LMFDB - Embedded newform 6975.2.a.bp.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6975,2,Mod(1,6975)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6975.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6975, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6975 = 3^{2} \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6975.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,-4,0,6,0,0,2,-9,0,0,2,0,4,-2,0,4,-19] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$55.6956554098$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.144209.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 6x^{3} - x^{2} + 6x - 1 $ x^5 - 6x^3 - x^2 + 6x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 775)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.871612$ of defining polynomial
Character	$\chi$	$=$	6975.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.58398 q^{2} +4.67698 q^{4} -2.11190 q^{7} -6.91727 q^{8} +4.70783 q^{11} -3.53665 q^{13} +5.45713 q^{14} +8.52017 q^{16} -6.45560 q^{17} +0.766680 q^{19} -12.1650 q^{22} +5.38481 q^{23} +9.13865 q^{26} -9.87733 q^{28} -8.03414 q^{29} -1.00000 q^{31} -8.18144 q^{32} +16.6812 q^{34} +11.5690 q^{37} -1.98109 q^{38} +6.47040 q^{41} -5.80082 q^{43} +22.0184 q^{44} -13.9143 q^{46} +8.68724 q^{47} -2.53986 q^{49} -16.5408 q^{52} -4.05063 q^{53} +14.6086 q^{56} +20.7601 q^{58} -9.36743 q^{59} +3.40129 q^{61} +2.58398 q^{62} +4.10039 q^{64} +7.82514 q^{67} -30.1927 q^{68} +3.15106 q^{71} +13.4359 q^{73} -29.8942 q^{74} +3.58574 q^{76} -9.94249 q^{77} -11.6812 q^{79} -16.7194 q^{82} -9.79474 q^{83} +14.9892 q^{86} -32.5653 q^{88} +0.211117 q^{89} +7.46907 q^{91} +25.1846 q^{92} -22.4477 q^{94} +7.13932 q^{97} +6.56296 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 4 q^{2} + 6 q^{4} + 2 q^{7} - 9 q^{8} + 2 q^{11} + 4 q^{13} - 2 q^{14} + 4 q^{16} - 19 q^{17} + 8 q^{19} - 10 q^{22} - 12 q^{23} + 16 q^{26} + 6 q^{28} - 6 q^{29} - 5 q^{31} - 7 q^{32} + 31 q^{34}+ \cdots - 10 q^{98}+O(q^{100}) $ 5 * q - 4 * q^2 + 6 * q^4 + 2 * q^7 - 9 * q^8 + 2 * q^11 + 4 * q^13 - 2 * q^14 + 4 * q^16 - 19 * q^17 + 8 * q^19 - 10 * q^22 - 12 * q^23 + 16 * q^26 + 6 * q^28 - 6 * q^29 - 5 * q^31 - 7 * q^32 + 31 * q^34 + 16 * q^37 - 14 * q^38 + 2 * q^41 + q^43 + 4 * q^44 - 11 * q^46 - 8 * q^47 - 9 * q^49 - 26 * q^52 - 3 * q^53 + 9 * q^56 + 5 * q^58 + 4 * q^59 - 5 * q^61 + 4 * q^62 - 5 * q^64 + 13 * q^67 - 30 * q^68 - 6 * q^71 + 19 * q^73 - 4 * q^74 - 5 * q^76 - 23 * q^77 - 6 * q^79 - 27 * q^82 - 18 * q^83 + 7 * q^86 - q^88 + 31 * q^89 + 8 * q^91 + 16 * q^92 - 14 * q^94 - q^97 - 10 * q^98

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 $

$2$	−2.58398	−1.82715	−0.913577	−	0.406667i	$-0.866691\pi$
$2$	−2.58398	−1.82715	−0.913577	+	0.406667i	$0.866691\pi$
$3$	0	0
$3$	0	0
$4$	4.67698	2.33849
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.11190	−0.798225	−0.399112	−	0.916902i	$-0.630682\pi$
$7$	−2.11190	−0.798225	−0.399112	+	0.916902i	$0.630682\pi$
$8$	−6.91727	−2.44562
$9$	0	0
$10$	0	0
$11$	4.70783	1.41946	0.709732	−	0.704472i	$-0.248816\pi$
$11$	4.70783	1.41946	0.709732	+	0.704472i	$0.248816\pi$
$12$	0	0
$13$	−3.53665	−0.980890	−0.490445	−	0.871472i	$-0.663166\pi$
$13$	−3.53665	−0.980890	−0.490445	+	0.871472i	$0.663166\pi$
$14$	5.45713	1.45848
$15$	0	0
$16$	8.52017	2.13004
$17$	−6.45560	−1.56571	−0.782856	−	0.622203i	$-0.786238\pi$
$17$	−6.45560	−1.56571	−0.782856	+	0.622203i	$0.786238\pi$
$18$	0	0
$19$	0.766680	0.175888	0.0879442	−	0.996125i	$-0.471970\pi$
$19$	0.766680	0.175888	0.0879442	+	0.996125i	$0.471970\pi$
$20$	0	0
$21$	0	0
$22$	−12.1650	−2.59358
$23$	5.38481	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$23$	5.38481	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$24$	0	0
$25$	0	0
$26$	9.13865	1.79224
$27$	0	0
$28$	−9.87733	−1.86664
$29$	−8.03414	−1.49190	−0.745951	−	0.666000i	$-0.768005\pi$
$29$	−8.03414	−1.49190	−0.745951	+	0.666000i	$0.768005\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	−8.18144	−1.44629
$33$	0	0
$34$	16.6812	2.86080
$35$	0	0
$36$	0	0
$37$	11.5690	1.90194	0.950969	−	0.309287i	$-0.100090\pi$
$37$	11.5690	1.90194	0.950969	+	0.309287i	$0.100090\pi$
$38$	−1.98109	−0.321375
$39$	0	0
$40$	0	0
$41$	6.47040	1.01051	0.505254	−	0.862971i	$-0.331399\pi$
$41$	6.47040	1.01051	0.505254	+	0.862971i	$0.331399\pi$
$42$	0	0
$43$	−5.80082	−0.884617	−0.442309	−	0.896863i	$-0.645840\pi$
$43$	−5.80082	−0.884617	−0.442309	+	0.896863i	$0.645840\pi$
$44$	22.0184	3.31940
$45$	0	0
$46$	−13.9143	−2.05155
$47$	8.68724	1.26716	0.633582	−	0.773675i	$-0.281584\pi$
$47$	8.68724	1.26716	0.633582	+	0.773675i	$0.281584\pi$
$48$	0	0
$49$	−2.53986	−0.362837
$50$	0	0
$51$	0	0
$52$	−16.5408	−2.29380
$53$	−4.05063	−0.556396	−0.278198	−	0.960524i	$-0.589737\pi$
$53$	−4.05063	−0.556396	−0.278198	+	0.960524i	$0.589737\pi$
$54$	0	0
$55$	0	0
$56$	14.6086	1.95216
$57$	0	0
$58$	20.7601	2.72593
$59$	−9.36743	−1.21954	−0.609768	−	0.792580i	$-0.708737\pi$
$59$	−9.36743	−1.21954	−0.609768	+	0.792580i	$0.708737\pi$
$60$	0	0
$61$	3.40129	0.435491	0.217745	−	0.976006i	$-0.430130\pi$
$61$	3.40129	0.435491	0.217745	+	0.976006i	$0.430130\pi$
$62$	2.58398	0.328166
$63$	0	0
$64$	4.10039	0.512548
$65$	0	0
$66$	0	0
$67$	7.82514	0.955993	0.477997	−	0.878362i	$-0.341363\pi$
$67$	7.82514	0.955993	0.477997	+	0.878362i	$0.341363\pi$
$68$	−30.1927	−3.66140
$69$	0	0
$70$	0	0
$71$	3.15106	0.373962	0.186981	−	0.982363i	$-0.440130\pi$
$71$	3.15106	0.373962	0.186981	+	0.982363i	$0.440130\pi$
$72$	0	0
$73$	13.4359	1.57255	0.786277	−	0.617874i	$-0.212006\pi$
$73$	13.4359	1.57255	0.786277	+	0.617874i	$0.212006\pi$
$74$	−29.8942	−3.47513
$75$	0	0
$76$	3.58574	0.411313
$77$	−9.94249	−1.13305
$78$	0	0
$79$	−11.6812	−1.31423	−0.657117	−	0.753789i	$-0.728224\pi$
$79$	−11.6812	−1.31423	−0.657117	+	0.753789i	$0.728224\pi$
$80$	0	0
$81$	0	0
$82$	−16.7194	−1.84635
$83$	−9.79474	−1.07511	−0.537556	−	0.843228i	$-0.680653\pi$
$83$	−9.79474	−1.07511	−0.537556	+	0.843228i	$0.680653\pi$
$84$	0	0
$85$	0	0
$86$	14.9892	1.61633
$87$	0	0
$88$	−32.5653	−3.47148
$89$	0.211117	0.0223784	0.0111892	−	0.999937i	$-0.496438\pi$
$89$	0.211117	0.0223784	0.0111892	+	0.999937i	$0.496438\pi$
$90$	0	0
$91$	7.46907	0.782971
$92$	25.1846	2.62568
$93$	0	0
$94$	−22.4477	−2.31530
$95$	0	0
$96$	0	0
$97$	7.13932	0.724888	0.362444	−	0.932006i	$-0.381942\pi$
$97$	7.13932	0.724888	0.362444	+	0.932006i	$0.381942\pi$
$98$	6.56296	0.662959
$99$	0	0
$100$	0	0
$101$	8.60641	0.856370	0.428185	−	0.903691i	$-0.359153\pi$
$101$	8.60641	0.856370	0.428185	+	0.903691i	$0.359153\pi$
$102$	0	0
$103$	1.57511	0.155200	0.0776002	−	0.996985i	$-0.475274\pi$
$103$	1.57511	0.155200	0.0776002	+	0.996985i	$0.475274\pi$
$104$	24.4640	2.39889
$105$	0	0
$106$	10.4668	1.01662
$107$	−1.24980	−0.120823	−0.0604116	−	0.998174i	$-0.519241\pi$
$107$	−1.24980	−0.120823	−0.0604116	+	0.998174i	$0.519241\pi$
$108$	0	0
$109$	10.6818	1.02313	0.511566	−	0.859244i	$-0.329066\pi$
$109$	10.6818	1.02313	0.511566	+	0.859244i	$0.329066\pi$
$110$	0	0
$111$	0	0
$112$	−17.9938	−1.70025
$113$	−0.755521	−0.0710734	−0.0355367	−	0.999368i	$-0.511314\pi$
$113$	−0.755521	−0.0710734	−0.0355367	+	0.999368i	$0.511314\pi$
$114$	0	0
$115$	0	0
$116$	−37.5755	−3.48880
$117$	0	0
$118$	24.2053	2.22828
$119$	13.6336	1.24979
$120$	0	0
$121$	11.1637	1.01488
$122$	−8.78888	−0.795708
$123$	0	0
$124$	−4.67698	−0.420005
$125$	0	0
$126$	0	0
$127$	−7.11477	−0.631334	−0.315667	−	0.948870i	$-0.602228\pi$
$127$	−7.11477	−0.631334	−0.315667	+	0.948870i	$0.602228\pi$
$128$	5.76754	0.509784
$129$	0	0
$130$	0	0
$131$	6.39521	0.558752	0.279376	−	0.960182i	$-0.409872\pi$
$131$	6.39521	0.558752	0.279376	+	0.960182i	$0.409872\pi$
$132$	0	0
$133$	−1.61915	−0.140398
$134$	−20.2200	−1.74675
$135$	0	0
$136$	44.6551	3.82914
$137$	−13.6264	−1.16418	−0.582089	−	0.813125i	$-0.697764\pi$
$137$	−13.6264	−1.16418	−0.582089	+	0.813125i	$0.697764\pi$
$138$	0	0
$139$	−0.912415	−0.0773900	−0.0386950	−	0.999251i	$-0.512320\pi$
$139$	−0.912415	−0.0773900	−0.0386950	+	0.999251i	$0.512320\pi$
$140$	0	0
$141$	0	0
$142$	−8.14230	−0.683287
$143$	−16.6499	−1.39234
$144$	0	0
$145$	0	0
$146$	−34.7182	−2.87330
$147$	0	0
$148$	54.1081	4.44766
$149$	6.88466	0.564013	0.282007	−	0.959412i	$-0.409000\pi$
$149$	6.88466	0.564013	0.282007	+	0.959412i	$0.409000\pi$
$150$	0	0
$151$	−15.4481	−1.25715	−0.628575	−	0.777749i	$-0.716362\pi$
$151$	−15.4481	−1.25715	−0.628575	+	0.777749i	$0.716362\pi$
$152$	−5.30333	−0.430157
$153$	0	0
$154$	25.6912	2.07026
$155$	0	0
$156$	0	0
$157$	2.73611	0.218365	0.109183	−	0.994022i	$-0.465177\pi$
$157$	2.73611	0.218365	0.109183	+	0.994022i	$0.465177\pi$
$158$	30.1840	2.40131
$159$	0	0
$160$	0	0
$161$	−11.3722	−0.896255
$162$	0	0
$163$	6.17569	0.483718	0.241859	−	0.970311i	$-0.422243\pi$
$163$	6.17569	0.483718	0.241859	+	0.970311i	$0.422243\pi$
$164$	30.2619	2.36306
$165$	0	0
$166$	25.3095	1.96440
$167$	−17.1559	−1.32756	−0.663782	−	0.747926i	$-0.731050\pi$
$167$	−17.1559	−1.32756	−0.663782	+	0.747926i	$0.731050\pi$
$168$	0	0
$169$	−0.492102	−0.0378540
$170$	0	0
$171$	0	0
$172$	−27.1303	−2.06867
$173$	6.07994	0.462250	0.231125	−	0.972924i	$-0.425759\pi$
$173$	6.07994	0.462250	0.231125	+	0.972924i	$0.425759\pi$
$174$	0	0
$175$	0	0
$176$	40.1115	3.02352
$177$	0	0
$178$	−0.545524	−0.0408888
$179$	−17.8901	−1.33717	−0.668583	−	0.743638i	$-0.733099\pi$
$179$	−17.8901	−1.33717	−0.668583	+	0.743638i	$0.733099\pi$
$180$	0	0
$181$	−18.3051	−1.36061	−0.680303	−	0.732931i	$-0.738152\pi$
$181$	−18.3051	−1.36061	−0.680303	+	0.732931i	$0.738152\pi$
$182$	−19.3000	−1.43061
$183$	0	0
$184$	−37.2482	−2.74597
$185$	0	0
$186$	0	0
$187$	−30.3918	−2.22247
$188$	40.6300	2.96325
$189$	0	0
$190$	0	0
$191$	−1.09600	−0.0793041	−0.0396521	−	0.999214i	$-0.512625\pi$
$191$	−1.09600	−0.0793041	−0.0396521	+	0.999214i	$0.512625\pi$
$192$	0	0
$193$	−15.5155	−1.11683	−0.558414	−	0.829562i	$-0.688590\pi$
$193$	−15.5155	−1.11683	−0.558414	+	0.829562i	$0.688590\pi$
$194$	−18.4479	−1.32448
$195$	0	0
$196$	−11.8789	−0.848490
$197$	−12.2418	−0.872194	−0.436097	−	0.899900i	$-0.643640\pi$
$197$	−12.2418	−0.872194	−0.436097	+	0.899900i	$0.643640\pi$
$198$	0	0
$199$	−6.49517	−0.460430	−0.230215	−	0.973140i	$-0.573943\pi$
$199$	−6.49517	−0.460430	−0.230215	+	0.973140i	$0.573943\pi$
$200$	0	0
$201$	0	0
$202$	−22.2388	−1.56472
$203$	16.9673	1.19087
$204$	0	0
$205$	0	0
$206$	−4.07006	−0.283575
$207$	0	0
$208$	−30.1329	−2.08934
$209$	3.60940	0.249667
$210$	0	0
$211$	8.40626	0.578711	0.289355	−	0.957222i	$-0.406559\pi$
$211$	8.40626	0.578711	0.289355	+	0.957222i	$0.406559\pi$
$212$	−18.9447	−1.30113
$213$	0	0
$214$	3.22947	0.220762
$215$	0	0
$216$	0	0
$217$	2.11190	0.143365
$218$	−27.6016	−1.86942
$219$	0	0
$220$	0	0
$221$	22.8312	1.53579
$222$	0	0
$223$	10.1283	0.678238	0.339119	−	0.940743i	$-0.389871\pi$
$223$	10.1283	0.678238	0.339119	+	0.940743i	$0.389871\pi$
$224$	17.2784	1.15446
$225$	0	0
$226$	1.95225	0.129862
$227$	3.73800	0.248100	0.124050	−	0.992276i	$-0.460412\pi$
$227$	3.73800	0.248100	0.124050	+	0.992276i	$0.460412\pi$
$228$	0	0
$229$	−19.7631	−1.30598	−0.652990	−	0.757367i	$-0.726485\pi$
$229$	−19.7631	−1.30598	−0.652990	+	0.757367i	$0.726485\pi$
$230$	0	0
$231$	0	0
$232$	55.5743	3.64863
$233$	−26.5881	−1.74184	−0.870921	−	0.491423i	$-0.836477\pi$
$233$	−26.5881	−1.74184	−0.870921	+	0.491423i	$0.836477\pi$
$234$	0	0
$235$	0	0
$236$	−43.8113	−2.85187
$237$	0	0
$238$	−35.2290	−2.28356
$239$	16.6224	1.07522	0.537608	−	0.843195i	$-0.319328\pi$
$239$	16.6224	1.07522	0.537608	+	0.843195i	$0.319328\pi$
$240$	0	0
$241$	−0.0903393	−0.00581926	−0.00290963	−	0.999996i	$-0.500926\pi$
$241$	−0.0903393	−0.00581926	−0.00290963	+	0.999996i	$0.500926\pi$
$242$	−28.8467	−1.85434
$243$	0	0
$244$	15.9078	1.01839
$245$	0	0
$246$	0	0
$247$	−2.71148	−0.172527
$248$	6.91727	0.439247
$249$	0	0
$250$	0	0
$251$	−3.67391	−0.231895	−0.115948	−	0.993255i	$-0.536990\pi$
$251$	−3.67391	−0.231895	−0.115948	+	0.993255i	$0.536990\pi$
$252$	0	0
$253$	25.3507	1.59379
$254$	18.3845	1.15354
$255$	0	0
$256$	−23.1040	−1.44400
$257$	31.5960	1.97090	0.985451	−	0.169960i	$-0.0543639\pi$
$257$	31.5960	1.97090	0.985451	+	0.169960i	$0.0543639\pi$
$258$	0	0
$259$	−24.4327	−1.51817
$260$	0	0
$261$	0	0
$262$	−16.5251	−1.02093
$263$	−20.6444	−1.27299	−0.636495	−	0.771281i	$-0.719616\pi$
$263$	−20.6444	−1.27299	−0.636495	+	0.771281i	$0.719616\pi$
$264$	0	0
$265$	0	0
$266$	4.18387	0.256530
$267$	0	0
$268$	36.5980	2.23558
$269$	21.8419	1.33172	0.665862	−	0.746075i	$-0.268064\pi$
$269$	21.8419	1.33172	0.665862	+	0.746075i	$0.268064\pi$
$270$	0	0
$271$	9.08392	0.551809	0.275905	−	0.961185i	$-0.411023\pi$
$271$	9.08392	0.551809	0.275905	+	0.961185i	$0.411023\pi$
$272$	−55.0028	−3.33503
$273$	0	0
$274$	35.2103	2.12713
$275$	0	0
$276$	0	0
$277$	−22.4910	−1.35135	−0.675676	−	0.737199i	$-0.736148\pi$
$277$	−22.4910	−1.35135	−0.675676	+	0.737199i	$0.736148\pi$
$278$	2.35767	0.141403
$279$	0	0
$280$	0	0
$281$	9.88551	0.589720	0.294860	−	0.955540i	$-0.404727\pi$
$281$	9.88551	0.589720	0.294860	+	0.955540i	$0.404727\pi$
$282$	0	0
$283$	−4.56044	−0.271090	−0.135545	−	0.990771i	$-0.543278\pi$
$283$	−4.56044	−0.271090	−0.135545	+	0.990771i	$0.543278\pi$
$284$	14.7375	0.874507
$285$	0	0
$286$	43.0232	2.54402
$287$	−13.6649	−0.806612
$288$	0	0
$289$	24.6747	1.45145
$290$	0	0
$291$	0	0
$292$	62.8394	3.67740
$293$	32.9820	1.92683	0.963415	−	0.268015i	$-0.0863676\pi$
$293$	32.9820	1.92683	0.963415	+	0.268015i	$0.0863676\pi$
$294$	0	0
$295$	0	0
$296$	−80.0261	−4.65143
$297$	0	0
$298$	−17.7899	−1.03054
$299$	−19.0442	−1.10135
$300$	0	0
$301$	12.2508	0.706123
$302$	39.9177	2.29701
$303$	0	0
$304$	6.53224	0.374650
$305$	0	0
$306$	0	0
$307$	−11.2816	−0.643877	−0.321939	−	0.946761i	$-0.604335\pi$
$307$	−11.2816	−0.643877	−0.321939	+	0.946761i	$0.604335\pi$
$308$	−46.5008	−2.64963
$309$	0	0
$310$	0	0
$311$	12.2410	0.694126	0.347063	−	0.937842i	$-0.387179\pi$
$311$	12.2410	0.694126	0.347063	+	0.937842i	$0.387179\pi$
$312$	0	0
$313$	−10.0264	−0.566728	−0.283364	−	0.959012i	$-0.591450\pi$
$313$	−10.0264	−0.566728	−0.283364	+	0.959012i	$0.591450\pi$
$314$	−7.07006	−0.398987
$315$	0	0
$316$	−54.6325	−3.07332
$317$	−2.30927	−0.129702	−0.0648509	−	0.997895i	$-0.520657\pi$
$317$	−2.30927	−0.129702	−0.0648509	+	0.997895i	$0.520657\pi$
$318$	0	0
$319$	−37.8234	−2.11770
$320$	0	0
$321$	0	0
$322$	29.3856	1.63759
$323$	−4.94937	−0.275391
$324$	0	0
$325$	0	0
$326$	−15.9579	−0.883826
$327$	0	0
$328$	−44.7575	−2.47132
$329$	−18.3466	−1.01148
$330$	0	0
$331$	9.04859	0.497356	0.248678	−	0.968586i	$-0.420004\pi$
$331$	9.04859	0.497356	0.248678	+	0.968586i	$0.420004\pi$
$332$	−45.8098	−2.51414
$333$	0	0
$334$	44.3306	2.42566
$335$	0	0
$336$	0	0
$337$	26.4888	1.44294	0.721468	−	0.692448i	$-0.243468\pi$
$337$	26.4888	1.44294	0.721468	+	0.692448i	$0.243468\pi$
$338$	1.27158	0.0691651
$339$	0	0
$340$	0	0
$341$	−4.70783	−0.254943
$342$	0	0
$343$	20.1473	1.08785
$344$	40.1259	2.16344
$345$	0	0
$346$	−15.7105	−0.844601
$347$	−37.0463	−1.98875	−0.994374	−	0.105927i	$-0.966219\pi$
$347$	−37.0463	−1.98875	−0.994374	+	0.105927i	$0.966219\pi$
$348$	0	0
$349$	−17.5804	−0.941059	−0.470529	−	0.882384i	$-0.655937\pi$
$349$	−17.5804	−0.941059	−0.470529	+	0.882384i	$0.655937\pi$
$350$	0	0
$351$	0	0
$352$	−38.5168	−2.05295
$353$	−24.2026	−1.28817	−0.644087	−	0.764952i	$-0.722763\pi$
$353$	−24.2026	−1.28817	−0.644087	+	0.764952i	$0.722763\pi$
$354$	0	0
$355$	0	0
$356$	0.987391	0.0523316
$357$	0	0
$358$	46.2276	2.44321
$359$	−13.0616	−0.689366	−0.344683	−	0.938719i	$-0.612014\pi$
$359$	−13.0616	−0.689366	−0.344683	+	0.938719i	$0.612014\pi$
$360$	0	0
$361$	−18.4122	−0.969063
$362$	47.3000	2.48604
$363$	0	0
$364$	34.9327	1.83097
$365$	0	0
$366$	0	0
$367$	−21.4926	−1.12191	−0.560953	−	0.827848i	$-0.689565\pi$
$367$	−21.4926	−1.12191	−0.560953	+	0.827848i	$0.689565\pi$
$368$	45.8795	2.39163
$369$	0	0
$370$	0	0
$371$	8.55454	0.444129
$372$	0	0
$373$	7.79176	0.403442	0.201721	−	0.979443i	$-0.435347\pi$
$373$	7.79176	0.403442	0.201721	+	0.979443i	$0.435347\pi$
$374$	78.5321	4.06080
$375$	0	0
$376$	−60.0920	−3.09901
$377$	28.4140	1.46339
$378$	0	0
$379$	23.8722	1.22623	0.613116	−	0.789993i	$-0.289916\pi$
$379$	23.8722	1.22623	0.613116	+	0.789993i	$0.289916\pi$
$380$	0	0
$381$	0	0
$382$	2.83206	0.144901
$383$	8.53896	0.436321	0.218160	−	0.975913i	$-0.429994\pi$
$383$	8.53896	0.436321	0.218160	+	0.975913i	$0.429994\pi$
$384$	0	0
$385$	0	0
$386$	40.0918	2.04062
$387$	0	0
$388$	33.3904	1.69514
$389$	−2.02322	−0.102581	−0.0512906	−	0.998684i	$-0.516333\pi$
$389$	−2.02322	−0.102581	−0.0512906	+	0.998684i	$0.516333\pi$
$390$	0	0
$391$	−34.7621	−1.75800
$392$	17.5689	0.887363
$393$	0	0
$394$	31.6327	1.59363
$395$	0	0
$396$	0	0
$397$	−8.41708	−0.422441	−0.211220	−	0.977438i	$-0.567744\pi$
$397$	−8.41708	−0.422441	−0.211220	+	0.977438i	$0.567744\pi$
$398$	16.7834	0.841277
$399$	0	0
$400$	0	0
$401$	−9.32972	−0.465904	−0.232952	−	0.972488i	$-0.574839\pi$
$401$	−9.32972	−0.465904	−0.232952	+	0.972488i	$0.574839\pi$
$402$	0	0
$403$	3.53665	0.176173
$404$	40.2520	2.00261
$405$	0	0
$406$	−43.8434	−2.17591
$407$	54.4650	2.69973
$408$	0	0
$409$	22.2516	1.10027	0.550136	−	0.835075i	$-0.314576\pi$
$409$	22.2516	1.10027	0.550136	+	0.835075i	$0.314576\pi$
$410$	0	0
$411$	0	0
$412$	7.36676	0.362934
$413$	19.7831	0.973463
$414$	0	0
$415$	0	0
$416$	28.9349	1.41865
$417$	0	0
$418$	−9.32663	−0.456180
$419$	6.87270	0.335753	0.167877	−	0.985808i	$-0.446309\pi$
$419$	6.87270	0.335753	0.167877	+	0.985808i	$0.446309\pi$
$420$	0	0
$421$	20.9630	1.02168	0.510838	−	0.859677i	$-0.329335\pi$
$421$	20.9630	1.02168	0.510838	+	0.859677i	$0.329335\pi$
$422$	−21.7216	−1.05739
$423$	0	0
$424$	28.0193	1.36074
$425$	0	0
$426$	0	0
$427$	−7.18320	−0.347619
$428$	−5.84531	−0.282544
$429$	0	0
$430$	0	0
$431$	−33.2926	−1.60365	−0.801825	−	0.597559i	$-0.796137\pi$
$431$	−33.2926	−1.60365	−0.801825	+	0.597559i	$0.796137\pi$
$432$	0	0
$433$	−0.696968	−0.0334941	−0.0167471	−	0.999860i	$-0.505331\pi$
$433$	−0.696968	−0.0334941	−0.0167471	+	0.999860i	$0.505331\pi$
$434$	−5.45713	−0.261951
$435$	0	0
$436$	49.9586	2.39258
$437$	4.12842	0.197489
$438$	0	0
$439$	−17.6634	−0.843030	−0.421515	−	0.906821i	$-0.638502\pi$
$439$	−17.6634	−0.843030	−0.421515	+	0.906821i	$0.638502\pi$
$440$	0	0
$441$	0	0
$442$	−58.9955	−2.80613
$443$	30.2630	1.43784	0.718920	−	0.695093i	$-0.244637\pi$
$443$	30.2630	1.43784	0.718920	+	0.695093i	$0.244637\pi$
$444$	0	0
$445$	0	0
$446$	−26.1713	−1.23925
$447$	0	0
$448$	−8.65963	−0.409129
$449$	−0.907674	−0.0428358	−0.0214179	−	0.999771i	$-0.506818\pi$
$449$	−0.907674	−0.0428358	−0.0214179	+	0.999771i	$0.506818\pi$
$450$	0	0
$451$	30.4616	1.43438
$452$	−3.53355	−0.166204
$453$	0	0
$454$	−9.65893	−0.453316
$455$	0	0
$456$	0	0
$457$	31.6382	1.47997	0.739987	−	0.672622i	$-0.234832\pi$
$457$	31.6382	1.47997	0.739987	+	0.672622i	$0.234832\pi$
$458$	51.0674	2.38622
$459$	0	0
$460$	0	0
$461$	4.12145	0.191955	0.0959775	−	0.995384i	$-0.469402\pi$
$461$	4.12145	0.191955	0.0959775	+	0.995384i	$0.469402\pi$
$462$	0	0
$463$	−1.55269	−0.0721598	−0.0360799	−	0.999349i	$-0.511487\pi$
$463$	−1.55269	−0.0721598	−0.0360799	+	0.999349i	$0.511487\pi$
$464$	−68.4522	−3.17782
$465$	0	0
$466$	68.7031	3.18261
$467$	0.497828	0.0230367	0.0115184	−	0.999934i	$-0.496334\pi$
$467$	0.497828	0.0230367	0.0115184	+	0.999934i	$0.496334\pi$
$468$	0	0
$469$	−16.5260	−0.763098
$470$	0	0
$471$	0	0
$472$	64.7970	2.98253
$473$	−27.3093	−1.25568
$474$	0	0
$475$	0	0
$476$	63.7641	2.92262
$477$	0	0
$478$	−42.9521	−1.96459
$479$	−16.6728	−0.761802	−0.380901	−	0.924616i	$-0.624386\pi$
$479$	−16.6728	−0.761802	−0.380901	+	0.924616i	$0.624386\pi$
$480$	0	0
$481$	−40.9156	−1.86559
$482$	0.233435	0.0106327
$483$	0	0
$484$	52.2122	2.37328
$485$	0	0
$486$	0	0
$487$	1.58398	0.0717772	0.0358886	−	0.999356i	$-0.488574\pi$
$487$	1.58398	0.0717772	0.0358886	+	0.999356i	$0.488574\pi$
$488$	−23.5276	−1.06505
$489$	0	0
$490$	0	0
$491$	−21.1723	−0.955494	−0.477747	−	0.878498i	$-0.658546\pi$
$491$	−21.1723	−0.955494	−0.477747	+	0.878498i	$0.658546\pi$
$492$	0	0
$493$	51.8652	2.33589
$494$	7.00642	0.315234
$495$	0	0
$496$	−8.52017	−0.382567
$497$	−6.65475	−0.298506
$498$	0	0
$499$	8.52633	0.381691	0.190845	−	0.981620i	$-0.438877\pi$
$499$	8.52633	0.381691	0.190845	+	0.981620i	$0.438877\pi$
$500$	0	0
$501$	0	0
$502$	9.49333	0.423708
$503$	27.2917	1.21688	0.608439	−	0.793601i	$-0.291796\pi$
$503$	27.2917	1.21688	0.608439	+	0.793601i	$0.291796\pi$
$504$	0	0
$505$	0	0
$506$	−65.5059	−2.91209
$507$	0	0
$508$	−33.2756	−1.47637
$509$	−27.4555	−1.21694	−0.608472	−	0.793576i	$-0.708217\pi$
$509$	−27.4555	−1.21694	−0.608472	+	0.793576i	$0.708217\pi$
$510$	0	0
$511$	−28.3753	−1.25525
$512$	48.1654	2.12863
$513$	0	0
$514$	−81.6435	−3.60114
$515$	0	0
$516$	0	0
$517$	40.8980	1.79869
$518$	63.1337	2.77394
$519$	0	0
$520$	0	0
$521$	−31.8719	−1.39633	−0.698167	−	0.715935i	$-0.746001\pi$
$521$	−31.8719	−1.39633	−0.698167	+	0.715935i	$0.746001\pi$
$522$	0	0
$523$	−25.7877	−1.12762	−0.563808	−	0.825906i	$-0.690664\pi$
$523$	−25.7877	−1.12762	−0.563808	+	0.825906i	$0.690664\pi$
$524$	29.9103	1.30664
$525$	0	0
$526$	53.3449	2.32595
$527$	6.45560	0.281210
$528$	0	0
$529$	5.99614	0.260702
$530$	0	0
$531$	0	0
$532$	−7.57275	−0.328320
$533$	−22.8836	−0.991197
$534$	0	0
$535$	0	0
$536$	−54.1286	−2.33800
$537$	0	0
$538$	−56.4392	−2.43327
$539$	−11.9572	−0.515034
$540$	0	0
$541$	−33.7515	−1.45109	−0.725546	−	0.688174i	$-0.758413\pi$
$541$	−33.7515	−1.45109	−0.725546	+	0.688174i	$0.758413\pi$
$542$	−23.4727	−1.00824
$543$	0	0
$544$	52.8161	2.26447
$545$	0	0
$546$	0	0
$547$	−4.64965	−0.198805	−0.0994023	−	0.995047i	$-0.531693\pi$
$547$	−4.64965	−0.198805	−0.0994023	+	0.995047i	$0.531693\pi$
$548$	−63.7301	−2.72242
$549$	0	0
$550$	0	0
$551$	−6.15961	−0.262408
$552$	0	0
$553$	24.6695	1.04905
$554$	58.1163	2.46913
$555$	0	0
$556$	−4.26734	−0.180976
$557$	−21.1741	−0.897175	−0.448587	−	0.893739i	$-0.648073\pi$
$557$	−21.1741	−0.897175	−0.448587	+	0.893739i	$0.648073\pi$
$558$	0	0
$559$	20.5155	0.867712
$560$	0	0
$561$	0	0
$562$	−25.5440	−1.07751
$563$	17.4396	0.734993	0.367497	−	0.930025i	$-0.380215\pi$
$563$	17.4396	0.734993	0.367497	+	0.930025i	$0.380215\pi$
$564$	0	0
$565$	0	0
$566$	11.7841	0.495322
$567$	0	0
$568$	−21.7968	−0.914572
$569$	3.10033	0.129973	0.0649864	−	0.997886i	$-0.479300\pi$
$569$	3.10033	0.129973	0.0649864	+	0.997886i	$0.479300\pi$
$570$	0	0
$571$	−0.0734149	−0.00307232	−0.00153616	−	0.999999i	$-0.500489\pi$
$571$	−0.0734149	−0.00307232	−0.00153616	+	0.999999i	$0.500489\pi$
$572$	−77.8714	−3.25597
$573$	0	0
$574$	35.3098	1.47380
$575$	0	0
$576$	0	0
$577$	−37.6211	−1.56619	−0.783094	−	0.621903i	$-0.786360\pi$
$577$	−37.6211	−1.56619	−0.783094	+	0.621903i	$0.786360\pi$
$578$	−63.7591	−2.65203
$579$	0	0
$580$	0	0
$581$	20.6856	0.858182
$582$	0	0
$583$	−19.0697	−0.789784
$584$	−92.9398	−3.84588
$585$	0	0
$586$	−85.2250	−3.52061
$587$	−1.80472	−0.0744887	−0.0372443	−	0.999306i	$-0.511858\pi$
$587$	−1.80472	−0.0744887	−0.0372443	+	0.999306i	$0.511858\pi$
$588$	0	0
$589$	−0.766680	−0.0315905
$590$	0	0
$591$	0	0
$592$	98.5701	4.05121
$593$	11.8238	0.485544	0.242772	−	0.970083i	$-0.421943\pi$
$593$	11.8238	0.485544	0.242772	+	0.970083i	$0.421943\pi$
$594$	0	0
$595$	0	0
$596$	32.1994	1.31894
$597$	0	0
$598$	49.2099	2.01234
$599$	−2.94551	−0.120350	−0.0601751	−	0.998188i	$-0.519166\pi$
$599$	−2.94551	−0.120350	−0.0601751	+	0.998188i	$0.519166\pi$
$600$	0	0
$601$	23.0418	0.939896	0.469948	−	0.882694i	$-0.344273\pi$
$601$	23.0418	0.939896	0.469948	+	0.882694i	$0.344273\pi$
$602$	−31.6558	−1.29020
$603$	0	0
$604$	−72.2505	−2.93983
$605$	0	0
$606$	0	0
$607$	40.1456	1.62946	0.814731	−	0.579840i	$-0.196885\pi$
$607$	40.1456	1.62946	0.814731	+	0.579840i	$0.196885\pi$
$608$	−6.27254	−0.254385
$609$	0	0
$610$	0	0
$611$	−30.7237	−1.24295
$612$	0	0
$613$	1.94946	0.0787378	0.0393689	−	0.999225i	$-0.487465\pi$
$613$	1.94946	0.0787378	0.0393689	+	0.999225i	$0.487465\pi$
$614$	29.1516	1.17646
$615$	0	0
$616$	68.7749	2.77102
$617$	−47.6120	−1.91679	−0.958394	−	0.285449i	$-0.907857\pi$
$617$	−47.6120	−1.91679	−0.958394	+	0.285449i	$0.907857\pi$
$618$	0	0
$619$	18.8037	0.755785	0.377893	−	0.925849i	$-0.376649\pi$
$619$	18.8037	0.755785	0.377893	+	0.925849i	$0.376649\pi$
$620$	0	0
$621$	0	0
$622$	−31.6307	−1.26827
$623$	−0.445860	−0.0178630
$624$	0	0
$625$	0	0
$626$	25.9082	1.03550
$627$	0	0
$628$	12.7967	0.510645
$629$	−74.6850	−2.97789
$630$	0	0
$631$	−27.4357	−1.09220	−0.546099	−	0.837721i	$-0.683888\pi$
$631$	−27.4357	−1.09220	−0.546099	+	0.837721i	$0.683888\pi$
$632$	80.8018	3.21412
$633$	0	0
$634$	5.96713	0.236985
$635$	0	0
$636$	0	0
$637$	8.98259	0.355903
$638$	97.7350	3.86937
$639$	0	0
$640$	0	0
$641$	28.9531	1.14358	0.571790	−	0.820400i	$-0.306249\pi$
$641$	28.9531	1.14358	0.571790	+	0.820400i	$0.306249\pi$
$642$	0	0
$643$	−30.1906	−1.19060	−0.595301	−	0.803502i	$-0.702967\pi$
$643$	−30.1906	−1.19060	−0.595301	+	0.803502i	$0.702967\pi$
$644$	−53.1875	−2.09588
$645$	0	0
$646$	12.7891	0.503181
$647$	−18.3546	−0.721595	−0.360797	−	0.932644i	$-0.617495\pi$
$647$	−18.3546	−0.721595	−0.360797	+	0.932644i	$0.617495\pi$
$648$	0	0
$649$	−44.1002	−1.73109
$650$	0	0
$651$	0	0
$652$	28.8836	1.13117
$653$	−22.9005	−0.896166	−0.448083	−	0.893992i	$-0.647893\pi$
$653$	−22.9005	−0.896166	−0.448083	+	0.893992i	$0.647893\pi$
$654$	0	0
$655$	0	0
$656$	55.1289	2.15242
$657$	0	0
$658$	47.4074	1.84813
$659$	0.736981	0.0287087	0.0143544	−	0.999897i	$-0.495431\pi$
$659$	0.736981	0.0287087	0.0143544	+	0.999897i	$0.495431\pi$
$660$	0	0
$661$	−24.4038	−0.949200	−0.474600	−	0.880202i	$-0.657407\pi$
$661$	−24.4038	−0.949200	−0.474600	+	0.880202i	$0.657407\pi$
$662$	−23.3814	−0.908745
$663$	0	0
$664$	67.7529	2.62932
$665$	0	0
$666$	0	0
$667$	−43.2623	−1.67512
$668$	−80.2378	−3.10449
$669$	0	0
$670$	0	0
$671$	16.0127	0.618163
$672$	0	0
$673$	−18.9137	−0.729070	−0.364535	−	0.931190i	$-0.618772\pi$
$673$	−18.9137	−0.729070	−0.364535	+	0.931190i	$0.618772\pi$
$674$	−68.4466	−2.63647
$675$	0	0
$676$	−2.30155	−0.0885212
$677$	−9.95364	−0.382550	−0.191275	−	0.981537i	$-0.561262\pi$
$677$	−9.95364	−0.382550	−0.191275	+	0.981537i	$0.561262\pi$
$678$	0	0
$679$	−15.0776	−0.578624
$680$	0	0
$681$	0	0
$682$	12.1650	0.465820
$683$	21.3600	0.817319	0.408660	−	0.912687i	$-0.365996\pi$
$683$	21.3600	0.817319	0.408660	+	0.912687i	$0.365996\pi$
$684$	0	0
$685$	0	0
$686$	−52.0602	−1.98767
$687$	0	0
$688$	−49.4240	−1.88427
$689$	14.3256	0.545764
$690$	0	0
$691$	−42.4198	−1.61372	−0.806862	−	0.590739i	$-0.798836\pi$
$691$	−42.4198	−1.61372	−0.806862	+	0.590739i	$0.798836\pi$
$692$	28.4358	1.08097
$693$	0	0
$694$	95.7270	3.63375
$695$	0	0
$696$	0	0
$697$	−41.7703	−1.58216
$698$	45.4276	1.71946
$699$	0	0
$700$	0	0
$701$	−11.7949	−0.445486	−0.222743	−	0.974877i	$-0.571501\pi$
$701$	−11.7949	−0.445486	−0.222743	+	0.974877i	$0.571501\pi$
$702$	0	0
$703$	8.86974	0.334529
$704$	19.3039	0.727544
$705$	0	0
$706$	62.5392	2.35369
$707$	−18.1759	−0.683576
$708$	0	0
$709$	40.8193	1.53300	0.766500	−	0.642245i	$-0.221997\pi$
$709$	40.8193	1.53300	0.766500	+	0.642245i	$0.221997\pi$
$710$	0	0
$711$	0	0
$712$	−1.46036	−0.0547292
$713$	−5.38481	−0.201663
$714$	0	0
$715$	0	0
$716$	−83.6714	−3.12695
$717$	0	0
$718$	33.7510	1.25958
$719$	40.5659	1.51285	0.756426	−	0.654080i	$-0.226944\pi$
$719$	40.5659	1.51285	0.756426	+	0.654080i	$0.226944\pi$
$720$	0	0
$721$	−3.32649	−0.123885
$722$	47.5769	1.77063
$723$	0	0
$724$	−85.6124	−3.18176
$725$	0	0
$726$	0	0
$727$	−20.5810	−0.763306	−0.381653	−	0.924306i	$-0.624645\pi$
$727$	−20.5810	−0.763306	−0.381653	+	0.924306i	$0.624645\pi$
$728$	−51.6656	−1.91485
$729$	0	0
$730$	0	0
$731$	37.4478	1.38506
$732$	0	0
$733$	−25.8781	−0.955831	−0.477916	−	0.878406i	$-0.658608\pi$
$733$	−25.8781	−0.955831	−0.477916	+	0.878406i	$0.658608\pi$
$734$	55.5366	2.04989
$735$	0	0
$736$	−44.0555	−1.62391
$737$	36.8394	1.35700
$738$	0	0
$739$	−26.4783	−0.974021	−0.487011	−	0.873396i	$-0.661913\pi$
$739$	−26.4783	−0.974021	−0.487011	+	0.873396i	$0.661913\pi$
$740$	0	0
$741$	0	0
$742$	−22.1048	−0.811492
$743$	−5.00968	−0.183787	−0.0918936	−	0.995769i	$-0.529292\pi$
$743$	−5.00968	−0.183787	−0.0918936	+	0.995769i	$0.529292\pi$
$744$	0	0
$745$	0	0
$746$	−20.1338	−0.737151
$747$	0	0
$748$	−142.142	−5.19723
$749$	2.63947	0.0964440
$750$	0	0
$751$	−21.0437	−0.767895	−0.383947	−	0.923355i	$-0.625436\pi$
$751$	−21.0437	−0.767895	−0.383947	+	0.923355i	$0.625436\pi$
$752$	74.0168	2.69911
$753$	0	0
$754$	−73.4212	−2.67384
$755$	0	0
$756$	0	0
$757$	−32.6609	−1.18708	−0.593541	−	0.804804i	$-0.702270\pi$
$757$	−32.6609	−1.18708	−0.593541	+	0.804804i	$0.702270\pi$
$758$	−61.6854	−2.24051
$759$	0	0
$760$	0	0
$761$	9.58422	0.347428	0.173714	−	0.984796i	$-0.444423\pi$
$761$	9.58422	0.347428	0.173714	+	0.984796i	$0.444423\pi$
$762$	0	0
$763$	−22.5589	−0.816689
$764$	−5.12599	−0.185452
$765$	0	0
$766$	−22.0646	−0.797225
$767$	33.1293	1.19623
$768$	0	0
$769$	−30.2404	−1.09050	−0.545248	−	0.838275i	$-0.683564\pi$
$769$	−30.2404	−1.09050	−0.545248	+	0.838275i	$0.683564\pi$
$770$	0	0
$771$	0	0
$772$	−72.5656	−2.61169
$773$	−8.88586	−0.319602	−0.159801	−	0.987149i	$-0.551085\pi$
$773$	−8.88586	−0.319602	−0.159801	+	0.987149i	$0.551085\pi$
$774$	0	0
$775$	0	0
$776$	−49.3846	−1.77280
$777$	0	0
$778$	5.22797	0.187432
$779$	4.96073	0.177737
$780$	0	0
$781$	14.8347	0.530826
$782$	89.8248	3.21213
$783$	0	0
$784$	−21.6400	−0.772858
$785$	0	0
$786$	0	0
$787$	32.3032	1.15148	0.575742	−	0.817632i	$-0.304713\pi$
$787$	32.3032	1.15148	0.575742	+	0.817632i	$0.304713\pi$
$788$	−57.2547	−2.03962
$789$	0	0
$790$	0	0
$791$	1.59559	0.0567326
$792$	0	0
$793$	−12.0292	−0.427169
$794$	21.7496	0.771864
$795$	0	0
$796$	−30.3778	−1.07671
$797$	24.1925	0.856942	0.428471	−	0.903556i	$-0.359052\pi$
$797$	24.1925	0.856942	0.428471	+	0.903556i	$0.359052\pi$
$798$	0	0
$799$	−56.0813	−1.98401
$800$	0	0
$801$	0	0
$802$	24.1079	0.851278
$803$	63.2539	2.23218
$804$	0	0
$805$	0	0
$806$	−9.13865	−0.321895
$807$	0	0
$808$	−59.5329	−2.09436
$809$	1.57516	0.0553796	0.0276898	−	0.999617i	$-0.491185\pi$
$809$	1.57516	0.0553796	0.0276898	+	0.999617i	$0.491185\pi$
$810$	0	0
$811$	−22.5306	−0.791157	−0.395579	−	0.918432i	$-0.629456\pi$
$811$	−22.5306	−0.791157	−0.395579	+	0.918432i	$0.629456\pi$
$812$	79.3559	2.78485
$813$	0	0
$814$	−140.737	−4.93282
$815$	0	0
$816$	0	0
$817$	−4.44737	−0.155594
$818$	−57.4979	−2.01037
$819$	0	0
$820$	0	0
$821$	24.5336	0.856228	0.428114	−	0.903725i	$-0.359178\pi$
$821$	24.5336	0.856228	0.428114	+	0.903725i	$0.359178\pi$
$822$	0	0
$823$	9.88629	0.344614	0.172307	−	0.985043i	$-0.444878\pi$
$823$	9.88629	0.344614	0.172307	+	0.985043i	$0.444878\pi$
$824$	−10.8955	−0.379562
$825$	0	0
$826$	−51.1193	−1.77867
$827$	−38.5489	−1.34047	−0.670237	−	0.742147i	$-0.733808\pi$
$827$	−38.5489	−1.34047	−0.670237	+	0.742147i	$0.733808\pi$
$828$	0	0
$829$	−19.0794	−0.662655	−0.331327	−	0.943516i	$-0.607496\pi$
$829$	−19.0794	−0.662655	−0.331327	+	0.943516i	$0.607496\pi$
$830$	0	0
$831$	0	0
$832$	−14.5016	−0.502754
$833$	16.3963	0.568098
$834$	0	0
$835$	0	0
$836$	16.8811	0.583844
$837$	0	0
$838$	−17.7590	−0.613473
$839$	−47.1714	−1.62854	−0.814269	−	0.580488i	$-0.802862\pi$
$839$	−47.1714	−1.62854	−0.814269	+	0.580488i	$0.802862\pi$
$840$	0	0
$841$	35.5474	1.22577
$842$	−54.1681	−1.86676
$843$	0	0
$844$	39.3159	1.35331
$845$	0	0
$846$	0	0
$847$	−23.5766	−0.810100
$848$	−34.5120	−1.18515
$849$	0	0
$850$	0	0
$851$	62.2970	2.13551
$852$	0	0
$853$	24.5249	0.839717	0.419858	−	0.907590i	$-0.362080\pi$
$853$	24.5249	0.839717	0.419858	+	0.907590i	$0.362080\pi$
$854$	18.5613	0.635154
$855$	0	0
$856$	8.64523	0.295488
$857$	−42.9761	−1.46803	−0.734017	−	0.679131i	$-0.762357\pi$
$857$	−42.9761	−1.46803	−0.734017	+	0.679131i	$0.762357\pi$
$858$	0	0
$859$	−48.9415	−1.66986	−0.834932	−	0.550353i	$-0.814493\pi$
$859$	−48.9415	−1.66986	−0.834932	+	0.550353i	$0.814493\pi$
$860$	0	0
$861$	0	0
$862$	86.0277	2.93011
$863$	−32.7812	−1.11589	−0.557943	−	0.829880i	$-0.688409\pi$
$863$	−32.7812	−1.11589	−0.557943	+	0.829880i	$0.688409\pi$
$864$	0	0
$865$	0	0
$866$	1.80096	0.0611989
$867$	0	0
$868$	9.87733	0.335258
$869$	−54.9929	−1.86551
$870$	0	0
$871$	−27.6748	−0.937725
$872$	−73.8889	−2.50219
$873$	0	0
$874$	−10.6678	−0.360843
$875$	0	0
$876$	0	0
$877$	1.30919	0.0442083	0.0221041	−	0.999756i	$-0.492963\pi$
$877$	1.30919	0.0442083	0.0221041	+	0.999756i	$0.492963\pi$
$878$	45.6421	1.54035
$879$	0	0
$880$	0	0
$881$	−32.1645	−1.08365	−0.541825	−	0.840491i	$-0.682267\pi$
$881$	−32.1645	−1.08365	−0.541825	+	0.840491i	$0.682267\pi$
$882$	0	0
$883$	−57.3931	−1.93143	−0.965715	−	0.259604i	$-0.916408\pi$
$883$	−57.3931	−1.93143	−0.965715	+	0.259604i	$0.916408\pi$
$884$	106.781	3.59143
$885$	0	0
$886$	−78.1992	−2.62715
$887$	−16.6057	−0.557566	−0.278783	−	0.960354i	$-0.589931\pi$
$887$	−16.6057	−0.557566	−0.278783	+	0.960354i	$0.589931\pi$
$888$	0	0
$889$	15.0257	0.503947
$890$	0	0
$891$	0	0
$892$	47.3696	1.58605
$893$	6.66033	0.222879
$894$	0	0
$895$	0	0
$896$	−12.1805	−0.406922
$897$	0	0
$898$	2.34541	0.0782675
$899$	8.03414	0.267954
$900$	0	0
$901$	26.1492	0.871156
$902$	−78.7122	−2.62083
$903$	0	0
$904$	5.22614	0.173819
$905$	0	0
$906$	0	0
$907$	−3.74422	−0.124325	−0.0621624	−	0.998066i	$-0.519800\pi$
$907$	−3.74422	−0.124325	−0.0621624	+	0.998066i	$0.519800\pi$
$908$	17.4825	0.580178
$909$	0	0
$910$	0	0
$911$	25.6118	0.848558	0.424279	−	0.905531i	$-0.360528\pi$
$911$	25.6118	0.848558	0.424279	+	0.905531i	$0.360528\pi$
$912$	0	0
$913$	−46.1120	−1.52608
$914$	−81.7527	−2.70414
$915$	0	0
$916$	−92.4314	−3.05402
$917$	−13.5061	−0.446010
$918$	0	0
$919$	−11.9450	−0.394030	−0.197015	−	0.980400i	$-0.563125\pi$
$919$	−11.9450	−0.394030	−0.197015	+	0.980400i	$0.563125\pi$
$920$	0	0
$921$	0	0
$922$	−10.6498	−0.350731
$923$	−11.1442	−0.366816
$924$	0	0
$925$	0	0
$926$	4.01214	0.131847
$927$	0	0
$928$	65.7309	2.15772
$929$	−13.3211	−0.437051	−0.218525	−	0.975831i	$-0.570125\pi$
$929$	−13.3211	−0.437051	−0.218525	+	0.975831i	$0.570125\pi$
$930$	0	0
$931$	−1.94726	−0.0638188
$932$	−124.352	−4.07328
$933$	0	0
$934$	−1.28638	−0.0420917
$935$	0	0
$936$	0	0
$937$	−17.3862	−0.567981	−0.283991	−	0.958827i	$-0.591658\pi$
$937$	−17.3862	−0.567981	−0.283991	+	0.958827i	$0.591658\pi$
$938$	42.7028	1.39430
$939$	0	0
$940$	0	0
$941$	−26.0346	−0.848702	−0.424351	−	0.905498i	$-0.639498\pi$
$941$	−26.0346	−0.848702	−0.424351	+	0.905498i	$0.639498\pi$
$942$	0	0
$943$	34.8419	1.13461
$944$	−79.8121	−2.59766
$945$	0	0
$946$	70.5668	2.29432
$947$	−1.30792	−0.0425018	−0.0212509	−	0.999774i	$-0.506765\pi$
$947$	−1.30792	−0.0425018	−0.0212509	+	0.999774i	$0.506765\pi$
$948$	0	0
$949$	−47.5181	−1.54250
$950$	0	0
$951$	0	0
$952$	−94.3073	−3.05652
$953$	−40.9822	−1.32755	−0.663773	−	0.747935i	$-0.731046\pi$
$953$	−40.9822	−1.32755	−0.663773	+	0.747935i	$0.731046\pi$
$954$	0	0
$955$	0	0
$956$	77.7428	2.51438
$957$	0	0
$958$	43.0824	1.39193
$959$	28.7776	0.929276
$960$	0	0
$961$	1.00000	0.0322581
$962$	105.725	3.40872
$963$	0	0
$964$	−0.422515	−0.0136083
$965$	0	0
$966$	0	0
$967$	−38.8300	−1.24869	−0.624344	−	0.781150i	$-0.714633\pi$
$967$	−38.8300	−1.24869	−0.624344	+	0.781150i	$0.714633\pi$
$968$	−77.2220	−2.48201
$969$	0	0
$970$	0	0
$971$	20.8747	0.669901	0.334950	−	0.942236i	$-0.391280\pi$
$971$	20.8747	0.669901	0.334950	+	0.942236i	$0.391280\pi$
$972$	0	0
$973$	1.92693	0.0617746
$974$	−4.09299	−0.131148
$975$	0	0
$976$	28.9796	0.927613
$977$	10.9170	0.349265	0.174633	−	0.984634i	$-0.444126\pi$
$977$	10.9170	0.349265	0.174633	+	0.984634i	$0.444126\pi$
$978$	0	0
$979$	0.993905	0.0317653
$980$	0	0
$981$	0	0
$982$	54.7090	1.74583
$983$	−23.2316	−0.740973	−0.370486	−	0.928838i	$-0.620809\pi$
$983$	−23.2316	−0.740973	−0.370486	+	0.928838i	$0.620809\pi$
$984$	0	0
$985$	0	0
$986$	−134.019	−4.26803
$987$	0	0
$988$	−12.6815	−0.403453
$989$	−31.2363	−0.993257
$990$	0	0
$991$	18.8123	0.597594	0.298797	−	0.954317i	$-0.403415\pi$
$991$	18.8123	0.597594	0.298797	+	0.954317i	$0.403415\pi$
$992$	8.18144	0.259761
$993$	0	0
$994$	17.1958	0.545416
$995$	0	0
$996$	0	0
$997$	−22.3280	−0.707134	−0.353567	−	0.935409i	$-0.615031\pi$
$997$	−22.3280	−0.707134	−0.353567	+	0.935409i	$0.615031\pi$
$998$	−22.0319	−0.697408
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6975.2.a.bp.1.1		5
3.2	odd	2		775.2.a.k.1.5	yes	5
5.4	even	2		6975.2.a.by.1.5		5
15.2	even	4		775.2.b.g.249.10		10
15.8	even	4		775.2.b.g.249.1		10
15.14	odd	2		775.2.a.h.1.1	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
775.2.a.h.1.1	✓	5	15.14	odd	2
775.2.a.k.1.5	yes	5	3.2	odd	2
775.2.b.g.249.1		10	15.8	even	4
775.2.b.g.249.10		10	15.2	even	4
6975.2.a.bp.1.1		5	1.1	even	1	trivial
6975.2.a.by.1.5		5	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6975 = 3^{2} \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6975.a (trivial)

\(f(q)\)	\(=\)	\(q-2.58398 q^{2} +4.67698 q^{4} -2.11190 q^{7} -6.91727 q^{8} +4.70783 q^{11} -3.53665 q^{13} +5.45713 q^{14} +8.52017 q^{16} -6.45560 q^{17} +0.766680 q^{19} -12.1650 q^{22} +5.38481 q^{23} +9.13865 q^{26} -9.87733 q^{28} -8.03414 q^{29} -1.00000 q^{31} -8.18144 q^{32} +16.6812 q^{34} +11.5690 q^{37} -1.98109 q^{38} +6.47040 q^{41} -5.80082 q^{43} +22.0184 q^{44} -13.9143 q^{46} +8.68724 q^{47} -2.53986 q^{49} -16.5408 q^{52} -4.05063 q^{53} +14.6086 q^{56} +20.7601 q^{58} -9.36743 q^{59} +3.40129 q^{61} +2.58398 q^{62} +4.10039 q^{64} +7.82514 q^{67} -30.1927 q^{68} +3.15106 q^{71} +13.4359 q^{73} -29.8942 q^{74} +3.58574 q^{76} -9.94249 q^{77} -11.6812 q^{79} -16.7194 q^{82} -9.79474 q^{83} +14.9892 q^{86} -32.5653 q^{88} +0.211117 q^{89} +7.46907 q^{91} +25.1846 q^{92} -22.4477 q^{94} +7.13932 q^{97} +6.56296 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 4 q^{2} + 6 q^{4} + 2 q^{7} - 9 q^{8} + 2 q^{11} + 4 q^{13} - 2 q^{14} + 4 q^{16} - 19 q^{17} + 8 q^{19} - 10 q^{22} - 12 q^{23} + 16 q^{26} + 6 q^{28} - 6 q^{29} - 5 q^{31} - 7 q^{32} + 31 q^{34}+ \cdots - 10 q^{98}+O(q^{100}) \) 5 * q - 4 * q^2 + 6 * q^4 + 2 * q^7 - 9 * q^8 + 2 * q^11 + 4 * q^13 - 2 * q^14 + 4 * q^16 - 19 * q^17 + 8 * q^19 - 10 * q^22 - 12 * q^23 + 16 * q^26 + 6 * q^28 - 6 * q^29 - 5 * q^31 - 7 * q^32 + 31 * q^34 + 16 * q^37 - 14 * q^38 + 2 * q^41 + q^43 + 4 * q^44 - 11 * q^46 - 8 * q^47 - 9 * q^49 - 26 * q^52 - 3 * q^53 + 9 * q^56 + 5 * q^58 + 4 * q^59 - 5 * q^61 + 4 * q^62 - 5 * q^64 + 13 * q^67 - 30 * q^68 - 6 * q^71 + 19 * q^73 - 4 * q^74 - 5 * q^76 - 23 * q^77 - 6 * q^79 - 27 * q^82 - 18 * q^83 + 7 * q^86 - q^88 + 31 * q^89 + 8 * q^91 + 16 * q^92 - 14 * q^94 - q^97 - 10 * q^98

Introduction

L-functions

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