	−4.95684	−0.588269	−0.294135	−	0.955764i	$-0.595031\pi$
$71$	−4.95684	−0.588269	−0.294135	+	0.955764i	$0.595031\pi$
$72$	0	0
$73$	−8.48963	−0.993636	−0.496818	−	0.867855i	$-0.665498\pi$
$73$	−8.48963	−0.993636	−0.496818	+	0.867855i	$0.665498\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−8.67282	−0.988360
$78$	0	0
$79$	−14.5905	−1.64156	−0.820778	−	0.571248i	$-0.806460\pi$
$79$	−14.5905	−1.64156	−0.820778	+	0.571248i	$0.806460\pi$
$80$	0	0
$81$	6.16246	0.684718
$82$	0	0
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0.489634	0.0524943
$88$	0	0
$89$	−9.54731	−1.01201	−0.506006	−	0.862530i	$-0.668879\pi$
$89$	−9.54731	−1.01201	−0.506006	+	0.862530i	$0.668879\pi$
$90$	0	0
$91$	−5.42801	−0.569010
$92$	0	0
$93$	0.816810	0.0846993
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−2.28797	−0.232308	−0.116154	−	0.993231i	$-0.537057\pi$
$97$	−2.28797	−0.232308	−0.116154	+	0.993231i	$0.537057\pi$
$98$	0	0
$99$	−7.14399	−0.717998
$100$	0	0
$101$	−9.67282	−0.962482	−0.481241	−	0.876588i	$-0.659814\pi$
$101$	−9.67282	−0.962482	−0.481241	+	0.876588i	$0.659814\pi$
$102$	0	0
$103$	1.51037	0.148821	0.0744104	−	0.997228i	$-0.476293\pi$
$103$	1.51037	0.148821	0.0744104	+	0.997228i	$0.476293\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	11.4465	1.10657	0.553286	−	0.832991i	$-0.313374\pi$
$107$	11.4465	1.10657	0.553286	+	0.832991i	$0.313374\pi$
$108$	0	0
$109$	15.1809	1.45407	0.727035	−	0.686601i	$-0.240898\pi$
$109$	15.1809	1.45407	0.727035	+	0.686601i	$0.240898\pi$
$110$	0	0
$111$	−0.489634	−0.0464740
$112$	0	0
$113$	−6.28797	−0.591523	−0.295761	−	0.955262i	$-0.595573\pi$
$113$	−6.28797	−0.591523	−0.295761	+	0.955262i	$0.595573\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−4.47116	−0.413359
$118$	0	0
$119$	3.24482	0.297452
$120$	0	0
$121$	−3.85601	−0.350547
$122$	0	0
$123$	−1.63362	−0.147299
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−8.48963	−0.753333	−0.376667	−	0.926349i	$-0.622930\pi$
$127$	−8.48963	−0.753333	−0.376667	+	0.926349i	$0.622930\pi$
$128$	0	0
$129$	4.85601	0.427548
$130$	0	0
$131$	13.9361	1.21760	0.608802	−	0.793322i	$-0.291650\pi$
$131$	13.9361	1.21760	0.608802	+	0.793322i	$0.291650\pi$
$132$	0	0
$133$	6.48963	0.562723
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−22.1625	−1.89347	−0.946733	−	0.322019i	$-0.895639\pi$
$137$	−22.1625	−1.89347	−0.946733	+	0.322019i	$0.895639\pi$
$138$	0	0
$139$	−13.4280	−1.13895	−0.569474	−	0.822009i	$-0.692853\pi$
$139$	−13.4280	−1.13895	−0.569474	+	0.822009i	$0.692853\pi$
$140$	0	0
$141$	−0.654353	−0.0551064
$142$	0	0
$143$	4.47116	0.373897
$144$	0	0
$145$	0	0
$146$	0	0
$147$	2.01847	0.166481
$148$	0	0
$149$	−5.77761	−0.473320	−0.236660	−	0.971593i	$-0.576053\pi$
$149$	−5.77761	−0.473320	−0.236660	+	0.971593i	$0.576053\pi$
$150$	0	0
$151$	−7.83528	−0.637626	−0.318813	−	0.947818i	$-0.603284\pi$
$151$	−7.83528	−0.637626	−0.318813	+	0.947818i	$0.603284\pi$
$152$	0	0
$153$	2.67282	0.216085
$154$	0	0
$155$	0	0
$156$	0	0
$157$	13.6913	1.09268	0.546342	−	0.837562i	$-0.316020\pi$
$157$	13.6913	1.09268	0.546342	+	0.837562i	$0.316020\pi$
$158$	0	0
$159$	−1.71598	−0.136086
$160$	0	0
$161$	−21.6521	−1.70642
$162$	0	0
$163$	−8.11930	−0.635953	−0.317976	−	0.948099i	$-0.603003\pi$
$163$	−8.11930	−0.635953	−0.317976	+	0.948099i	$0.603003\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2.67282	−0.206829	−0.103415	−	0.994638i	$-0.532977\pi$
$167$	−2.67282	−0.206829	−0.103415	+	0.994638i	$0.532977\pi$
$168$	0	0
$169$	−10.2017	−0.784743
$170$	0	0
$171$	5.34565	0.408792
$172$	0	0
$173$	−9.91369	−0.753724	−0.376862	−	0.926269i	$-0.622997\pi$
$173$	−9.91369	−0.753724	−0.376862	+	0.926269i	$0.622997\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−1.79834	−0.135171
$178$	0	0
$179$	−7.34565	−0.549039	−0.274520	−	0.961581i	$-0.588519\pi$
$179$	−7.34565	−0.549039	−0.274520	+	0.961581i	$0.588519\pi$
$180$	0	0
$181$	−18.1233	−1.34709	−0.673545	−	0.739146i	$-0.735229\pi$
$181$	−18.1233	−1.34709	−0.673545	+	0.739146i	$0.735229\pi$
$182$	0	0
$183$	−2.12325	−0.156955
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−2.67282	−0.195456
$188$	0	0
$189$	10.5288	0.765860
$190$	0	0
$191$	−1.42405	−0.103041	−0.0515205	−	0.998672i	$-0.516407\pi$
$191$	−1.42405	−0.103041	−0.0515205	+	0.998672i	$0.516407\pi$
$192$	0	0
$193$	−21.8353	−1.57174	−0.785869	−	0.618393i	$-0.787784\pi$
$193$	−21.8353	−1.57174	−0.785869	+	0.618393i	$0.787784\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4.69129	0.334241	0.167120	−	0.985936i	$-0.446553\pi$
$197$	4.69129	0.334241	0.167120	+	0.985936i	$0.446553\pi$
$198$	0	0
$199$	17.3641	1.23091	0.615455	−	0.788172i	$-0.288972\pi$
$199$	17.3641	1.23091	0.615455	+	0.788172i	$0.288972\pi$
$200$	0	0
$201$	4.03694	0.284744
$202$	0	0
$203$	−2.77761	−0.194950
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−17.8353	−1.23964
$208$	0	0
$209$	−5.34565	−0.369766
$210$	0	0
$211$	1.33113	0.0916387	0.0458194	−	0.998950i	$-0.485410\pi$
$211$	1.33113	0.0916387	0.0458194	+	0.998950i	$0.485410\pi$
$212$	0	0
$213$	−2.83528	−0.194270
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−4.63362	−0.314551
$218$	0	0
$219$	−4.85601	−0.328139
$220$	0	0
$221$	−1.67282	−0.112526
$222$	0	0
$223$	27.4689	1.83945	0.919727	−	0.392559i	$-0.128410\pi$
$223$	27.4689	1.83945	0.919727	+	0.392559i	$0.128410\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−10.4073	−0.690755	−0.345378	−	0.938464i	$-0.612249\pi$
$227$	−10.4073	−0.690755	−0.345378	+	0.938464i	$0.612249\pi$
$228$	0	0
$229$	5.58651	0.369167	0.184584	−	0.982817i	$-0.440906\pi$
$229$	5.58651	0.369167	0.184584	+	0.982817i	$0.440906\pi$
$230$	0	0
$231$	−4.96080	−0.326396
$232$	0	0
$233$	−8.12325	−0.532172	−0.266086	−	0.963949i	$-0.585731\pi$
$233$	−8.12325	−0.532172	−0.266086	+	0.963949i	$0.585731\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−8.34565	−0.542108
$238$	0	0
$239$	−10.1153	−0.654308	−0.327154	−	0.944971i	$-0.606090\pi$
$239$	−10.1153	−0.654308	−0.327154	+	0.944971i	$0.606090\pi$
$240$	0	0
$241$	−11.9216	−0.767937	−0.383969	−	0.923346i	$-0.625443\pi$
$241$	−11.9216	−0.767937	−0.383969	+	0.923346i	$0.625443\pi$
$242$	0	0
$243$	13.2593	0.850587
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−3.34565	−0.212878
$248$	0	0
$249$	2.28797	0.144994
$250$	0	0
$251$	−28.0369	−1.76968	−0.884838	−	0.465899i	$-0.845731\pi$
$251$	−28.0369	−1.76968	−0.884838	+	0.465899i	$0.845731\pi$
$252$	0	0
$253$	17.8353	1.12129
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−15.8824	−0.990716	−0.495358	−	0.868689i	$-0.664963\pi$
$257$	−15.8824	−0.990716	−0.495358	+	0.868689i	$0.664963\pi$
$258$	0	0
$259$	2.77761	0.172592
$260$	0	0
$261$	−2.28797	−0.141622
$262$	0	0
$263$	−2.69129	−0.165952	−0.0829762	−	0.996552i	$-0.526443\pi$
$263$	−2.69129	−0.165952	−0.0829762	+	0.996552i	$0.526443\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−5.46100	−0.334208
$268$	0	0
$269$	−13.6336	−0.831257	−0.415628	−	0.909535i	$-0.636438\pi$
$269$	−13.6336	−0.831257	−0.415628	+	0.909535i	$0.636438\pi$
$270$	0	0
$271$	7.34565	0.446216	0.223108	−	0.974794i	$-0.428380\pi$
$271$	7.34565	0.446216	0.223108	+	0.974794i	$0.428380\pi$
$272$	0	0
$273$	−3.10478	−0.187910
$274$	0	0
$275$	0	0
$276$	0	0
$277$	21.7569	1.30724	0.653622	−	0.756821i	$-0.273249\pi$
$277$	21.7569	1.30724	0.653622	+	0.756821i	$0.273249\pi$
$278$	0	0
$279$	−3.81681	−0.228506
$280$	0	0
$281$	−31.9793	−1.90772	−0.953862	−	0.300247i	$-0.902931\pi$
$281$	−31.9793	−1.90772	−0.953862	+	0.300247i	$0.902931\pi$
$282$	0	0
$283$	10.3064	0.612655	0.306327	−	0.951926i	$-0.400900\pi$
$283$	10.3064	0.612655	0.306327	+	0.951926i	$0.400900\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	9.26724	0.547028
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−1.30871	−0.0767177
$292$	0	0
$293$	6.57595	0.384171	0.192085	−	0.981378i	$-0.438475\pi$
$293$	6.57595	0.384171	0.192085	+	0.981378i	$0.438475\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−8.67282	−0.503248
$298$	0	0
$299$	11.1625	0.645542
$300$	0	0
$301$	−27.5473	−1.58780
$302$	0	0
$303$	−5.53279	−0.317850
$304$	0	0
$305$	0	0
$306$	0	0
$307$	7.26724	0.414763	0.207382	−	0.978260i	$-0.433506\pi$
$307$	7.26724	0.414763	0.207382	+	0.978260i	$0.433506\pi$
$308$	0	0
$309$	0.863919	0.0491466
$310$	0	0
$311$	5.55126	0.314783	0.157392	−	0.987536i	$-0.449692\pi$
$311$	5.55126	0.314783	0.157392	+	0.987536i	$0.449692\pi$
$312$	0	0
$313$	−22.5266	−1.27328	−0.636639	−	0.771162i	$-0.719676\pi$
$313$	−22.5266	−1.27328	−0.636639	+	0.771162i	$0.719676\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−29.9506	−1.68219	−0.841097	−	0.540884i	$-0.818090\pi$
$317$	−29.9506	−1.68219	−0.841097	+	0.540884i	$0.818090\pi$
$318$	0	0
$319$	2.28797	0.128102
$320$	0	0
$321$	6.54731	0.365435
$322$	0	0
$323$	2.00000	0.111283
$324$	0	0
$325$	0	0
$326$	0	0
$327$	8.68339	0.480193
$328$	0	0
$329$	3.71203	0.204651
$330$	0	0
$331$	−21.1809	−1.16421	−0.582105	−	0.813114i	$-0.697771\pi$
$331$	−21.1809	−1.16421	−0.582105	+	0.813114i	$0.697771\pi$
$332$	0	0
$333$	2.28797	0.125380
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−19.4610	−1.06011	−0.530054	−	0.847964i	$-0.677828\pi$
$337$	−19.4610	−1.06011	−0.530054	+	0.847964i	$0.677828\pi$
$338$	0	0
$339$	−3.59668	−0.195345
$340$	0	0
$341$	3.81681	0.206692
$342$	0	0
$343$	11.2633	0.608160
$344$	0	0
$345$	0	0
$346$	0	0
$347$	8.10083	0.434875	0.217438	−	0.976074i	$-0.430230\pi$
$347$	8.10083	0.434875	0.217438	+	0.976074i	$0.430230\pi$
$348$	0	0
$349$	−15.6913	−0.839936	−0.419968	−	0.907539i	$-0.637959\pi$
$349$	−15.6913	−0.839936	−0.419968	+	0.907539i	$0.637959\pi$
$350$	0	0
$351$	−5.42801	−0.289726
$352$	0	0
$353$	22.9585	1.22196	0.610980	−	0.791646i	$-0.290776\pi$
$353$	22.9585	1.22196	0.610980	+	0.791646i	$0.290776\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	1.85601	0.0982306
$358$	0	0
$359$	10.5759	0.558177	0.279089	−	0.960265i	$-0.409968\pi$
$359$	10.5759	0.558177	0.279089	+	0.960265i	$0.409968\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	−2.20561	−0.115765
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−15.2554	−0.796324	−0.398162	−	0.917315i	$-0.630352\pi$
$367$	−15.2554	−0.796324	−0.398162	+	0.917315i	$0.630352\pi$
$368$	0	0
$369$	7.63362	0.397390
$370$	0	0
$371$	9.73445	0.505388
$372$	0	0
$373$	29.4297	1.52381	0.761906	−	0.647688i	$-0.224264\pi$
$373$	29.4297	1.52381	0.761906	+	0.647688i	$0.224264\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.43196	0.0737497
$378$	0	0
$379$	−12.7552	−0.655190	−0.327595	−	0.944818i	$-0.606238\pi$
$379$	−12.7552	−0.655190	−0.327595	+	0.944818i	$0.606238\pi$
$380$	0	0
$381$	−4.85601	−0.248781
$382$	0	0
$383$	37.3042	1.90615	0.953077	−	0.302727i	$-0.0978970\pi$
$383$	37.3042	1.90615	0.953077	+	0.302727i	$0.0978970\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−22.6913	−1.15346
$388$	0	0
$389$	−15.4320	−0.782431	−0.391216	−	0.920299i	$-0.627945\pi$
$389$	−15.4320	−0.782431	−0.391216	+	0.920299i	$0.627945\pi$
$390$	0	0
$391$	−6.67282	−0.337459
$392$	0	0
$393$	7.97136	0.402102
$394$	0	0
$395$	0	0
$396$	0	0
$397$	20.5759	1.03268	0.516339	−	0.856385i	$-0.327295\pi$
$397$	20.5759	1.03268	0.516339	+	0.856385i	$0.327295\pi$
$398$	0	0
$399$	3.71203	0.185834
$400$	0	0
$401$	25.5473	1.27577	0.637886	−	0.770131i	$-0.279809\pi$
$401$	25.5473	1.27577	0.637886	+	0.770131i	$0.279809\pi$
$402$	0	0
$403$	2.38880	0.118995
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2.28797	−0.113411
$408$	0	0
$409$	10.1809	0.503415	0.251707	−	0.967803i	$-0.419008\pi$
$409$	10.1809	0.503415	0.251707	+	0.967803i	$0.419008\pi$
$410$	0	0
$411$	−12.6768	−0.625299
$412$	0	0
$413$	10.2017	0.501991
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−7.68073	−0.376127
$418$	0	0
$419$	−25.5737	−1.24936	−0.624678	−	0.780882i	$-0.714770\pi$
$419$	−25.5737	−1.24936	−0.624678	+	0.780882i	$0.714770\pi$
$420$	0	0
$421$	13.5944	0.662551	0.331276	−	0.943534i	$-0.392521\pi$
$421$	13.5944	0.662551	0.331276	+	0.943534i	$0.392521\pi$
$422$	0	0
$423$	3.05767	0.148669
$424$	0	0
$425$	0	0
$426$	0	0
$427$	12.0448	0.582891
$428$	0	0
$429$	2.55748	0.123476
$430$	0	0
$431$	−23.7160	−1.14236	−0.571179	−	0.820825i	$-0.693514\pi$
$431$	−23.7160	−1.14236	−0.571179	+	0.820825i	$0.693514\pi$
$432$	0	0
$433$	21.5473	1.03550	0.517749	−	0.855533i	$-0.326770\pi$
$433$	21.5473	1.03550	0.517749	+	0.855533i	$0.326770\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−13.3456	−0.638409
$438$	0	0
$439$	12.0224	0.573799	0.286899	−	0.957961i	$-0.407375\pi$
$439$	12.0224	0.573799	0.286899	+	0.957961i	$0.407375\pi$
$440$	0	0
$441$	−9.43196	−0.449141
$442$	0	0
$443$	−32.5187	−1.54501	−0.772504	−	0.635009i	$-0.780996\pi$
$443$	−32.5187	−1.54501	−0.772504	+	0.635009i	$0.780996\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−3.30475	−0.156309
$448$	0	0
$449$	14.8145	0.699142	0.349571	−	0.936910i	$-0.386327\pi$
$449$	14.8145	0.699142	0.349571	+	0.936910i	$0.386327\pi$
$450$	0	0
$451$	−7.63362	−0.359453
$452$	0	0
$453$	−4.48173	−0.210570
$454$	0	0
$455$	0	0
$456$	0	0
$457$	6.65209	0.311172	0.155586	−	0.987822i	$-0.450273\pi$
$457$	6.65209	0.311172	0.155586	+	0.987822i	$0.450273\pi$
$458$	0	0
$459$	3.24482	0.151455
$460$	0	0
$461$	33.2672	1.54941	0.774705	−	0.632323i	$-0.217898\pi$
$461$	33.2672	1.54941	0.774705	+	0.632323i	$0.217898\pi$
$462$	0	0
$463$	−3.91369	−0.181884	−0.0909422	−	0.995856i	$-0.528988\pi$
$463$	−3.91369	−0.181884	−0.0909422	+	0.995856i	$0.528988\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	10.8560	0.502356	0.251178	−	0.967941i	$-0.419182\pi$
$467$	10.8560	0.502356	0.251178	+	0.967941i	$0.419182\pi$
$468$	0	0
$469$	−22.9009	−1.05746
$470$	0	0
$471$	7.83133	0.360849
$472$	0	0
$473$	22.6913	1.04335
$474$	0	0
$475$	0	0
$476$	0	0
$477$	8.01847	0.367141
$478$	0	0
$479$	3.40558	0.155605	0.0778025	−	0.996969i	$-0.475210\pi$
$479$	3.40558	0.155605	0.0778025	+	0.996969i	$0.475210\pi$
$480$	0	0
$481$	−1.43196	−0.0652917
$482$	0	0
$483$	−12.3849	−0.563530
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−4.92385	−0.223121	−0.111561	−	0.993758i	$-0.535585\pi$
$487$	−4.92385	−0.223121	−0.111561	+	0.993758i	$0.535585\pi$
$488$	0	0
$489$	−4.64419	−0.210017
$490$	0	0
$491$	−13.5552	−0.611738	−0.305869	−	0.952074i	$-0.598947\pi$
$491$	−13.5552	−0.611738	−0.305869	+	0.952074i	$0.598947\pi$
$492$	0	0
$493$	−0.856013	−0.0385529
$494$	0	0
$495$	0	0
$496$	0	0
$497$	16.0841	0.721468
$498$	0	0
$499$	−16.0040	−0.716435	−0.358218	−	0.933638i	$-0.616615\pi$
$499$	−16.0040	−0.716435	−0.358218	+	0.933638i	$0.616615\pi$
$500$	0	0
$501$	−1.52884	−0.0683034
$502$	0	0
$503$	−7.28571	−0.324854	−0.162427	−	0.986721i	$-0.551932\pi$
$503$	−7.28571	−0.324854	−0.162427	+	0.986721i	$0.551932\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−5.83528	−0.259154
$508$	0	0
$509$	5.89296	0.261201	0.130600	−	0.991435i	$-0.458310\pi$
$509$	5.89296	0.261201	0.130600	+	0.991435i	$0.458310\pi$
$510$	0	0
$511$	27.5473	1.21862
$512$	0	0
$513$	6.48963	0.286524
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−3.05767	−0.134476
$518$	0	0
$519$	−5.67056	−0.248910
$520$	0	0
$521$	10.3664	0.454159	0.227080	−	0.973876i	$-0.427082\pi$
$521$	10.3664	0.454159	0.227080	+	0.973876i	$0.427082\pi$
$522$	0	0
$523$	−6.07841	−0.265790	−0.132895	−	0.991130i	$-0.542427\pi$
$523$	−6.07841	−0.265790	−0.132895	+	0.991130i	$0.542427\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.42801	−0.0622050
$528$	0	0
$529$	21.5266	0.935938
$530$	0	0
$531$	8.40332	0.364673
$532$	0	0
$533$	−4.77761	−0.206941
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−4.20166	−0.181315
$538$	0	0
$539$	9.43196	0.406263
$540$	0	0
$541$	28.1602	1.21070	0.605351	−	0.795959i	$-0.293033\pi$
$541$	28.1602	1.21070	0.605351	+	0.795959i	$0.293033\pi$
$542$	0	0
$543$	−10.3664	−0.444864
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−8.39671	−0.359017	−0.179509	−	0.983756i	$-0.557451\pi$
$547$	−8.39671	−0.359017	−0.179509	+	0.983756i	$0.557451\pi$
$548$	0	0
$549$	9.92159	0.423443
$550$	0	0
$551$	−1.71203	−0.0729348
$552$	0	0
$553$	47.3434	2.01325
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−8.61515	−0.365036	−0.182518	−	0.983203i	$-0.558425\pi$
$557$	−8.61515	−0.365036	−0.182518	+	0.983203i	$0.558425\pi$
$558$	0	0
$559$	14.2017	0.600666
$560$	0	0
$561$	−1.52884	−0.0645476
$562$	0	0
$563$	30.2465	1.27474	0.637369	−	0.770559i	$-0.280023\pi$
$563$	30.2465	1.27474	0.637369	+	0.770559i	$0.280023\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−19.9960	−0.839755
$568$	0	0
$569$	−35.3826	−1.48332	−0.741658	−	0.670779i	$-0.765960\pi$
$569$	−35.3826	−1.48332	−0.741658	+	0.670779i	$0.765960\pi$
$570$	0	0
$571$	−42.6952	−1.78674	−0.893370	−	0.449321i	$-0.851666\pi$
$571$	−42.6952	−1.78674	−0.893370	+	0.449321i	$0.851666\pi$
$572$	0	0
$573$	−0.814549	−0.0340283
$574$	0	0
$575$	0	0
$576$	0	0
$577$	28.9872	1.20675	0.603376	−	0.797457i	$-0.293822\pi$
$577$	28.9872	1.20675	0.603376	+	0.797457i	$0.293822\pi$
$578$	0	0
$579$	−12.4896	−0.519051
$580$	0	0
$581$	−12.9793	−0.538471
$582$	0	0
$583$	−8.01847	−0.332091
$584$	0	0
$585$	0	0
$586$	0	0
$587$	14.6544	0.604850	0.302425	−	0.953173i	$-0.402204\pi$
$587$	14.6544	0.604850	0.302425	+	0.953173i	$0.402204\pi$
$588$	0	0
$589$	−2.85601	−0.117680
$590$	0	0
$591$	2.68339	0.110380
$592$	0	0
$593$	17.6050	0.722950	0.361475	−	0.932382i	$-0.382273\pi$
$593$	17.6050	0.722950	0.361475	+	0.932382i	$0.382273\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	9.93216	0.406496
$598$	0	0
$599$	39.5473	1.61586	0.807930	−	0.589279i	$-0.200588\pi$
$599$	39.5473	1.61586	0.807930	+	0.589279i	$0.200588\pi$
$600$	0	0
$601$	13.3456	0.544380	0.272190	−	0.962243i	$-0.412252\pi$
$601$	13.3456	0.544380	0.272190	+	0.962243i	$0.412252\pi$
$602$	0	0
$603$	−18.8639	−0.768198
$604$	0	0
$605$	0	0
$606$	0	0
$607$	32.1272	1.30400	0.652002	−	0.758218i	$-0.273930\pi$
$607$	32.1272	1.30400	0.652002	+	0.758218i	$0.273930\pi$
$608$	0	0
$609$	−1.58877	−0.0643803
$610$	0	0
$611$	−1.91369	−0.0774195
$612$	0	0
$613$	−20.5843	−0.831390	−0.415695	−	0.909504i	$-0.636462\pi$
$613$	−20.5843	−0.831390	−0.415695	+	0.909504i	$0.636462\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	37.1025	1.49369	0.746846	−	0.664997i	$-0.231567\pi$
$617$	37.1025	1.49369	0.746846	+	0.664997i	$0.231567\pi$
$618$	0	0
$619$	−10.4633	−0.420554	−0.210277	−	0.977642i	$-0.567437\pi$
$619$	−10.4633	−0.420554	−0.210277	+	0.977642i	$0.567437\pi$
$620$	0	0
$621$	−21.6521	−0.868869
$622$	0	0
$623$	30.9793	1.24116
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−3.05767	−0.122112
$628$	0	0
$629$	0.856013	0.0341315
$630$	0	0
$631$	−1.39502	−0.0555348	−0.0277674	−	0.999614i	$-0.508840\pi$
$631$	−1.39502	−0.0555348	−0.0277674	+	0.999614i	$0.508840\pi$
$632$	0	0
$633$	0.761397	0.0302628
$634$	0	0
$635$	0	0
$636$	0	0
$637$	5.90312	0.233890
$638$	0	0
$639$	13.2488	0.524113
$640$	0	0
$641$	12.6992	0.501588	0.250794	−	0.968040i	$-0.419308\pi$
$641$	12.6992	0.501588	0.250794	+	0.968040i	$0.419308\pi$
$642$	0	0
$643$	−19.2448	−0.758941	−0.379471	−	0.925204i	$-0.623894\pi$
$643$	−19.2448	−0.758941	−0.379471	+	0.925204i	$0.623894\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−32.8930	−1.29316	−0.646578	−	0.762848i	$-0.723800\pi$
$647$	−32.8930	−1.29316	−0.646578	+	0.762848i	$0.723800\pi$
$648$	0	0
$649$	−8.40332	−0.329859
$650$	0	0
$651$	−2.65040	−0.103877
$652$	0	0
$653$	5.79834	0.226907	0.113453	−	0.993543i	$-0.463809\pi$
$653$	5.79834	0.226907	0.113453	+	0.993543i	$0.463809\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	22.6913	0.885272
$658$	0	0
$659$	−28.5187	−1.11093	−0.555465	−	0.831540i	$-0.687460\pi$
$659$	−28.5187	−1.11093	−0.555465	+	0.831540i	$0.687460\pi$
$660$	0	0
$661$	−44.0083	−1.71172	−0.855862	−	0.517204i	$-0.826973\pi$
$661$	−44.0083	−1.71172	−0.855862	+	0.517204i	$0.826973\pi$
$662$	0	0
$663$	−0.956844	−0.0371607
$664$	0	0
$665$	0	0
$666$	0	0
$667$	5.71203	0.221171
$668$	0	0
$669$	15.7120	0.607462
$670$	0	0
$671$	−9.92159	−0.383019
$672$	0	0
$673$	−11.9216	−0.459544	−0.229772	−	0.973245i	$-0.573798\pi$
$673$	−11.9216	−0.459544	−0.229772	+	0.973245i	$0.573798\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	4.61289	0.177288	0.0886439	−	0.996063i	$-0.471747\pi$
$677$	4.61289	0.177288	0.0886439	+	0.996063i	$0.471747\pi$
$678$	0	0
$679$	7.42405	0.284909
$680$	0	0
$681$	−5.95289	−0.228115
$682$	0	0
$683$	21.7345	0.831646	0.415823	−	0.909446i	$-0.363494\pi$
$683$	21.7345	0.831646	0.415823	+	0.909446i	$0.363494\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	3.19545	0.121914
$688$	0	0
$689$	−5.01847	−0.191189
$690$	0	0
$691$	−23.5249	−0.894929	−0.447464	−	0.894302i	$-0.647673\pi$
$691$	−23.5249	−0.894929	−0.447464	+	0.894302i	$0.647673\pi$
$692$	0	0
$693$	23.1809	0.880571
$694$	0	0
$695$	0	0
$696$	0	0
$697$	2.85601	0.108179
$698$	0	0
$699$	−4.64645	−0.175745
$700$	0	0
$701$	34.5266	1.30405	0.652025	−	0.758197i	$-0.273920\pi$
$701$	34.5266	1.30405	0.652025	+	0.758197i	$0.273920\pi$
$702$	0	0
$703$	1.71203	0.0645703
$704$	0	0
$705$	0	0
$706$	0	0
$707$	31.3865	1.18041
$708$	0	0
$709$	−48.1681	−1.80899	−0.904496	−	0.426483i	$-0.859752\pi$
$709$	−48.1681	−1.80899	−0.904496	+	0.426483i	$0.859752\pi$
$710$	0	0
$711$	38.9977	1.46253
$712$	0	0
$713$	9.52884	0.356858
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−5.78591	−0.216079
$718$	0	0
$719$	26.2840	0.980229	0.490114	−	0.871658i	$-0.336955\pi$
$719$	26.2840	0.980229	0.490114	+	0.871658i	$0.336955\pi$
$720$	0	0
$721$	−4.90086	−0.182518
$722$	0	0
$723$	−6.81907	−0.253604
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−30.2096	−1.12041	−0.560205	−	0.828354i	$-0.689278\pi$
$727$	−30.2096	−1.12041	−0.560205	+	0.828354i	$0.689278\pi$
$728$	0	0
$729$	−10.9031	−0.403819
$730$	0	0
$731$	−8.48963	−0.314000
$732$	0	0
$733$	−23.4896	−0.867609	−0.433805	−	0.901007i	$-0.642829\pi$
$733$	−23.4896	−0.867609	−0.433805	+	0.901007i	$0.642829\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	18.8639	0.694861
$738$	0	0
$739$	−6.66226	−0.245075	−0.122538	−	0.992464i	$-0.539103\pi$
$739$	−6.66226	−0.245075	−0.122538	+	0.992464i	$0.539103\pi$
$740$	0	0
$741$	−1.91369	−0.0703011
$742$	0	0
$743$	−21.2554	−0.779784	−0.389892	−	0.920861i	$-0.627488\pi$
$743$	−21.2554	−0.779784	−0.389892	+	0.920861i	$0.627488\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−10.6913	−0.391174
$748$	0	0
$749$	−37.1417	−1.35713
$750$	0	0
$751$	−31.8458	−1.16207	−0.581036	−	0.813878i	$-0.697352\pi$
$751$	−31.8458	−1.16207	−0.581036	+	0.813878i	$0.697352\pi$
$752$	0	0
$753$	−16.0369	−0.584419
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−4.41123	−0.160329	−0.0801644	−	0.996782i	$-0.525545\pi$
$757$	−4.41123	−0.160329	−0.0801644	+	0.996782i	$0.525545\pi$
$758$	0	0
$759$	10.2017	0.370297
$760$	0	0
$761$	29.0969	1.05476	0.527380	−	0.849629i	$-0.323174\pi$
$761$	29.0969	1.05476	0.527380	+	0.849629i	$0.323174\pi$
$762$	0	0
$763$	−49.2593	−1.78331
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−5.25934	−0.189904
$768$	0	0
$769$	9.11269	0.328612	0.164306	−	0.986409i	$-0.447462\pi$
$769$	9.11269	0.328612	0.164306	+	0.986409i	$0.447462\pi$
$770$	0	0
$771$	−9.08462	−0.327175
$772$	0	0
$773$	−29.7467	−1.06991	−0.534957	−	0.844879i	$-0.679672\pi$
$773$	−29.7467	−1.06991	−0.534957	+	0.844879i	$0.679672\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	1.58877	0.0569969
$778$	0	0
$779$	5.71203	0.204655
$780$	0	0
$781$	−13.2488	−0.474078
$782$	0	0
$783$	−2.77761	−0.0992636
$784$	0	0
$785$	0	0
$786$	0	0
$787$	17.2554	0.615088	0.307544	−	0.951534i	$-0.400493\pi$
$787$	17.2554	0.615088	0.307544	+	0.951534i	$0.400493\pi$
$788$	0	0
$789$	−1.53940	−0.0548042
$790$	0	0
$791$	20.4033	0.725459
$792$	0	0
$793$	−6.20957	−0.220508
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−25.7776	−0.913090	−0.456545	−	0.889700i	$-0.650913\pi$
$797$	−25.7776	−0.913090	−0.456545	+	0.889700i	$0.650913\pi$
$798$	0	0
$799$	1.14399	0.0404713
$800$	0	0
$801$	25.5183	0.901644
$802$	0	0
$803$	−22.6913	−0.800758
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−7.79834	−0.274515
$808$	0	0
$809$	−14.9423	−0.525344	−0.262672	−	0.964885i	$-0.584604\pi$
$809$	−14.9423	−0.525344	−0.262672	+	0.964885i	$0.584604\pi$
$810$	0	0
$811$	−28.3658	−0.996058	−0.498029	−	0.867160i	$-0.665943\pi$
$811$	−28.3658	−0.996058	−0.498029	+	0.867160i	$0.665943\pi$
$812$	0	0
$813$	4.20166	0.147359
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−16.9793	−0.594029
$818$	0	0
$819$	14.5081	0.506954
$820$	0	0
$821$	47.0162	1.64088	0.820439	−	0.571735i	$-0.193729\pi$
$821$	47.0162	1.64088	0.820439	+	0.571735i	$0.193729\pi$
$822$	0	0
$823$	−27.3681	−0.953991	−0.476995	−	0.878906i	$-0.658274\pi$
$823$	−27.3681	−0.953991	−0.476995	+	0.878906i	$0.658274\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−54.0139	−1.87825	−0.939125	−	0.343577i	$-0.888361\pi$
$827$	−54.0139	−1.87825	−0.939125	+	0.343577i	$0.888361\pi$
$828$	0	0
$829$	−23.9793	−0.832834	−0.416417	−	0.909174i	$-0.636714\pi$
$829$	−23.9793	−0.832834	−0.416417	+	0.909174i	$0.636714\pi$
$830$	0	0
$831$	12.4448	0.431705
$832$	0	0
$833$	−3.52884	−0.122267
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−4.63362	−0.160161
$838$	0	0
$839$	15.2554	0.526674	0.263337	−	0.964704i	$-0.415177\pi$
$839$	15.2554	0.526674	0.263337	+	0.964704i	$0.415177\pi$
$840$	0	0
$841$	−28.2672	−0.974732
$842$	0	0
$843$	−18.2919	−0.630007
$844$	0	0
$845$	0	0
$846$	0	0
$847$	12.5121	0.429919
$848$	0	0
$849$	5.89522	0.202323
$850$	0	0
$851$	−5.71203	−0.195806
$852$	0	0
$853$	34.1233	1.16836	0.584179	−	0.811625i	$-0.301417\pi$
$853$	34.1233	1.16836	0.584179	+	0.811625i	$0.301417\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−29.0656	−0.992861	−0.496431	−	0.868076i	$-0.665356\pi$
$857$	−29.0656	−0.992861	−0.496431	+	0.868076i	$0.665356\pi$
$858$	0	0
$859$	9.71993	0.331640	0.165820	−	0.986156i	$-0.446973\pi$
$859$	9.71993	0.331640	0.165820	+	0.986156i	$0.446973\pi$
$860$	0	0
$861$	5.30080	0.180651
$862$	0	0
$863$	11.2303	0.382284	0.191142	−	0.981562i	$-0.438781\pi$
$863$	11.2303	0.382284	0.191142	+	0.981562i	$0.438781\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0.571993	0.0194259
$868$	0	0
$869$	−38.9977	−1.32291
$870$	0	0
$871$	11.8062	0.400039
$872$	0	0
$873$	6.11535	0.206973
$874$	0	0
$875$	0	0
$876$	0	0
$877$	11.8767	0.401049	0.200525	−	0.979689i	$-0.435735\pi$
$877$	11.8767	0.401049	0.200525	+	0.979689i	$0.435735\pi$
$878$	0	0
$879$	3.76140	0.126869
$880$	0	0
$881$	−14.1233	−0.475825	−0.237912	−	0.971287i	$-0.576463\pi$
$881$	−14.1233	−0.475825	−0.237912	+	0.971287i	$0.576463\pi$
$882$	0	0
$883$	−2.53900	−0.0854443	−0.0427221	−	0.999087i	$-0.513603\pi$
$883$	−2.53900	−0.0854443	−0.0427221	+	0.999087i	$0.513603\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−29.8471	−1.00217	−0.501084	−	0.865398i	$-0.667065\pi$
$887$	−29.8471	−1.00217	−0.501084	+	0.865398i	$0.667065\pi$
$888$	0	0
$889$	27.5473	0.923907
$890$	0	0
$891$	16.4712	0.551805
$892$	0	0
$893$	2.28797	0.0765641
$894$	0	0
$895$	0	0
$896$	0	0
$897$	6.38485	0.213184
$898$	0	0
$899$	1.22239	0.0407691
$900$	0	0
$901$	3.00000	0.0999445
$902$	0	0
$903$	−15.7569	−0.524356
$904$	0	0
$905$	0	0
$906$	0	0
$907$	10.8824	0.361344	0.180672	−	0.983543i	$-0.442173\pi$
$907$	10.8824	0.361344	0.180672	+	0.983543i	$0.442173\pi$
$908$	0	0
$909$	25.8538	0.857515
$910$	0	0
$911$	−5.73445	−0.189991	−0.0949954	−	0.995478i	$-0.530284\pi$
$911$	−5.73445	−0.189991	−0.0949954	+	0.995478i	$0.530284\pi$
$912$	0	0
$913$	10.6913	0.353830
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−45.2201	−1.49330
$918$	0	0
$919$	16.1233	0.531857	0.265929	−	0.963993i	$-0.414321\pi$
$919$	16.1233	0.531857	0.265929	+	0.963993i	$0.414321\pi$
$920$	0	0
$921$	4.15681	0.136972
$922$	0	0
$923$	−8.29193	−0.272932
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−4.03694	−0.132591
$928$	0	0
$929$	−2.77761	−0.0911303	−0.0455652	−	0.998961i	$-0.514509\pi$
$929$	−2.77761	−0.0911303	−0.0455652	+	0.998961i	$0.514509\pi$
$930$	0	0
$931$	−7.05767	−0.231306
$932$	0	0
$933$	3.17528	0.103954
$934$	0	0
$935$	0	0
$936$	0	0
$937$	30.7411	1.00427	0.502133	−	0.864790i	$-0.332549\pi$
$937$	30.7411	1.00427	0.502133	+	0.864790i	$0.332549\pi$
$938$	0	0
$939$	−12.8850	−0.420488
$940$	0	0
$941$	52.7776	1.72050	0.860250	−	0.509872i	$-0.170307\pi$
$941$	52.7776	1.72050	0.860250	+	0.509872i	$0.170307\pi$
$942$	0	0
$943$	−19.0577	−0.620603
$944$	0	0
$945$	0	0
$946$	0	0
$947$	61.3641	1.99407	0.997033	−	0.0769758i	$-0.0245264\pi$
$947$	61.3641	1.99407	0.997033	+	0.0769758i	$0.0245264\pi$
$948$	0	0
$949$	−14.2017	−0.461005
$950$	0	0
$951$	−17.1316	−0.555529
$952$	0	0
$953$	−17.5002	−0.566887	−0.283444	−	0.958989i	$-0.591477\pi$
$953$	−17.5002	−0.566887	−0.283444	+	0.958989i	$0.591477\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	1.30871	0.0423044
$958$	0	0
$959$	71.9131	2.32220
$960$	0	0
$961$	−28.9608	−0.934219
$962$	0	0
$963$	−30.5944	−0.985891
$964$	0	0
$965$	0	0
$966$	0	0
$967$	11.1440	0.358366	0.179183	−	0.983816i	$-0.442655\pi$
$967$	11.1440	0.358366	0.179183	+	0.983816i	$0.442655\pi$
$968$	0	0
$969$	1.14399	0.0367501
$970$	0	0
$971$	−33.9216	−1.08860	−0.544298	−	0.838892i	$-0.683204\pi$
$971$	−33.9216	−1.08860	−0.544298	+	0.838892i	$0.683204\pi$
$972$	0	0
$973$	43.5714	1.39684
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−24.3377	−0.778633	−0.389317	−	0.921104i	$-0.627289\pi$
$977$	−24.3377	−0.778633	−0.389317	+	0.921104i	$0.627289\pi$
$978$	0	0
$979$	−25.5183	−0.815568
$980$	0	0
$981$	−40.5759	−1.29549
$982$	0	0
$983$	−19.6191	−0.625752	−0.312876	−	0.949794i	$-0.601292\pi$
$983$	−19.6191	−0.625752	−0.312876	+	0.949794i	$0.601292\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	2.12325	0.0675839
$988$	0	0
$989$	56.6498	1.80136
$990$	0	0
$991$	42.2011	1.34056	0.670281	−	0.742107i	$-0.266174\pi$
$991$	42.2011	1.34056	0.670281	+	0.742107i	$0.266174\pi$
$992$	0	0
$993$	−12.1153	−0.384469
$994$	0	0
$995$	0	0
$996$	0	0
$997$	16.7697	0.531102	0.265551	−	0.964097i	$-0.414446\pi$
$997$	16.7697	0.531102	0.265551	+	0.964097i	$0.414446\pi$
$998$	0	0
$999$	2.77761	0.0878796

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6800.2.a.bq.1.2		3
4.3	odd	2		3400.2.a.m.1.2	✓	3
5.4	even	2		6800.2.a.bj.1.2		3
20.3	even	4		3400.2.e.l.2449.3		6
20.7	even	4		3400.2.e.l.2449.4		6
20.19	odd	2		3400.2.a.q.1.2	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3400.2.a.m.1.2	✓	3	4.3	odd	2
3400.2.a.q.1.2	yes	3	20.19	odd	2
3400.2.e.l.2449.3		6	20.3	even	4
3400.2.e.l.2449.4		6	20.7	even	4
6800.2.a.bj.1.2		3	5.4	even	2
6800.2.a.bq.1.2		3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6800 = 2^{4} \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6800.a (trivial)

\(f(q)\)	\(=\)	\(q+0.571993 q^{3} -3.24482 q^{7} -2.67282 q^{9} +O(q^{10})\)	\(q+0.571993 q^{3} -3.24482 q^{7} -2.67282 q^{9} +2.67282 q^{11} +1.67282 q^{13} -1.00000 q^{17} -2.00000 q^{19} -1.85601 q^{21} +6.67282 q^{23} -3.24482 q^{27} +0.856013 q^{29} +1.42801 q^{31} +1.52884 q^{33} -0.856013 q^{37} +0.956844 q^{39} -2.85601 q^{41} +8.48963 q^{43} -1.14399 q^{47} +3.52884 q^{49} -0.571993 q^{51} -3.00000 q^{53} -1.14399 q^{57} -3.14399 q^{59} -3.71203 q^{61} +8.67282 q^{63} +7.05767 q^{67} +3.81681 q^{69} -4.95684 q^{71} -8.48963 q^{73} -8.67282 q^{77} -14.5905 q^{79} +6.16246 q^{81} +4.00000 q^{83} +0.489634 q^{87} -9.54731 q^{89} -5.42801 q^{91} +0.816810 q^{93} -2.28797 q^{97} -7.14399 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{3} + q^{7} + 2 q^{9}+O(q^{10}) \) 3 * q + q^3 + q^7 + 2 * q^9	\( 3 q + q^{3} + q^{7} + 2 q^{9} - 2 q^{11} - 5 q^{13} - 3 q^{17} - 6 q^{19} - 7 q^{21} + 10 q^{23} + q^{27} + 4 q^{29} + 5 q^{31} - 4 q^{33} - 4 q^{37} - 5 q^{39} - 10 q^{41} + 4 q^{43} - 2 q^{47} + 2 q^{49} - q^{51} - 9 q^{53} - 2 q^{57} - 8 q^{59} - 14 q^{61} + 16 q^{63} + 4 q^{67} - 7 q^{71} - 4 q^{73} - 16 q^{77} - 13 q^{79} - 13 q^{81} + 12 q^{83} - 20 q^{87} + 10 q^{89} - 17 q^{91} - 9 q^{93} - 4 q^{97} - 20 q^{99}+O(q^{100}) \) 3 * q + q^3 + q^7 + 2 * q^9 - 2 * q^11 - 5 * q^13 - 3 * q^17 - 6 * q^19 - 7 * q^21 + 10 * q^23 + q^27 + 4 * q^29 + 5 * q^31 - 4 * q^33 - 4 * q^37 - 5 * q^39 - 10 * q^41 + 4 * q^43 - 2 * q^47 + 2 * q^49 - q^51 - 9 * q^53 - 2 * q^57 - 8 * q^59 - 14 * q^61 + 16 * q^63 + 4 * q^67 - 7 * q^71 - 4 * q^73 - 16 * q^77 - 13 * q^79 - 13 * q^81 + 12 * q^83 - 20 * q^87 + 10 * q^89 - 17 * q^91 - 9 * q^93 - 4 * q^97 - 20 * q^99

Introduction

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