LMFDB - Embedded newform 6900.2.a.ba.1.4

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:	$0$
Dimension:	$4$
Coefficient field:	4.4.175557.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 8x^{2} + 3x + 2 $ x^4 - 2x^3 - 8x^2 + 3*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.672035$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +4.30400 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +4.30400 q^{7} +1.00000 q^{9} -0.672035 q^{11} -3.64807 q^{13} +7.56447 q^{17} +5.22040 q^{19} -4.30400 q^{21} +1.00000 q^{23} -1.00000 q^{27} -1.58844 q^{29} +10.5004 q^{31} +0.672035 q^{33} -1.56447 q^{37} +3.64807 q^{39} -4.63196 q^{41} -8.29614 q^{43} +8.22040 q^{47} +11.5244 q^{49} -7.56447 q^{51} -5.52440 q^{53} -5.22040 q^{57} -2.26048 q^{59} +1.89244 q^{61} +4.30400 q^{63} +10.2800 q^{67} -1.00000 q^{69} -6.91641 q^{71} +5.32010 q^{73} -2.89244 q^{77} -10.4168 q^{79} +1.00000 q^{81} +15.1050 q^{83} +1.58844 q^{87} +5.53226 q^{89} -15.7013 q^{91} -10.5004 q^{93} +8.71556 q^{97} -0.672035 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + 3 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 + 3 * q^7 + 4 * q^9	$ 4 q - 4 q^{3} + 3 q^{7} + 4 q^{9} - 2 q^{11} + q^{13} + 10 q^{17} + 2 q^{19} - 3 q^{21} + 4 q^{23} - 4 q^{27} - q^{29} - 6 q^{31} + 2 q^{33} + 14 q^{37} - q^{39} - 5 q^{41} - 2 q^{43} + 14 q^{47} + 13 q^{49} - 10 q^{51} + 11 q^{53} - 2 q^{57} - 3 q^{59} - 12 q^{61} + 3 q^{63} + 12 q^{67} - 4 q^{69} - 23 q^{71} + 5 q^{73} + 8 q^{77} + 11 q^{79} + 4 q^{81} + 5 q^{83} + q^{87} + 6 q^{89} - 19 q^{91} + 6 q^{93} + 26 q^{97} - 2 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 + 3 * q^7 + 4 * q^9 - 2 * q^11 + q^13 + 10 * q^17 + 2 * q^19 - 3 * q^21 + 4 * q^23 - 4 * q^27 - q^29 - 6 * q^31 + 2 * q^33 + 14 * q^37 - q^39 - 5 * q^41 - 2 * q^43 + 14 * q^47 + 13 * q^49 - 10 * q^51 + 11 * q^53 - 2 * q^57 - 3 * q^59 - 12 * q^61 + 3 * q^63 + 12 * q^67 - 4 * q^69 - 23 * q^71 + 5 * q^73 + 8 * q^77 + 11 * q^79 + 4 * q^81 + 5 * q^83 + q^87 + 6 * q^89 - 19 * q^91 + 6 * q^93 + 26 * 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
$7$	4.30400	1.62676	0.813379	−	0.581734i	$-0.197625\pi$
$7$	4.30400	1.62676	0.813379	+	0.581734i	$0.197625\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.672035	−0.202626	−0.101313	−	0.994855i	$-0.532304\pi$
$11$	−0.672035	−0.202626	−0.101313	+	0.994855i	$0.532304\pi$
$12$	0	0
$13$	−3.64807	−1.01179	−0.505896	−	0.862594i	$-0.668838\pi$
$13$	−3.64807	−1.01179	−0.505896	+	0.862594i	$0.668838\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	7.56447	1.83465	0.917327	−	0.398134i	$-0.130342\pi$
$17$	7.56447	1.83465	0.917327	+	0.398134i	$0.130342\pi$
$18$	0	0
$19$	5.22040	1.19764	0.598821	−	0.800883i	$-0.295636\pi$
$19$	5.22040	1.19764	0.598821	+	0.800883i	$0.295636\pi$
$20$	0	0
$21$	−4.30400	−0.939209
$22$	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
$29$	−1.58844	−0.294966	−0.147483	−	0.989065i	$-0.547117\pi$
$29$	−1.58844	−0.294966	−0.147483	+	0.989065i	$0.547117\pi$
$30$	0	0
$31$	10.5004	1.88593	0.942967	−	0.332886i	$-0.108023\pi$
$31$	10.5004	1.88593	0.942967	+	0.332886i	$0.108023\pi$
$32$	0	0
$33$	0.672035	0.116986
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−1.56447	−0.257198	−0.128599	−	0.991697i	$-0.541048\pi$
$37$	−1.56447	−0.257198	−0.128599	+	0.991697i	$0.541048\pi$
$38$	0	0
$39$	3.64807	0.584159
$40$	0	0
$41$	−4.63196	−0.723391	−0.361696	−	0.932296i	$-0.617802\pi$
$41$	−4.63196	−0.723391	−0.361696	+	0.932296i	$0.617802\pi$
$42$	0	0
$43$	−8.29614	−1.26515	−0.632575	−	0.774499i	$-0.718002\pi$
$43$	−8.29614	−1.26515	−0.632575	+	0.774499i	$0.718002\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.22040	1.19907	0.599535	−	0.800349i	$-0.295352\pi$
$47$	8.22040	1.19907	0.599535	+	0.800349i	$0.295352\pi$
$48$	0	0
$49$	11.5244	1.64634
$50$	0	0
$51$	−7.56447	−1.05924
$52$	0	0
$53$	−5.52440	−0.758835	−0.379418	−	0.925226i	$-0.623876\pi$
$53$	−5.52440	−0.758835	−0.379418	+	0.925226i	$0.623876\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−5.22040	−0.691459
$58$	0	0
$59$	−2.26048	−0.294289	−0.147144	−	0.989115i	$-0.547008\pi$
$59$	−2.26048	−0.294289	−0.147144	+	0.989115i	$0.547008\pi$
$60$	0	0
$61$	1.89244	0.242302	0.121151	−	0.992634i	$-0.461341\pi$
$61$	1.89244	0.242302	0.121151	+	0.992634i	$0.461341\pi$
$62$	0	0
$63$	4.30400	0.542253
$64$	0	0
$65$	0	0
$66$	0	0
$67$	10.2800	1.25591	0.627953	−	0.778251i	$-0.283893\pi$
$67$	10.2800	1.25591	0.627953	+	0.778251i	$0.283893\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−6.91641	−0.820826	−0.410413	−	0.911900i	$-0.634616\pi$
$71$	−6.91641	−0.820826	−0.410413	+	0.911900i	$0.634616\pi$
$72$	0	0
$73$	5.32010	0.622671	0.311336	−	0.950300i	$-0.399224\pi$
$73$	5.32010	0.622671	0.311336	+	0.950300i	$0.399224\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.89244	−0.329624
$78$	0	0
$79$	−10.4168	−1.17199	−0.585993	−	0.810316i	$-0.699295\pi$
$79$	−10.4168	−1.17199	−0.585993	+	0.810316i	$0.699295\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	15.1050	1.65799	0.828994	−	0.559258i	$-0.188914\pi$
$83$	15.1050	1.65799	0.828994	+	0.559258i	$0.188914\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1.58844	0.170299
$88$	0	0
$89$	5.53226	0.586419	0.293209	−	0.956048i	$-0.405277\pi$
$89$	5.53226	0.586419	0.293209	+	0.956048i	$0.405277\pi$
$90$	0	0
$91$	−15.7013	−1.64594
$92$	0	0
$93$	−10.5004	−1.08884
$94$	0	0
$95$	0	0
$96$	0	0
$97$	8.71556	0.884931	0.442465	−	0.896786i	$-0.354104\pi$
$97$	8.71556	0.884931	0.442465	+	0.896786i	$0.354104\pi$
$98$	0	0
$99$	−0.672035	−0.0675421
$100$	0	0
$101$	1.13681	0.113117	0.0565584	−	0.998399i	$-0.481987\pi$
$101$	1.13681	0.113117	0.0565584	+	0.998399i	$0.481987\pi$
$102$	0	0
$103$	10.8524	1.06932	0.534658	−	0.845069i	$-0.320441\pi$
$103$	10.8524	1.06932	0.534658	+	0.845069i	$0.320441\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−8.22040	−0.794696	−0.397348	−	0.917668i	$-0.630069\pi$
$107$	−8.22040	−0.794696	−0.397348	+	0.917668i	$0.630069\pi$
$108$	0	0
$109$	−11.8044	−1.13066	−0.565330	−	0.824865i	$-0.691251\pi$
$109$	−11.8044	−1.13066	−0.565330	+	0.824865i	$0.691251\pi$
$110$	0	0
$111$	1.56447	0.148493
$112$	0	0
$113$	9.99214	0.939981	0.469991	−	0.882671i	$-0.344257\pi$
$113$	9.99214	0.939981	0.469991	+	0.882671i	$0.344257\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−3.64807	−0.337264
$118$	0	0
$119$	32.5575	2.98454
$120$	0	0
$121$	−10.5484	−0.958943
$122$	0	0
$123$	4.63196	0.417650
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−3.95207	−0.350689	−0.175345	−	0.984507i	$-0.556104\pi$
$127$	−3.95207	−0.350689	−0.175345	+	0.984507i	$0.556104\pi$
$128$	0	0
$129$	8.29614	0.730434
$130$	0	0
$131$	−14.2204	−1.24244	−0.621221	−	0.783635i	$-0.713363\pi$
$131$	−14.2204	−1.24244	−0.621221	+	0.783635i	$0.713363\pi$
$132$	0	0
$133$	22.4686	1.94828
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−17.0810	−1.45933	−0.729665	−	0.683805i	$-0.760324\pi$
$137$	−17.0810	−1.45933	−0.729665	+	0.683805i	$0.760324\pi$
$138$	0	0
$139$	−0.180331	−0.0152955	−0.00764776	−	0.999971i	$-0.502434\pi$
$139$	−0.180331	−0.0152955	−0.00764776	+	0.999971i	$0.502434\pi$
$140$	0	0
$141$	−8.22040	−0.692283
$142$	0	0
$143$	2.45163	0.205016
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−11.5244	−0.950517
$148$	0	0
$149$	−20.7805	−1.70240	−0.851201	−	0.524840i	$-0.824125\pi$
$149$	−20.7805	−1.70240	−0.851201	+	0.524840i	$0.824125\pi$
$150$	0	0
$151$	14.4926	1.17939	0.589695	−	0.807626i	$-0.299248\pi$
$151$	14.4926	1.17939	0.589695	+	0.807626i	$0.299248\pi$
$152$	0	0
$153$	7.56447	0.611552
$154$	0	0
$155$	0	0
$156$	0	0
$157$	20.7609	1.65690	0.828451	−	0.560062i	$-0.189223\pi$
$157$	20.7609	1.65690	0.828451	+	0.560062i	$0.189223\pi$
$158$	0	0
$159$	5.52440	0.438114
$160$	0	0
$161$	4.30400	0.339203
$162$	0	0
$163$	−9.02925	−0.707225	−0.353613	−	0.935392i	$-0.615047\pi$
$163$	−9.02925	−0.707225	−0.353613	+	0.935392i	$0.615047\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.82495	0.141219	0.0706095	−	0.997504i	$-0.477506\pi$
$167$	1.82495	0.141219	0.0706095	+	0.997504i	$0.477506\pi$
$168$	0	0
$169$	0.308409	0.0237237
$170$	0	0
$171$	5.22040	0.399214
$172$	0	0
$173$	4.47263	0.340048	0.170024	−	0.985440i	$-0.445615\pi$
$173$	4.47263	0.340048	0.170024	+	0.985440i	$0.445615\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2.26048	0.169908
$178$	0	0
$179$	−7.52440	−0.562400	−0.281200	−	0.959649i	$-0.590732\pi$
$179$	−7.52440	−0.562400	−0.281200	+	0.959649i	$0.590732\pi$
$180$	0	0
$181$	9.41684	0.699948	0.349974	−	0.936759i	$-0.386190\pi$
$181$	9.41684	0.699948	0.349974	+	0.936759i	$0.386190\pi$
$182$	0	0
$183$	−1.89244	−0.139893
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−5.08359	−0.371749
$188$	0	0
$189$	−4.30400	−0.313070
$190$	0	0
$191$	−13.9760	−1.01127	−0.505635	−	0.862747i	$-0.668742\pi$
$191$	−13.9760	−1.01127	−0.505635	+	0.862747i	$0.668742\pi$
$192$	0	0
$193$	11.1964	0.805937	0.402969	−	0.915214i	$-0.367978\pi$
$193$	11.1964	0.805937	0.402969	+	0.915214i	$0.367978\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	20.0292	1.42702	0.713512	−	0.700643i	$-0.247103\pi$
$197$	20.0292	1.42702	0.713512	+	0.700643i	$0.247103\pi$
$198$	0	0
$199$	6.64462	0.471025	0.235512	−	0.971871i	$-0.424323\pi$
$199$	6.64462	0.471025	0.235512	+	0.971871i	$0.424323\pi$
$200$	0	0
$201$	−10.2800	−0.725098
$202$	0	0
$203$	−6.83665	−0.479839
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−3.50830	−0.242674
$210$	0	0
$211$	−2.35232	−0.161940	−0.0809701	−	0.996717i	$-0.525802\pi$
$211$	−2.35232	−0.161940	−0.0809701	+	0.996717i	$0.525802\pi$
$212$	0	0
$213$	6.91641	0.473904
$214$	0	0
$215$	0	0
$216$	0	0
$217$	45.1939	3.06796
$218$	0	0
$219$	−5.32010	−0.359499
$220$	0	0
$221$	−27.5957	−1.85629
$222$	0	0
$223$	−12.0971	−0.810083	−0.405042	−	0.914298i	$-0.632743\pi$
$223$	−12.0971	−0.810083	−0.405042	+	0.914298i	$0.632743\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−25.9573	−1.72285	−0.861425	−	0.507885i	$-0.830427\pi$
$227$	−25.9573	−1.72285	−0.861425	+	0.507885i	$0.830427\pi$
$228$	0	0
$229$	3.29230	0.217561	0.108781	−	0.994066i	$-0.465305\pi$
$229$	3.29230	0.217561	0.108781	+	0.994066i	$0.465305\pi$
$230$	0	0
$231$	2.89244	0.190309
$232$	0	0
$233$	−22.1616	−1.45186	−0.725929	−	0.687770i	$-0.758590\pi$
$233$	−22.1616	−1.45186	−0.725929	+	0.687770i	$0.758590\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	10.4168	0.676647
$238$	0	0
$239$	3.39200	0.219410	0.109705	−	0.993964i	$-0.465009\pi$
$239$	3.39200	0.219410	0.109705	+	0.993964i	$0.465009\pi$
$240$	0	0
$241$	−3.59189	−0.231374	−0.115687	−	0.993286i	$-0.536907\pi$
$241$	−3.59189	−0.231374	−0.115687	+	0.993286i	$0.536907\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−19.0444	−1.21177
$248$	0	0
$249$	−15.1050	−0.957239
$250$	0	0
$251$	31.0462	1.95962	0.979810	−	0.199930i	$-0.0640714\pi$
$251$	31.0462	1.95962	0.979810	+	0.199930i	$0.0640714\pi$
$252$	0	0
$253$	−0.672035	−0.0422505
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4.81526	0.300368	0.150184	−	0.988658i	$-0.452013\pi$
$257$	4.81526	0.300368	0.150184	+	0.988658i	$0.452013\pi$
$258$	0	0
$259$	−6.73350	−0.418399
$260$	0	0
$261$	−1.58844	−0.0983220
$262$	0	0
$263$	31.1602	1.92142	0.960710	−	0.277554i	$-0.0895236\pi$
$263$	31.1602	1.92142	0.960710	+	0.277554i	$0.0895236\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−5.53226	−0.338569
$268$	0	0
$269$	−5.18675	−0.316241	−0.158121	−	0.987420i	$-0.550544\pi$
$269$	−5.18675	−0.316241	−0.158121	+	0.987420i	$0.550544\pi$
$270$	0	0
$271$	−5.65152	−0.343305	−0.171653	−	0.985158i	$-0.554911\pi$
$271$	−5.65152	−0.343305	−0.171653	+	0.985158i	$0.554911\pi$
$272$	0	0
$273$	15.7013	0.950285
$274$	0	0
$275$	0	0
$276$	0	0
$277$	28.6437	1.72103	0.860515	−	0.509425i	$-0.170142\pi$
$277$	28.6437	1.72103	0.860515	+	0.509425i	$0.170142\pi$
$278$	0	0
$279$	10.5004	0.628645
$280$	0	0
$281$	−14.0854	−0.840266	−0.420133	−	0.907463i	$-0.638017\pi$
$281$	−14.0854	−0.840266	−0.420133	+	0.907463i	$0.638017\pi$
$282$	0	0
$283$	23.6695	1.40700	0.703502	−	0.710694i	$-0.251619\pi$
$283$	23.6695	1.40700	0.703502	+	0.710694i	$0.251619\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−19.9360	−1.17678
$288$	0	0
$289$	40.2213	2.36596
$290$	0	0
$291$	−8.71556	−0.510915
$292$	0	0
$293$	18.7962	1.09809	0.549043	−	0.835794i	$-0.314992\pi$
$293$	18.7962	1.09809	0.549043	+	0.835794i	$0.314992\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0.672035	0.0389954
$298$	0	0
$299$	−3.64807	−0.210973
$300$	0	0
$301$	−35.7066	−2.05809
$302$	0	0
$303$	−1.13681	−0.0653080
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−8.23306	−0.469886	−0.234943	−	0.972009i	$-0.575490\pi$
$307$	−8.23306	−0.469886	−0.234943	+	0.972009i	$0.575490\pi$
$308$	0	0
$309$	−10.8524	−0.617370
$310$	0	0
$311$	−23.9251	−1.35667	−0.678335	−	0.734753i	$-0.737298\pi$
$311$	−23.9251	−1.35667	−0.678335	+	0.734753i	$0.737298\pi$
$312$	0	0
$313$	−8.54051	−0.482738	−0.241369	−	0.970433i	$-0.577596\pi$
$313$	−8.54051	−0.482738	−0.241369	+	0.970433i	$0.577596\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−15.8079	−0.887859	−0.443930	−	0.896062i	$-0.646416\pi$
$317$	−15.8079	−0.887859	−0.443930	+	0.896062i	$0.646416\pi$
$318$	0	0
$319$	1.06749	0.0597679
$320$	0	0
$321$	8.22040	0.458818
$322$	0	0
$323$	39.4896	2.19726
$324$	0	0
$325$	0	0
$326$	0	0
$327$	11.8044	0.652787
$328$	0	0
$329$	35.3806	1.95060
$330$	0	0
$331$	−10.4652	−0.575217	−0.287609	−	0.957748i	$-0.592860\pi$
$331$	−10.4652	−0.575217	−0.287609	+	0.957748i	$0.592860\pi$
$332$	0	0
$333$	−1.56447	−0.0857327
$334$	0	0
$335$	0	0
$336$	0	0
$337$	10.1867	0.554907	0.277454	−	0.960739i	$-0.410509\pi$
$337$	10.1867	0.554907	0.277454	+	0.960739i	$0.410509\pi$
$338$	0	0
$339$	−9.99214	−0.542699
$340$	0	0
$341$	−7.05666	−0.382140
$342$	0	0
$343$	19.4730	1.05144
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4.52785	0.243068	0.121534	−	0.992587i	$-0.461219\pi$
$347$	4.52785	0.243068	0.121534	+	0.992587i	$0.461219\pi$
$348$	0	0
$349$	16.7512	0.896672	0.448336	−	0.893865i	$-0.352017\pi$
$349$	16.7512	0.896672	0.448336	+	0.893865i	$0.352017\pi$
$350$	0	0
$351$	3.64807	0.194720
$352$	0	0
$353$	−14.6084	−0.777526	−0.388763	−	0.921338i	$-0.627097\pi$
$353$	−14.6084	−0.777526	−0.388763	+	0.921338i	$0.627097\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−32.5575	−1.72312
$358$	0	0
$359$	10.4326	0.550610	0.275305	−	0.961357i	$-0.411221\pi$
$359$	10.4326	0.550610	0.275305	+	0.961357i	$0.411221\pi$
$360$	0	0
$361$	8.25262	0.434348
$362$	0	0
$363$	10.5484	0.553646
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−17.4251	−0.909582	−0.454791	−	0.890598i	$-0.650286\pi$
$367$	−17.4251	−0.909582	−0.454791	+	0.890598i	$0.650286\pi$
$368$	0	0
$369$	−4.63196	−0.241130
$370$	0	0
$371$	−23.7770	−1.23444
$372$	0	0
$373$	13.5308	0.700599	0.350300	−	0.936638i	$-0.386080\pi$
$373$	13.5308	0.700599	0.350300	+	0.936638i	$0.386080\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	5.79474	0.298444
$378$	0	0
$379$	18.1372	0.931645	0.465823	−	0.884878i	$-0.345758\pi$
$379$	18.1372	0.931645	0.465823	+	0.884878i	$0.345758\pi$
$380$	0	0
$381$	3.95207	0.202471
$382$	0	0
$383$	27.9202	1.42666	0.713329	−	0.700829i	$-0.247187\pi$
$383$	27.9202	1.42666	0.713329	+	0.700829i	$0.247187\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−8.29614	−0.421716
$388$	0	0
$389$	−33.3093	−1.68885	−0.844424	−	0.535676i	$-0.820057\pi$
$389$	−33.3093	−1.68885	−0.844424	+	0.535676i	$0.820057\pi$
$390$	0	0
$391$	7.56447	0.382552
$392$	0	0
$393$	14.2204	0.717324
$394$	0	0
$395$	0	0
$396$	0	0
$397$	24.6739	1.23835	0.619173	−	0.785255i	$-0.287468\pi$
$397$	24.6739	1.23835	0.619173	+	0.785255i	$0.287468\pi$
$398$	0	0
$399$	−22.4686	−1.12484
$400$	0	0
$401$	−4.04535	−0.202015	−0.101008	−	0.994886i	$-0.532207\pi$
$401$	−4.04535	−0.202015	−0.101008	+	0.994886i	$0.532207\pi$
$402$	0	0
$403$	−38.3063	−1.90817
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.05138	0.0521151
$408$	0	0
$409$	0.914573	0.0452227	0.0226114	−	0.999744i	$-0.492802\pi$
$409$	0.914573	0.0452227	0.0226114	+	0.999744i	$0.492802\pi$
$410$	0	0
$411$	17.0810	0.842544
$412$	0	0
$413$	−9.72909	−0.478737
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0.180331	0.00883087
$418$	0	0
$419$	−15.0117	−0.733369	−0.366685	−	0.930345i	$-0.619507\pi$
$419$	−15.0117	−0.733369	−0.366685	+	0.930345i	$0.619507\pi$
$420$	0	0
$421$	16.9696	0.827049	0.413524	−	0.910493i	$-0.364298\pi$
$421$	16.9696	0.827049	0.413524	+	0.910493i	$0.364298\pi$
$422$	0	0
$423$	8.22040	0.399690
$424$	0	0
$425$	0	0
$426$	0	0
$427$	8.14506	0.394167
$428$	0	0
$429$	−2.45163	−0.118366
$430$	0	0
$431$	27.1733	1.30889	0.654447	−	0.756108i	$-0.272902\pi$
$431$	27.1733	1.30889	0.654447	+	0.756108i	$0.272902\pi$
$432$	0	0
$433$	23.1094	1.11057	0.555283	−	0.831661i	$-0.312610\pi$
$433$	23.1094	1.11057	0.555283	+	0.831661i	$0.312610\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5.22040	0.249726
$438$	0	0
$439$	31.1485	1.48664	0.743319	−	0.668938i	$-0.233251\pi$
$439$	31.1485	1.48664	0.743319	+	0.668938i	$0.233251\pi$
$440$	0	0
$441$	11.5244	0.548781
$442$	0	0
$443$	−13.1681	−0.625633	−0.312817	−	0.949814i	$-0.601273\pi$
$443$	−13.1681	−0.625633	−0.312817	+	0.949814i	$0.601273\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	20.7805	0.982882
$448$	0	0
$449$	29.6759	1.40049	0.700245	−	0.713902i	$-0.253074\pi$
$449$	29.6759	1.40049	0.700245	+	0.713902i	$0.253074\pi$
$450$	0	0
$451$	3.11284	0.146578
$452$	0	0
$453$	−14.4926	−0.680921
$454$	0	0
$455$	0	0
$456$	0	0
$457$	31.7047	1.48308	0.741542	−	0.670906i	$-0.234095\pi$
$457$	31.7047	1.48308	0.741542	+	0.670906i	$0.234095\pi$
$458$	0	0
$459$	−7.56447	−0.353079
$460$	0	0
$461$	33.8723	1.57759	0.788795	−	0.614656i	$-0.210705\pi$
$461$	33.8723	1.57759	0.788795	+	0.614656i	$0.210705\pi$
$462$	0	0
$463$	0.781429	0.0363161	0.0181580	−	0.999835i	$-0.494220\pi$
$463$	0.781429	0.0363161	0.0181580	+	0.999835i	$0.494220\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−19.3802	−0.896810	−0.448405	−	0.893831i	$-0.648008\pi$
$467$	−19.3802	−0.896810	−0.448405	+	0.893831i	$0.648008\pi$
$468$	0	0
$469$	44.2452	2.04306
$470$	0	0
$471$	−20.7609	−0.956612
$472$	0	0
$473$	5.57530	0.256352
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−5.52440	−0.252945
$478$	0	0
$479$	−2.07718	−0.0949088	−0.0474544	−	0.998873i	$-0.515111\pi$
$479$	−2.07718	−0.0949088	−0.0474544	+	0.998873i	$0.515111\pi$
$480$	0	0
$481$	5.70731	0.260231
$482$	0	0
$483$	−4.30400	−0.195839
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−19.5518	−0.885977	−0.442989	−	0.896527i	$-0.646082\pi$
$487$	−19.5518	−0.885977	−0.442989	+	0.896527i	$0.646082\pi$
$488$	0	0
$489$	9.02925	0.408317
$490$	0	0
$491$	−3.54012	−0.159763	−0.0798817	−	0.996804i	$-0.525454\pi$
$491$	−3.54012	−0.159763	−0.0798817	+	0.996804i	$0.525454\pi$
$492$	0	0
$493$	−12.0157	−0.541161
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−29.7682	−1.33529
$498$	0	0
$499$	26.0196	1.16480	0.582398	−	0.812904i	$-0.302115\pi$
$499$	26.0196	1.16480	0.582398	+	0.812904i	$0.302115\pi$
$500$	0	0
$501$	−1.82495	−0.0815328
$502$	0	0
$503$	−14.4560	−0.644559	−0.322280	−	0.946645i	$-0.604449\pi$
$503$	−14.4560	−0.644559	−0.322280	+	0.946645i	$0.604449\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−0.308409	−0.0136969
$508$	0	0
$509$	−20.7814	−0.921121	−0.460560	−	0.887628i	$-0.652351\pi$
$509$	−20.7814	−0.921121	−0.460560	+	0.887628i	$0.652351\pi$
$510$	0	0
$511$	22.8977	1.01294
$512$	0	0
$513$	−5.22040	−0.230486
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−5.52440	−0.242963
$518$	0	0
$519$	−4.47263	−0.196327
$520$	0	0
$521$	36.8885	1.61612	0.808058	−	0.589103i	$-0.200519\pi$
$521$	36.8885	1.61612	0.808058	+	0.589103i	$0.200519\pi$
$522$	0	0
$523$	20.4827	0.895646	0.447823	−	0.894122i	$-0.352199\pi$
$523$	20.4827	0.895646	0.447823	+	0.894122i	$0.352199\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	79.4303	3.46004
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−2.26048	−0.0980963
$532$	0	0
$533$	16.8977	0.731922
$534$	0	0
$535$	0	0
$536$	0	0
$537$	7.52440	0.324702
$538$	0	0
$539$	−7.74481	−0.333592
$540$	0	0
$541$	−17.1245	−0.736241	−0.368121	−	0.929778i	$-0.619999\pi$
$541$	−17.1245	−0.736241	−0.368121	+	0.929778i	$0.619999\pi$
$542$	0	0
$543$	−9.41684	−0.404115
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−33.5557	−1.43474	−0.717368	−	0.696694i	$-0.754653\pi$
$547$	−33.5557	−1.43474	−0.717368	+	0.696694i	$0.754653\pi$
$548$	0	0
$549$	1.89244	0.0807673
$550$	0	0
$551$	−8.29230	−0.353264
$552$	0	0
$553$	−44.8341	−1.90654
$554$	0	0
$555$	0	0
$556$	0	0
$557$	20.4086	0.864740	0.432370	−	0.901696i	$-0.357677\pi$
$557$	20.4086	0.864740	0.432370	+	0.901696i	$0.357677\pi$
$558$	0	0
$559$	30.2649	1.28007
$560$	0	0
$561$	5.08359	0.214630
$562$	0	0
$563$	5.17889	0.218264	0.109132	−	0.994027i	$-0.465193\pi$
$563$	5.17889	0.218264	0.109132	+	0.994027i	$0.465193\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	4.30400	0.180751
$568$	0	0
$569$	−6.05321	−0.253764	−0.126882	−	0.991918i	$-0.540497\pi$
$569$	−6.05321	−0.253764	−0.126882	+	0.991918i	$0.540497\pi$
$570$	0	0
$571$	45.3263	1.89684	0.948422	−	0.317009i	$-0.102679\pi$
$571$	45.3263	1.89684	0.948422	+	0.317009i	$0.102679\pi$
$572$	0	0
$573$	13.9760	0.583857
$574$	0	0
$575$	0	0
$576$	0	0
$577$	4.55133	0.189474	0.0947372	−	0.995502i	$-0.469799\pi$
$577$	4.55133	0.189474	0.0947372	+	0.995502i	$0.469799\pi$
$578$	0	0
$579$	−11.1964	−0.465308
$580$	0	0
$581$	65.0118	2.69714
$582$	0	0
$583$	3.71259	0.153760
$584$	0	0
$585$	0	0
$586$	0	0
$587$	46.5148	1.91987	0.959936	−	0.280218i	$-0.0904068\pi$
$587$	46.5148	1.91987	0.959936	+	0.280218i	$0.0904068\pi$
$588$	0	0
$589$	54.8165	2.25868
$590$	0	0
$591$	−20.0292	−0.823893
$592$	0	0
$593$	−7.81574	−0.320954	−0.160477	−	0.987040i	$-0.551303\pi$
$593$	−7.81574	−0.320954	−0.160477	+	0.987040i	$0.551303\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−6.64462	−0.271946
$598$	0	0
$599$	44.4461	1.81602	0.908009	−	0.418951i	$-0.137602\pi$
$599$	44.4461	1.81602	0.908009	+	0.418951i	$0.137602\pi$
$600$	0	0
$601$	−19.3367	−0.788760	−0.394380	−	0.918947i	$-0.629041\pi$
$601$	−19.3367	−0.788760	−0.394380	+	0.918947i	$0.629041\pi$
$602$	0	0
$603$	10.2800	0.418635
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−18.4172	−0.747532	−0.373766	−	0.927523i	$-0.621934\pi$
$607$	−18.4172	−0.747532	−0.373766	+	0.927523i	$0.621934\pi$
$608$	0	0
$609$	6.83665	0.277035
$610$	0	0
$611$	−29.9886	−1.21321
$612$	0	0
$613$	46.8170	1.89092	0.945460	−	0.325737i	$-0.105612\pi$
$613$	46.8170	1.89092	0.945460	+	0.325737i	$0.105612\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	35.9338	1.44664	0.723320	−	0.690513i	$-0.242615\pi$
$617$	35.9338	1.44664	0.723320	+	0.690513i	$0.242615\pi$
$618$	0	0
$619$	−17.3880	−0.698882	−0.349441	−	0.936958i	$-0.613628\pi$
$619$	−17.3880	−0.698882	−0.349441	+	0.936958i	$0.613628\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	23.8108	0.953961
$624$	0	0
$625$	0	0
$626$	0	0
$627$	3.50830	0.140108
$628$	0	0
$629$	−11.8344	−0.471870
$630$	0	0
$631$	−8.19002	−0.326040	−0.163020	−	0.986623i	$-0.552123\pi$
$631$	−8.19002	−0.326040	−0.163020	+	0.986623i	$0.552123\pi$
$632$	0	0
$633$	2.35232	0.0934962
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−42.0418	−1.66576
$638$	0	0
$639$	−6.91641	−0.273609
$640$	0	0
$641$	−4.98773	−0.197003	−0.0985017	−	0.995137i	$-0.531405\pi$
$641$	−4.98773	−0.197003	−0.0985017	+	0.995137i	$0.531405\pi$
$642$	0	0
$643$	−19.3514	−0.763143	−0.381572	−	0.924339i	$-0.624617\pi$
$643$	−19.3514	−0.763143	−0.381572	+	0.924339i	$0.624617\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−40.4964	−1.59208	−0.796039	−	0.605245i	$-0.793075\pi$
$647$	−40.4964	−1.59208	−0.796039	+	0.605245i	$0.793075\pi$
$648$	0	0
$649$	1.51912	0.0596307
$650$	0	0
$651$	−45.1939	−1.77129
$652$	0	0
$653$	26.5683	1.03970	0.519849	−	0.854258i	$-0.325988\pi$
$653$	26.5683	1.03970	0.519849	+	0.854258i	$0.325988\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	5.32010	0.207557
$658$	0	0
$659$	6.82879	0.266012	0.133006	−	0.991115i	$-0.457537\pi$
$659$	6.82879	0.266012	0.133006	+	0.991115i	$0.457537\pi$
$660$	0	0
$661$	−6.43457	−0.250276	−0.125138	−	0.992139i	$-0.539937\pi$
$661$	−6.43457	−0.250276	−0.125138	+	0.992139i	$0.539937\pi$
$662$	0	0
$663$	27.5957	1.07173
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1.58844	−0.0615047
$668$	0	0
$669$	12.0971	0.467702
$670$	0	0
$671$	−1.27179	−0.0490968
$672$	0	0
$673$	−30.6058	−1.17977	−0.589884	−	0.807488i	$-0.700826\pi$
$673$	−30.6058	−1.17977	−0.589884	+	0.807488i	$0.700826\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	37.1280	1.42694	0.713472	−	0.700683i	$-0.247121\pi$
$677$	37.1280	1.42694	0.713472	+	0.700683i	$0.247121\pi$
$678$	0	0
$679$	37.5117	1.43957
$680$	0	0
$681$	25.9573	0.994688
$682$	0	0
$683$	−4.58699	−0.175516	−0.0877582	−	0.996142i	$-0.527970\pi$
$683$	−4.58699	−0.175516	−0.0877582	+	0.996142i	$0.527970\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−3.29230	−0.125609
$688$	0	0
$689$	20.1534	0.767783
$690$	0	0
$691$	7.89885	0.300487	0.150243	−	0.988649i	$-0.451994\pi$
$691$	7.89885	0.300487	0.150243	+	0.988649i	$0.451994\pi$
$692$	0	0
$693$	−2.89244	−0.109875
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−35.0384	−1.32717
$698$	0	0
$699$	22.1616	0.838230
$700$	0	0
$701$	−27.9416	−1.05534	−0.527670	−	0.849449i	$-0.676934\pi$
$701$	−27.9416	−1.05534	−0.527670	+	0.849449i	$0.676934\pi$
$702$	0	0
$703$	−8.16719	−0.308031
$704$	0	0
$705$	0	0
$706$	0	0
$707$	4.89283	0.184014
$708$	0	0
$709$	1.07190	0.0402560	0.0201280	−	0.999797i	$-0.493593\pi$
$709$	1.07190	0.0402560	0.0201280	+	0.999797i	$0.493593\pi$
$710$	0	0
$711$	−10.4168	−0.390662
$712$	0	0
$713$	10.5004	0.393244
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−3.39200	−0.126677
$718$	0	0
$719$	−32.8284	−1.22429	−0.612146	−	0.790744i	$-0.709694\pi$
$719$	−32.8284	−1.22429	−0.612146	+	0.790744i	$0.709694\pi$
$720$	0	0
$721$	46.7086	1.73952
$722$	0	0
$723$	3.59189	0.133584
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−7.51961	−0.278887	−0.139443	−	0.990230i	$-0.544531\pi$
$727$	−7.51961	−0.278887	−0.139443	+	0.990230i	$0.544531\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−62.7559	−2.32111
$732$	0	0
$733$	−11.0366	−0.407647	−0.203823	−	0.979008i	$-0.565337\pi$
$733$	−11.0366	−0.407647	−0.203823	+	0.979008i	$0.565337\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−6.90855	−0.254480
$738$	0	0
$739$	3.73952	0.137561	0.0687803	−	0.997632i	$-0.478089\pi$
$739$	3.73952	0.137561	0.0687803	+	0.997632i	$0.478089\pi$
$740$	0	0
$741$	19.0444	0.699613
$742$	0	0
$743$	−9.88641	−0.362697	−0.181349	−	0.983419i	$-0.558046\pi$
$743$	−9.88641	−0.362697	−0.181349	+	0.983419i	$0.558046\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	15.1050	0.552662
$748$	0	0
$749$	−35.3806	−1.29278
$750$	0	0
$751$	−52.2974	−1.90836	−0.954180	−	0.299235i	$-0.903269\pi$
$751$	−52.2974	−1.90836	−0.954180	+	0.299235i	$0.903269\pi$
$752$	0	0
$753$	−31.0462	−1.13139
$754$	0	0
$755$	0	0
$756$	0	0
$757$	40.6939	1.47905	0.739523	−	0.673132i	$-0.235051\pi$
$757$	40.6939	1.47905	0.739523	+	0.673132i	$0.235051\pi$
$758$	0	0
$759$	0.672035	0.0243933
$760$	0	0
$761$	12.3524	0.447775	0.223887	−	0.974615i	$-0.428125\pi$
$761$	12.3524	0.447775	0.223887	+	0.974615i	$0.428125\pi$
$762$	0	0
$763$	−50.8063	−1.83931
$764$	0	0
$765$	0	0
$766$	0	0
$767$	8.24637	0.297759
$768$	0	0
$769$	−8.28003	−0.298586	−0.149293	−	0.988793i	$-0.547700\pi$
$769$	−8.28003	−0.298586	−0.149293	+	0.988793i	$0.547700\pi$
$770$	0	0
$771$	−4.81526	−0.173417
$772$	0	0
$773$	11.8171	0.425031	0.212516	−	0.977158i	$-0.431834\pi$
$773$	11.8171	0.425031	0.212516	+	0.977158i	$0.431834\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	6.73350	0.241563
$778$	0	0
$779$	−24.1807	−0.866364
$780$	0	0
$781$	4.64807	0.166321
$782$	0	0
$783$	1.58844	0.0567662
$784$	0	0
$785$	0	0
$786$	0	0
$787$	24.5609	0.875503	0.437751	−	0.899096i	$-0.355775\pi$
$787$	24.5609	0.875503	0.437751	+	0.899096i	$0.355775\pi$
$788$	0	0
$789$	−31.1602	−1.10933
$790$	0	0
$791$	43.0062	1.52912
$792$	0	0
$793$	−6.90375	−0.245159
$794$	0	0
$795$	0	0
$796$	0	0
$797$	14.7970	0.524135	0.262068	−	0.965050i	$-0.415596\pi$
$797$	14.7970	0.524135	0.262068	+	0.965050i	$0.415596\pi$
$798$	0	0
$799$	62.1830	2.19988
$800$	0	0
$801$	5.53226	0.195473
$802$	0	0
$803$	−3.57530	−0.126170
$804$	0	0
$805$	0	0
$806$	0	0
$807$	5.18675	0.182582
$808$	0	0
$809$	−27.9438	−0.982452	−0.491226	−	0.871032i	$-0.663451\pi$
$809$	−27.9438	−0.982452	−0.491226	+	0.871032i	$0.663451\pi$
$810$	0	0
$811$	−37.2940	−1.30957	−0.654786	−	0.755815i	$-0.727241\pi$
$811$	−37.2940	−1.30957	−0.654786	+	0.755815i	$0.727241\pi$
$812$	0	0
$813$	5.65152	0.198207
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−43.3092	−1.51520
$818$	0	0
$819$	−15.7013	−0.548647
$820$	0	0
$821$	−3.38376	−0.118094	−0.0590470	−	0.998255i	$-0.518806\pi$
$821$	−3.38376	−0.118094	−0.0590470	+	0.998255i	$0.518806\pi$
$822$	0	0
$823$	−54.5344	−1.90095	−0.950475	−	0.310802i	$-0.899402\pi$
$823$	−54.5344	−1.90095	−0.950475	+	0.310802i	$0.899402\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−39.3201	−1.36729	−0.683647	−	0.729813i	$-0.739607\pi$
$827$	−39.3201	−1.36729	−0.683647	+	0.729813i	$0.739607\pi$
$828$	0	0
$829$	−49.4943	−1.71901	−0.859504	−	0.511128i	$-0.829228\pi$
$829$	−49.4943	−1.71901	−0.859504	+	0.511128i	$0.829228\pi$
$830$	0	0
$831$	−28.6437	−0.993638
$832$	0	0
$833$	87.1760	3.02047
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−10.5004	−0.362948
$838$	0	0
$839$	−36.0371	−1.24414	−0.622070	−	0.782962i	$-0.713708\pi$
$839$	−36.0371	−1.24414	−0.622070	+	0.782962i	$0.713708\pi$
$840$	0	0
$841$	−26.4769	−0.912995
$842$	0	0
$843$	14.0854	0.485128
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−45.4002	−1.55997
$848$	0	0
$849$	−23.6695	−0.812334
$850$	0	0
$851$	−1.56447	−0.0536295
$852$	0	0
$853$	4.36507	0.149457	0.0747286	−	0.997204i	$-0.476191\pi$
$853$	4.36507	0.149457	0.0747286	+	0.997204i	$0.476191\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−51.6385	−1.76394	−0.881969	−	0.471308i	$-0.843782\pi$
$857$	−51.6385	−1.76394	−0.881969	+	0.471308i	$0.843782\pi$
$858$	0	0
$859$	−24.8064	−0.846385	−0.423192	−	0.906040i	$-0.639091\pi$
$859$	−24.8064	−0.846385	−0.423192	+	0.906040i	$0.639091\pi$
$860$	0	0
$861$	19.9360	0.679416
$862$	0	0
$863$	−32.9477	−1.12155	−0.560776	−	0.827968i	$-0.689497\pi$
$863$	−32.9477	−1.12155	−0.560776	+	0.827968i	$0.689497\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−40.2213	−1.36599
$868$	0	0
$869$	7.00049	0.237475
$870$	0	0
$871$	−37.5023	−1.27072
$872$	0	0
$873$	8.71556	0.294977
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−23.4691	−0.792495	−0.396248	−	0.918144i	$-0.629688\pi$
$877$	−23.4691	−0.792495	−0.396248	+	0.918144i	$0.629688\pi$
$878$	0	0
$879$	−18.7962	−0.633980
$880$	0	0
$881$	20.9824	0.706917	0.353458	−	0.935450i	$-0.385006\pi$
$881$	20.9824	0.706917	0.353458	+	0.935450i	$0.385006\pi$
$882$	0	0
$883$	−55.7940	−1.87762	−0.938809	−	0.344439i	$-0.888069\pi$
$883$	−55.7940	−1.87762	−0.938809	+	0.344439i	$0.888069\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−55.7607	−1.87226	−0.936130	−	0.351655i	$-0.885619\pi$
$887$	−55.7607	−1.87226	−0.936130	+	0.351655i	$0.885619\pi$
$888$	0	0
$889$	−17.0097	−0.570487
$890$	0	0
$891$	−0.672035	−0.0225140
$892$	0	0
$893$	42.9138	1.43606
$894$	0	0
$895$	0	0
$896$	0	0
$897$	3.64807	0.121805
$898$	0	0
$899$	−16.6793	−0.556287
$900$	0	0
$901$	−41.7892	−1.39220
$902$	0	0
$903$	35.7066	1.18824
$904$	0	0
$905$	0	0
$906$	0	0
$907$	46.1719	1.53311	0.766556	−	0.642177i	$-0.221969\pi$
$907$	46.1719	1.53311	0.766556	+	0.642177i	$0.221969\pi$
$908$	0	0
$909$	1.13681	0.0377056
$910$	0	0
$911$	−56.1216	−1.85939	−0.929695	−	0.368329i	$-0.879930\pi$
$911$	−56.1216	−1.85939	−0.929695	+	0.368329i	$0.879930\pi$
$912$	0	0
$913$	−10.1511	−0.335952
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−61.2046	−2.02115
$918$	0	0
$919$	−33.1860	−1.09471	−0.547353	−	0.836902i	$-0.684364\pi$
$919$	−33.1860	−1.09471	−0.547353	+	0.836902i	$0.684364\pi$
$920$	0	0
$921$	8.23306	0.271289
$922$	0	0
$923$	25.2315	0.830506
$924$	0	0
$925$	0	0
$926$	0	0
$927$	10.8524	0.356438
$928$	0	0
$929$	−16.4090	−0.538361	−0.269181	−	0.963090i	$-0.586753\pi$
$929$	−16.4090	−0.538361	−0.269181	+	0.963090i	$0.586753\pi$
$930$	0	0
$931$	60.1620	1.97173
$932$	0	0
$933$	23.9251	0.783274
$934$	0	0
$935$	0	0
$936$	0	0
$937$	4.27140	0.139541	0.0697703	−	0.997563i	$-0.477773\pi$
$937$	4.27140	0.139541	0.0697703	+	0.997563i	$0.477773\pi$
$938$	0	0
$939$	8.54051	0.278709
$940$	0	0
$941$	19.8963	0.648600	0.324300	−	0.945954i	$-0.394871\pi$
$941$	19.8963	0.648600	0.324300	+	0.945954i	$0.394871\pi$
$942$	0	0
$943$	−4.63196	−0.150837
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−41.9032	−1.36167	−0.680835	−	0.732437i	$-0.738383\pi$
$947$	−41.9032	−1.36167	−0.680835	+	0.732437i	$0.738383\pi$
$948$	0	0
$949$	−19.4081	−0.630014
$950$	0	0
$951$	15.8079	0.512606
$952$	0	0
$953$	−53.3885	−1.72942	−0.864711	−	0.502270i	$-0.832498\pi$
$953$	−53.3885	−1.72942	−0.864711	+	0.502270i	$0.832498\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−1.06749	−0.0345070
$958$	0	0
$959$	−73.5167	−2.37398
$960$	0	0
$961$	79.2592	2.55675
$962$	0	0
$963$	−8.22040	−0.264899
$964$	0	0
$965$	0	0
$966$	0	0
$967$	58.2152	1.87208	0.936038	−	0.351899i	$-0.114464\pi$
$967$	58.2152	1.87208	0.936038	+	0.351899i	$0.114464\pi$
$968$	0	0
$969$	−39.4896	−1.26859
$970$	0	0
$971$	−42.3742	−1.35985	−0.679926	−	0.733280i	$-0.737988\pi$
$971$	−42.3742	−1.35985	−0.679926	+	0.733280i	$0.737988\pi$
$972$	0	0
$973$	−0.776146	−0.0248821
$974$	0	0
$975$	0	0
$976$	0	0
$977$	37.7866	1.20890	0.604451	−	0.796643i	$-0.293393\pi$
$977$	37.7866	1.20890	0.604451	+	0.796643i	$0.293393\pi$
$978$	0	0
$979$	−3.71788	−0.118824
$980$	0	0
$981$	−11.8044	−0.376887
$982$	0	0
$983$	−38.8498	−1.23912	−0.619558	−	0.784951i	$-0.712688\pi$
$983$	−38.8498	−1.23912	−0.619558	+	0.784951i	$0.712688\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−35.3806	−1.12618
$988$	0	0
$989$	−8.29614	−0.263802
$990$	0	0
$991$	31.7227	1.00770	0.503852	−	0.863790i	$-0.331916\pi$
$991$	31.7227	1.00770	0.503852	+	0.863790i	$0.331916\pi$
$992$	0	0
$993$	10.4652	0.332102
$994$	0	0
$995$	0	0
$996$	0	0
$997$	52.9046	1.67551	0.837753	−	0.546049i	$-0.183869\pi$
$997$	52.9046	1.67551	0.837753	+	0.546049i	$0.183869\pi$
$998$	0	0
$999$	1.56447	0.0494978

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6900.2.a.ba.1.4	✓	4
5.2	odd	4		6900.2.f.s.6349.8		8
5.3	odd	4		6900.2.f.s.6349.1		8
5.4	even	2		6900.2.a.bb.1.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6900.2.a.ba.1.4	✓	4	1.1	even	1	trivial
6900.2.a.bb.1.1	yes	4	5.4	even	2
6900.2.f.s.6349.1		8	5.3	odd	4
6900.2.f.s.6349.8		8	5.2	odd	4

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +4.30400 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +4.30400 q^{7} +1.00000 q^{9} -0.672035 q^{11} -3.64807 q^{13} +7.56447 q^{17} +5.22040 q^{19} -4.30400 q^{21} +1.00000 q^{23} -1.00000 q^{27} -1.58844 q^{29} +10.5004 q^{31} +0.672035 q^{33} -1.56447 q^{37} +3.64807 q^{39} -4.63196 q^{41} -8.29614 q^{43} +8.22040 q^{47} +11.5244 q^{49} -7.56447 q^{51} -5.52440 q^{53} -5.22040 q^{57} -2.26048 q^{59} +1.89244 q^{61} +4.30400 q^{63} +10.2800 q^{67} -1.00000 q^{69} -6.91641 q^{71} +5.32010 q^{73} -2.89244 q^{77} -10.4168 q^{79} +1.00000 q^{81} +15.1050 q^{83} +1.58844 q^{87} +5.53226 q^{89} -15.7013 q^{91} -10.5004 q^{93} +8.71556 q^{97} -0.672035 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 3 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 + 3 * q^7 + 4 * q^9	\( 4 q - 4 q^{3} + 3 q^{7} + 4 q^{9} - 2 q^{11} + q^{13} + 10 q^{17} + 2 q^{19} - 3 q^{21} + 4 q^{23} - 4 q^{27} - q^{29} - 6 q^{31} + 2 q^{33} + 14 q^{37} - q^{39} - 5 q^{41} - 2 q^{43} + 14 q^{47} + 13 q^{49} - 10 q^{51} + 11 q^{53} - 2 q^{57} - 3 q^{59} - 12 q^{61} + 3 q^{63} + 12 q^{67} - 4 q^{69} - 23 q^{71} + 5 q^{73} + 8 q^{77} + 11 q^{79} + 4 q^{81} + 5 q^{83} + q^{87} + 6 q^{89} - 19 q^{91} + 6 q^{93} + 26 q^{97} - 2 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 + 3 * q^7 + 4 * q^9 - 2 * q^11 + q^13 + 10 * q^17 + 2 * q^19 - 3 * q^21 + 4 * q^23 - 4 * q^27 - q^29 - 6 * q^31 + 2 * q^33 + 14 * q^37 - q^39 - 5 * q^41 - 2 * q^43 + 14 * q^47 + 13 * q^49 - 10 * q^51 + 11 * q^53 - 2 * q^57 - 3 * q^59 - 12 * q^61 + 3 * q^63 + 12 * q^67 - 4 * q^69 - 23 * q^71 + 5 * q^73 + 8 * q^77 + 11 * q^79 + 4 * q^81 + 5 * q^83 + q^87 + 6 * q^89 - 19 * q^91 + 6 * q^93 + 26 * q^97 - 2 * q^99

Introduction

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