LMFDB - Embedded newform 6975.2.a.z.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 = [3,-2,0,2,0,0,-8,-6,0,0,3,0,-4,10,0,4,-7] 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:	$3$
Coefficient field:	3.3.148.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.48119$ 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)

$f(q)$	$=$	$q-2.48119 q^{2} +4.15633 q^{4} -2.80606 q^{7} -5.35026 q^{8} -0.675131 q^{11} -6.63752 q^{13} +6.96239 q^{14} +4.96239 q^{16} +2.83146 q^{17} +3.00000 q^{19} +1.67513 q^{22} -1.32487 q^{23} +16.4690 q^{26} -11.6629 q^{28} +0.906679 q^{29} +1.00000 q^{31} -1.61213 q^{32} -7.02539 q^{34} -0.712742 q^{37} -7.44358 q^{38} -4.21933 q^{41} -0.775746 q^{43} -2.80606 q^{44} +3.28726 q^{46} +3.03761 q^{47} +0.873992 q^{49} -27.5877 q^{52} +14.0557 q^{53} +15.0132 q^{56} -2.24965 q^{58} +9.18172 q^{59} +7.92478 q^{61} -2.48119 q^{62} -5.92478 q^{64} -12.1187 q^{67} +11.7685 q^{68} +10.2374 q^{71} +2.12601 q^{73} +1.76845 q^{74} +12.4690 q^{76} +1.89446 q^{77} -1.86177 q^{79} +10.4690 q^{82} +7.95017 q^{83} +1.92478 q^{86} +3.61213 q^{88} -6.67513 q^{89} +18.6253 q^{91} -5.50659 q^{92} -7.53690 q^{94} +4.50659 q^{97} -2.16854 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{2} + 2 q^{4} - 8 q^{7} - 6 q^{8} + 3 q^{11} - 4 q^{13} + 10 q^{14} + 4 q^{16} - 7 q^{17} + 9 q^{19} - 9 q^{23} + 18 q^{26} - 4 q^{28} + 9 q^{29} + 3 q^{31} - 4 q^{32} - 6 q^{34} - 8 q^{37} - 6 q^{38}+ \cdots - 22 q^{98}+O(q^{100}) $ 3 * q - 2 * q^2 + 2 * q^4 - 8 * q^7 - 6 * q^8 + 3 * q^11 - 4 * q^13 + 10 * q^14 + 4 * q^16 - 7 * q^17 + 9 * q^19 - 9 * q^23 + 18 * q^26 - 4 * q^28 + 9 * q^29 + 3 * q^31 - 4 * q^32 - 6 * q^34 - 8 * q^37 - 6 * q^38 + 2 * q^41 - 4 * q^43 - 8 * q^44 + 4 * q^46 + 20 * q^47 + 11 * q^49 - 30 * q^52 + 25 * q^53 + 4 * q^56 + 10 * q^58 + 2 * q^59 + 2 * q^61 - 2 * q^62 + 4 * q^64 - 15 * q^67 + 24 * q^68 - 12 * q^71 - 2 * q^73 - 6 * q^74 + 6 * q^76 - 14 * q^77 + 12 * q^79 - 13 * q^83 - 16 * q^86 + 10 * q^88 - 15 * q^89 + 14 * q^91 + 4 * q^92 - 7 * q^97 - 22 * 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 $

$2$	−2.48119	−1.75447	−0.877235	−	0.480062i	$-0.840614\pi$
$2$	−2.48119	−1.75447	−0.877235	+	0.480062i	$0.840614\pi$
$3$	0	0
$3$	0	0
$4$	4.15633	2.07816
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.80606	−1.06059	−0.530296	−	0.847812i	$-0.677919\pi$
$7$	−2.80606	−1.06059	−0.530296	+	0.847812i	$0.677919\pi$
$8$	−5.35026	−1.89160
$9$	0	0
$10$	0	0
$11$	−0.675131	−0.203560	−0.101780	−	0.994807i	$-0.532454\pi$
$11$	−0.675131	−0.203560	−0.101780	+	0.994807i	$0.532454\pi$
$12$	0	0
$13$	−6.63752	−1.84092	−0.920458	−	0.390841i	$-0.872184\pi$
$13$	−6.63752	−1.84092	−0.920458	+	0.390841i	$0.872184\pi$
$14$	6.96239	1.86078
$15$	0	0
$16$	4.96239	1.24060
$17$	2.83146	0.686729	0.343364	−	0.939202i	$-0.388433\pi$
$17$	2.83146	0.686729	0.343364	+	0.939202i	$0.388433\pi$
$18$	0	0
$19$	3.00000	0.688247	0.344124	−	0.938924i	$-0.388176\pi$
$19$	3.00000	0.688247	0.344124	+	0.938924i	$0.388176\pi$
$20$	0	0
$21$	0	0
$22$	1.67513	0.357139
$23$	−1.32487	−0.276254	−0.138127	−	0.990415i	$-0.544108\pi$
$23$	−1.32487	−0.276254	−0.138127	+	0.990415i	$0.544108\pi$
$24$	0	0
$25$	0	0
$26$	16.4690	3.22983
$27$	0	0
$28$	−11.6629	−2.20408
$29$	0.906679	0.168366	0.0841830	−	0.996450i	$-0.473172\pi$
$29$	0.906679	0.168366	0.0841830	+	0.996450i	$0.473172\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	−1.61213	−0.284986
$33$	0	0
$34$	−7.02539	−1.20484
$35$	0	0
$36$	0	0
$37$	−0.712742	−0.117174	−0.0585871	−	0.998282i	$-0.518660\pi$
$37$	−0.712742	−0.117174	−0.0585871	+	0.998282i	$0.518660\pi$
$38$	−7.44358	−1.20751
$39$	0	0
$40$	0	0
$41$	−4.21933	−0.658949	−0.329474	−	0.944165i	$-0.606871\pi$
$41$	−4.21933	−0.658949	−0.329474	+	0.944165i	$0.606871\pi$
$42$	0	0
$43$	−0.775746	−0.118300	−0.0591501	−	0.998249i	$-0.518839\pi$
$43$	−0.775746	−0.118300	−0.0591501	+	0.998249i	$0.518839\pi$
$44$	−2.80606	−0.423030
$45$	0	0
$46$	3.28726	0.484680
$47$	3.03761	0.443081	0.221541	−	0.975151i	$-0.428891\pi$
$47$	3.03761	0.443081	0.221541	+	0.975151i	$0.428891\pi$
$48$	0	0
$49$	0.873992	0.124856
$50$	0	0
$51$	0	0
$52$	−27.5877	−3.82572
$53$	14.0557	1.93070	0.965350	−	0.260958i	$-0.0840385\pi$
$53$	14.0557	1.93070	0.965350	+	0.260958i	$0.0840385\pi$
$54$	0	0
$55$	0	0
$56$	15.0132	2.00622
$57$	0	0
$58$	−2.24965	−0.295393
$59$	9.18172	1.19536	0.597679	−	0.801736i	$-0.296090\pi$
$59$	9.18172	1.19536	0.597679	+	0.801736i	$0.296090\pi$
$60$	0	0
$61$	7.92478	1.01466	0.507332	−	0.861751i	$-0.330632\pi$
$61$	7.92478	1.01466	0.507332	+	0.861751i	$0.330632\pi$
$62$	−2.48119	−0.315112
$63$	0	0
$64$	−5.92478	−0.740597
$65$	0	0
$66$	0	0
$67$	−12.1187	−1.48054	−0.740268	−	0.672312i	$-0.765302\pi$
$67$	−12.1187	−1.48054	−0.740268	+	0.672312i	$0.765302\pi$
$68$	11.7685	1.42713
$69$	0	0
$70$	0	0
$71$	10.2374	1.21496	0.607480	−	0.794335i	$-0.292181\pi$
$71$	10.2374	1.21496	0.607480	+	0.794335i	$0.292181\pi$
$72$	0	0
$73$	2.12601	0.248830	0.124415	−	0.992230i	$-0.460295\pi$
$73$	2.12601	0.248830	0.124415	+	0.992230i	$0.460295\pi$
$74$	1.76845	0.205578
$75$	0	0
$76$	12.4690	1.43029
$77$	1.89446	0.215894
$78$	0	0
$79$	−1.86177	−0.209466	−0.104733	−	0.994500i	$-0.533399\pi$
$79$	−1.86177	−0.209466	−0.104733	+	0.994500i	$0.533399\pi$
$80$	0	0
$81$	0	0
$82$	10.4690	1.15610
$83$	7.95017	0.872645	0.436322	−	0.899790i	$-0.356281\pi$
$83$	7.95017	0.872645	0.436322	+	0.899790i	$0.356281\pi$
$84$	0	0
$85$	0	0
$86$	1.92478	0.207554
$87$	0	0
$88$	3.61213	0.385054
$89$	−6.67513	−0.707562	−0.353781	−	0.935328i	$-0.615104\pi$
$89$	−6.67513	−0.707562	−0.353781	+	0.935328i	$0.615104\pi$
$90$	0	0
$91$	18.6253	1.95246
$92$	−5.50659	−0.574101
$93$	0	0
$94$	−7.53690	−0.777372
$95$	0	0
$96$	0	0
$97$	4.50659	0.457575	0.228787	−	0.973476i	$-0.426524\pi$
$97$	4.50659	0.457575	0.228787	+	0.973476i	$0.426524\pi$
$98$	−2.16854	−0.219056
$99$	0	0
$100$	0	0
$101$	11.3503	1.12939	0.564697	−	0.825299i	$-0.308993\pi$
$101$	11.3503	1.12939	0.564697	+	0.825299i	$0.308993\pi$
$102$	0	0
$103$	−5.51388	−0.543299	−0.271649	−	0.962396i	$-0.587569\pi$
$103$	−5.51388	−0.543299	−0.271649	+	0.962396i	$0.587569\pi$
$104$	35.5125	3.48228
$105$	0	0
$106$	−34.8749	−3.38735
$107$	2.81828	0.272454	0.136227	−	0.990678i	$-0.456502\pi$
$107$	2.81828	0.272454	0.136227	+	0.990678i	$0.456502\pi$
$108$	0	0
$109$	−6.08110	−0.582464	−0.291232	−	0.956652i	$-0.594065\pi$
$109$	−6.08110	−0.582464	−0.291232	+	0.956652i	$0.594065\pi$
$110$	0	0
$111$	0	0
$112$	−13.9248	−1.31577
$113$	16.4993	1.55212	0.776061	−	0.630657i	$-0.217215\pi$
$113$	16.4993	1.55212	0.776061	+	0.630657i	$0.217215\pi$
$114$	0	0
$115$	0	0
$116$	3.76845	0.349892
$117$	0	0
$118$	−22.7816	−2.09722
$119$	−7.94525	−0.728339
$120$	0	0
$121$	−10.5442	−0.958563
$122$	−19.6629	−1.78020
$123$	0	0
$124$	4.15633	0.373249
$125$	0	0
$126$	0	0
$127$	7.53690	0.668792	0.334396	−	0.942433i	$-0.391468\pi$
$127$	7.53690	0.668792	0.334396	+	0.942433i	$0.391468\pi$
$128$	17.9248	1.58434
$129$	0	0
$130$	0	0
$131$	−2.99508	−0.261681	−0.130840	−	0.991403i	$-0.541768\pi$
$131$	−2.99508	−0.261681	−0.130840	+	0.991403i	$0.541768\pi$
$132$	0	0
$133$	−8.41819	−0.729950
$134$	30.0689	2.59756
$135$	0	0
$136$	−15.1490	−1.29902
$137$	−15.4060	−1.31622	−0.658110	−	0.752921i	$-0.728644\pi$
$137$	−15.4060	−1.31622	−0.658110	+	0.752921i	$0.728644\pi$
$138$	0	0
$139$	−9.15140	−0.776212	−0.388106	−	0.921615i	$-0.626871\pi$
$139$	−9.15140	−0.776212	−0.388106	+	0.921615i	$0.626871\pi$
$140$	0	0
$141$	0	0
$142$	−25.4010	−2.13161
$143$	4.48119	0.374736
$144$	0	0
$145$	0	0
$146$	−5.27504	−0.436565
$147$	0	0
$148$	−2.96239	−0.243507
$149$	−18.6072	−1.52436	−0.762181	−	0.647364i	$-0.775871\pi$
$149$	−18.6072	−1.52436	−0.762181	+	0.647364i	$0.775871\pi$
$150$	0	0
$151$	15.5877	1.26851	0.634254	−	0.773125i	$-0.281307\pi$
$151$	15.5877	1.26851	0.634254	+	0.773125i	$0.281307\pi$
$152$	−16.0508	−1.30189
$153$	0	0
$154$	−4.70052	−0.378779
$155$	0	0
$156$	0	0
$157$	3.18664	0.254322	0.127161	−	0.991882i	$-0.459414\pi$
$157$	3.18664	0.254322	0.127161	+	0.991882i	$0.459414\pi$
$158$	4.61942	0.367501
$159$	0	0
$160$	0	0
$161$	3.71767	0.292993
$162$	0	0
$163$	−11.2619	−0.882097	−0.441049	−	0.897483i	$-0.645393\pi$
$163$	−11.2619	−0.882097	−0.441049	+	0.897483i	$0.645393\pi$
$164$	−17.5369	−1.36940
$165$	0	0
$166$	−19.7259	−1.53103
$167$	−18.2374	−1.41125	−0.705627	−	0.708583i	$-0.749335\pi$
$167$	−18.2374	−1.41125	−0.705627	+	0.708583i	$0.749335\pi$
$168$	0	0
$169$	31.0567	2.38897
$170$	0	0
$171$	0	0
$172$	−3.22425	−0.245847
$173$	−7.64481	−0.581224	−0.290612	−	0.956841i	$-0.593859\pi$
$173$	−7.64481	−0.581224	−0.290612	+	0.956841i	$0.593859\pi$
$174$	0	0
$175$	0	0
$176$	−3.35026	−0.252535
$177$	0	0
$178$	16.5623	1.24140
$179$	−8.99271	−0.672146	−0.336073	−	0.941836i	$-0.609099\pi$
$179$	−8.99271	−0.672146	−0.336073	+	0.941836i	$0.609099\pi$
$180$	0	0
$181$	17.2628	1.28314	0.641568	−	0.767066i	$-0.278284\pi$
$181$	17.2628	1.28314	0.641568	+	0.767066i	$0.278284\pi$
$182$	−46.2130	−3.42553
$183$	0	0
$184$	7.08840	0.522564
$185$	0	0
$186$	0	0
$187$	−1.91160	−0.139790
$188$	12.6253	0.920795
$189$	0	0
$190$	0	0
$191$	13.1065	0.948353	0.474176	−	0.880430i	$-0.342746\pi$
$191$	13.1065	0.948353	0.474176	+	0.880430i	$0.342746\pi$
$192$	0	0
$193$	−14.7685	−1.06306	−0.531528	−	0.847041i	$-0.678382\pi$
$193$	−14.7685	−1.06306	−0.531528	+	0.847041i	$0.678382\pi$
$194$	−11.1817	−0.802801
$195$	0	0
$196$	3.63259	0.259471
$197$	−6.43866	−0.458735	−0.229368	−	0.973340i	$-0.573666\pi$
$197$	−6.43866	−0.458735	−0.229368	+	0.973340i	$0.573666\pi$
$198$	0	0
$199$	−20.4264	−1.44799	−0.723996	−	0.689804i	$-0.757697\pi$
$199$	−20.4264	−1.44799	−0.723996	+	0.689804i	$0.757697\pi$
$200$	0	0
$201$	0	0
$202$	−28.1622	−1.98149
$203$	−2.54420	−0.178568
$204$	0	0
$205$	0	0
$206$	13.6810	0.953201
$207$	0	0
$208$	−32.9380	−2.28384
$209$	−2.02539	−0.140099
$210$	0	0
$211$	−1.31265	−0.0903666	−0.0451833	−	0.998979i	$-0.514387\pi$
$211$	−1.31265	−0.0903666	−0.0451833	+	0.998979i	$0.514387\pi$
$212$	58.4201	4.01231
$213$	0	0
$214$	−6.99271	−0.478012
$215$	0	0
$216$	0	0
$217$	−2.80606	−0.190488
$218$	15.0884	1.02192
$219$	0	0
$220$	0	0
$221$	−18.7938	−1.26421
$222$	0	0
$223$	22.2882	1.49253	0.746265	−	0.665649i	$-0.231845\pi$
$223$	22.2882	1.49253	0.746265	+	0.665649i	$0.231845\pi$
$224$	4.52373	0.302254
$225$	0	0
$226$	−40.9380	−2.72315
$227$	−29.1998	−1.93806	−0.969030	−	0.246943i	$-0.920574\pi$
$227$	−29.1998	−1.93806	−0.969030	+	0.246943i	$0.920574\pi$
$228$	0	0
$229$	15.8011	1.04417	0.522084	−	0.852894i	$-0.325155\pi$
$229$	15.8011	1.04417	0.522084	+	0.852894i	$0.325155\pi$
$230$	0	0
$231$	0	0
$232$	−4.85097	−0.318482
$233$	−2.06063	−0.134997	−0.0674983	−	0.997719i	$-0.521502\pi$
$233$	−2.06063	−0.134997	−0.0674983	+	0.997719i	$0.521502\pi$
$234$	0	0
$235$	0	0
$236$	38.1622	2.48415
$237$	0	0
$238$	19.7137	1.27785
$239$	−22.1065	−1.42995	−0.714975	−	0.699150i	$-0.753562\pi$
$239$	−22.1065	−1.42995	−0.714975	+	0.699150i	$0.753562\pi$
$240$	0	0
$241$	−20.0484	−1.29143	−0.645716	−	0.763578i	$-0.723441\pi$
$241$	−20.0484	−1.29143	−0.645716	+	0.763578i	$0.723441\pi$
$242$	26.1622	1.68177
$243$	0	0
$244$	32.9380	2.10864
$245$	0	0
$246$	0	0
$247$	−19.9126	−1.26701
$248$	−5.35026	−0.339742
$249$	0	0
$250$	0	0
$251$	4.39280	0.277271	0.138635	−	0.990343i	$-0.455728\pi$
$251$	4.39280	0.277271	0.138635	+	0.990343i	$0.455728\pi$
$252$	0	0
$253$	0.894460	0.0562342
$254$	−18.7005	−1.17338
$255$	0	0
$256$	−32.6253	−2.03908
$257$	11.7708	0.734244	0.367122	−	0.930173i	$-0.380343\pi$
$257$	11.7708	0.734244	0.367122	+	0.930173i	$0.380343\pi$
$258$	0	0
$259$	2.00000	0.124274
$260$	0	0
$261$	0	0
$262$	7.43136	0.459111
$263$	11.0376	0.680608	0.340304	−	0.940315i	$-0.389470\pi$
$263$	11.0376	0.680608	0.340304	+	0.940315i	$0.389470\pi$
$264$	0	0
$265$	0	0
$266$	20.8872	1.28067
$267$	0	0
$268$	−50.3693	−3.07680
$269$	−25.6629	−1.56470	−0.782348	−	0.622842i	$-0.785978\pi$
$269$	−25.6629	−1.56470	−0.782348	+	0.622842i	$0.785978\pi$
$270$	0	0
$271$	−8.76353	−0.532346	−0.266173	−	0.963925i	$-0.585759\pi$
$271$	−8.76353	−0.532346	−0.266173	+	0.963925i	$0.585759\pi$
$272$	14.0508	0.851954
$273$	0	0
$274$	38.2252	2.30927
$275$	0	0
$276$	0	0
$277$	10.8872	0.654146	0.327073	−	0.944999i	$-0.393938\pi$
$277$	10.8872	0.654146	0.327073	+	0.944999i	$0.393938\pi$
$278$	22.7064	1.36184
$279$	0	0
$280$	0	0
$281$	12.7005	0.757650	0.378825	−	0.925468i	$-0.376328\pi$
$281$	12.7005	0.757650	0.378825	+	0.925468i	$0.376328\pi$
$282$	0	0
$283$	11.1260	0.661373	0.330686	−	0.943741i	$-0.392720\pi$
$283$	11.1260	0.661373	0.330686	+	0.943741i	$0.392720\pi$
$284$	42.5501	2.52488
$285$	0	0
$286$	−11.1187	−0.657463
$287$	11.8397	0.698876
$288$	0	0
$289$	−8.98286	−0.528403
$290$	0	0
$291$	0	0
$292$	8.83638	0.517110
$293$	−25.3684	−1.48204	−0.741018	−	0.671485i	$-0.765657\pi$
$293$	−25.3684	−1.48204	−0.741018	+	0.671485i	$0.765657\pi$
$294$	0	0
$295$	0	0
$296$	3.81336	0.221647
$297$	0	0
$298$	46.1681	2.67445
$299$	8.79384	0.508561
$300$	0	0
$301$	2.17679	0.125468
$302$	−38.6761	−2.22556
$303$	0	0
$304$	14.8872	0.853838
$305$	0	0
$306$	0	0
$307$	26.4445	1.50927	0.754635	−	0.656145i	$-0.227814\pi$
$307$	26.4445	1.50927	0.754635	+	0.656145i	$0.227814\pi$
$308$	7.87399	0.448662
$309$	0	0
$310$	0	0
$311$	0.456757	0.0259003	0.0129501	−	0.999916i	$-0.495878\pi$
$311$	0.456757	0.0259003	0.0129501	+	0.999916i	$0.495878\pi$
$312$	0	0
$313$	−13.0254	−0.736239	−0.368119	−	0.929779i	$-0.619998\pi$
$313$	−13.0254	−0.736239	−0.368119	+	0.929779i	$0.619998\pi$
$314$	−7.90668	−0.446200
$315$	0	0
$316$	−7.73813	−0.435304
$317$	−20.9199	−1.17498	−0.587488	−	0.809233i	$-0.699883\pi$
$317$	−20.9199	−1.17498	−0.587488	+	0.809233i	$0.699883\pi$
$318$	0	0
$319$	−0.612127	−0.0342725
$320$	0	0
$321$	0	0
$322$	−9.22425	−0.514048
$323$	8.49437	0.472639
$324$	0	0
$325$	0	0
$326$	27.9429	1.54761
$327$	0	0
$328$	22.5745	1.24647
$329$	−8.52373	−0.469928
$330$	0	0
$331$	−12.5623	−0.690486	−0.345243	−	0.938513i	$-0.612204\pi$
$331$	−12.5623	−0.690486	−0.345243	+	0.938513i	$0.612204\pi$
$332$	33.0435	1.81350
$333$	0	0
$334$	45.2506	2.47600
$335$	0	0
$336$	0	0
$337$	16.0992	0.876979	0.438490	−	0.898736i	$-0.355514\pi$
$337$	16.0992	0.876979	0.438490	+	0.898736i	$0.355514\pi$
$338$	−77.0576	−4.19138
$339$	0	0
$340$	0	0
$341$	−0.675131	−0.0365604
$342$	0	0
$343$	17.1900	0.928171
$344$	4.15045	0.223777
$345$	0	0
$346$	18.9683	1.01974
$347$	−6.23155	−0.334527	−0.167264	−	0.985912i	$-0.553493\pi$
$347$	−6.23155	−0.334527	−0.167264	+	0.985912i	$0.553493\pi$
$348$	0	0
$349$	9.75131	0.521976	0.260988	−	0.965342i	$-0.415952\pi$
$349$	9.75131	0.521976	0.260988	+	0.965342i	$0.415952\pi$
$350$	0	0
$351$	0	0
$352$	1.08840	0.0580117
$353$	14.2424	0.758044	0.379022	−	0.925388i	$-0.376260\pi$
$353$	14.2424	0.758044	0.379022	+	0.925388i	$0.376260\pi$
$354$	0	0
$355$	0	0
$356$	−27.7440	−1.47043
$357$	0	0
$358$	22.3127	1.17926
$359$	26.2882	1.38744	0.693719	−	0.720245i	$-0.255971\pi$
$359$	26.2882	1.38744	0.693719	+	0.720245i	$0.255971\pi$
$360$	0	0
$361$	−10.0000	−0.526316
$362$	−42.8324	−2.25122
$363$	0	0
$364$	77.4128	4.05753
$365$	0	0
$366$	0	0
$367$	7.58769	0.396074	0.198037	−	0.980195i	$-0.436543\pi$
$367$	7.58769	0.396074	0.198037	+	0.980195i	$0.436543\pi$
$368$	−6.57452	−0.342720
$369$	0	0
$370$	0	0
$371$	−39.4412	−2.04769
$372$	0	0
$373$	1.99271	0.103178	0.0515892	−	0.998668i	$-0.483571\pi$
$373$	1.99271	0.103178	0.0515892	+	0.998668i	$0.483571\pi$
$374$	4.74306	0.245258
$375$	0	0
$376$	−16.2520	−0.838134
$377$	−6.01810	−0.309948
$378$	0	0
$379$	−8.08110	−0.415098	−0.207549	−	0.978225i	$-0.566549\pi$
$379$	−8.08110	−0.415098	−0.207549	+	0.978225i	$0.566549\pi$
$380$	0	0
$381$	0	0
$382$	−32.5198	−1.66386
$383$	−3.10791	−0.158807	−0.0794034	−	0.996843i	$-0.525302\pi$
$383$	−3.10791	−0.158807	−0.0794034	+	0.996843i	$0.525302\pi$
$384$	0	0
$385$	0	0
$386$	36.6434	1.86510
$387$	0	0
$388$	18.7308	0.950914
$389$	24.1768	1.22581	0.612906	−	0.790156i	$-0.290000\pi$
$389$	24.1768	1.22581	0.612906	+	0.790156i	$0.290000\pi$
$390$	0	0
$391$	−3.75131	−0.189712
$392$	−4.67609	−0.236178
$393$	0	0
$394$	15.9756	0.804837
$395$	0	0
$396$	0	0
$397$	−0.531024	−0.0266514	−0.0133257	−	0.999911i	$-0.504242\pi$
$397$	−0.531024	−0.0266514	−0.0133257	+	0.999911i	$0.504242\pi$
$398$	50.6820	2.54046
$399$	0	0
$400$	0	0
$401$	−26.7113	−1.33390	−0.666950	−	0.745102i	$-0.732401\pi$
$401$	−26.7113	−1.33390	−0.666950	+	0.745102i	$0.732401\pi$
$402$	0	0
$403$	−6.63752	−0.330638
$404$	47.1754	2.34706
$405$	0	0
$406$	6.31265	0.313292
$407$	0.481194	0.0238519
$408$	0	0
$409$	−12.8119	−0.633510	−0.316755	−	0.948507i	$-0.602593\pi$
$409$	−12.8119	−0.633510	−0.316755	+	0.948507i	$0.602593\pi$
$410$	0	0
$411$	0	0
$412$	−22.9175	−1.12906
$413$	−25.7645	−1.26779
$414$	0	0
$415$	0	0
$416$	10.7005	0.524636
$417$	0	0
$418$	5.02539	0.245800
$419$	−24.8119	−1.21214	−0.606071	−	0.795410i	$-0.707255\pi$
$419$	−24.8119	−1.21214	−0.606071	+	0.795410i	$0.707255\pi$
$420$	0	0
$421$	19.8192	0.965931	0.482965	−	0.875640i	$-0.339560\pi$
$421$	19.8192	0.965931	0.482965	+	0.875640i	$0.339560\pi$
$422$	3.25694	0.158545
$423$	0	0
$424$	−75.2017	−3.65212
$425$	0	0
$426$	0	0
$427$	−22.2374	−1.07614
$428$	11.7137	0.566203
$429$	0	0
$430$	0	0
$431$	2.41231	0.116197	0.0580985	−	0.998311i	$-0.481496\pi$
$431$	2.41231	0.116197	0.0580985	+	0.998311i	$0.481496\pi$
$432$	0	0
$433$	−16.7757	−0.806191	−0.403095	−	0.915158i	$-0.632066\pi$
$433$	−16.7757	−0.806191	−0.403095	+	0.915158i	$0.632066\pi$
$434$	6.96239	0.334205
$435$	0	0
$436$	−25.2750	−1.21045
$437$	−3.97461	−0.190131
$438$	0	0
$439$	−16.8568	−0.804533	−0.402267	−	0.915523i	$-0.631778\pi$
$439$	−16.8568	−0.804533	−0.402267	+	0.915523i	$0.631778\pi$
$440$	0	0
$441$	0	0
$442$	46.6312	2.21802
$443$	31.4699	1.49518	0.747591	−	0.664160i	$-0.231210\pi$
$443$	31.4699	1.49518	0.747591	+	0.664160i	$0.231210\pi$
$444$	0	0
$445$	0	0
$446$	−55.3014	−2.61860
$447$	0	0
$448$	16.6253	0.785472
$449$	27.3757	1.29194	0.645969	−	0.763364i	$-0.276454\pi$
$449$	27.3757	1.29194	0.645969	+	0.763364i	$0.276454\pi$
$450$	0	0
$451$	2.84860	0.134135
$452$	68.5764	3.22556
$453$	0	0
$454$	72.4504	3.40027
$455$	0	0
$456$	0	0
$457$	11.2022	0.524016	0.262008	−	0.965066i	$-0.415615\pi$
$457$	11.2022	0.524016	0.262008	+	0.965066i	$0.415615\pi$
$458$	−39.2057	−1.83196
$459$	0	0
$460$	0	0
$461$	−30.9370	−1.44088	−0.720440	−	0.693518i	$-0.756060\pi$
$461$	−30.9370	−1.44088	−0.720440	+	0.693518i	$0.756060\pi$
$462$	0	0
$463$	−2.46310	−0.114470	−0.0572349	−	0.998361i	$-0.518228\pi$
$463$	−2.46310	−0.114470	−0.0572349	+	0.998361i	$0.518228\pi$
$464$	4.49929	0.208874
$465$	0	0
$466$	5.11283	0.236847
$467$	11.7381	0.543176	0.271588	−	0.962414i	$-0.412451\pi$
$467$	11.7381	0.543176	0.271588	+	0.962414i	$0.412451\pi$
$468$	0	0
$469$	34.0059	1.57025
$470$	0	0
$471$	0	0
$472$	−49.1246	−2.26114
$473$	0.523730	0.0240811
$474$	0	0
$475$	0	0
$476$	−33.0230	−1.51361
$477$	0	0
$478$	54.8505	2.50880
$479$	−20.2981	−0.927442	−0.463721	−	0.885981i	$-0.653486\pi$
$479$	−20.2981	−0.927442	−0.463721	+	0.885981i	$0.653486\pi$
$480$	0	0
$481$	4.73084	0.215708
$482$	49.7440	2.26578
$483$	0	0
$484$	−43.8251	−1.99205
$485$	0	0
$486$	0	0
$487$	5.86177	0.265622	0.132811	−	0.991141i	$-0.457600\pi$
$487$	5.86177	0.265622	0.132811	+	0.991141i	$0.457600\pi$
$488$	−42.3996	−1.91934
$489$	0	0
$490$	0	0
$491$	13.7635	0.621139	0.310570	−	0.950551i	$-0.399480\pi$
$491$	13.7635	0.621139	0.310570	+	0.950551i	$0.399480\pi$
$492$	0	0
$493$	2.56722	0.115622
$494$	49.4069	2.22292
$495$	0	0
$496$	4.96239	0.222818
$497$	−28.7269	−1.28858
$498$	0	0
$499$	−20.6229	−0.923209	−0.461605	−	0.887086i	$-0.652726\pi$
$499$	−20.6229	−0.923209	−0.461605	+	0.887086i	$0.652726\pi$
$500$	0	0
$501$	0	0
$502$	−10.8994	−0.486463
$503$	9.67750	0.431498	0.215749	−	0.976449i	$-0.430781\pi$
$503$	9.67750	0.431498	0.215749	+	0.976449i	$0.430781\pi$
$504$	0	0
$505$	0	0
$506$	−2.21933	−0.0986612
$507$	0	0
$508$	31.3258	1.38986
$509$	−0.232503	−0.0103055	−0.00515276	−	0.999987i	$-0.501640\pi$
$509$	−0.232503	−0.0103055	−0.00515276	+	0.999987i	$0.501640\pi$
$510$	0	0
$511$	−5.96571	−0.263908
$512$	45.1002	1.99316
$513$	0	0
$514$	−29.2057	−1.28821
$515$	0	0
$516$	0	0
$517$	−2.05079	−0.0901934
$518$	−4.96239	−0.218035
$519$	0	0
$520$	0	0
$521$	23.4436	1.02708	0.513541	−	0.858065i	$-0.328333\pi$
$521$	23.4436	1.02708	0.513541	+	0.858065i	$0.328333\pi$
$522$	0	0
$523$	−32.6761	−1.42883	−0.714413	−	0.699725i	$-0.753306\pi$
$523$	−32.6761	−1.42883	−0.714413	+	0.699725i	$0.753306\pi$
$524$	−12.4485	−0.543816
$525$	0	0
$526$	−27.3865	−1.19411
$527$	2.83146	0.123340
$528$	0	0
$529$	−21.2447	−0.923684
$530$	0	0
$531$	0	0
$532$	−34.9887	−1.51695
$533$	28.0059	1.21307
$534$	0	0
$535$	0	0
$536$	64.8383	2.80059
$537$	0	0
$538$	63.6747	2.74521
$539$	−0.590059	−0.0254156
$540$	0	0
$541$	−44.3317	−1.90597	−0.952984	−	0.303019i	$-0.902005\pi$
$541$	−44.3317	−1.90597	−0.952984	+	0.303019i	$0.902005\pi$
$542$	21.7440	0.933985
$543$	0	0
$544$	−4.56467	−0.195708
$545$	0	0
$546$	0	0
$547$	−32.7685	−1.40108	−0.700539	−	0.713614i	$-0.747057\pi$
$547$	−32.7685	−1.40108	−0.700539	+	0.713614i	$0.747057\pi$
$548$	−64.0322	−2.73532
$549$	0	0
$550$	0	0
$551$	2.72004	0.115877
$552$	0	0
$553$	5.22425	0.222158
$554$	−27.0132	−1.14768
$555$	0	0
$556$	−38.0362	−1.61309
$557$	22.3235	0.945875	0.472938	−	0.881096i	$-0.343194\pi$
$557$	22.3235	0.945875	0.472938	+	0.881096i	$0.343194\pi$
$558$	0	0
$559$	5.14903	0.217781
$560$	0	0
$561$	0	0
$562$	−31.5125	−1.32927
$563$	−31.8578	−1.34265	−0.671323	−	0.741165i	$-0.734274\pi$
$563$	−31.8578	−1.34265	−0.671323	+	0.741165i	$0.734274\pi$
$564$	0	0
$565$	0	0
$566$	−27.6058	−1.16036
$567$	0	0
$568$	−54.7729	−2.29822
$569$	25.1138	1.05282	0.526412	−	0.850229i	$-0.323537\pi$
$569$	25.1138	1.05282	0.526412	+	0.850229i	$0.323537\pi$
$570$	0	0
$571$	12.3634	0.517394	0.258697	−	0.965959i	$-0.416707\pi$
$571$	12.3634	0.517394	0.258697	+	0.965959i	$0.416707\pi$
$572$	18.6253	0.778763
$573$	0	0
$574$	−29.3766	−1.22616
$575$	0	0
$576$	0	0
$577$	43.1813	1.79766	0.898830	−	0.438298i	$-0.144419\pi$
$577$	43.1813	1.79766	0.898830	+	0.438298i	$0.144419\pi$
$578$	22.2882	0.927067
$579$	0	0
$580$	0	0
$581$	−22.3087	−0.925520
$582$	0	0
$583$	−9.48944	−0.393013
$584$	−11.3747	−0.470688
$585$	0	0
$586$	62.9438	2.60019
$587$	43.8651	1.81051	0.905253	−	0.424873i	$-0.139681\pi$
$587$	43.8651	1.81051	0.905253	+	0.424873i	$0.139681\pi$
$588$	0	0
$589$	3.00000	0.123613
$590$	0	0
$591$	0	0
$592$	−3.53690	−0.145366
$593$	37.9067	1.55664	0.778320	−	0.627867i	$-0.216072\pi$
$593$	37.9067	1.55664	0.778320	+	0.627867i	$0.216072\pi$
$594$	0	0
$595$	0	0
$596$	−77.3376	−3.16787
$597$	0	0
$598$	−21.8192	−0.892255
$599$	−9.96731	−0.407253	−0.203627	−	0.979049i	$-0.565273\pi$
$599$	−9.96731	−0.407253	−0.203627	+	0.979049i	$0.565273\pi$
$600$	0	0
$601$	−26.8143	−1.09378	−0.546889	−	0.837205i	$-0.684188\pi$
$601$	−26.8143	−1.09378	−0.546889	+	0.837205i	$0.684188\pi$
$602$	−5.40105	−0.220130
$603$	0	0
$604$	64.7875	2.63617
$605$	0	0
$606$	0	0
$607$	−29.9029	−1.21372	−0.606861	−	0.794808i	$-0.707571\pi$
$607$	−29.9029	−1.21372	−0.606861	+	0.794808i	$0.707571\pi$
$608$	−4.83638	−0.196141
$609$	0	0
$610$	0	0
$611$	−20.1622	−0.815675
$612$	0	0
$613$	1.52705	0.0616772	0.0308386	−	0.999524i	$-0.490182\pi$
$613$	1.52705	0.0616772	0.0308386	+	0.999524i	$0.490182\pi$
$614$	−65.6140	−2.64797
$615$	0	0
$616$	−10.1359	−0.408385
$617$	−3.44358	−0.138633	−0.0693167	−	0.997595i	$-0.522082\pi$
$617$	−3.44358	−0.138633	−0.0693167	+	0.997595i	$0.522082\pi$
$618$	0	0
$619$	3.51247	0.141178	0.0705890	−	0.997505i	$-0.477512\pi$
$619$	3.51247	0.141178	0.0705890	+	0.997505i	$0.477512\pi$
$620$	0	0
$621$	0	0
$622$	−1.13330	−0.0454413
$623$	18.7308	0.750435
$624$	0	0
$625$	0	0
$626$	32.3185	1.29171
$627$	0	0
$628$	13.2447	0.528522
$629$	−2.01810	−0.0804669
$630$	0	0
$631$	16.0122	0.637436	0.318718	−	0.947850i	$-0.396748\pi$
$631$	16.0122	0.637436	0.318718	+	0.947850i	$0.396748\pi$
$632$	9.96097	0.396226
$633$	0	0
$634$	51.9062	2.06146
$635$	0	0
$636$	0	0
$637$	−5.80114	−0.229849
$638$	1.51881	0.0601301
$639$	0	0
$640$	0	0
$641$	−17.1197	−0.676186	−0.338093	−	0.941113i	$-0.609782\pi$
$641$	−17.1197	−0.676186	−0.338093	+	0.941113i	$0.609782\pi$
$642$	0	0
$643$	20.4509	0.806504	0.403252	−	0.915089i	$-0.367880\pi$
$643$	20.4509	0.806504	0.403252	+	0.915089i	$0.367880\pi$
$644$	15.4518	0.608887
$645$	0	0
$646$	−21.0762	−0.829231
$647$	−42.5901	−1.67439	−0.837194	−	0.546906i	$-0.815805\pi$
$647$	−42.5901	−1.67439	−0.837194	+	0.546906i	$0.815805\pi$
$648$	0	0
$649$	−6.19886	−0.243327
$650$	0	0
$651$	0	0
$652$	−46.8080	−1.83314
$653$	−40.3815	−1.58025	−0.790126	−	0.612945i	$-0.789985\pi$
$653$	−40.3815	−1.58025	−0.790126	+	0.612945i	$0.789985\pi$
$654$	0	0
$655$	0	0
$656$	−20.9380	−0.817490
$657$	0	0
$658$	21.1490	0.824475
$659$	4.06537	0.158364	0.0791822	−	0.996860i	$-0.474769\pi$
$659$	4.06537	0.158364	0.0791822	+	0.996860i	$0.474769\pi$
$660$	0	0
$661$	5.40360	0.210176	0.105088	−	0.994463i	$-0.466488\pi$
$661$	5.40360	0.210176	0.105088	+	0.994463i	$0.466488\pi$
$662$	31.1695	1.21144
$663$	0	0
$664$	−42.5355	−1.65070
$665$	0	0
$666$	0	0
$667$	−1.20123	−0.0465118
$668$	−75.8007	−2.93282
$669$	0	0
$670$	0	0
$671$	−5.35026	−0.206545
$672$	0	0
$673$	−38.0484	−1.46666	−0.733329	−	0.679874i	$-0.762035\pi$
$673$	−38.0484	−1.46666	−0.733329	+	0.679874i	$0.762035\pi$
$674$	−39.9452	−1.53863
$675$	0	0
$676$	129.082	4.96468
$677$	−33.1451	−1.27387	−0.636934	−	0.770918i	$-0.719798\pi$
$677$	−33.1451	−1.27387	−0.636934	+	0.770918i	$0.719798\pi$
$678$	0	0
$679$	−12.6458	−0.485300
$680$	0	0
$681$	0	0
$682$	1.67513	0.0641441
$683$	25.0494	0.958488	0.479244	−	0.877682i	$-0.340911\pi$
$683$	25.0494	0.958488	0.479244	+	0.877682i	$0.340911\pi$
$684$	0	0
$685$	0	0
$686$	−42.6516	−1.62845
$687$	0	0
$688$	−3.84955	−0.146763
$689$	−93.2950	−3.55426
$690$	0	0
$691$	9.95254	0.378612	0.189306	−	0.981918i	$-0.439376\pi$
$691$	9.95254	0.378612	0.189306	+	0.981918i	$0.439376\pi$
$692$	−31.7743	−1.20788
$693$	0	0
$694$	15.4617	0.586917
$695$	0	0
$696$	0	0
$697$	−11.9468	−0.452519
$698$	−24.1949	−0.915790
$699$	0	0
$700$	0	0
$701$	16.0444	0.605990	0.302995	−	0.952992i	$-0.402013\pi$
$701$	16.0444	0.605990	0.302995	+	0.952992i	$0.402013\pi$
$702$	0	0
$703$	−2.13823	−0.0806448
$704$	4.00000	0.150756
$705$	0	0
$706$	−35.3380	−1.32996
$707$	−31.8496	−1.19783
$708$	0	0
$709$	−12.5115	−0.469880	−0.234940	−	0.972010i	$-0.575489\pi$
$709$	−12.5115	−0.469880	−0.234940	+	0.972010i	$0.575489\pi$
$710$	0	0
$711$	0	0
$712$	35.7137	1.33843
$713$	−1.32487	−0.0496167
$714$	0	0
$715$	0	0
$716$	−37.3766	−1.39683
$717$	0	0
$718$	−65.2262	−2.43422
$719$	−46.2784	−1.72589	−0.862946	−	0.505296i	$-0.831383\pi$
$719$	−46.2784	−1.72589	−0.862946	+	0.505296i	$0.831383\pi$
$720$	0	0
$721$	15.4723	0.576219
$722$	24.8119	0.923405
$723$	0	0
$724$	71.7499	2.66656
$725$	0	0
$726$	0	0
$727$	13.9697	0.518107	0.259053	−	0.965863i	$-0.416589\pi$
$727$	13.9697	0.518107	0.259053	+	0.965863i	$0.416589\pi$
$728$	−99.6502	−3.69328
$729$	0	0
$730$	0	0
$731$	−2.19649	−0.0812402
$732$	0	0
$733$	−30.2506	−1.11733	−0.558666	−	0.829393i	$-0.688687\pi$
$733$	−30.2506	−1.11733	−0.558666	+	0.829393i	$0.688687\pi$
$734$	−18.8265	−0.694900
$735$	0	0
$736$	2.13586	0.0787287
$737$	8.18172	0.301377
$738$	0	0
$739$	−28.5477	−1.05014	−0.525072	−	0.851058i	$-0.675962\pi$
$739$	−28.5477	−1.05014	−0.525072	+	0.851058i	$0.675962\pi$
$740$	0	0
$741$	0	0
$742$	97.8613	3.59260
$743$	44.3888	1.62847	0.814234	−	0.580537i	$-0.197157\pi$
$743$	44.3888	1.62847	0.814234	+	0.580537i	$0.197157\pi$
$744$	0	0
$745$	0	0
$746$	−4.94429	−0.181023
$747$	0	0
$748$	−7.94525	−0.290507
$749$	−7.90828	−0.288962
$750$	0	0
$751$	46.6688	1.70297	0.851484	−	0.524380i	$-0.175703\pi$
$751$	46.6688	1.70297	0.851484	+	0.524380i	$0.175703\pi$
$752$	15.0738	0.549685
$753$	0	0
$754$	14.9321	0.543794
$755$	0	0
$756$	0	0
$757$	−33.9995	−1.23573	−0.617867	−	0.786283i	$-0.712003\pi$
$757$	−33.9995	−1.23573	−0.617867	+	0.786283i	$0.712003\pi$
$758$	20.0508	0.728277
$759$	0	0
$760$	0	0
$761$	−28.8930	−1.04737	−0.523686	−	0.851911i	$-0.675444\pi$
$761$	−28.8930	−1.04737	−0.523686	+	0.851911i	$0.675444\pi$
$762$	0	0
$763$	17.0640	0.617757
$764$	54.4749	1.97083
$765$	0	0
$766$	7.71133	0.278622
$767$	−60.9438	−2.20055
$768$	0	0
$769$	−4.96968	−0.179211	−0.0896057	−	0.995977i	$-0.528561\pi$
$769$	−4.96968	−0.179211	−0.0896057	+	0.995977i	$0.528561\pi$
$770$	0	0
$771$	0	0
$772$	−61.3825	−2.20920
$773$	−6.23647	−0.224310	−0.112155	−	0.993691i	$-0.535775\pi$
$773$	−6.23647	−0.224310	−0.112155	+	0.993691i	$0.535775\pi$
$774$	0	0
$775$	0	0
$776$	−24.1114	−0.865550
$777$	0	0
$778$	−59.9873	−2.15065
$779$	−12.6580	−0.453519
$780$	0	0
$781$	−6.91160	−0.247317
$782$	9.30773	0.332844
$783$	0	0
$784$	4.33709	0.154896
$785$	0	0
$786$	0	0
$787$	−24.0000	−0.855508	−0.427754	−	0.903895i	$-0.640695\pi$
$787$	−24.0000	−0.855508	−0.427754	+	0.903895i	$0.640695\pi$
$788$	−26.7612	−0.953327
$789$	0	0
$790$	0	0
$791$	−46.2981	−1.64617
$792$	0	0
$793$	−52.6009	−1.86791
$794$	1.31757	0.0467590
$795$	0	0
$796$	−84.8989	−3.00916
$797$	44.1779	1.56486	0.782431	−	0.622737i	$-0.213979\pi$
$797$	44.1779	1.56486	0.782431	+	0.622737i	$0.213979\pi$
$798$	0	0
$799$	8.60086	0.304277
$800$	0	0
$801$	0	0
$802$	66.2760	2.34029
$803$	−1.43533	−0.0506518
$804$	0	0
$805$	0	0
$806$	16.4690	0.580095
$807$	0	0
$808$	−60.7269	−2.13636
$809$	−16.2471	−0.571217	−0.285609	−	0.958346i	$-0.592196\pi$
$809$	−16.2471	−0.571217	−0.285609	+	0.958346i	$0.592196\pi$
$810$	0	0
$811$	−10.2776	−0.360895	−0.180448	−	0.983585i	$-0.557755\pi$
$811$	−10.2776	−0.360895	−0.180448	+	0.983585i	$0.557755\pi$
$812$	−10.5745	−0.371093
$813$	0	0
$814$	−1.19394	−0.0418475
$815$	0	0
$816$	0	0
$817$	−2.32724	−0.0814198
$818$	31.7889	1.11147
$819$	0	0
$820$	0	0
$821$	−36.3742	−1.26947	−0.634735	−	0.772730i	$-0.718891\pi$
$821$	−36.3742	−1.26947	−0.634735	+	0.772730i	$0.718891\pi$
$822$	0	0
$823$	−39.9267	−1.39176	−0.695878	−	0.718160i	$-0.744985\pi$
$823$	−39.9267	−1.39176	−0.695878	+	0.718160i	$0.744985\pi$
$824$	29.5007	1.02771
$825$	0	0
$826$	63.9267	2.22429
$827$	−21.4469	−0.745782	−0.372891	−	0.927875i	$-0.621633\pi$
$827$	−21.4469	−0.745782	−0.372891	+	0.927875i	$0.621633\pi$
$828$	0	0
$829$	−13.1006	−0.455003	−0.227502	−	0.973778i	$-0.573056\pi$
$829$	−13.1006	−0.455003	−0.227502	+	0.973778i	$0.573056\pi$
$830$	0	0
$831$	0	0
$832$	39.3258	1.36338
$833$	2.47467	0.0857422
$834$	0	0
$835$	0	0
$836$	−8.41819	−0.291149
$837$	0	0
$838$	61.5633	2.12667
$839$	−50.0870	−1.72919	−0.864597	−	0.502465i	$-0.832426\pi$
$839$	−50.0870	−1.72919	−0.864597	+	0.502465i	$0.832426\pi$
$840$	0	0
$841$	−28.1779	−0.971653
$842$	−49.1754	−1.69470
$843$	0	0
$844$	−5.45580	−0.187796
$845$	0	0
$846$	0	0
$847$	29.5877	1.01665
$848$	69.7499	2.39522
$849$	0	0
$850$	0	0
$851$	0.944290	0.0323699
$852$	0	0
$853$	−51.1206	−1.75034	−0.875168	−	0.483818i	$-0.839250\pi$
$853$	−51.1206	−1.75034	−0.875168	+	0.483818i	$0.839250\pi$
$854$	55.1754	1.88806
$855$	0	0
$856$	−15.0785	−0.515374
$857$	−2.98049	−0.101811	−0.0509057	−	0.998703i	$-0.516211\pi$
$857$	−2.98049	−0.101811	−0.0509057	+	0.998703i	$0.516211\pi$
$858$	0	0
$859$	−40.5501	−1.38355	−0.691775	−	0.722113i	$-0.743171\pi$
$859$	−40.5501	−1.38355	−0.691775	+	0.722113i	$0.743171\pi$
$860$	0	0
$861$	0	0
$862$	−5.98541	−0.203864
$863$	−53.3522	−1.81613	−0.908065	−	0.418830i	$-0.862440\pi$
$863$	−53.3522	−1.81613	−0.908065	+	0.418830i	$0.862440\pi$
$864$	0	0
$865$	0	0
$866$	41.6239	1.41444
$867$	0	0
$868$	−11.6629	−0.395865
$869$	1.25694	0.0426388
$870$	0	0
$871$	80.4382	2.72554
$872$	32.5355	1.10179
$873$	0	0
$874$	9.86177	0.333579
$875$	0	0
$876$	0	0
$877$	−42.8529	−1.44704	−0.723519	−	0.690304i	$-0.757477\pi$
$877$	−42.8529	−1.44704	−0.723519	+	0.690304i	$0.757477\pi$
$878$	41.8251	1.41153
$879$	0	0
$880$	0	0
$881$	18.3488	0.618188	0.309094	−	0.951031i	$-0.399974\pi$
$881$	18.3488	0.618188	0.309094	+	0.951031i	$0.399974\pi$
$882$	0	0
$883$	55.9342	1.88233	0.941167	−	0.337941i	$-0.109730\pi$
$883$	55.9342	1.88233	0.941167	+	0.337941i	$0.109730\pi$
$884$	−78.1133	−2.62724
$885$	0	0
$886$	−78.0830	−2.62325
$887$	54.9805	1.84606	0.923032	−	0.384723i	$-0.125703\pi$
$887$	54.9805	1.84606	0.923032	+	0.384723i	$0.125703\pi$
$888$	0	0
$889$	−21.1490	−0.709316
$890$	0	0
$891$	0	0
$892$	92.6371	3.10172
$893$	9.11283	0.304949
$894$	0	0
$895$	0	0
$896$	−50.2981	−1.68034
$897$	0	0
$898$	−67.9243	−2.26666
$899$	0.906679	0.0302394
$900$	0	0
$901$	39.7981	1.32587
$902$	−7.06793	−0.235336
$903$	0	0
$904$	−88.2755	−2.93600
$905$	0	0
$906$	0	0
$907$	−34.6893	−1.15184	−0.575919	−	0.817507i	$-0.695356\pi$
$907$	−34.6893	−1.15184	−0.575919	+	0.817507i	$0.695356\pi$
$908$	−121.364	−4.02760
$909$	0	0
$910$	0	0
$911$	−12.5306	−0.415156	−0.207578	−	0.978218i	$-0.566558\pi$
$911$	−12.5306	−0.415156	−0.207578	+	0.978218i	$0.566558\pi$
$912$	0	0
$913$	−5.36741	−0.177635
$914$	−27.7948	−0.919370
$915$	0	0
$916$	65.6747	2.16995
$917$	8.40437	0.277537
$918$	0	0
$919$	−23.7962	−0.784965	−0.392482	−	0.919760i	$-0.628384\pi$
$919$	−23.7962	−0.784965	−0.392482	+	0.919760i	$0.628384\pi$
$920$	0	0
$921$	0	0
$922$	76.7607	2.52798
$923$	−67.9511	−2.23664
$924$	0	0
$925$	0	0
$926$	6.11142	0.200834
$927$	0	0
$928$	−1.46168	−0.0479820
$929$	−58.4963	−1.91920	−0.959600	−	0.281367i	$-0.909212\pi$
$929$	−58.4963	−1.91920	−0.959600	+	0.281367i	$0.909212\pi$
$930$	0	0
$931$	2.62198	0.0859318
$932$	−8.56467	−0.280545
$933$	0	0
$934$	−29.1246	−0.952986
$935$	0	0
$936$	0	0
$937$	−25.8872	−0.845697	−0.422848	−	0.906200i	$-0.638970\pi$
$937$	−25.8872	−0.845697	−0.422848	+	0.906200i	$0.638970\pi$
$938$	−84.3752	−2.75495
$939$	0	0
$940$	0	0
$941$	22.2022	0.723771	0.361885	−	0.932223i	$-0.382133\pi$
$941$	22.2022	0.723771	0.361885	+	0.932223i	$0.382133\pi$
$942$	0	0
$943$	5.59006	0.182037
$944$	45.5633	1.48296
$945$	0	0
$946$	−1.29948	−0.0422496
$947$	4.54912	0.147827	0.0739133	−	0.997265i	$-0.476451\pi$
$947$	4.54912	0.147827	0.0739133	+	0.997265i	$0.476451\pi$
$948$	0	0
$949$	−14.1114	−0.458076
$950$	0	0
$951$	0	0
$952$	42.5091	1.37773
$953$	37.7400	1.22252	0.611260	−	0.791430i	$-0.290663\pi$
$953$	37.7400	1.22252	0.611260	+	0.791430i	$0.290663\pi$
$954$	0	0
$955$	0	0
$956$	−91.8818	−2.97167
$957$	0	0
$958$	50.3634	1.62717
$959$	43.2301	1.39597
$960$	0	0
$961$	1.00000	0.0322581
$962$	−11.7381	−0.378453
$963$	0	0
$964$	−83.3277	−2.68381
$965$	0	0
$966$	0	0
$967$	12.6474	0.406712	0.203356	−	0.979105i	$-0.434815\pi$
$967$	12.6474	0.406712	0.203356	+	0.979105i	$0.434815\pi$
$968$	56.4142	1.81322
$969$	0	0
$970$	0	0
$971$	43.8677	1.40778	0.703890	−	0.710309i	$-0.251445\pi$
$971$	43.8677	1.40778	0.703890	+	0.710309i	$0.251445\pi$
$972$	0	0
$973$	25.6794	0.823244
$974$	−14.5442	−0.466026
$975$	0	0
$976$	39.3258	1.25879
$977$	−34.1197	−1.09158	−0.545792	−	0.837920i	$-0.683771\pi$
$977$	−34.1197	−1.09158	−0.545792	+	0.837920i	$0.683771\pi$
$978$	0	0
$979$	4.50659	0.144031
$980$	0	0
$981$	0	0
$982$	−34.1500	−1.08977
$983$	38.0557	1.21379	0.606894	−	0.794783i	$-0.292415\pi$
$983$	38.0557	1.21379	0.606894	+	0.794783i	$0.292415\pi$
$984$	0	0
$985$	0	0
$986$	−6.36977	−0.202855
$987$	0	0
$988$	−82.7631	−2.63304
$989$	1.02776	0.0326809
$990$	0	0
$991$	17.2365	0.547535	0.273767	−	0.961796i	$-0.411730\pi$
$991$	17.2365	0.547535	0.273767	+	0.961796i	$0.411730\pi$
$992$	−1.61213	−0.0511851
$993$	0	0
$994$	71.2769	2.26077
$995$	0	0
$996$	0	0
$997$	−17.6048	−0.557551	−0.278775	−	0.960356i	$-0.589928\pi$
$997$	−17.6048	−0.557551	−0.278775	+	0.960356i	$0.589928\pi$
$998$	51.1695	1.61974
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6975.2.a.z.1.1	✓	3
3.2	odd	2		6975.2.a.bg.1.3	yes	3
5.4	even	2		6975.2.a.bh.1.3	yes	3
15.14	odd	2		6975.2.a.ba.1.1	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6975.2.a.z.1.1	✓	3	1.1	even	1	trivial
6975.2.a.ba.1.1	yes	3	15.14	odd	2
6975.2.a.bg.1.3	yes	3	3.2	odd	2
6975.2.a.bh.1.3	yes	3	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)

\(f(q)\)	\(=\)	\(q-2.48119 q^{2} +4.15633 q^{4} -2.80606 q^{7} -5.35026 q^{8} -0.675131 q^{11} -6.63752 q^{13} +6.96239 q^{14} +4.96239 q^{16} +2.83146 q^{17} +3.00000 q^{19} +1.67513 q^{22} -1.32487 q^{23} +16.4690 q^{26} -11.6629 q^{28} +0.906679 q^{29} +1.00000 q^{31} -1.61213 q^{32} -7.02539 q^{34} -0.712742 q^{37} -7.44358 q^{38} -4.21933 q^{41} -0.775746 q^{43} -2.80606 q^{44} +3.28726 q^{46} +3.03761 q^{47} +0.873992 q^{49} -27.5877 q^{52} +14.0557 q^{53} +15.0132 q^{56} -2.24965 q^{58} +9.18172 q^{59} +7.92478 q^{61} -2.48119 q^{62} -5.92478 q^{64} -12.1187 q^{67} +11.7685 q^{68} +10.2374 q^{71} +2.12601 q^{73} +1.76845 q^{74} +12.4690 q^{76} +1.89446 q^{77} -1.86177 q^{79} +10.4690 q^{82} +7.95017 q^{83} +1.92478 q^{86} +3.61213 q^{88} -6.67513 q^{89} +18.6253 q^{91} -5.50659 q^{92} -7.53690 q^{94} +4.50659 q^{97} -2.16854 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{2} + 2 q^{4} - 8 q^{7} - 6 q^{8} + 3 q^{11} - 4 q^{13} + 10 q^{14} + 4 q^{16} - 7 q^{17} + 9 q^{19} - 9 q^{23} + 18 q^{26} - 4 q^{28} + 9 q^{29} + 3 q^{31} - 4 q^{32} - 6 q^{34} - 8 q^{37} - 6 q^{38}+ \cdots - 22 q^{98}+O(q^{100}) \) 3 * q - 2 * q^2 + 2 * q^4 - 8 * q^7 - 6 * q^8 + 3 * q^11 - 4 * q^13 + 10 * q^14 + 4 * q^16 - 7 * q^17 + 9 * q^19 - 9 * q^23 + 18 * q^26 - 4 * q^28 + 9 * q^29 + 3 * q^31 - 4 * q^32 - 6 * q^34 - 8 * q^37 - 6 * q^38 + 2 * q^41 - 4 * q^43 - 8 * q^44 + 4 * q^46 + 20 * q^47 + 11 * q^49 - 30 * q^52 + 25 * q^53 + 4 * q^56 + 10 * q^58 + 2 * q^59 + 2 * q^61 - 2 * q^62 + 4 * q^64 - 15 * q^67 + 24 * q^68 - 12 * q^71 - 2 * q^73 - 6 * q^74 + 6 * q^76 - 14 * q^77 + 12 * q^79 - 13 * q^83 - 16 * q^86 + 10 * q^88 - 15 * q^89 + 14 * q^91 + 4 * q^92 - 7 * q^97 - 22 * q^98

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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