LMFDB - Embedded newform 6900.2.a.j.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6900,2,Mod(1,6900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6900, 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("6900.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6900.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:	$55.0967773947$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{15}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 15 $ x^2 - 15
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1380)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.87298$ of defining polynomial
Character	$\chi$	$=$	6900.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -3.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -3.00000 q^{7} +1.00000 q^{9} +4.87298 q^{11} +2.87298 q^{13} -3.87298 q^{17} -2.87298 q^{19} +3.00000 q^{21} +1.00000 q^{23} -1.00000 q^{27} -5.87298 q^{29} +3.00000 q^{31} -4.87298 q^{33} -1.00000 q^{37} -2.87298 q^{39} -5.87298 q^{41} -3.74597 q^{43} +6.87298 q^{47} +2.00000 q^{49} +3.87298 q^{51} +3.87298 q^{53} +2.87298 q^{57} +5.87298 q^{59} +8.87298 q^{61} -3.00000 q^{63} -10.7460 q^{67} -1.00000 q^{69} -13.6190 q^{71} +2.87298 q^{73} -14.6190 q^{77} +4.00000 q^{79} +1.00000 q^{81} +0.127017 q^{83} +5.87298 q^{87} +1.74597 q^{89} -8.61895 q^{91} -3.00000 q^{93} -8.00000 q^{97} +4.87298 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 6 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 6 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} - 6 q^{7} + 2 q^{9} + 2 q^{11} - 2 q^{13} + 2 q^{19} + 6 q^{21} + 2 q^{23} - 2 q^{27} - 4 q^{29} + 6 q^{31} - 2 q^{33} - 2 q^{37} + 2 q^{39} - 4 q^{41} + 8 q^{43} + 6 q^{47} + 4 q^{49} - 2 q^{57} + 4 q^{59} + 10 q^{61} - 6 q^{63} - 6 q^{67} - 2 q^{69} - 4 q^{71} - 2 q^{73} - 6 q^{77} + 8 q^{79} + 2 q^{81} + 8 q^{83} + 4 q^{87} - 12 q^{89} + 6 q^{91} - 6 q^{93} - 16 q^{97} + 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 6 * q^7 + 2 * q^9 + 2 * q^11 - 2 * q^13 + 2 * q^19 + 6 * q^21 + 2 * q^23 - 2 * q^27 - 4 * q^29 + 6 * q^31 - 2 * q^33 - 2 * q^37 + 2 * q^39 - 4 * q^41 + 8 * q^43 + 6 * q^47 + 4 * q^49 - 2 * q^57 + 4 * q^59 + 10 * q^61 - 6 * q^63 - 6 * q^67 - 2 * q^69 - 4 * q^71 - 2 * q^73 - 6 * q^77 + 8 * q^79 + 2 * q^81 + 8 * q^83 + 4 * q^87 - 12 * q^89 + 6 * q^91 - 6 * q^93 - 16 * q^97 + 2 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.87298	1.46926	0.734630	−	0.678468i	$-0.237356\pi$
$11$	4.87298	1.46926	0.734630	+	0.678468i	$0.237356\pi$
$12$	0	0
$13$	2.87298	0.796822	0.398411	−	0.917207i	$-0.369562\pi$
$13$	2.87298	0.796822	0.398411	+	0.917207i	$0.369562\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.87298	−0.939336	−0.469668	−	0.882843i	$-0.655626\pi$
$17$	−3.87298	−0.939336	−0.469668	+	0.882843i	$0.655626\pi$
$18$	0	0
$19$	−2.87298	−0.659108	−0.329554	−	0.944137i	$-0.606898\pi$
$19$	−2.87298	−0.659108	−0.329554	+	0.944137i	$0.606898\pi$
$20$	0	0
$21$	3.00000	0.654654
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−5.87298	−1.09059	−0.545293	−	0.838246i	$-0.683582\pi$
$29$	−5.87298	−1.09059	−0.545293	+	0.838246i	$0.683582\pi$
$30$	0	0
$31$	3.00000	0.538816	0.269408	−	0.963026i	$-0.413172\pi$
$31$	3.00000	0.538816	0.269408	+	0.963026i	$0.413172\pi$
$32$	0	0
$33$	−4.87298	−0.848278
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−1.00000	−0.164399	−0.0821995	−	0.996616i	$-0.526194\pi$
$37$	−1.00000	−0.164399	−0.0821995	+	0.996616i	$0.526194\pi$
$38$	0	0
$39$	−2.87298	−0.460046
$40$	0	0
$41$	−5.87298	−0.917206	−0.458603	−	0.888641i	$-0.651650\pi$
$41$	−5.87298	−0.917206	−0.458603	+	0.888641i	$0.651650\pi$
$42$	0	0
$43$	−3.74597	−0.571255	−0.285627	−	0.958341i	$-0.592202\pi$
$43$	−3.74597	−0.571255	−0.285627	+	0.958341i	$0.592202\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.87298	1.00253	0.501264	−	0.865295i	$-0.332869\pi$
$47$	6.87298	1.00253	0.501264	+	0.865295i	$0.332869\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	0	0
$51$	3.87298	0.542326
$52$	0	0
$53$	3.87298	0.531995	0.265998	−	0.963974i	$-0.414299\pi$
$53$	3.87298	0.531995	0.265998	+	0.963974i	$0.414299\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.87298	0.380536
$58$	0	0
$59$	5.87298	0.764597	0.382299	−	0.924039i	$-0.375133\pi$
$59$	5.87298	0.764597	0.382299	+	0.924039i	$0.375133\pi$
$60$	0	0
$61$	8.87298	1.13607	0.568035	−	0.823005i	$-0.307704\pi$
$61$	8.87298	1.13607	0.568035	+	0.823005i	$0.307704\pi$
$62$	0	0
$63$	−3.00000	−0.377964
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−10.7460	−1.31283	−0.656414	−	0.754401i	$-0.727928\pi$
$67$	−10.7460	−1.31283	−0.656414	+	0.754401i	$0.727928\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−13.6190	−1.61627	−0.808136	−	0.588996i	$-0.799523\pi$
$71$	−13.6190	−1.61627	−0.808136	+	0.588996i	$0.799523\pi$
$72$	0	0
$73$	2.87298	0.336257	0.168129	−	0.985765i	$-0.446228\pi$
$73$	2.87298	0.336257	0.168129	+	0.985765i	$0.446228\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−14.6190	−1.66598
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0.127017	0.0139419	0.00697094	−	0.999976i	$-0.497781\pi$
$83$	0.127017	0.0139419	0.00697094	+	0.999976i	$0.497781\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	5.87298	0.629650
$88$	0	0
$89$	1.74597	0.185072	0.0925360	−	0.995709i	$-0.470503\pi$
$89$	1.74597	0.185072	0.0925360	+	0.995709i	$0.470503\pi$
$90$	0	0
$91$	−8.61895	−0.903511
$92$	0	0
$93$	−3.00000	−0.311086
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−8.00000	−0.812277	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000	−0.812277	−0.406138	+	0.913812i	$0.633125\pi$
$98$	0	0
$99$	4.87298	0.489753
$100$	0	0
$101$	−3.87298	−0.385376	−0.192688	−	0.981260i	$-0.561721\pi$
$101$	−3.87298	−0.385376	−0.192688	+	0.981260i	$0.561721\pi$
$102$	0	0
$103$	6.00000	0.591198	0.295599	−	0.955312i	$-0.404481\pi$
$103$	6.00000	0.591198	0.295599	+	0.955312i	$0.404481\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.61895	0.543204	0.271602	−	0.962410i	$-0.412447\pi$
$107$	5.61895	0.543204	0.271602	+	0.962410i	$0.412447\pi$
$108$	0	0
$109$	16.6190	1.59181	0.795903	−	0.605424i	$-0.206996\pi$
$109$	16.6190	1.59181	0.795903	+	0.605424i	$0.206996\pi$
$110$	0	0
$111$	1.00000	0.0949158
$112$	0	0
$113$	3.87298	0.364340	0.182170	−	0.983267i	$-0.441688\pi$
$113$	3.87298	0.364340	0.182170	+	0.983267i	$0.441688\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.87298	0.265607
$118$	0	0
$119$	11.6190	1.06511
$120$	0	0
$121$	12.7460	1.15872
$122$	0	0
$123$	5.87298	0.529549
$124$	0	0
$125$	0	0
$126$	0	0
$127$	6.61895	0.587337	0.293668	−	0.955907i	$-0.405124\pi$
$127$	6.61895	0.587337	0.293668	+	0.955907i	$0.405124\pi$
$128$	0	0
$129$	3.74597	0.329814
$130$	0	0
$131$	11.4919	1.00405	0.502027	−	0.864852i	$-0.332588\pi$
$131$	11.4919	1.00405	0.502027	+	0.864852i	$0.332588\pi$
$132$	0	0
$133$	8.61895	0.747358
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−8.00000	−0.683486	−0.341743	−	0.939793i	$-0.611017\pi$
$137$	−8.00000	−0.683486	−0.341743	+	0.939793i	$0.611017\pi$
$138$	0	0
$139$	−20.4919	−1.73810	−0.869052	−	0.494722i	$-0.835270\pi$
$139$	−20.4919	−1.73810	−0.869052	+	0.494722i	$0.835270\pi$
$140$	0	0
$141$	−6.87298	−0.578810
$142$	0	0
$143$	14.0000	1.17074
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−2.00000	−0.164957
$148$	0	0
$149$	−10.8730	−0.890750	−0.445375	−	0.895344i	$-0.646930\pi$
$149$	−10.8730	−0.890750	−0.445375	+	0.895344i	$0.646930\pi$
$150$	0	0
$151$	3.74597	0.304842	0.152421	−	0.988316i	$-0.451293\pi$
$151$	3.74597	0.304842	0.152421	+	0.988316i	$0.451293\pi$
$152$	0	0
$153$	−3.87298	−0.313112
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−17.0000	−1.35675	−0.678374	−	0.734717i	$-0.737315\pi$
$157$	−17.0000	−1.35675	−0.678374	+	0.734717i	$0.737315\pi$
$158$	0	0
$159$	−3.87298	−0.307148
$160$	0	0
$161$	−3.00000	−0.236433
$162$	0	0
$163$	22.0000	1.72317	0.861586	−	0.507611i	$-0.169471\pi$
$163$	22.0000	1.72317	0.861586	+	0.507611i	$0.169471\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	9.12702	0.706270	0.353135	−	0.935572i	$-0.385116\pi$
$167$	9.12702	0.706270	0.353135	+	0.935572i	$0.385116\pi$
$168$	0	0
$169$	−4.74597	−0.365074
$170$	0	0
$171$	−2.87298	−0.219703
$172$	0	0
$173$	−23.4919	−1.78606	−0.893029	−	0.449999i	$-0.851425\pi$
$173$	−23.4919	−1.78606	−0.893029	+	0.449999i	$0.851425\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−5.87298	−0.441440
$178$	0	0
$179$	−15.7460	−1.17691	−0.588454	−	0.808530i	$-0.700263\pi$
$179$	−15.7460	−1.17691	−0.588454	+	0.808530i	$0.700263\pi$
$180$	0	0
$181$	−21.7460	−1.61636	−0.808182	−	0.588932i	$-0.799549\pi$
$181$	−21.7460	−1.61636	−0.808182	+	0.588932i	$0.799549\pi$
$182$	0	0
$183$	−8.87298	−0.655910
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−18.8730	−1.38013
$188$	0	0
$189$	3.00000	0.218218
$190$	0	0
$191$	−18.8730	−1.36560	−0.682801	−	0.730605i	$-0.739238\pi$
$191$	−18.8730	−1.36560	−0.682801	+	0.730605i	$0.739238\pi$
$192$	0	0
$193$	−9.74597	−0.701530	−0.350765	−	0.936464i	$-0.614078\pi$
$193$	−9.74597	−0.701530	−0.350765	+	0.936464i	$0.614078\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−4.25403	−0.303087	−0.151544	−	0.988451i	$-0.548424\pi$
$197$	−4.25403	−0.303087	−0.151544	+	0.988451i	$0.548424\pi$
$198$	0	0
$199$	19.2379	1.36374	0.681869	−	0.731474i	$-0.261167\pi$
$199$	19.2379	1.36374	0.681869	+	0.731474i	$0.261167\pi$
$200$	0	0
$201$	10.7460	0.757962
$202$	0	0
$203$	17.6190	1.23661
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−14.0000	−0.968400
$210$	0	0
$211$	1.00000	0.0688428	0.0344214	−	0.999407i	$-0.489041\pi$
$211$	1.00000	0.0688428	0.0344214	+	0.999407i	$0.489041\pi$
$212$	0	0
$213$	13.6190	0.933155
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−9.00000	−0.610960
$218$	0	0
$219$	−2.87298	−0.194138
$220$	0	0
$221$	−11.1270	−0.748484
$222$	0	0
$223$	2.00000	0.133930	0.0669650	−	0.997755i	$-0.478668\pi$
$223$	2.00000	0.133930	0.0669650	+	0.997755i	$0.478668\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−7.49193	−0.497257	−0.248629	−	0.968599i	$-0.579980\pi$
$227$	−7.49193	−0.497257	−0.248629	+	0.968599i	$0.579980\pi$
$228$	0	0
$229$	−5.74597	−0.379704	−0.189852	−	0.981813i	$-0.560801\pi$
$229$	−5.74597	−0.379704	−0.189852	+	0.981813i	$0.560801\pi$
$230$	0	0
$231$	14.6190	0.961856
$232$	0	0
$233$	−15.4919	−1.01491	−0.507455	−	0.861678i	$-0.669414\pi$
$233$	−15.4919	−1.01491	−0.507455	+	0.861678i	$0.669414\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−4.00000	−0.259828
$238$	0	0
$239$	−17.8730	−1.15611	−0.578054	−	0.815999i	$-0.696188\pi$
$239$	−17.8730	−1.15611	−0.578054	+	0.815999i	$0.696188\pi$
$240$	0	0
$241$	−5.12702	−0.330260	−0.165130	−	0.986272i	$-0.552804\pi$
$241$	−5.12702	−0.330260	−0.165130	+	0.986272i	$0.552804\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−8.25403	−0.525192
$248$	0	0
$249$	−0.127017	−0.00804935
$250$	0	0
$251$	2.00000	0.126239	0.0631194	−	0.998006i	$-0.479895\pi$
$251$	2.00000	0.126239	0.0631194	+	0.998006i	$0.479895\pi$
$252$	0	0
$253$	4.87298	0.306362
$254$	0	0
$255$	0	0
$256$	0	0
$257$	16.6190	1.03666	0.518331	−	0.855180i	$-0.326554\pi$
$257$	16.6190	1.03666	0.518331	+	0.855180i	$0.326554\pi$
$258$	0	0
$259$	3.00000	0.186411
$260$	0	0
$261$	−5.87298	−0.363529
$262$	0	0
$263$	21.8730	1.34875	0.674373	−	0.738391i	$-0.264414\pi$
$263$	21.8730	1.34875	0.674373	+	0.738391i	$0.264414\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−1.74597	−0.106851
$268$	0	0
$269$	−23.3649	−1.42458	−0.712292	−	0.701883i	$-0.752343\pi$
$269$	−23.3649	−1.42458	−0.712292	+	0.701883i	$0.752343\pi$
$270$	0	0
$271$	−7.25403	−0.440651	−0.220326	−	0.975426i	$-0.570712\pi$
$271$	−7.25403	−0.440651	−0.220326	+	0.975426i	$0.570712\pi$
$272$	0	0
$273$	8.61895	0.521643
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−14.0000	−0.841178	−0.420589	−	0.907251i	$-0.638177\pi$
$277$	−14.0000	−0.841178	−0.420589	+	0.907251i	$0.638177\pi$
$278$	0	0
$279$	3.00000	0.179605
$280$	0	0
$281$	−16.3649	−0.976249	−0.488125	−	0.872774i	$-0.662319\pi$
$281$	−16.3649	−0.976249	−0.488125	+	0.872774i	$0.662319\pi$
$282$	0	0
$283$	−22.2379	−1.32191	−0.660953	−	0.750427i	$-0.729848\pi$
$283$	−22.2379	−1.32191	−0.660953	+	0.750427i	$0.729848\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	17.6190	1.04001
$288$	0	0
$289$	−2.00000	−0.117647
$290$	0	0
$291$	8.00000	0.468968
$292$	0	0
$293$	13.6190	0.795628	0.397814	−	0.917466i	$-0.369769\pi$
$293$	13.6190	0.795628	0.397814	+	0.917466i	$0.369769\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−4.87298	−0.282759
$298$	0	0
$299$	2.87298	0.166149
$300$	0	0
$301$	11.2379	0.647742
$302$	0	0
$303$	3.87298	0.222497
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−9.12702	−0.520906	−0.260453	−	0.965486i	$-0.583872\pi$
$307$	−9.12702	−0.520906	−0.260453	+	0.965486i	$0.583872\pi$
$308$	0	0
$309$	−6.00000	−0.341328
$310$	0	0
$311$	−4.00000	−0.226819	−0.113410	−	0.993548i	$-0.536177\pi$
$311$	−4.00000	−0.226819	−0.113410	+	0.993548i	$0.536177\pi$
$312$	0	0
$313$	−7.00000	−0.395663	−0.197832	−	0.980236i	$-0.563390\pi$
$313$	−7.00000	−0.395663	−0.197832	+	0.980236i	$0.563390\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−22.3649	−1.25614	−0.628069	−	0.778157i	$-0.716155\pi$
$317$	−22.3649	−1.25614	−0.628069	+	0.778157i	$0.716155\pi$
$318$	0	0
$319$	−28.6190	−1.60235
$320$	0	0
$321$	−5.61895	−0.313619
$322$	0	0
$323$	11.1270	0.619124
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−16.6190	−0.919030
$328$	0	0
$329$	−20.6190	−1.13676
$330$	0	0
$331$	−22.2379	−1.22231	−0.611153	−	0.791513i	$-0.709294\pi$
$331$	−22.2379	−1.22231	−0.611153	+	0.791513i	$0.709294\pi$
$332$	0	0
$333$	−1.00000	−0.0547997
$334$	0	0
$335$	0	0
$336$	0	0
$337$	1.49193	0.0812708	0.0406354	−	0.999174i	$-0.487062\pi$
$337$	1.49193	0.0812708	0.0406354	+	0.999174i	$0.487062\pi$
$338$	0	0
$339$	−3.87298	−0.210352
$340$	0	0
$341$	14.6190	0.791661
$342$	0	0
$343$	15.0000	0.809924
$344$	0	0
$345$	0	0
$346$	0	0
$347$	25.7460	1.38212	0.691058	−	0.722799i	$-0.257145\pi$
$347$	25.7460	1.38212	0.691058	+	0.722799i	$0.257145\pi$
$348$	0	0
$349$	−14.7460	−0.789334	−0.394667	−	0.918824i	$-0.629140\pi$
$349$	−14.7460	−0.789334	−0.394667	+	0.918824i	$0.629140\pi$
$350$	0	0
$351$	−2.87298	−0.153349
$352$	0	0
$353$	1.12702	0.0599850	0.0299925	−	0.999550i	$-0.490452\pi$
$353$	1.12702	0.0599850	0.0299925	+	0.999550i	$0.490452\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−11.6190	−0.614940
$358$	0	0
$359$	−21.1270	−1.11504	−0.557521	−	0.830163i	$-0.688247\pi$
$359$	−21.1270	−1.11504	−0.557521	+	0.830163i	$0.688247\pi$
$360$	0	0
$361$	−10.7460	−0.565577
$362$	0	0
$363$	−12.7460	−0.668990
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−25.9839	−1.35635	−0.678173	−	0.734902i	$-0.737228\pi$
$367$	−25.9839	−1.35635	−0.678173	+	0.734902i	$0.737228\pi$
$368$	0	0
$369$	−5.87298	−0.305735
$370$	0	0
$371$	−11.6190	−0.603226
$372$	0	0
$373$	−1.74597	−0.0904027	−0.0452014	−	0.998978i	$-0.514393\pi$
$373$	−1.74597	−0.0904027	−0.0452014	+	0.998978i	$0.514393\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−16.8730	−0.869003
$378$	0	0
$379$	−12.0000	−0.616399	−0.308199	−	0.951322i	$-0.599726\pi$
$379$	−12.0000	−0.616399	−0.308199	+	0.951322i	$0.599726\pi$
$380$	0	0
$381$	−6.61895	−0.339099
$382$	0	0
$383$	25.6190	1.30907	0.654534	−	0.756033i	$-0.272865\pi$
$383$	25.6190	1.30907	0.654534	+	0.756033i	$0.272865\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−3.74597	−0.190418
$388$	0	0
$389$	3.74597	0.189928	0.0949640	−	0.995481i	$-0.469726\pi$
$389$	3.74597	0.189928	0.0949640	+	0.995481i	$0.469726\pi$
$390$	0	0
$391$	−3.87298	−0.195865
$392$	0	0
$393$	−11.4919	−0.579691
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−19.4919	−0.978272	−0.489136	−	0.872208i	$-0.662688\pi$
$397$	−19.4919	−0.978272	−0.489136	+	0.872208i	$0.662688\pi$
$398$	0	0
$399$	−8.61895	−0.431487
$400$	0	0
$401$	27.2379	1.36020	0.680098	−	0.733121i	$-0.261937\pi$
$401$	27.2379	1.36020	0.680098	+	0.733121i	$0.261937\pi$
$402$	0	0
$403$	8.61895	0.429340
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−4.87298	−0.241545
$408$	0	0
$409$	22.2379	1.09959	0.549797	−	0.835299i	$-0.314705\pi$
$409$	22.2379	1.09959	0.549797	+	0.835299i	$0.314705\pi$
$410$	0	0
$411$	8.00000	0.394611
$412$	0	0
$413$	−17.6190	−0.866972
$414$	0	0
$415$	0	0
$416$	0	0
$417$	20.4919	1.00349
$418$	0	0
$419$	2.61895	0.127944	0.0639720	−	0.997952i	$-0.479623\pi$
$419$	2.61895	0.127944	0.0639720	+	0.997952i	$0.479623\pi$
$420$	0	0
$421$	−38.8730	−1.89455	−0.947277	−	0.320417i	$-0.896177\pi$
$421$	−38.8730	−1.89455	−0.947277	+	0.320417i	$0.896177\pi$
$422$	0	0
$423$	6.87298	0.334176
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−26.6190	−1.28818
$428$	0	0
$429$	−14.0000	−0.675926
$430$	0	0
$431$	5.74597	0.276773	0.138387	−	0.990378i	$-0.455808\pi$
$431$	5.74597	0.276773	0.138387	+	0.990378i	$0.455808\pi$
$432$	0	0
$433$	−19.0000	−0.913082	−0.456541	−	0.889702i	$-0.650912\pi$
$433$	−19.0000	−0.913082	−0.456541	+	0.889702i	$0.650912\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.87298	−0.137433
$438$	0	0
$439$	−26.0000	−1.24091	−0.620456	−	0.784241i	$-0.713053\pi$
$439$	−26.0000	−1.24091	−0.620456	+	0.784241i	$0.713053\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	0	0
$443$	24.6190	1.16968	0.584841	−	0.811148i	$-0.301157\pi$
$443$	24.6190	1.16968	0.584841	+	0.811148i	$0.301157\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	10.8730	0.514274
$448$	0	0
$449$	−21.8730	−1.03225	−0.516125	−	0.856513i	$-0.672626\pi$
$449$	−21.8730	−1.03225	−0.516125	+	0.856513i	$0.672626\pi$
$450$	0	0
$451$	−28.6190	−1.34761
$452$	0	0
$453$	−3.74597	−0.176001
$454$	0	0
$455$	0	0
$456$	0	0
$457$	17.0000	0.795226	0.397613	−	0.917553i	$-0.369839\pi$
$457$	17.0000	0.795226	0.397613	+	0.917553i	$0.369839\pi$
$458$	0	0
$459$	3.87298	0.180775
$460$	0	0
$461$	31.7460	1.47856	0.739279	−	0.673400i	$-0.235167\pi$
$461$	31.7460	1.47856	0.739279	+	0.673400i	$0.235167\pi$
$462$	0	0
$463$	−34.8730	−1.62068	−0.810342	−	0.585957i	$-0.800719\pi$
$463$	−34.8730	−1.62068	−0.810342	+	0.585957i	$0.800719\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	5.61895	0.260014	0.130007	−	0.991513i	$-0.458500\pi$
$467$	5.61895	0.260014	0.130007	+	0.991513i	$0.458500\pi$
$468$	0	0
$469$	32.2379	1.48861
$470$	0	0
$471$	17.0000	0.783319
$472$	0	0
$473$	−18.2540	−0.839321
$474$	0	0
$475$	0	0
$476$	0	0
$477$	3.87298	0.177332
$478$	0	0
$479$	−34.8730	−1.59339	−0.796694	−	0.604383i	$-0.793420\pi$
$479$	−34.8730	−1.59339	−0.796694	+	0.604383i	$0.793420\pi$
$480$	0	0
$481$	−2.87298	−0.130997
$482$	0	0
$483$	3.00000	0.136505
$484$	0	0
$485$	0	0
$486$	0	0
$487$	26.1109	1.18320	0.591599	−	0.806233i	$-0.298497\pi$
$487$	26.1109	1.18320	0.591599	+	0.806233i	$0.298497\pi$
$488$	0	0
$489$	−22.0000	−0.994874
$490$	0	0
$491$	28.1270	1.26935	0.634677	−	0.772777i	$-0.281133\pi$
$491$	28.1270	1.26935	0.634677	+	0.772777i	$0.281133\pi$
$492$	0	0
$493$	22.7460	1.02443
$494$	0	0
$495$	0	0
$496$	0	0
$497$	40.8569	1.83268
$498$	0	0
$499$	3.25403	0.145671	0.0728353	−	0.997344i	$-0.476795\pi$
$499$	3.25403	0.145671	0.0728353	+	0.997344i	$0.476795\pi$
$500$	0	0
$501$	−9.12702	−0.407765
$502$	0	0
$503$	37.1109	1.65469	0.827346	−	0.561692i	$-0.189849\pi$
$503$	37.1109	1.65469	0.827346	+	0.561692i	$0.189849\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	4.74597	0.210776
$508$	0	0
$509$	20.0000	0.886484	0.443242	−	0.896402i	$-0.353828\pi$
$509$	20.0000	0.886484	0.443242	+	0.896402i	$0.353828\pi$
$510$	0	0
$511$	−8.61895	−0.381280
$512$	0	0
$513$	2.87298	0.126845
$514$	0	0
$515$	0	0
$516$	0	0
$517$	33.4919	1.47297
$518$	0	0
$519$	23.4919	1.03118
$520$	0	0
$521$	40.8730	1.79068	0.895339	−	0.445385i	$-0.146933\pi$
$521$	40.8730	1.79068	0.895339	+	0.445385i	$0.146933\pi$
$522$	0	0
$523$	18.9839	0.830107	0.415053	−	0.909797i	$-0.363763\pi$
$523$	18.9839	0.830107	0.415053	+	0.909797i	$0.363763\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−11.6190	−0.506129
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	5.87298	0.254866
$532$	0	0
$533$	−16.8730	−0.730850
$534$	0	0
$535$	0	0
$536$	0	0
$537$	15.7460	0.679489
$538$	0	0
$539$	9.74597	0.419789
$540$	0	0
$541$	−40.0000	−1.71973	−0.859867	−	0.510518i	$-0.829454\pi$
$541$	−40.0000	−1.71973	−0.859867	+	0.510518i	$0.829454\pi$
$542$	0	0
$543$	21.7460	0.933209
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−16.0000	−0.684111	−0.342055	−	0.939680i	$-0.611123\pi$
$547$	−16.0000	−0.684111	−0.342055	+	0.939680i	$0.611123\pi$
$548$	0	0
$549$	8.87298	0.378690
$550$	0	0
$551$	16.8730	0.718813
$552$	0	0
$553$	−12.0000	−0.510292
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−28.1270	−1.19178	−0.595890	−	0.803066i	$-0.703201\pi$
$557$	−28.1270	−1.19178	−0.595890	+	0.803066i	$0.703201\pi$
$558$	0	0
$559$	−10.7621	−0.455188
$560$	0	0
$561$	18.8730	0.796818
$562$	0	0
$563$	−33.1109	−1.39546	−0.697729	−	0.716362i	$-0.745806\pi$
$563$	−33.1109	−1.39546	−0.697729	+	0.716362i	$0.745806\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−3.00000	−0.125988
$568$	0	0
$569$	4.25403	0.178338	0.0891692	−	0.996016i	$-0.471579\pi$
$569$	4.25403	0.178338	0.0891692	+	0.996016i	$0.471579\pi$
$570$	0	0
$571$	36.3649	1.52182	0.760912	−	0.648855i	$-0.224752\pi$
$571$	36.3649	1.52182	0.760912	+	0.648855i	$0.224752\pi$
$572$	0	0
$573$	18.8730	0.788430
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−8.00000	−0.333044	−0.166522	−	0.986038i	$-0.553254\pi$
$577$	−8.00000	−0.333044	−0.166522	+	0.986038i	$0.553254\pi$
$578$	0	0
$579$	9.74597	0.405029
$580$	0	0
$581$	−0.381050	−0.0158086
$582$	0	0
$583$	18.8730	0.781639
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−13.4919	−0.556872	−0.278436	−	0.960455i	$-0.589816\pi$
$587$	−13.4919	−0.556872	−0.278436	+	0.960455i	$0.589816\pi$
$588$	0	0
$589$	−8.61895	−0.355138
$590$	0	0
$591$	4.25403	0.174988
$592$	0	0
$593$	−44.6190	−1.83228	−0.916140	−	0.400858i	$-0.868712\pi$
$593$	−44.6190	−1.83228	−0.916140	+	0.400858i	$0.868712\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−19.2379	−0.787355
$598$	0	0
$599$	21.7460	0.888516	0.444258	−	0.895899i	$-0.353467\pi$
$599$	21.7460	0.888516	0.444258	+	0.895899i	$0.353467\pi$
$600$	0	0
$601$	−12.7460	−0.519919	−0.259959	−	0.965620i	$-0.583709\pi$
$601$	−12.7460	−0.519919	−0.259959	+	0.965620i	$0.583709\pi$
$602$	0	0
$603$	−10.7460	−0.437610
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−31.1270	−1.26341	−0.631703	−	0.775210i	$-0.717644\pi$
$607$	−31.1270	−1.26341	−0.631703	+	0.775210i	$0.717644\pi$
$608$	0	0
$609$	−17.6190	−0.713956
$610$	0	0
$611$	19.7460	0.798836
$612$	0	0
$613$	−48.9839	−1.97844	−0.989220	−	0.146438i	$-0.953219\pi$
$613$	−48.9839	−1.97844	−0.989220	+	0.146438i	$0.953219\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	17.6190	0.709312	0.354656	−	0.934997i	$-0.384598\pi$
$617$	17.6190	0.709312	0.354656	+	0.934997i	$0.384598\pi$
$618$	0	0
$619$	−16.0000	−0.643094	−0.321547	−	0.946894i	$-0.604203\pi$
$619$	−16.0000	−0.643094	−0.321547	+	0.946894i	$0.604203\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	−5.23790	−0.209852
$624$	0	0
$625$	0	0
$626$	0	0
$627$	14.0000	0.559106
$628$	0	0
$629$	3.87298	0.154426
$630$	0	0
$631$	32.3649	1.28843	0.644213	−	0.764846i	$-0.277185\pi$
$631$	32.3649	1.28843	0.644213	+	0.764846i	$0.277185\pi$
$632$	0	0
$633$	−1.00000	−0.0397464
$634$	0	0
$635$	0	0
$636$	0	0
$637$	5.74597	0.227663
$638$	0	0
$639$	−13.6190	−0.538757
$640$	0	0
$641$	−10.1109	−0.399356	−0.199678	−	0.979862i	$-0.563990\pi$
$641$	−10.1109	−0.399356	−0.199678	+	0.979862i	$0.563990\pi$
$642$	0	0
$643$	22.4919	0.886995	0.443498	−	0.896276i	$-0.353737\pi$
$643$	22.4919	0.886995	0.443498	+	0.896276i	$0.353737\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−27.8569	−1.09517	−0.547583	−	0.836751i	$-0.684452\pi$
$647$	−27.8569	−1.09517	−0.547583	+	0.836751i	$0.684452\pi$
$648$	0	0
$649$	28.6190	1.12339
$650$	0	0
$651$	9.00000	0.352738
$652$	0	0
$653$	10.1109	0.395669	0.197835	−	0.980235i	$-0.436609\pi$
$653$	10.1109	0.395669	0.197835	+	0.980235i	$0.436609\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	2.87298	0.112086
$658$	0	0
$659$	28.8730	1.12473	0.562366	−	0.826889i	$-0.309891\pi$
$659$	28.8730	1.12473	0.562366	+	0.826889i	$0.309891\pi$
$660$	0	0
$661$	−43.2379	−1.68176	−0.840880	−	0.541222i	$-0.817962\pi$
$661$	−43.2379	−1.68176	−0.840880	+	0.541222i	$0.817962\pi$
$662$	0	0
$663$	11.1270	0.432138
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−5.87298	−0.227403
$668$	0	0
$669$	−2.00000	−0.0773245
$670$	0	0
$671$	43.2379	1.66918
$672$	0	0
$673$	−19.1270	−0.737292	−0.368646	−	0.929570i	$-0.620179\pi$
$673$	−19.1270	−0.737292	−0.368646	+	0.929570i	$0.620179\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	7.11088	0.273293	0.136647	−	0.990620i	$-0.456367\pi$
$677$	7.11088	0.273293	0.136647	+	0.990620i	$0.456367\pi$
$678$	0	0
$679$	24.0000	0.921035
$680$	0	0
$681$	7.49193	0.287092
$682$	0	0
$683$	−17.1270	−0.655347	−0.327674	−	0.944791i	$-0.606265\pi$
$683$	−17.1270	−0.655347	−0.327674	+	0.944791i	$0.606265\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	5.74597	0.219222
$688$	0	0
$689$	11.1270	0.423906
$690$	0	0
$691$	46.7298	1.77769	0.888843	−	0.458211i	$-0.151510\pi$
$691$	46.7298	1.77769	0.888843	+	0.458211i	$0.151510\pi$
$692$	0	0
$693$	−14.6190	−0.555328
$694$	0	0
$695$	0	0
$696$	0	0
$697$	22.7460	0.861565
$698$	0	0
$699$	15.4919	0.585959
$700$	0	0
$701$	29.8569	1.12768	0.563839	−	0.825885i	$-0.309324\pi$
$701$	29.8569	1.12768	0.563839	+	0.825885i	$0.309324\pi$
$702$	0	0
$703$	2.87298	0.108357
$704$	0	0
$705$	0	0
$706$	0	0
$707$	11.6190	0.436976
$708$	0	0
$709$	−6.61895	−0.248580	−0.124290	−	0.992246i	$-0.539665\pi$
$709$	−6.61895	−0.248580	−0.124290	+	0.992246i	$0.539665\pi$
$710$	0	0
$711$	4.00000	0.150012
$712$	0	0
$713$	3.00000	0.112351
$714$	0	0
$715$	0	0
$716$	0	0
$717$	17.8730	0.667479
$718$	0	0
$719$	−39.3649	−1.46806	−0.734032	−	0.679115i	$-0.762364\pi$
$719$	−39.3649	−1.46806	−0.734032	+	0.679115i	$0.762364\pi$
$720$	0	0
$721$	−18.0000	−0.670355
$722$	0	0
$723$	5.12702	0.190676
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−11.2540	−0.417389	−0.208694	−	0.977981i	$-0.566921\pi$
$727$	−11.2540	−0.417389	−0.208694	+	0.977981i	$0.566921\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	14.5081	0.536600
$732$	0	0
$733$	−22.7460	−0.840141	−0.420071	−	0.907491i	$-0.637995\pi$
$733$	−22.7460	−0.840141	−0.420071	+	0.907491i	$0.637995\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−52.3649	−1.92889
$738$	0	0
$739$	13.2540	0.487557	0.243779	−	0.969831i	$-0.421613\pi$
$739$	13.2540	0.487557	0.243779	+	0.969831i	$0.421613\pi$
$740$	0	0
$741$	8.25403	0.303219
$742$	0	0
$743$	36.0000	1.32071	0.660356	−	0.750953i	$-0.270405\pi$
$743$	36.0000	1.32071	0.660356	+	0.750953i	$0.270405\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0.127017	0.00464730
$748$	0	0
$749$	−16.8569	−0.615936
$750$	0	0
$751$	−29.8569	−1.08949	−0.544746	−	0.838601i	$-0.683374\pi$
$751$	−29.8569	−1.08949	−0.544746	+	0.838601i	$0.683374\pi$
$752$	0	0
$753$	−2.00000	−0.0728841
$754$	0	0
$755$	0	0
$756$	0	0
$757$	4.74597	0.172495	0.0862475	−	0.996274i	$-0.472512\pi$
$757$	4.74597	0.172495	0.0862475	+	0.996274i	$0.472512\pi$
$758$	0	0
$759$	−4.87298	−0.176878
$760$	0	0
$761$	−14.8569	−0.538560	−0.269280	−	0.963062i	$-0.586786\pi$
$761$	−14.8569	−0.538560	−0.269280	+	0.963062i	$0.586786\pi$
$762$	0	0
$763$	−49.8569	−1.80494
$764$	0	0
$765$	0	0
$766$	0	0
$767$	16.8730	0.609248
$768$	0	0
$769$	28.8730	1.04119	0.520593	−	0.853805i	$-0.325711\pi$
$769$	28.8730	1.04119	0.520593	+	0.853805i	$0.325711\pi$
$770$	0	0
$771$	−16.6190	−0.598517
$772$	0	0
$773$	−4.25403	−0.153007	−0.0765035	−	0.997069i	$-0.524376\pi$
$773$	−4.25403	−0.153007	−0.0765035	+	0.997069i	$0.524376\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−3.00000	−0.107624
$778$	0	0
$779$	16.8730	0.604537
$780$	0	0
$781$	−66.3649	−2.37472
$782$	0	0
$783$	5.87298	0.209883
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−25.0000	−0.891154	−0.445577	−	0.895244i	$-0.647001\pi$
$787$	−25.0000	−0.891154	−0.445577	+	0.895244i	$0.647001\pi$
$788$	0	0
$789$	−21.8730	−0.778699
$790$	0	0
$791$	−11.6190	−0.413122
$792$	0	0
$793$	25.4919	0.905245
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−11.8730	−0.420563	−0.210281	−	0.977641i	$-0.567438\pi$
$797$	−11.8730	−0.420563	−0.210281	+	0.977641i	$0.567438\pi$
$798$	0	0
$799$	−26.6190	−0.941711
$800$	0	0
$801$	1.74597	0.0616907
$802$	0	0
$803$	14.0000	0.494049
$804$	0	0
$805$	0	0
$806$	0	0
$807$	23.3649	0.822484
$808$	0	0
$809$	24.8569	0.873920	0.436960	−	0.899481i	$-0.356055\pi$
$809$	24.8569	0.873920	0.436960	+	0.899481i	$0.356055\pi$
$810$	0	0
$811$	28.4919	1.00049	0.500244	−	0.865885i	$-0.333244\pi$
$811$	28.4919	1.00049	0.500244	+	0.865885i	$0.333244\pi$
$812$	0	0
$813$	7.25403	0.254410
$814$	0	0
$815$	0	0
$816$	0	0
$817$	10.7621	0.376518
$818$	0	0
$819$	−8.61895	−0.301170
$820$	0	0
$821$	−3.49193	−0.121869	−0.0609347	−	0.998142i	$-0.519408\pi$
$821$	−3.49193	−0.121869	−0.0609347	+	0.998142i	$0.519408\pi$
$822$	0	0
$823$	−44.0000	−1.53374	−0.766872	−	0.641800i	$-0.778188\pi$
$823$	−44.0000	−1.53374	−0.766872	+	0.641800i	$0.778188\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	15.1109	0.525457	0.262728	−	0.964870i	$-0.415378\pi$
$827$	15.1109	0.525457	0.262728	+	0.964870i	$0.415378\pi$
$828$	0	0
$829$	3.50807	0.121840	0.0609201	−	0.998143i	$-0.480597\pi$
$829$	3.50807	0.121840	0.0609201	+	0.998143i	$0.480597\pi$
$830$	0	0
$831$	14.0000	0.485655
$832$	0	0
$833$	−7.74597	−0.268382
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−3.00000	−0.103695
$838$	0	0
$839$	18.0000	0.621429	0.310715	−	0.950503i	$-0.399432\pi$
$839$	18.0000	0.621429	0.310715	+	0.950503i	$0.399432\pi$
$840$	0	0
$841$	5.49193	0.189377
$842$	0	0
$843$	16.3649	0.563638
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−38.2379	−1.31387
$848$	0	0
$849$	22.2379	0.763203
$850$	0	0
$851$	−1.00000	−0.0342796
$852$	0	0
$853$	43.7460	1.49783	0.748917	−	0.662664i	$-0.230574\pi$
$853$	43.7460	1.49783	0.748917	+	0.662664i	$0.230574\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	14.5081	0.495586	0.247793	−	0.968813i	$-0.420295\pi$
$857$	14.5081	0.495586	0.247793	+	0.968813i	$0.420295\pi$
$858$	0	0
$859$	29.0000	0.989467	0.494734	−	0.869045i	$-0.335266\pi$
$859$	29.0000	0.989467	0.494734	+	0.869045i	$0.335266\pi$
$860$	0	0
$861$	−17.6190	−0.600452
$862$	0	0
$863$	−8.25403	−0.280971	−0.140485	−	0.990083i	$-0.544866\pi$
$863$	−8.25403	−0.280971	−0.140485	+	0.990083i	$0.544866\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	2.00000	0.0679236
$868$	0	0
$869$	19.4919	0.661219
$870$	0	0
$871$	−30.8730	−1.04609
$872$	0	0
$873$	−8.00000	−0.270759
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−54.7298	−1.84810	−0.924048	−	0.382277i	$-0.875140\pi$
$877$	−54.7298	−1.84810	−0.924048	+	0.382277i	$0.875140\pi$
$878$	0	0
$879$	−13.6190	−0.459356
$880$	0	0
$881$	4.11088	0.138499	0.0692496	−	0.997599i	$-0.477940\pi$
$881$	4.11088	0.138499	0.0692496	+	0.997599i	$0.477940\pi$
$882$	0	0
$883$	−55.3488	−1.86263	−0.931317	−	0.364209i	$-0.881340\pi$
$883$	−55.3488	−1.86263	−0.931317	+	0.364209i	$0.881340\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−19.7460	−0.663005	−0.331502	−	0.943454i	$-0.607555\pi$
$887$	−19.7460	−0.663005	−0.331502	+	0.943454i	$0.607555\pi$
$888$	0	0
$889$	−19.8569	−0.665977
$890$	0	0
$891$	4.87298	0.163251
$892$	0	0
$893$	−19.7460	−0.660774
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−2.87298	−0.0959261
$898$	0	0
$899$	−17.6190	−0.587625
$900$	0	0
$901$	−15.0000	−0.499722
$902$	0	0
$903$	−11.2379	−0.373974
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−30.7460	−1.02090	−0.510452	−	0.859907i	$-0.670522\pi$
$907$	−30.7460	−1.02090	−0.510452	+	0.859907i	$0.670522\pi$
$908$	0	0
$909$	−3.87298	−0.128459
$910$	0	0
$911$	−16.0000	−0.530104	−0.265052	−	0.964234i	$-0.585389\pi$
$911$	−16.0000	−0.530104	−0.265052	+	0.964234i	$0.585389\pi$
$912$	0	0
$913$	0.618950	0.0204843
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−34.4758	−1.13849
$918$	0	0
$919$	−44.0000	−1.45143	−0.725713	−	0.687998i	$-0.758490\pi$
$919$	−44.0000	−1.45143	−0.725713	+	0.687998i	$0.758490\pi$
$920$	0	0
$921$	9.12702	0.300745
$922$	0	0
$923$	−39.1270	−1.28788
$924$	0	0
$925$	0	0
$926$	0	0
$927$	6.00000	0.197066
$928$	0	0
$929$	31.3649	1.02905	0.514525	−	0.857476i	$-0.327968\pi$
$929$	31.3649	1.02905	0.514525	+	0.857476i	$0.327968\pi$
$930$	0	0
$931$	−5.74597	−0.188316
$932$	0	0
$933$	4.00000	0.130954
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−54.4758	−1.77965	−0.889823	−	0.456305i	$-0.849173\pi$
$937$	−54.4758	−1.77965	−0.889823	+	0.456305i	$0.849173\pi$
$938$	0	0
$939$	7.00000	0.228436
$940$	0	0
$941$	−16.1109	−0.525200	−0.262600	−	0.964905i	$-0.584580\pi$
$941$	−16.1109	−0.525200	−0.262600	+	0.964905i	$0.584580\pi$
$942$	0	0
$943$	−5.87298	−0.191251
$944$	0	0
$945$	0	0
$946$	0	0
$947$	25.2379	0.820122	0.410061	−	0.912058i	$-0.365507\pi$
$947$	25.2379	0.820122	0.410061	+	0.912058i	$0.365507\pi$
$948$	0	0
$949$	8.25403	0.267937
$950$	0	0
$951$	22.3649	0.725232
$952$	0	0
$953$	−8.98387	−0.291016	−0.145508	−	0.989357i	$-0.546482\pi$
$953$	−8.98387	−0.291016	−0.145508	+	0.989357i	$0.546482\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	28.6190	0.925119
$958$	0	0
$959$	24.0000	0.775000
$960$	0	0
$961$	−22.0000	−0.709677
$962$	0	0
$963$	5.61895	0.181068
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−21.3810	−0.687568	−0.343784	−	0.939049i	$-0.611709\pi$
$967$	−21.3810	−0.687568	−0.343784	+	0.939049i	$0.611709\pi$
$968$	0	0
$969$	−11.1270	−0.357451
$970$	0	0
$971$	53.7460	1.72479	0.862395	−	0.506236i	$-0.168963\pi$
$971$	53.7460	1.72479	0.862395	+	0.506236i	$0.168963\pi$
$972$	0	0
$973$	61.4758	1.97082
$974$	0	0
$975$	0	0
$976$	0	0
$977$	51.8730	1.65956	0.829782	−	0.558088i	$-0.188465\pi$
$977$	51.8730	1.65956	0.829782	+	0.558088i	$0.188465\pi$
$978$	0	0
$979$	8.50807	0.271919
$980$	0	0
$981$	16.6190	0.530602
$982$	0	0
$983$	−7.36492	−0.234904	−0.117452	−	0.993079i	$-0.537473\pi$
$983$	−7.36492	−0.234904	−0.117452	+	0.993079i	$0.537473\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	20.6190	0.656308
$988$	0	0
$989$	−3.74597	−0.119115
$990$	0	0
$991$	−11.5081	−0.365566	−0.182783	−	0.983153i	$-0.558511\pi$
$991$	−11.5081	−0.365566	−0.182783	+	0.983153i	$0.558511\pi$
$992$	0	0
$993$	22.2379	0.705698
$994$	0	0
$995$	0	0
$996$	0	0
$997$	41.7460	1.32211	0.661054	−	0.750338i	$-0.270109\pi$
$997$	41.7460	1.32211	0.661054	+	0.750338i	$0.270109\pi$
$998$	0	0
$999$	1.00000	0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6900.2.a.j.1.2		2
5.2	odd	4		6900.2.f.o.6349.4		4
5.3	odd	4		6900.2.f.o.6349.2		4
5.4	even	2		1380.2.a.i.1.2	✓	2
15.14	odd	2		4140.2.a.p.1.1		2
20.19	odd	2		5520.2.a.bj.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.2.a.i.1.2	✓	2	5.4	even	2
4140.2.a.p.1.1		2	15.14	odd	2
5520.2.a.bj.1.1		2	20.19	odd	2
6900.2.a.j.1.2		2	1.1	even	1	trivial
6900.2.f.o.6349.2		4	5.3	odd	4
6900.2.f.o.6349.4		4	5.2	odd	4

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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:	\( N \)	\(=\)	\( 6900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6900.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -3.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -3.00000 q^{7} +1.00000 q^{9} +4.87298 q^{11} +2.87298 q^{13} -3.87298 q^{17} -2.87298 q^{19} +3.00000 q^{21} +1.00000 q^{23} -1.00000 q^{27} -5.87298 q^{29} +3.00000 q^{31} -4.87298 q^{33} -1.00000 q^{37} -2.87298 q^{39} -5.87298 q^{41} -3.74597 q^{43} +6.87298 q^{47} +2.00000 q^{49} +3.87298 q^{51} +3.87298 q^{53} +2.87298 q^{57} +5.87298 q^{59} +8.87298 q^{61} -3.00000 q^{63} -10.7460 q^{67} -1.00000 q^{69} -13.6190 q^{71} +2.87298 q^{73} -14.6190 q^{77} +4.00000 q^{79} +1.00000 q^{81} +0.127017 q^{83} +5.87298 q^{87} +1.74597 q^{89} -8.61895 q^{91} -3.00000 q^{93} -8.00000 q^{97} +4.87298 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 6 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 6 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} - 6 q^{7} + 2 q^{9} + 2 q^{11} - 2 q^{13} + 2 q^{19} + 6 q^{21} + 2 q^{23} - 2 q^{27} - 4 q^{29} + 6 q^{31} - 2 q^{33} - 2 q^{37} + 2 q^{39} - 4 q^{41} + 8 q^{43} + 6 q^{47} + 4 q^{49} - 2 q^{57} + 4 q^{59} + 10 q^{61} - 6 q^{63} - 6 q^{67} - 2 q^{69} - 4 q^{71} - 2 q^{73} - 6 q^{77} + 8 q^{79} + 2 q^{81} + 8 q^{83} + 4 q^{87} - 12 q^{89} + 6 q^{91} - 6 q^{93} - 16 q^{97} + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 6 * q^7 + 2 * q^9 + 2 * q^11 - 2 * q^13 + 2 * q^19 + 6 * q^21 + 2 * q^23 - 2 * q^27 - 4 * q^29 + 6 * q^31 - 2 * q^33 - 2 * q^37 + 2 * q^39 - 4 * q^41 + 8 * q^43 + 6 * q^47 + 4 * q^49 - 2 * q^57 + 4 * q^59 + 10 * q^61 - 6 * q^63 - 6 * q^67 - 2 * q^69 - 4 * q^71 - 2 * q^73 - 6 * q^77 + 8 * q^79 + 2 * q^81 + 8 * q^83 + 4 * q^87 - 12 * q^89 + 6 * q^91 - 6 * q^93 - 16 * q^97 + 2 * q^99

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