LMFDB - Embedded newform 6975.2.a.bt.1.2

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,-1,0,3,0,0,-2,-3,0,0,6,0,-4,8,0,-5,-12] 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:	5.5.582992.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 6x^{3} + 4x^{2} + 7x - 3 $ x^5 - x^4 - 6x^3 + 4x^2 + 7*x - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1395)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.36680$ 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)

$f(q)$	$=$	$q-1.36680 q^{2} -0.131850 q^{4} -2.67055 q^{7} +2.91382 q^{8} +2.30375 q^{11} +3.08572 q^{13} +3.65012 q^{14} -3.71892 q^{16} -3.28332 q^{17} +5.71622 q^{19} -3.14877 q^{22} -3.43560 q^{23} -4.21757 q^{26} +0.352113 q^{28} +3.08349 q^{29} -1.00000 q^{31} -0.744613 q^{32} +4.48765 q^{34} -2.62968 q^{37} -7.81295 q^{38} -9.47249 q^{41} -8.24280 q^{43} -0.303750 q^{44} +4.69579 q^{46} -6.54432 q^{47} +0.131850 q^{49} -0.406852 q^{52} +5.46944 q^{53} -7.78151 q^{56} -4.21452 q^{58} +11.6447 q^{59} -6.98262 q^{61} +1.36680 q^{62} +8.45557 q^{64} +13.6470 q^{67} +0.432905 q^{68} +8.78420 q^{71} -7.19795 q^{73} +3.59426 q^{74} -0.753684 q^{76} -6.15228 q^{77} -16.9786 q^{79} +12.9470 q^{82} -0.438294 q^{83} +11.2663 q^{86} +6.71271 q^{88} +11.6439 q^{89} -8.24057 q^{91} +0.452984 q^{92} +8.94480 q^{94} +0.582140 q^{97} -0.180213 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{2} + 3 q^{4} - 2 q^{7} - 3 q^{8} + 6 q^{11} - 4 q^{13} + 8 q^{14} - 5 q^{16} - 12 q^{17} - 4 q^{19} + 4 q^{22} - 8 q^{23} + 2 q^{26} - 6 q^{28} + 14 q^{29} - 5 q^{31} - 7 q^{32} - 2 q^{34} - 4 q^{37}+ \cdots + 5 q^{98}+O(q^{100}) $ 5 * q - q^2 + 3 * q^4 - 2 * q^7 - 3 * q^8 + 6 * q^11 - 4 * q^13 + 8 * q^14 - 5 * q^16 - 12 * q^17 - 4 * q^19 + 4 * q^22 - 8 * q^23 + 2 * q^26 - 6 * q^28 + 14 * q^29 - 5 * q^31 - 7 * q^32 - 2 * q^34 - 4 * q^37 - 2 * q^38 + 16 * q^41 + 2 * q^43 + 4 * q^44 - 8 * q^46 - 2 * q^47 - 3 * q^49 + 2 * q^52 - 14 * q^53 + 12 * q^56 - 20 * q^58 + 10 * q^59 - 14 * q^61 + q^62 - 5 * q^64 + 2 * q^67 - 26 * q^68 + 12 * q^71 - 10 * q^73 + 10 * q^74 - 32 * q^76 - 26 * q^77 - 12 * q^79 + 6 * q^82 - 12 * q^83 - 10 * q^86 - 14 * q^88 + 14 * q^89 - 16 * q^91 + 14 * q^92 + 4 * q^94 - 8 * q^97 + 5 * 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 $

$2$	−1.36680	−0.966476	−0.483238	−	0.875489i	$-0.660539\pi$
$2$	−1.36680	−0.966476	−0.483238	+	0.875489i	$0.660539\pi$
$3$	0	0
$3$	0	0
$4$	−0.131850	−0.0659250
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.67055	−1.00937	−0.504687	−	0.863302i	$-0.668392\pi$
$7$	−2.67055	−1.00937	−0.504687	+	0.863302i	$0.668392\pi$
$8$	2.91382	1.03019
$9$	0	0
$10$	0	0
$11$	2.30375	0.694607	0.347303	−	0.937753i	$-0.387097\pi$
$11$	2.30375	0.694607	0.347303	+	0.937753i	$0.387097\pi$
$12$	0	0
$13$	3.08572	0.855824	0.427912	−	0.903820i	$-0.359249\pi$
$13$	3.08572	0.855824	0.427912	+	0.903820i	$0.359249\pi$
$14$	3.65012	0.975535
$15$	0	0
$16$	−3.71892	−0.929729
$17$	−3.28332	−0.796321	−0.398161	−	0.917316i	$-0.630351\pi$
$17$	−3.28332	−0.796321	−0.398161	+	0.917316i	$0.630351\pi$
$18$	0	0
$19$	5.71622	1.31139	0.655696	−	0.755025i	$-0.272375\pi$
$19$	5.71622	1.31139	0.655696	+	0.755025i	$0.272375\pi$
$20$	0	0
$21$	0	0
$22$	−3.14877	−0.671320
$23$	−3.43560	−0.716372	−0.358186	−	0.933650i	$-0.616605\pi$
$23$	−3.43560	−0.716372	−0.358186	+	0.933650i	$0.616605\pi$
$24$	0	0
$25$	0	0
$26$	−4.21757	−0.827133
$27$	0	0
$28$	0.352113	0.0665430
$29$	3.08349	0.572589	0.286295	−	0.958142i	$-0.407576\pi$
$29$	3.08349	0.572589	0.286295	+	0.958142i	$0.407576\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	−0.744613	−0.131630
$33$	0	0
$34$	4.48765	0.769625
$35$	0	0
$36$	0	0
$37$	−2.62968	−0.432317	−0.216159	−	0.976358i	$-0.569353\pi$
$37$	−2.62968	−0.432317	−0.216159	+	0.976358i	$0.569353\pi$
$38$	−7.81295	−1.26743
$39$	0	0
$40$	0	0
$41$	−9.47249	−1.47935	−0.739677	−	0.672962i	$-0.765022\pi$
$41$	−9.47249	−1.47935	−0.739677	+	0.672962i	$0.765022\pi$
$42$	0	0
$43$	−8.24280	−1.25702	−0.628508	−	0.777803i	$-0.716334\pi$
$43$	−8.24280	−1.25702	−0.628508	+	0.777803i	$0.716334\pi$
$44$	−0.303750	−0.0457920
$45$	0	0
$46$	4.69579	0.692356
$47$	−6.54432	−0.954587	−0.477294	−	0.878744i	$-0.658382\pi$
$47$	−6.54432	−0.954587	−0.477294	+	0.878744i	$0.658382\pi$
$48$	0	0
$49$	0.131850	0.0188357
$50$	0	0
$51$	0	0
$52$	−0.406852	−0.0564203
$53$	5.46944	0.751286	0.375643	−	0.926765i	$-0.377422\pi$
$53$	5.46944	0.751286	0.375643	+	0.926765i	$0.377422\pi$
$54$	0	0
$55$	0	0
$56$	−7.78151	−1.03985
$57$	0	0
$58$	−4.21452	−0.553393
$59$	11.6447	1.51601	0.758007	−	0.652246i	$-0.226173\pi$
$59$	11.6447	1.51601	0.758007	+	0.652246i	$0.226173\pi$
$60$	0	0
$61$	−6.98262	−0.894032	−0.447016	−	0.894526i	$-0.647513\pi$
$61$	−6.98262	−0.894032	−0.447016	+	0.894526i	$0.647513\pi$
$62$	1.36680	0.173584
$63$	0	0
$64$	8.45557	1.05695
$65$	0	0
$66$	0	0
$67$	13.6470	1.66724	0.833621	−	0.552337i	$-0.186264\pi$
$67$	13.6470	1.66724	0.833621	+	0.552337i	$0.186264\pi$
$68$	0.432905	0.0524975
$69$	0	0
$70$	0	0
$71$	8.78420	1.04249	0.521246	−	0.853406i	$-0.325467\pi$
$71$	8.78420	1.04249	0.521246	+	0.853406i	$0.325467\pi$
$72$	0	0
$73$	−7.19795	−0.842457	−0.421228	−	0.906955i	$-0.638401\pi$
$73$	−7.19795	−0.842457	−0.421228	+	0.906955i	$0.638401\pi$
$74$	3.59426	0.417824
$75$	0	0
$76$	−0.753684	−0.0864535
$77$	−6.15228	−0.701118
$78$	0	0
$79$	−16.9786	−1.91025	−0.955123	−	0.296209i	$-0.904277\pi$
$79$	−16.9786	−1.91025	−0.955123	+	0.296209i	$0.904277\pi$
$80$	0	0
$81$	0	0
$82$	12.9470	1.42976
$83$	−0.438294	−0.0481090	−0.0240545	−	0.999711i	$-0.507658\pi$
$83$	−0.438294	−0.0481090	−0.0240545	+	0.999711i	$0.507658\pi$
$84$	0	0
$85$	0	0
$86$	11.2663	1.21487
$87$	0	0
$88$	6.71271	0.715577
$89$	11.6439	1.23425	0.617126	−	0.786864i	$-0.288297\pi$
$89$	11.6439	1.23425	0.617126	+	0.786864i	$0.288297\pi$
$90$	0	0
$91$	−8.24057	−0.863847
$92$	0.452984	0.0472269
$93$	0	0
$94$	8.94480	0.922585
$95$	0	0
$96$	0	0
$97$	0.582140	0.0591074	0.0295537	−	0.999563i	$-0.490591\pi$
$97$	0.582140	0.0591074	0.0295537	+	0.999563i	$0.490591\pi$
$98$	−0.180213	−0.0182043
$99$	0	0
$100$	0	0
$101$	14.8356	1.47620	0.738099	−	0.674692i	$-0.235724\pi$
$101$	14.8356	1.47620	0.738099	+	0.674692i	$0.235724\pi$
$102$	0	0
$103$	3.94543	0.388755	0.194377	−	0.980927i	$-0.437731\pi$
$103$	3.94543	0.388755	0.194377	+	0.980927i	$0.437731\pi$
$104$	8.99122	0.881662
$105$	0	0
$106$	−7.47565	−0.726099
$107$	−0.949067	−0.0917498	−0.0458749	−	0.998947i	$-0.514608\pi$
$107$	−0.949067	−0.0917498	−0.0458749	+	0.998947i	$0.514608\pi$
$108$	0	0
$109$	−3.53121	−0.338228	−0.169114	−	0.985596i	$-0.554091\pi$
$109$	−3.53121	−0.338228	−0.169114	+	0.985596i	$0.554091\pi$
$110$	0	0
$111$	0	0
$112$	9.93156	0.938444
$113$	8.79363	0.827235	0.413617	−	0.910451i	$-0.364265\pi$
$113$	8.79363	0.827235	0.413617	+	0.910451i	$0.364265\pi$
$114$	0	0
$115$	0	0
$116$	−0.406558	−0.0377480
$117$	0	0
$118$	−15.9160	−1.46519
$119$	8.76827	0.803786
$120$	0	0
$121$	−5.69274	−0.517522
$122$	9.54386	0.864060
$123$	0	0
$124$	0.131850	0.0118405
$125$	0	0
$126$	0	0
$127$	−2.22283	−0.197244	−0.0986222	−	0.995125i	$-0.531444\pi$
$127$	−2.22283	−0.197244	−0.0986222	+	0.995125i	$0.531444\pi$
$128$	−10.0679	−0.889882
$129$	0	0
$130$	0	0
$131$	−16.6465	−1.45441	−0.727206	−	0.686420i	$-0.759181\pi$
$131$	−16.6465	−1.45441	−0.727206	+	0.686420i	$0.759181\pi$
$132$	0	0
$133$	−15.2655	−1.32368
$134$	−18.6527	−1.61135
$135$	0	0
$136$	−9.56699	−0.820362
$137$	2.56886	0.219473	0.109736	−	0.993961i	$-0.464999\pi$
$137$	2.56886	0.219473	0.109736	+	0.993961i	$0.464999\pi$
$138$	0	0
$139$	−19.8248	−1.68152	−0.840758	−	0.541412i	$-0.817890\pi$
$139$	−19.8248	−1.68152	−0.840758	+	0.541412i	$0.817890\pi$
$140$	0	0
$141$	0	0
$142$	−12.0063	−1.00754
$143$	7.10872	0.594461
$144$	0	0
$145$	0	0
$146$	9.83818	0.814214
$147$	0	0
$148$	0.346724	0.0285005
$149$	7.91908	0.648757	0.324378	−	0.945927i	$-0.394845\pi$
$149$	7.91908	0.648757	0.324378	+	0.945927i	$0.394845\pi$
$150$	0	0
$151$	16.5279	1.34502	0.672510	−	0.740088i	$-0.265216\pi$
$151$	16.5279	1.34502	0.672510	+	0.740088i	$0.265216\pi$
$152$	16.6560	1.35098
$153$	0	0
$154$	8.40896	0.677613
$155$	0	0
$156$	0	0
$157$	16.3469	1.30462	0.652312	−	0.757950i	$-0.273799\pi$
$157$	16.3469	1.30462	0.652312	+	0.757950i	$0.273799\pi$
$158$	23.2065	1.84621
$159$	0	0
$160$	0	0
$161$	9.17495	0.723087
$162$	0	0
$163$	9.72706	0.761882	0.380941	−	0.924599i	$-0.375600\pi$
$163$	9.72706	0.761882	0.380941	+	0.924599i	$0.375600\pi$
$164$	1.24895	0.0975265
$165$	0	0
$166$	0.599061	0.0464962
$167$	−4.41435	−0.341592	−0.170796	−	0.985306i	$-0.554634\pi$
$167$	−4.41435	−0.341592	−0.170796	+	0.985306i	$0.554634\pi$
$168$	0	0
$169$	−3.47834	−0.267565
$170$	0	0
$171$	0	0
$172$	1.08681	0.0828688
$173$	19.5726	1.48807	0.744037	−	0.668138i	$-0.232909\pi$
$173$	19.5726	1.48807	0.744037	+	0.668138i	$0.232909\pi$
$174$	0	0
$175$	0	0
$176$	−8.56745	−0.645796
$177$	0	0
$178$	−15.9149	−1.19287
$179$	−1.06019	−0.0792423	−0.0396211	−	0.999215i	$-0.512615\pi$
$179$	−1.06019	−0.0792423	−0.0396211	+	0.999215i	$0.512615\pi$
$180$	0	0
$181$	−13.5324	−1.00585	−0.502927	−	0.864329i	$-0.667743\pi$
$181$	−13.5324	−1.00585	−0.502927	+	0.864329i	$0.667743\pi$
$182$	11.2632	0.834887
$183$	0	0
$184$	−10.0107	−0.738000
$185$	0	0
$186$	0	0
$187$	−7.56394	−0.553130
$188$	0.862869	0.0629312
$189$	0	0
$190$	0	0
$191$	−3.23830	−0.234315	−0.117157	−	0.993113i	$-0.537378\pi$
$191$	−3.23830	−0.234315	−0.117157	+	0.993113i	$0.537378\pi$
$192$	0	0
$193$	4.36891	0.314481	0.157240	−	0.987560i	$-0.449740\pi$
$193$	4.36891	0.314481	0.157240	+	0.987560i	$0.449740\pi$
$194$	−0.795671	−0.0571258
$195$	0	0
$196$	−0.0173844	−0.00124175
$197$	−23.5486	−1.67777	−0.838885	−	0.544309i	$-0.816792\pi$
$197$	−23.5486	−1.67777	−0.838885	+	0.544309i	$0.816792\pi$
$198$	0	0
$199$	−14.0829	−0.998310	−0.499155	−	0.866513i	$-0.666356\pi$
$199$	−14.0829	−0.998310	−0.499155	+	0.866513i	$0.666356\pi$
$200$	0	0
$201$	0	0
$202$	−20.2774	−1.42671
$203$	−8.23461	−0.577957
$204$	0	0
$205$	0	0
$206$	−5.39263	−0.375722
$207$	0	0
$208$	−11.4755	−0.795684
$209$	13.1687	0.910901
$210$	0	0
$211$	−5.55948	−0.382730	−0.191365	−	0.981519i	$-0.561291\pi$
$211$	−5.55948	−0.382730	−0.191365	+	0.981519i	$0.561291\pi$
$212$	−0.721146	−0.0495285
$213$	0	0
$214$	1.29719	0.0886739
$215$	0	0
$216$	0	0
$217$	2.67055	0.181289
$218$	4.82646	0.326889
$219$	0	0
$220$	0	0
$221$	−10.1314	−0.681511
$222$	0	0
$223$	−0.914452	−0.0612362	−0.0306181	−	0.999531i	$-0.509748\pi$
$223$	−0.914452	−0.0612362	−0.0306181	+	0.999531i	$0.509748\pi$
$224$	1.98853	0.132864
$225$	0	0
$226$	−12.0192	−0.799502
$227$	19.4609	1.29166	0.645831	−	0.763480i	$-0.276511\pi$
$227$	19.4609	1.29166	0.645831	+	0.763480i	$0.276511\pi$
$228$	0	0
$229$	−7.18882	−0.475051	−0.237525	−	0.971381i	$-0.576336\pi$
$229$	−7.18882	−0.475051	−0.237525	+	0.971381i	$0.576336\pi$
$230$	0	0
$231$	0	0
$232$	8.98472	0.589876
$233$	21.0742	1.38062	0.690310	−	0.723514i	$-0.257474\pi$
$233$	21.0742	1.38062	0.690310	+	0.723514i	$0.257474\pi$
$234$	0	0
$235$	0	0
$236$	−1.53536	−0.0999433
$237$	0	0
$238$	−11.9845	−0.776839
$239$	−5.04721	−0.326477	−0.163238	−	0.986587i	$-0.552194\pi$
$239$	−5.04721	−0.326477	−0.163238	+	0.986587i	$0.552194\pi$
$240$	0	0
$241$	−24.2481	−1.56196	−0.780978	−	0.624558i	$-0.785279\pi$
$241$	−24.2481	−1.56196	−0.780978	+	0.624558i	$0.785279\pi$
$242$	7.78085	0.500172
$243$	0	0
$244$	0.920658	0.0589391
$245$	0	0
$246$	0	0
$247$	17.6386	1.12232
$248$	−2.91382	−0.185028
$249$	0	0
$250$	0	0
$251$	−2.25924	−0.142602	−0.0713009	−	0.997455i	$-0.522715\pi$
$251$	−2.25924	−0.142602	−0.0713009	+	0.997455i	$0.522715\pi$
$252$	0	0
$253$	−7.91476	−0.497597
$254$	3.03817	0.190632
$255$	0	0
$256$	−3.15035	−0.196897
$257$	7.65751	0.477662	0.238831	−	0.971061i	$-0.423236\pi$
$257$	7.65751	0.477662	0.238831	+	0.971061i	$0.423236\pi$
$258$	0	0
$259$	7.02271	0.436370
$260$	0	0
$261$	0	0
$262$	22.7525	1.40565
$263$	−1.08473	−0.0668873	−0.0334437	−	0.999441i	$-0.510647\pi$
$263$	−1.08473	−0.0668873	−0.0334437	+	0.999441i	$0.510647\pi$
$264$	0	0
$265$	0	0
$266$	20.8649	1.27931
$267$	0	0
$268$	−1.79935	−0.109913
$269$	−3.97131	−0.242135	−0.121068	−	0.992644i	$-0.538632\pi$
$269$	−3.97131	−0.242135	−0.121068	+	0.992644i	$0.538632\pi$
$270$	0	0
$271$	15.6019	0.947750	0.473875	−	0.880592i	$-0.342855\pi$
$271$	15.6019	0.947750	0.473875	+	0.880592i	$0.342855\pi$
$272$	12.2104	0.740363
$273$	0	0
$274$	−3.51113	−0.212115
$275$	0	0
$276$	0	0
$277$	−9.38214	−0.563718	−0.281859	−	0.959456i	$-0.590951\pi$
$277$	−9.38214	−0.563718	−0.281859	+	0.959456i	$0.590951\pi$
$278$	27.0966	1.62514
$279$	0	0
$280$	0	0
$281$	−20.6193	−1.23005	−0.615023	−	0.788509i	$-0.710853\pi$
$281$	−20.6193	−1.23005	−0.615023	+	0.788509i	$0.710853\pi$
$282$	0	0
$283$	−9.37478	−0.557273	−0.278636	−	0.960397i	$-0.589882\pi$
$283$	−9.37478	−0.557273	−0.278636	+	0.960397i	$0.589882\pi$
$284$	−1.15820	−0.0687264
$285$	0	0
$286$	−9.71622	−0.574532
$287$	25.2968	1.49322
$288$	0	0
$289$	−6.21984	−0.365873
$290$	0	0
$291$	0	0
$292$	0.949051	0.0555390
$293$	−16.9315	−0.989150	−0.494575	−	0.869135i	$-0.664676\pi$
$293$	−16.9315	−0.989150	−0.494575	+	0.869135i	$0.664676\pi$
$294$	0	0
$295$	0	0
$296$	−7.66242	−0.445369
$297$	0	0
$298$	−10.8238	−0.627007
$299$	−10.6013	−0.613089
$300$	0	0
$301$	22.0128	1.26880
$302$	−22.5904	−1.29993
$303$	0	0
$304$	−21.2581	−1.21924
$305$	0	0
$306$	0	0
$307$	−17.5702	−1.00278	−0.501392	−	0.865220i	$-0.667179\pi$
$307$	−17.5702	−1.00278	−0.501392	+	0.865220i	$0.667179\pi$
$308$	0.811179	0.0462212
$309$	0	0
$310$	0	0
$311$	−13.0871	−0.742103	−0.371052	−	0.928612i	$-0.621003\pi$
$311$	−13.0871	−0.742103	−0.371052	+	0.928612i	$0.621003\pi$
$312$	0	0
$313$	−21.8024	−1.23235	−0.616174	−	0.787610i	$-0.711318\pi$
$313$	−21.8024	−1.23235	−0.616174	+	0.787610i	$0.711318\pi$
$314$	−22.3430	−1.26089
$315$	0	0
$316$	2.23864	0.125933
$317$	−23.3546	−1.31172	−0.655862	−	0.754881i	$-0.727695\pi$
$317$	−23.3546	−1.31172	−0.655862	+	0.754881i	$0.727695\pi$
$318$	0	0
$319$	7.10358	0.397724
$320$	0	0
$321$	0	0
$322$	−12.5403	−0.698846
$323$	−18.7682	−1.04429
$324$	0	0
$325$	0	0
$326$	−13.2950	−0.736340
$327$	0	0
$328$	−27.6011	−1.52402
$329$	17.4770	0.963536
$330$	0	0
$331$	−28.3357	−1.55747	−0.778735	−	0.627353i	$-0.784138\pi$
$331$	−28.3357	−1.55747	−0.778735	+	0.627353i	$0.784138\pi$
$332$	0.0577891	0.00317159
$333$	0	0
$334$	6.03354	0.330141
$335$	0	0
$336$	0	0
$337$	−7.85146	−0.427696	−0.213848	−	0.976867i	$-0.568600\pi$
$337$	−7.85146	−0.427696	−0.213848	+	0.976867i	$0.568600\pi$
$338$	4.75421	0.258595
$339$	0	0
$340$	0	0
$341$	−2.30375	−0.124755
$342$	0	0
$343$	18.3418	0.990362
$344$	−24.0180	−1.29497
$345$	0	0
$346$	−26.7518	−1.43819
$347$	−26.6717	−1.43181	−0.715907	−	0.698196i	$-0.753986\pi$
$347$	−26.6717	−1.43181	−0.715907	+	0.698196i	$0.753986\pi$
$348$	0	0
$349$	23.9245	1.28065	0.640326	−	0.768103i	$-0.278799\pi$
$349$	23.9245	1.28065	0.640326	+	0.768103i	$0.278799\pi$
$350$	0	0
$351$	0	0
$352$	−1.71540	−0.0914313
$353$	−27.3233	−1.45427	−0.727135	−	0.686494i	$-0.759149\pi$
$353$	−27.3233	−1.45427	−0.727135	+	0.686494i	$0.759149\pi$
$354$	0	0
$355$	0	0
$356$	−1.53525	−0.0813681
$357$	0	0
$358$	1.44907	0.0765857
$359$	−15.3108	−0.808072	−0.404036	−	0.914743i	$-0.632393\pi$
$359$	−15.3108	−0.808072	−0.404036	+	0.914743i	$0.632393\pi$
$360$	0	0
$361$	13.6752	0.719747
$362$	18.4961	0.972133
$363$	0	0
$364$	1.08652	0.0569491
$365$	0	0
$366$	0	0
$367$	−12.4980	−0.652391	−0.326196	−	0.945302i	$-0.605767\pi$
$367$	−12.4980	−0.652391	−0.326196	+	0.945302i	$0.605767\pi$
$368$	12.7767	0.666032
$369$	0	0
$370$	0	0
$371$	−14.6064	−0.758328
$372$	0	0
$373$	−26.2682	−1.36011	−0.680057	−	0.733159i	$-0.738045\pi$
$373$	−26.2682	−1.36011	−0.680057	+	0.733159i	$0.738045\pi$
$374$	10.3384	0.534586
$375$	0	0
$376$	−19.0690	−0.983407
$377$	9.51477	0.490036
$378$	0	0
$379$	9.78391	0.502566	0.251283	−	0.967914i	$-0.419148\pi$
$379$	9.78391	0.502566	0.251283	+	0.967914i	$0.419148\pi$
$380$	0	0
$381$	0	0
$382$	4.42611	0.226460
$383$	14.7863	0.755545	0.377772	−	0.925898i	$-0.376690\pi$
$383$	14.7863	0.755545	0.377772	+	0.925898i	$0.376690\pi$
$384$	0	0
$385$	0	0
$386$	−5.97144	−0.303938
$387$	0	0
$388$	−0.0767552	−0.00389666
$389$	3.38114	0.171430	0.0857152	−	0.996320i	$-0.472682\pi$
$389$	3.38114	0.171430	0.0857152	+	0.996320i	$0.472682\pi$
$390$	0	0
$391$	11.2802	0.570462
$392$	0.384187	0.0194044
$393$	0	0
$394$	32.1863	1.62152
$395$	0	0
$396$	0	0
$397$	28.0035	1.40546	0.702729	−	0.711458i	$-0.251965\pi$
$397$	28.0035	1.40546	0.702729	+	0.711458i	$0.251965\pi$
$398$	19.2485	0.964842
$399$	0	0
$400$	0	0
$401$	19.2602	0.961809	0.480904	−	0.876773i	$-0.340308\pi$
$401$	19.2602	0.961809	0.480904	+	0.876773i	$0.340308\pi$
$402$	0	0
$403$	−3.08572	−0.153711
$404$	−1.95608	−0.0973185
$405$	0	0
$406$	11.2551	0.558581
$407$	−6.05813	−0.300291
$408$	0	0
$409$	5.08671	0.251521	0.125761	−	0.992061i	$-0.459863\pi$
$409$	5.08671	0.251521	0.125761	+	0.992061i	$0.459863\pi$
$410$	0	0
$411$	0	0
$412$	−0.520205	−0.0256287
$413$	−31.0979	−1.53023
$414$	0	0
$415$	0	0
$416$	−2.29767	−0.112652
$417$	0	0
$418$	−17.9991	−0.880363
$419$	10.0382	0.490397	0.245199	−	0.969473i	$-0.421147\pi$
$419$	10.0382	0.490397	0.245199	+	0.969473i	$0.421147\pi$
$420$	0	0
$421$	10.7083	0.521890	0.260945	−	0.965354i	$-0.415966\pi$
$421$	10.7083	0.521890	0.260945	+	0.965354i	$0.415966\pi$
$422$	7.59871	0.369899
$423$	0	0
$424$	15.9370	0.773967
$425$	0	0
$426$	0	0
$427$	18.6474	0.902413
$428$	0.125135	0.00604861
$429$	0	0
$430$	0	0
$431$	−13.8111	−0.665259	−0.332629	−	0.943058i	$-0.607936\pi$
$431$	−13.8111	−0.665259	−0.332629	+	0.943058i	$0.607936\pi$
$432$	0	0
$433$	−31.7078	−1.52378	−0.761890	−	0.647707i	$-0.775728\pi$
$433$	−31.7078	−1.52378	−0.761890	+	0.647707i	$0.775728\pi$
$434$	−3.65012	−0.175211
$435$	0	0
$436$	0.465590	0.0222977
$437$	−19.6386	−0.939444
$438$	0	0
$439$	16.1478	0.770692	0.385346	−	0.922772i	$-0.374082\pi$
$439$	16.1478	0.770692	0.385346	+	0.922772i	$0.374082\pi$
$440$	0	0
$441$	0	0
$442$	13.8476	0.658664
$443$	8.76811	0.416586	0.208293	−	0.978067i	$-0.433209\pi$
$443$	8.76811	0.416586	0.208293	+	0.978067i	$0.433209\pi$
$444$	0	0
$445$	0	0
$446$	1.24988	0.0591833
$447$	0	0
$448$	−22.5810	−1.06685
$449$	9.62892	0.454417	0.227208	−	0.973846i	$-0.427040\pi$
$449$	9.62892	0.454417	0.227208	+	0.973846i	$0.427040\pi$
$450$	0	0
$451$	−21.8222	−1.02757
$452$	−1.15944	−0.0545355
$453$	0	0
$454$	−26.5991	−1.24836
$455$	0	0
$456$	0	0
$457$	−12.2693	−0.573932	−0.286966	−	0.957941i	$-0.592647\pi$
$457$	−12.2693	−0.573932	−0.286966	+	0.957941i	$0.592647\pi$
$458$	9.82570	0.459125
$459$	0	0
$460$	0	0
$461$	−19.5174	−0.909018	−0.454509	−	0.890742i	$-0.650185\pi$
$461$	−19.5174	−0.909018	−0.454509	+	0.890742i	$0.650185\pi$
$462$	0	0
$463$	23.6187	1.09765	0.548827	−	0.835936i	$-0.315074\pi$
$463$	23.6187	1.09765	0.548827	+	0.835936i	$0.315074\pi$
$464$	−11.4672	−0.532353
$465$	0	0
$466$	−28.8043	−1.33434
$467$	24.6174	1.13916	0.569579	−	0.821937i	$-0.307106\pi$
$467$	24.6174	1.13916	0.569579	+	0.821937i	$0.307106\pi$
$468$	0	0
$469$	−36.4449	−1.68287
$470$	0	0
$471$	0	0
$472$	33.9306	1.56178
$473$	−18.9894	−0.873131
$474$	0	0
$475$	0	0
$476$	−1.15610	−0.0529896
$477$	0	0
$478$	6.89854	0.315532
$479$	13.2081	0.603493	0.301747	−	0.953388i	$-0.402430\pi$
$479$	13.2081	0.603493	0.301747	+	0.953388i	$0.402430\pi$
$480$	0	0
$481$	−8.11447	−0.369988
$482$	33.1423	1.50959
$483$	0	0
$484$	0.750588	0.0341176
$485$	0	0
$486$	0	0
$487$	10.2560	0.464744	0.232372	−	0.972627i	$-0.425351\pi$
$487$	10.2560	0.464744	0.232372	+	0.972627i	$0.425351\pi$
$488$	−20.3461	−0.921024
$489$	0	0
$490$	0	0
$491$	−3.55948	−0.160637	−0.0803184	−	0.996769i	$-0.525594\pi$
$491$	−3.55948	−0.160637	−0.0803184	+	0.996769i	$0.525594\pi$
$492$	0	0
$493$	−10.1241	−0.455965
$494$	−24.1086	−1.08470
$495$	0	0
$496$	3.71892	0.166984
$497$	−23.4587	−1.05226
$498$	0	0
$499$	41.8778	1.87471	0.937353	−	0.348380i	$-0.113268\pi$
$499$	41.8778	1.87471	0.937353	+	0.348380i	$0.113268\pi$
$500$	0	0
$501$	0	0
$502$	3.08793	0.137821
$503$	30.5737	1.36322	0.681608	−	0.731718i	$-0.261281\pi$
$503$	30.5737	1.36322	0.681608	+	0.731718i	$0.261281\pi$
$504$	0	0
$505$	0	0
$506$	10.8179	0.480915
$507$	0	0
$508$	0.293081	0.0130034
$509$	23.1169	1.02464	0.512319	−	0.858795i	$-0.328787\pi$
$509$	23.1169	1.02464	0.512319	+	0.858795i	$0.328787\pi$
$510$	0	0
$511$	19.2225	0.850354
$512$	24.4416	1.08018
$513$	0	0
$514$	−10.4663	−0.461649
$515$	0	0
$516$	0	0
$517$	−15.0765	−0.663063
$518$	−9.59866	−0.421741
$519$	0	0
$520$	0	0
$521$	21.6691	0.949340	0.474670	−	0.880164i	$-0.342567\pi$
$521$	21.6691	0.949340	0.474670	+	0.880164i	$0.342567\pi$
$522$	0	0
$523$	43.6537	1.90884	0.954421	−	0.298464i	$-0.0964743\pi$
$523$	43.6537	1.90884	0.954421	+	0.298464i	$0.0964743\pi$
$524$	2.19484	0.0958821
$525$	0	0
$526$	1.48261	0.0646450
$527$	3.28332	0.143023
$528$	0	0
$529$	−11.1967	−0.486811
$530$	0	0
$531$	0	0
$532$	2.01275	0.0872639
$533$	−29.2294	−1.26607
$534$	0	0
$535$	0	0
$536$	39.7648	1.71758
$537$	0	0
$538$	5.42800	0.234018
$539$	0.303750	0.0130834
$540$	0	0
$541$	−9.43196	−0.405511	−0.202756	−	0.979229i	$-0.564990\pi$
$541$	−9.43196	−0.405511	−0.202756	+	0.979229i	$0.564990\pi$
$542$	−21.3248	−0.915978
$543$	0	0
$544$	2.44480	0.104820
$545$	0	0
$546$	0	0
$547$	−38.1984	−1.63324	−0.816622	−	0.577173i	$-0.804156\pi$
$547$	−38.1984	−1.63324	−0.816622	+	0.577173i	$0.804156\pi$
$548$	−0.338705	−0.0144687
$549$	0	0
$550$	0	0
$551$	17.6259	0.750888
$552$	0	0
$553$	45.3423	1.92815
$554$	12.8235	0.544820
$555$	0	0
$556$	2.61390	0.110854
$557$	−6.96658	−0.295184	−0.147592	−	0.989048i	$-0.547152\pi$
$557$	−6.96658	−0.295184	−0.147592	+	0.989048i	$0.547152\pi$
$558$	0	0
$559$	−25.4350	−1.07578
$560$	0	0
$561$	0	0
$562$	28.1826	1.18881
$563$	−34.9958	−1.47490	−0.737448	−	0.675404i	$-0.763969\pi$
$563$	−34.9958	−1.47490	−0.737448	+	0.675404i	$0.763969\pi$
$564$	0	0
$565$	0	0
$566$	12.8135	0.538590
$567$	0	0
$568$	25.5956	1.07397
$569$	−24.8518	−1.04184	−0.520920	−	0.853605i	$-0.674411\pi$
$569$	−24.8518	−1.04184	−0.520920	+	0.853605i	$0.674411\pi$
$570$	0	0
$571$	−38.9715	−1.63091	−0.815453	−	0.578823i	$-0.803512\pi$
$571$	−38.9715	−1.63091	−0.815453	+	0.578823i	$0.803512\pi$
$572$	−0.937286	−0.0391899
$573$	0	0
$574$	−34.5757	−1.44316
$575$	0	0
$576$	0	0
$577$	−19.1873	−0.798779	−0.399389	−	0.916781i	$-0.630778\pi$
$577$	−19.1873	−0.798779	−0.399389	+	0.916781i	$0.630778\pi$
$578$	8.50129	0.353607
$579$	0	0
$580$	0	0
$581$	1.17049	0.0485600
$582$	0	0
$583$	12.6002	0.521848
$584$	−20.9735	−0.867891
$585$	0	0
$586$	23.1420	0.955989
$587$	−34.1466	−1.40938	−0.704690	−	0.709516i	$-0.748914\pi$
$587$	−34.1466	−1.40938	−0.704690	+	0.709516i	$0.748914\pi$
$588$	0	0
$589$	−5.71622	−0.235533
$590$	0	0
$591$	0	0
$592$	9.77957	0.401938
$593$	30.7276	1.26183	0.630915	−	0.775852i	$-0.282679\pi$
$593$	30.7276	1.26183	0.630915	+	0.775852i	$0.282679\pi$
$594$	0	0
$595$	0	0
$596$	−1.04413	−0.0427693
$597$	0	0
$598$	14.4899	0.592535
$599$	−7.66883	−0.313340	−0.156670	−	0.987651i	$-0.550076\pi$
$599$	−7.66883	−0.313340	−0.156670	+	0.987651i	$0.550076\pi$
$600$	0	0
$601$	−25.7864	−1.05185	−0.525925	−	0.850531i	$-0.676281\pi$
$601$	−25.7864	−1.05185	−0.525925	+	0.850531i	$0.676281\pi$
$602$	−30.0872	−1.22626
$603$	0	0
$604$	−2.17920	−0.0886705
$605$	0	0
$606$	0	0
$607$	−27.8762	−1.13146	−0.565731	−	0.824590i	$-0.691406\pi$
$607$	−27.8762	−1.13146	−0.565731	+	0.824590i	$0.691406\pi$
$608$	−4.25637	−0.172619
$609$	0	0
$610$	0	0
$611$	−20.1939	−0.816959
$612$	0	0
$613$	−23.6288	−0.954359	−0.477180	−	0.878806i	$-0.658341\pi$
$613$	−23.6288	−0.954359	−0.477180	+	0.878806i	$0.658341\pi$
$614$	24.0150	0.969167
$615$	0	0
$616$	−17.9266	−0.722285
$617$	24.0035	0.966346	0.483173	−	0.875525i	$-0.339484\pi$
$617$	24.0035	0.966346	0.483173	+	0.875525i	$0.339484\pi$
$618$	0	0
$619$	−4.58341	−0.184223	−0.0921114	−	0.995749i	$-0.529362\pi$
$619$	−4.58341	−0.184223	−0.0921114	+	0.995749i	$0.529362\pi$
$620$	0	0
$621$	0	0
$622$	17.8875	0.717224
$623$	−31.0957	−1.24582
$624$	0	0
$625$	0	0
$626$	29.7996	1.19103
$627$	0	0
$628$	−2.15534	−0.0860074
$629$	8.63408	0.344264
$630$	0	0
$631$	29.4443	1.17216	0.586079	−	0.810254i	$-0.300671\pi$
$631$	29.4443	1.17216	0.586079	+	0.810254i	$0.300671\pi$
$632$	−49.4727	−1.96792
$633$	0	0
$634$	31.9211	1.26775
$635$	0	0
$636$	0	0
$637$	0.406852	0.0161201
$638$	−9.70920	−0.384391
$639$	0	0
$640$	0	0
$641$	−39.5698	−1.56291	−0.781457	−	0.623960i	$-0.785523\pi$
$641$	−39.5698	−1.56291	−0.781457	+	0.623960i	$0.785523\pi$
$642$	0	0
$643$	−29.9180	−1.17985	−0.589926	−	0.807457i	$-0.700843\pi$
$643$	−29.9180	−1.17985	−0.589926	+	0.807457i	$0.700843\pi$
$644$	−1.20972	−0.0476696
$645$	0	0
$646$	25.6524	1.00928
$647$	−4.62388	−0.181783	−0.0908917	−	0.995861i	$-0.528972\pi$
$647$	−4.62388	−0.181783	−0.0908917	+	0.995861i	$0.528972\pi$
$648$	0	0
$649$	26.8265	1.05303
$650$	0	0
$651$	0	0
$652$	−1.28251	−0.0502271
$653$	3.14698	0.123151	0.0615754	−	0.998102i	$-0.480388\pi$
$653$	3.14698	0.123151	0.0615754	+	0.998102i	$0.480388\pi$
$654$	0	0
$655$	0	0
$656$	35.2274	1.37540
$657$	0	0
$658$	−23.8875	−0.931234
$659$	−40.4047	−1.57394	−0.786972	−	0.616989i	$-0.788352\pi$
$659$	−40.4047	−1.57394	−0.786972	+	0.616989i	$0.788352\pi$
$660$	0	0
$661$	−27.2285	−1.05907	−0.529533	−	0.848289i	$-0.677633\pi$
$661$	−27.2285	−1.05907	−0.529533	+	0.848289i	$0.677633\pi$
$662$	38.7293	1.50526
$663$	0	0
$664$	−1.27711	−0.0495614
$665$	0	0
$666$	0	0
$667$	−10.5936	−0.410187
$668$	0.582032	0.0225195
$669$	0	0
$670$	0	0
$671$	−16.0862	−0.621001
$672$	0	0
$673$	13.2128	0.509315	0.254658	−	0.967031i	$-0.418037\pi$
$673$	13.2128	0.509315	0.254658	+	0.967031i	$0.418037\pi$
$674$	10.7314	0.413358
$675$	0	0
$676$	0.458620	0.0176392
$677$	−34.1108	−1.31098	−0.655492	−	0.755202i	$-0.727539\pi$
$677$	−34.1108	−1.31098	−0.655492	+	0.755202i	$0.727539\pi$
$678$	0	0
$679$	−1.55464	−0.0596614
$680$	0	0
$681$	0	0
$682$	3.14877	0.120573
$683$	−16.4049	−0.627715	−0.313858	−	0.949470i	$-0.601621\pi$
$683$	−16.4049	−0.627715	−0.313858	+	0.949470i	$0.601621\pi$
$684$	0	0
$685$	0	0
$686$	−25.0696	−0.957160
$687$	0	0
$688$	30.6543	1.16868
$689$	16.8772	0.642968
$690$	0	0
$691$	−12.6442	−0.481009	−0.240505	−	0.970648i	$-0.577313\pi$
$691$	−12.6442	−0.481009	−0.240505	+	0.970648i	$0.577313\pi$
$692$	−2.58064	−0.0981014
$693$	0	0
$694$	36.4550	1.38381
$695$	0	0
$696$	0	0
$697$	31.1012	1.17804
$698$	−32.7001	−1.23772
$699$	0	0
$700$	0	0
$701$	12.5324	0.473343	0.236672	−	0.971590i	$-0.423943\pi$
$701$	12.5324	0.473343	0.236672	+	0.971590i	$0.423943\pi$
$702$	0	0
$703$	−15.0319	−0.566937
$704$	19.4795	0.734162
$705$	0	0
$706$	37.3455	1.40552
$707$	−39.6193	−1.49004
$708$	0	0
$709$	−18.8301	−0.707179	−0.353589	−	0.935401i	$-0.615039\pi$
$709$	−18.8301	−0.707179	−0.353589	+	0.935401i	$0.615039\pi$
$710$	0	0
$711$	0	0
$712$	33.9282	1.27151
$713$	3.43560	0.128664
$714$	0	0
$715$	0	0
$716$	0.139786	0.00522405
$717$	0	0
$718$	20.9268	0.780982
$719$	−10.3328	−0.385348	−0.192674	−	0.981263i	$-0.561716\pi$
$719$	−10.3328	−0.385348	−0.192674	+	0.981263i	$0.561716\pi$
$720$	0	0
$721$	−10.5365	−0.392399
$722$	−18.6913	−0.695617
$723$	0	0
$724$	1.78424	0.0663109
$725$	0	0
$726$	0	0
$727$	−8.64381	−0.320581	−0.160290	−	0.987070i	$-0.551243\pi$
$727$	−8.64381	−0.320581	−0.160290	+	0.987070i	$0.551243\pi$
$728$	−24.0115	−0.889927
$729$	0	0
$730$	0	0
$731$	27.0637	1.00099
$732$	0	0
$733$	−29.4206	−1.08668	−0.543338	−	0.839514i	$-0.682840\pi$
$733$	−29.4206	−1.08668	−0.543338	+	0.839514i	$0.682840\pi$
$734$	17.0823	0.630520
$735$	0	0
$736$	2.55819	0.0942963
$737$	31.4392	1.15808
$738$	0	0
$739$	−46.3000	−1.70317	−0.851587	−	0.524214i	$-0.824359\pi$
$739$	−46.3000	−1.70317	−0.851587	+	0.524214i	$0.824359\pi$
$740$	0	0
$741$	0	0
$742$	19.9641	0.732906
$743$	25.9844	0.953275	0.476637	−	0.879100i	$-0.341856\pi$
$743$	25.9844	0.953275	0.476637	+	0.879100i	$0.341856\pi$
$744$	0	0
$745$	0	0
$746$	35.9034	1.31452
$747$	0	0
$748$	0.997306	0.0364651
$749$	2.53453	0.0926098
$750$	0	0
$751$	−31.5291	−1.15051	−0.575257	−	0.817973i	$-0.695098\pi$
$751$	−31.5291	−1.15051	−0.575257	+	0.817973i	$0.695098\pi$
$752$	24.3378	0.887507
$753$	0	0
$754$	−13.0048	−0.473608
$755$	0	0
$756$	0	0
$757$	−40.0058	−1.45403	−0.727017	−	0.686619i	$-0.759094\pi$
$757$	−40.0058	−1.45403	−0.727017	+	0.686619i	$0.759094\pi$
$758$	−13.3727	−0.485717
$759$	0	0
$760$	0	0
$761$	29.5502	1.07120	0.535598	−	0.844473i	$-0.320086\pi$
$761$	29.5502	1.07120	0.535598	+	0.844473i	$0.320086\pi$
$762$	0	0
$763$	9.43027	0.341399
$764$	0.426970	0.0154472
$765$	0	0
$766$	−20.2100	−0.730216
$767$	35.9324	1.29744
$768$	0	0
$769$	−11.0734	−0.399316	−0.199658	−	0.979866i	$-0.563983\pi$
$769$	−11.0734	−0.399316	−0.199658	+	0.979866i	$0.563983\pi$
$770$	0	0
$771$	0	0
$772$	−0.576041	−0.0207322
$773$	10.0373	0.361018	0.180509	−	0.983573i	$-0.442226\pi$
$773$	10.0373	0.361018	0.180509	+	0.983573i	$0.442226\pi$
$774$	0	0
$775$	0	0
$776$	1.69625	0.0608918
$777$	0	0
$778$	−4.62135	−0.165683
$779$	−54.1469	−1.94001
$780$	0	0
$781$	20.2366	0.724122
$782$	−15.4178	−0.551338
$783$	0	0
$784$	−0.490339	−0.0175121
$785$	0	0
$786$	0	0
$787$	14.7132	0.524468	0.262234	−	0.965004i	$-0.415541\pi$
$787$	14.7132	0.524468	0.262234	+	0.965004i	$0.415541\pi$
$788$	3.10489	0.110607
$789$	0	0
$790$	0	0
$791$	−23.4838	−0.834989
$792$	0	0
$793$	−21.5464	−0.765135
$794$	−38.2753	−1.35834
$795$	0	0
$796$	1.85683	0.0658137
$797$	−47.3422	−1.67695	−0.838473	−	0.544944i	$-0.816551\pi$
$797$	−47.3422	−1.67695	−0.838473	+	0.544944i	$0.816551\pi$
$798$	0	0
$799$	21.4871	0.760158
$800$	0	0
$801$	0	0
$802$	−26.3249	−0.929565
$803$	−16.5823	−0.585176
$804$	0	0
$805$	0	0
$806$	4.21757	0.148558
$807$	0	0
$808$	43.2283	1.52077
$809$	−41.4758	−1.45821	−0.729105	−	0.684402i	$-0.760063\pi$
$809$	−41.4758	−1.45821	−0.729105	+	0.684402i	$0.760063\pi$
$810$	0	0
$811$	3.85504	0.135369	0.0676844	−	0.997707i	$-0.478439\pi$
$811$	3.85504	0.135369	0.0676844	+	0.997707i	$0.478439\pi$
$812$	1.08573	0.0381018
$813$	0	0
$814$	8.28028	0.290224
$815$	0	0
$816$	0	0
$817$	−47.1177	−1.64844
$818$	−6.95252	−0.243089
$819$	0	0
$820$	0	0
$821$	37.7001	1.31574	0.657872	−	0.753130i	$-0.271457\pi$
$821$	37.7001	1.31574	0.657872	+	0.753130i	$0.271457\pi$
$822$	0	0
$823$	−43.3812	−1.51217	−0.756087	−	0.654471i	$-0.772892\pi$
$823$	−43.3812	−1.51217	−0.756087	+	0.654471i	$0.772892\pi$
$824$	11.4963	0.400491
$825$	0	0
$826$	42.5046	1.47893
$827$	17.2935	0.601355	0.300678	−	0.953726i	$-0.402787\pi$
$827$	17.2935	0.601355	0.300678	+	0.953726i	$0.402787\pi$
$828$	0	0
$829$	−48.1238	−1.67141	−0.835705	−	0.549179i	$-0.814941\pi$
$829$	−48.1238	−1.67141	−0.835705	+	0.549179i	$0.814941\pi$
$830$	0	0
$831$	0	0
$832$	26.0915	0.904560
$833$	−0.432905	−0.0149993
$834$	0	0
$835$	0	0
$836$	−1.73630	−0.0600512
$837$	0	0
$838$	−13.7202	−0.473957
$839$	−16.0757	−0.554993	−0.277497	−	0.960727i	$-0.589505\pi$
$839$	−16.0757	−0.554993	−0.277497	+	0.960727i	$0.589505\pi$
$840$	0	0
$841$	−19.4921	−0.672142
$842$	−14.6361	−0.504394
$843$	0	0
$844$	0.733017	0.0252315
$845$	0	0
$846$	0	0
$847$	15.2028	0.522373
$848$	−20.3404	−0.698492
$849$	0	0
$850$	0	0
$851$	9.03454	0.309700
$852$	0	0
$853$	−14.0675	−0.481663	−0.240831	−	0.970567i	$-0.577420\pi$
$853$	−14.0675	−0.481663	−0.240831	+	0.970567i	$0.577420\pi$
$854$	−25.4874	−0.872160
$855$	0	0
$856$	−2.76541	−0.0945197
$857$	−22.7742	−0.777951	−0.388976	−	0.921248i	$-0.627171\pi$
$857$	−22.7742	−0.777951	−0.388976	+	0.921248i	$0.627171\pi$
$858$	0	0
$859$	26.0013	0.887153	0.443576	−	0.896237i	$-0.353709\pi$
$859$	26.0013	0.887153	0.443576	+	0.896237i	$0.353709\pi$
$860$	0	0
$861$	0	0
$862$	18.8771	0.642956
$863$	33.3478	1.13517	0.567586	−	0.823314i	$-0.307877\pi$
$863$	33.3478	1.13517	0.567586	+	0.823314i	$0.307877\pi$
$864$	0	0
$865$	0	0
$866$	43.3383	1.47270
$867$	0	0
$868$	−0.352113	−0.0119515
$869$	−39.1145	−1.32687
$870$	0	0
$871$	42.1107	1.42687
$872$	−10.2893	−0.348440
$873$	0	0
$874$	26.8422	0.907950
$875$	0	0
$876$	0	0
$877$	15.3075	0.516896	0.258448	−	0.966025i	$-0.416789\pi$
$877$	15.3075	0.516896	0.258448	+	0.966025i	$0.416789\pi$
$878$	−22.0708	−0.744855
$879$	0	0
$880$	0	0
$881$	33.5411	1.13003	0.565015	−	0.825081i	$-0.308871\pi$
$881$	33.5411	1.13003	0.565015	+	0.825081i	$0.308871\pi$
$882$	0	0
$883$	23.5127	0.791265	0.395633	−	0.918409i	$-0.370525\pi$
$883$	23.5127	0.791265	0.395633	+	0.918409i	$0.370525\pi$
$884$	1.33582	0.0449286
$885$	0	0
$886$	−11.9843	−0.402620
$887$	41.7310	1.40119	0.700594	−	0.713560i	$-0.252918\pi$
$887$	41.7310	1.40119	0.700594	+	0.713560i	$0.252918\pi$
$888$	0	0
$889$	5.93619	0.199093
$890$	0	0
$891$	0	0
$892$	0.120571	0.00403700
$893$	−37.4088	−1.25184
$894$	0	0
$895$	0	0
$896$	26.8868	0.898224
$897$	0	0
$898$	−13.1608	−0.439183
$899$	−3.08349	−0.102840
$900$	0	0
$901$	−17.9579	−0.598265
$902$	29.8267	0.993121
$903$	0	0
$904$	25.6230	0.852209
$905$	0	0
$906$	0	0
$907$	−0.319266	−0.0106011	−0.00530053	−	0.999986i	$-0.501687\pi$
$907$	−0.319266	−0.0106011	−0.00530053	+	0.999986i	$0.501687\pi$
$908$	−2.56591	−0.0851529
$909$	0	0
$910$	0	0
$911$	−16.8988	−0.559883	−0.279942	−	0.960017i	$-0.590315\pi$
$911$	−16.8988	−0.559883	−0.279942	+	0.960017i	$0.590315\pi$
$912$	0	0
$913$	−1.00972	−0.0334168
$914$	16.7697	0.554691
$915$	0	0
$916$	0.947847	0.0313177
$917$	44.4553	1.46804
$918$	0	0
$919$	34.4372	1.13598	0.567989	−	0.823036i	$-0.307722\pi$
$919$	34.4372	1.13598	0.567989	+	0.823036i	$0.307722\pi$
$920$	0	0
$921$	0	0
$922$	26.6765	0.878544
$923$	27.1056	0.892191
$924$	0	0
$925$	0	0
$926$	−32.2821	−1.06086
$927$	0	0
$928$	−2.29601	−0.0753701
$929$	60.7373	1.99273	0.996363	−	0.0852126i	$-0.0271570\pi$
$929$	60.7373	1.99273	0.996363	+	0.0852126i	$0.0271570\pi$
$930$	0	0
$931$	0.753684	0.0247010
$932$	−2.77864	−0.0910174
$933$	0	0
$934$	−33.6471	−1.10097
$935$	0	0
$936$	0	0
$937$	−35.9108	−1.17316	−0.586578	−	0.809893i	$-0.699525\pi$
$937$	−35.9108	−1.17316	−0.586578	+	0.809893i	$0.699525\pi$
$938$	49.8130	1.62645
$939$	0	0
$940$	0	0
$941$	25.6354	0.835689	0.417845	−	0.908519i	$-0.362786\pi$
$941$	25.6354	0.835689	0.417845	+	0.908519i	$0.362786\pi$
$942$	0	0
$943$	32.5437	1.05977
$944$	−43.3058	−1.40948
$945$	0	0
$946$	25.9547	0.843860
$947$	−28.0792	−0.912450	−0.456225	−	0.889864i	$-0.650799\pi$
$947$	−28.0792	−0.912450	−0.456225	+	0.889864i	$0.650799\pi$
$948$	0	0
$949$	−22.2109	−0.720995
$950$	0	0
$951$	0	0
$952$	25.5491	0.828052
$953$	−39.4284	−1.27721	−0.638605	−	0.769535i	$-0.720488\pi$
$953$	−39.4284	−1.27721	−0.638605	+	0.769535i	$0.720488\pi$
$954$	0	0
$955$	0	0
$956$	0.665475	0.0215230
$957$	0	0
$958$	−18.0529	−0.583262
$959$	−6.86028	−0.221530
$960$	0	0
$961$	1.00000	0.0322581
$962$	11.0909	0.357584
$963$	0	0
$964$	3.19711	0.102972
$965$	0	0
$966$	0	0
$967$	−43.9689	−1.41394	−0.706972	−	0.707241i	$-0.749940\pi$
$967$	−43.9689	−1.41394	−0.706972	+	0.707241i	$0.749940\pi$
$968$	−16.5876	−0.533146
$969$	0	0
$970$	0	0
$971$	20.7897	0.667174	0.333587	−	0.942719i	$-0.391741\pi$
$971$	20.7897	0.667174	0.333587	+	0.942719i	$0.391741\pi$
$972$	0	0
$973$	52.9431	1.69728
$974$	−14.0179	−0.449163
$975$	0	0
$976$	25.9678	0.831208
$977$	5.70313	0.182459	0.0912297	−	0.995830i	$-0.470920\pi$
$977$	5.70313	0.182459	0.0912297	+	0.995830i	$0.470920\pi$
$978$	0	0
$979$	26.8247	0.857320
$980$	0	0
$981$	0	0
$982$	4.86510	0.155252
$983$	−54.5310	−1.73927	−0.869634	−	0.493697i	$-0.835645\pi$
$983$	−54.5310	−1.73927	−0.869634	+	0.493697i	$0.835645\pi$
$984$	0	0
$985$	0	0
$986$	13.8376	0.440679
$987$	0	0
$988$	−2.32566	−0.0739890
$989$	28.3190	0.900491
$990$	0	0
$991$	−24.3033	−0.772021	−0.386011	−	0.922494i	$-0.626147\pi$
$991$	−24.3033	−0.772021	−0.386011	+	0.922494i	$0.626147\pi$
$992$	0.744613	0.0236415
$993$	0	0
$994$	32.0634	1.01699
$995$	0	0
$996$	0	0
$997$	−19.2783	−0.610551	−0.305276	−	0.952264i	$-0.598749\pi$
$997$	−19.2783	−0.610551	−0.305276	+	0.952264i	$0.598749\pi$
$998$	−57.2387	−1.81186
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6975.2.a.bt.1.2		5
3.2	odd	2		6975.2.a.bu.1.4		5
5.4	even	2		1395.2.a.p.1.4	yes	5
15.14	odd	2		1395.2.a.o.1.2	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1395.2.a.o.1.2	✓	5	15.14	odd	2
1395.2.a.p.1.4	yes	5	5.4	even	2
6975.2.a.bt.1.2		5	1.1	even	1	trivial
6975.2.a.bu.1.4		5	3.2	odd	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)

\(f(q)\)	\(=\)	\(q-1.36680 q^{2} -0.131850 q^{4} -2.67055 q^{7} +2.91382 q^{8} +2.30375 q^{11} +3.08572 q^{13} +3.65012 q^{14} -3.71892 q^{16} -3.28332 q^{17} +5.71622 q^{19} -3.14877 q^{22} -3.43560 q^{23} -4.21757 q^{26} +0.352113 q^{28} +3.08349 q^{29} -1.00000 q^{31} -0.744613 q^{32} +4.48765 q^{34} -2.62968 q^{37} -7.81295 q^{38} -9.47249 q^{41} -8.24280 q^{43} -0.303750 q^{44} +4.69579 q^{46} -6.54432 q^{47} +0.131850 q^{49} -0.406852 q^{52} +5.46944 q^{53} -7.78151 q^{56} -4.21452 q^{58} +11.6447 q^{59} -6.98262 q^{61} +1.36680 q^{62} +8.45557 q^{64} +13.6470 q^{67} +0.432905 q^{68} +8.78420 q^{71} -7.19795 q^{73} +3.59426 q^{74} -0.753684 q^{76} -6.15228 q^{77} -16.9786 q^{79} +12.9470 q^{82} -0.438294 q^{83} +11.2663 q^{86} +6.71271 q^{88} +11.6439 q^{89} -8.24057 q^{91} +0.452984 q^{92} +8.94480 q^{94} +0.582140 q^{97} -0.180213 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{2} + 3 q^{4} - 2 q^{7} - 3 q^{8} + 6 q^{11} - 4 q^{13} + 8 q^{14} - 5 q^{16} - 12 q^{17} - 4 q^{19} + 4 q^{22} - 8 q^{23} + 2 q^{26} - 6 q^{28} + 14 q^{29} - 5 q^{31} - 7 q^{32} - 2 q^{34} - 4 q^{37}+ \cdots + 5 q^{98}+O(q^{100}) \) 5 * q - q^2 + 3 * q^4 - 2 * q^7 - 3 * q^8 + 6 * q^11 - 4 * q^13 + 8 * q^14 - 5 * q^16 - 12 * q^17 - 4 * q^19 + 4 * q^22 - 8 * q^23 + 2 * q^26 - 6 * q^28 + 14 * q^29 - 5 * q^31 - 7 * q^32 - 2 * q^34 - 4 * q^37 - 2 * q^38 + 16 * q^41 + 2 * q^43 + 4 * q^44 - 8 * q^46 - 2 * q^47 - 3 * q^49 + 2 * q^52 - 14 * q^53 + 12 * q^56 - 20 * q^58 + 10 * q^59 - 14 * q^61 + q^62 - 5 * q^64 + 2 * q^67 - 26 * q^68 + 12 * q^71 - 10 * q^73 + 10 * q^74 - 32 * q^76 - 26 * q^77 - 12 * q^79 + 6 * q^82 - 12 * q^83 - 10 * q^86 - 14 * q^88 + 14 * q^89 - 16 * q^91 + 14 * q^92 + 4 * q^94 - 8 * q^97 + 5 * q^98

Introduction

L-functions

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