LMFDB - Embedded newform 4275.2.a.bf.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4275,2,Mod(1,4275)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4275, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4275.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4275 = 3^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4275.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$34.1360468641$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.837.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 1 $ x^3 - 6*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1425)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.52892$ of defining polynomial
Character	$\chi$	$=$	4275.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.52892 q^{2} +4.39543 q^{4} +4.92434 q^{7} -6.05784 q^{8} +1.13349 q^{11} -4.00000 q^{13} -12.4533 q^{14} +6.52892 q^{16} -6.79085 q^{17} -1.00000 q^{19} -2.86651 q^{22} +1.92434 q^{23} +10.1157 q^{26} +21.6446 q^{28} +5.00000 q^{29} +5.13349 q^{31} -4.39543 q^{32} +17.1735 q^{34} -9.05784 q^{37} +2.52892 q^{38} -3.86651 q^{41} -4.00000 q^{43} +4.98218 q^{44} -4.86651 q^{46} +4.26698 q^{47} +17.2492 q^{49} -17.5817 q^{52} -13.1157 q^{53} -29.8309 q^{56} -12.6446 q^{58} -9.19133 q^{59} +0.733016 q^{61} -12.9822 q^{62} -1.94216 q^{64} -4.86651 q^{67} -29.8487 q^{68} -11.1913 q^{71} -11.1157 q^{73} +22.9065 q^{74} -4.39543 q^{76} +5.58170 q^{77} -13.7730 q^{79} +9.77808 q^{82} +0.866508 q^{83} +10.1157 q^{86} -6.86651 q^{88} +10.8487 q^{89} -19.6974 q^{91} +8.45831 q^{92} -10.7909 q^{94} +9.32482 q^{97} -43.6217 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 6 q^{4} - 3 q^{8} + 3 q^{11} - 12 q^{13} - 15 q^{14} + 12 q^{16} - 6 q^{17} - 3 q^{19} - 9 q^{22} - 9 q^{23} + 27 q^{28} + 15 q^{29} + 15 q^{31} - 6 q^{32} + 6 q^{34} - 12 q^{37} - 12 q^{41} - 12 q^{43}+ \cdots - 57 q^{98}+O(q^{100}) $ 3 * q + 6 * q^4 - 3 * q^8 + 3 * q^11 - 12 * q^13 - 15 * q^14 + 12 * q^16 - 6 * q^17 - 3 * q^19 - 9 * q^22 - 9 * q^23 + 27 * q^28 + 15 * q^29 + 15 * q^31 - 6 * q^32 + 6 * q^34 - 12 * q^37 - 12 * q^41 - 12 * q^43 - 15 * q^44 - 15 * q^46 + 12 * q^47 + 21 * q^49 - 24 * q^52 - 9 * q^53 - 30 * q^56 - 12 * q^59 + 3 * q^61 - 9 * q^62 - 21 * q^64 - 15 * q^67 - 60 * q^68 - 18 * q^71 - 3 * q^73 + 24 * q^74 - 6 * q^76 - 12 * q^77 + 3 * q^79 - 9 * q^82 + 3 * q^83 - 21 * q^88 + 3 * q^89 + 9 * q^92 - 18 * q^94 + 12 * q^97 - 57 * 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 $

$2$	−2.52892	−1.78822	−0.894108	−	0.447852i	$-0.852189\pi$
$2$	−2.52892	−1.78822	−0.894108	+	0.447852i	$0.852189\pi$
$3$	0	0
$3$	0	0
$4$	4.39543	2.19771
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.92434	1.86123	0.930614	−	0.366003i	$-0.119274\pi$
$7$	4.92434	1.86123	0.930614	+	0.366003i	$0.119274\pi$
$8$	−6.05784	−2.14177
$9$	0	0
$10$	0	0
$11$	1.13349	0.341761	0.170880	−	0.985292i	$-0.445339\pi$
$11$	1.13349	0.341761	0.170880	+	0.985292i	$0.445339\pi$
$12$	0	0
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	−12.4533	−3.32827
$15$	0	0
$16$	6.52892	1.63223
$17$	−6.79085	−1.64702	−0.823512	−	0.567299i	$-0.807988\pi$
$17$	−6.79085	−1.64702	−0.823512	+	0.567299i	$0.807988\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	−2.86651	−0.611142
$23$	1.92434	0.401253	0.200627	−	0.979668i	$-0.435702\pi$
$23$	1.92434	0.401253	0.200627	+	0.979668i	$0.435702\pi$
$24$	0	0
$25$	0	0
$26$	10.1157	1.98385
$27$	0	0
$28$	21.6446	4.09044
$29$	5.00000	0.928477	0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000	0.928477	0.464238	+	0.885710i	$0.346328\pi$
$30$	0	0
$31$	5.13349	0.922002	0.461001	−	0.887400i	$-0.347490\pi$
$31$	5.13349	0.922002	0.461001	+	0.887400i	$0.347490\pi$
$32$	−4.39543	−0.777009
$33$	0	0
$34$	17.1735	2.94523
$35$	0	0
$36$	0	0
$37$	−9.05784	−1.48910	−0.744550	−	0.667567i	$-0.767336\pi$
$37$	−9.05784	−1.48910	−0.744550	+	0.667567i	$0.767336\pi$
$38$	2.52892	0.410245
$39$	0	0
$40$	0	0
$41$	−3.86651	−0.603847	−0.301924	−	0.953332i	$-0.597629\pi$
$41$	−3.86651	−0.603847	−0.301924	+	0.953332i	$0.597629\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	4.98218	0.751092
$45$	0	0
$46$	−4.86651	−0.717527
$47$	4.26698	0.622404	0.311202	−	0.950344i	$-0.399268\pi$
$47$	4.26698	0.622404	0.311202	+	0.950344i	$0.399268\pi$
$48$	0	0
$49$	17.2492	2.46417
$50$	0	0
$51$	0	0
$52$	−17.5817	−2.43814
$53$	−13.1157	−1.80158	−0.900788	−	0.434259i	$-0.857010\pi$
$53$	−13.1157	−1.80158	−0.900788	+	0.434259i	$0.857010\pi$
$54$	0	0
$55$	0	0
$56$	−29.8309	−3.98632
$57$	0	0
$58$	−12.6446	−1.66032
$59$	−9.19133	−1.19661	−0.598304	−	0.801269i	$-0.704159\pi$
$59$	−9.19133	−1.19661	−0.598304	+	0.801269i	$0.704159\pi$
$60$	0	0
$61$	0.733016	0.0938531	0.0469266	−	0.998898i	$-0.485057\pi$
$61$	0.733016	0.0938531	0.0469266	+	0.998898i	$0.485057\pi$
$62$	−12.9822	−1.64874
$63$	0	0
$64$	−1.94216	−0.242771
$65$	0	0
$66$	0	0
$67$	−4.86651	−0.594539	−0.297269	−	0.954794i	$-0.596076\pi$
$67$	−4.86651	−0.594539	−0.297269	+	0.954794i	$0.596076\pi$
$68$	−29.8487	−3.61969
$69$	0	0
$70$	0	0
$71$	−11.1913	−1.32817	−0.664083	−	0.747659i	$-0.731178\pi$
$71$	−11.1913	−1.32817	−0.664083	+	0.747659i	$0.731178\pi$
$72$	0	0
$73$	−11.1157	−1.30099	−0.650495	−	0.759510i	$-0.725439\pi$
$73$	−11.1157	−1.30099	−0.650495	+	0.759510i	$0.725439\pi$
$74$	22.9065	2.66283
$75$	0	0
$76$	−4.39543	−0.504190
$77$	5.58170	0.636094
$78$	0	0
$79$	−13.7730	−1.54959	−0.774794	−	0.632214i	$-0.782146\pi$
$79$	−13.7730	−1.54959	−0.774794	+	0.632214i	$0.782146\pi$
$80$	0	0
$81$	0	0
$82$	9.77808	1.07981
$83$	0.866508	0.0951116	0.0475558	−	0.998869i	$-0.484857\pi$
$83$	0.866508	0.0951116	0.0475558	+	0.998869i	$0.484857\pi$
$84$	0	0
$85$	0	0
$86$	10.1157	1.09080
$87$	0	0
$88$	−6.86651	−0.731972
$89$	10.8487	1.14996	0.574979	−	0.818168i	$-0.305010\pi$
$89$	10.8487	1.14996	0.574979	+	0.818168i	$0.305010\pi$
$90$	0	0
$91$	−19.6974	−2.06485
$92$	8.45831	0.881840
$93$	0	0
$94$	−10.7909	−1.11299
$95$	0	0
$96$	0	0
$97$	9.32482	0.946792	0.473396	−	0.880850i	$-0.343028\pi$
$97$	9.32482	0.946792	0.473396	+	0.880850i	$0.343028\pi$
$98$	−43.6217	−4.40646
$99$	0	0
$100$	0	0
$101$	2.79085	0.277700	0.138850	−	0.990313i	$-0.455659\pi$
$101$	2.79085	0.277700	0.138850	+	0.990313i	$0.455659\pi$
$102$	0	0
$103$	−15.0979	−1.48764	−0.743818	−	0.668382i	$-0.766987\pi$
$103$	−15.0979	−1.48764	−0.743818	+	0.668382i	$0.766987\pi$
$104$	24.2313	2.37608
$105$	0	0
$106$	33.1685	3.22161
$107$	13.0400	1.26063	0.630313	−	0.776341i	$-0.282927\pi$
$107$	13.0400	1.26063	0.630313	+	0.776341i	$0.282927\pi$
$108$	0	0
$109$	10.6395	1.01908	0.509542	−	0.860446i	$-0.329815\pi$
$109$	10.6395	1.01908	0.509542	+	0.860446i	$0.329815\pi$
$110$	0	0
$111$	0	0
$112$	32.1506	3.03795
$113$	−0.332540	−0.0312828	−0.0156414	−	0.999878i	$-0.504979\pi$
$113$	−0.332540	−0.0312828	−0.0156414	+	0.999878i	$0.504979\pi$
$114$	0	0
$115$	0	0
$116$	21.9771	2.04053
$117$	0	0
$118$	23.2441	2.13979
$119$	−33.4405	−3.06548
$120$	0	0
$121$	−9.71520	−0.883200
$122$	−1.85374	−0.167830
$123$	0	0
$124$	22.5639	2.02630
$125$	0	0
$126$	0	0
$127$	−8.45831	−0.750554	−0.375277	−	0.926913i	$-0.622452\pi$
$127$	−8.45831	−0.750554	−0.375277	+	0.926913i	$0.622452\pi$
$128$	13.7024	1.21113
$129$	0	0
$130$	0	0
$131$	−10.9822	−0.959518	−0.479759	−	0.877400i	$-0.659276\pi$
$131$	−10.9822	−0.959518	−0.479759	+	0.877400i	$0.659276\pi$
$132$	0	0
$133$	−4.92434	−0.426995
$134$	12.3070	1.06316
$135$	0	0
$136$	41.1379	3.52754
$137$	4.90652	0.419193	0.209596	−	0.977788i	$-0.432785\pi$
$137$	4.90652	0.419193	0.209596	+	0.977788i	$0.432785\pi$
$138$	0	0
$139$	2.92434	0.248040	0.124020	−	0.992280i	$-0.460421\pi$
$139$	2.92434	0.248040	0.124020	+	0.992280i	$0.460421\pi$
$140$	0	0
$141$	0	0
$142$	28.3019	2.37505
$143$	−4.53397	−0.379149
$144$	0	0
$145$	0	0
$146$	28.1106	2.32645
$147$	0	0
$148$	−39.8130	−3.27261
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	4.67518	0.380461	0.190230	−	0.981739i	$-0.439077\pi$
$151$	4.67518	0.380461	0.190230	+	0.981739i	$0.439077\pi$
$152$	6.05784	0.491355
$153$	0	0
$154$	−14.1157	−1.13747
$155$	0	0
$156$	0	0
$157$	13.4482	1.07328	0.536642	−	0.843810i	$-0.319693\pi$
$157$	13.4482	1.07328	0.536642	+	0.843810i	$0.319693\pi$
$158$	34.8309	2.77100
$159$	0	0
$160$	0	0
$161$	9.47613	0.746824
$162$	0	0
$163$	−21.5740	−1.68980	−0.844902	−	0.534920i	$-0.820342\pi$
$163$	−21.5740	−1.68980	−0.844902	+	0.534920i	$0.820342\pi$
$164$	−16.9950	−1.32708
$165$	0	0
$166$	−2.19133	−0.170080
$167$	−9.07566	−0.702295	−0.351148	−	0.936320i	$-0.614208\pi$
$167$	−9.07566	−0.702295	−0.351148	+	0.936320i	$0.614208\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	0	0
$172$	−17.5817	−1.34059
$173$	6.46603	0.491603	0.245802	−	0.969320i	$-0.420949\pi$
$173$	6.46603	0.491603	0.245802	+	0.969320i	$0.420949\pi$
$174$	0	0
$175$	0	0
$176$	7.40048	0.557832
$177$	0	0
$178$	−27.4354	−2.05637
$179$	11.3070	0.845125	0.422562	−	0.906334i	$-0.361131\pi$
$179$	11.3070	0.845125	0.422562	+	0.906334i	$0.361131\pi$
$180$	0	0
$181$	−22.6496	−1.68353	−0.841767	−	0.539841i	$-0.818484\pi$
$181$	−22.6496	−1.68353	−0.841767	+	0.539841i	$0.818484\pi$
$182$	49.8130	3.69239
$183$	0	0
$184$	−11.6574	−0.859392
$185$	0	0
$186$	0	0
$187$	−7.69738	−0.562888
$188$	18.7552	1.36786
$189$	0	0
$190$	0	0
$191$	−24.2969	−1.75806	−0.879031	−	0.476765i	$-0.841809\pi$
$191$	−24.2969	−1.75806	−0.879031	+	0.476765i	$0.841809\pi$
$192$	0	0
$193$	−6.67518	−0.480490	−0.240245	−	0.970712i	$-0.577228\pi$
$193$	−6.67518	−0.480490	−0.240245	+	0.970712i	$0.577228\pi$
$194$	−23.5817	−1.69307
$195$	0	0
$196$	75.8174	5.41553
$197$	23.2892	1.65929	0.829643	−	0.558295i	$-0.188544\pi$
$197$	23.2892	1.65929	0.829643	+	0.558295i	$0.188544\pi$
$198$	0	0
$199$	25.0400	1.77504	0.887520	−	0.460770i	$-0.152427\pi$
$199$	25.0400	1.77504	0.887520	+	0.460770i	$0.152427\pi$
$200$	0	0
$201$	0	0
$202$	−7.05784	−0.496588
$203$	24.6217	1.72811
$204$	0	0
$205$	0	0
$206$	38.1812	2.66021
$207$	0	0
$208$	−26.1157	−1.81080
$209$	−1.13349	−0.0784053
$210$	0	0
$211$	−11.7730	−0.810489	−0.405244	−	0.914208i	$-0.632814\pi$
$211$	−11.7730	−0.810489	−0.405244	+	0.914208i	$0.632814\pi$
$212$	−57.6490	−3.95935
$213$	0	0
$214$	−32.9771	−2.25427
$215$	0	0
$216$	0	0
$217$	25.2791	1.71606
$218$	−26.9065	−1.82234
$219$	0	0
$220$	0	0
$221$	27.1634	1.82721
$222$	0	0
$223$	3.92434	0.262794	0.131397	−	0.991330i	$-0.458054\pi$
$223$	3.92434	0.262794	0.131397	+	0.991330i	$0.458054\pi$
$224$	−21.6446	−1.44619
$225$	0	0
$226$	0.840967	0.0559403
$227$	−1.60962	−0.106834	−0.0534172	−	0.998572i	$-0.517011\pi$
$227$	−1.60962	−0.106834	−0.0534172	+	0.998572i	$0.517011\pi$
$228$	0	0
$229$	−1.93444	−0.127832	−0.0639158	−	0.997955i	$-0.520359\pi$
$229$	−1.93444	−0.127832	−0.0639158	+	0.997955i	$0.520359\pi$
$230$	0	0
$231$	0	0
$232$	−30.2892	−1.98858
$233$	−1.84869	−0.121112	−0.0605558	−	0.998165i	$-0.519287\pi$
$233$	−1.84869	−0.121112	−0.0605558	+	0.998165i	$0.519287\pi$
$234$	0	0
$235$	0	0
$236$	−40.3998	−2.62980
$237$	0	0
$238$	84.5683	5.48175
$239$	−2.25688	−0.145986	−0.0729929	−	0.997332i	$-0.523255\pi$
$239$	−2.25688	−0.145986	−0.0729929	+	0.997332i	$0.523255\pi$
$240$	0	0
$241$	−19.0578	−1.22762	−0.613812	−	0.789453i	$-0.710365\pi$
$241$	−19.0578	−1.22762	−0.613812	+	0.789453i	$0.710365\pi$
$242$	24.5689	1.57935
$243$	0	0
$244$	3.22192	0.206262
$245$	0	0
$246$	0	0
$247$	4.00000	0.254514
$248$	−31.0979	−1.97472
$249$	0	0
$250$	0	0
$251$	−26.7909	−1.69102	−0.845512	−	0.533957i	$-0.820705\pi$
$251$	−26.7909	−1.69102	−0.845512	+	0.533957i	$0.820705\pi$
$252$	0	0
$253$	2.18123	0.137133
$254$	21.3904	1.34215
$255$	0	0
$256$	−30.7680	−1.92300
$257$	16.3147	1.01768	0.508842	−	0.860860i	$-0.330074\pi$
$257$	16.3147	1.01768	0.508842	+	0.860860i	$0.330074\pi$
$258$	0	0
$259$	−44.6039	−2.77155
$260$	0	0
$261$	0	0
$262$	27.7730	1.71582
$263$	1.92434	0.118660	0.0593301	−	0.998238i	$-0.481104\pi$
$263$	1.92434	0.118660	0.0593301	+	0.998238i	$0.481104\pi$
$264$	0	0
$265$	0	0
$266$	12.4533	0.763558
$267$	0	0
$268$	−21.3904	−1.30663
$269$	−19.5817	−1.19392	−0.596959	−	0.802272i	$-0.703624\pi$
$269$	−19.5817	−1.19392	−0.596959	+	0.802272i	$0.703624\pi$
$270$	0	0
$271$	11.1913	0.679825	0.339912	−	0.940457i	$-0.389603\pi$
$271$	11.1913	0.679825	0.339912	+	0.940457i	$0.389603\pi$
$272$	−44.3369	−2.68832
$273$	0	0
$274$	−12.4082	−0.749607
$275$	0	0
$276$	0	0
$277$	−16.3147	−0.980257	−0.490128	−	0.871650i	$-0.663050\pi$
$277$	−16.3147	−0.980257	−0.490128	+	0.871650i	$0.663050\pi$
$278$	−7.39543	−0.443548
$279$	0	0
$280$	0	0
$281$	2.75084	0.164101	0.0820506	−	0.996628i	$-0.473853\pi$
$281$	2.75084	0.164101	0.0820506	+	0.996628i	$0.473853\pi$
$282$	0	0
$283$	20.2313	1.20263	0.601314	−	0.799013i	$-0.294644\pi$
$283$	20.2313	1.20263	0.601314	+	0.799013i	$0.294644\pi$
$284$	−49.1907	−2.91893
$285$	0	0
$286$	11.4660	0.678001
$287$	−19.0400	−1.12390
$288$	0	0
$289$	29.1157	1.71269
$290$	0	0
$291$	0	0
$292$	−48.8581	−2.85920
$293$	22.9822	1.34263	0.671317	−	0.741171i	$-0.265729\pi$
$293$	22.9822	1.34263	0.671317	+	0.741171i	$0.265729\pi$
$294$	0	0
$295$	0	0
$296$	54.8709	3.18931
$297$	0	0
$298$	0	0
$299$	−7.69738	−0.445151
$300$	0	0
$301$	−19.6974	−1.13534
$302$	−11.8231	−0.680346
$303$	0	0
$304$	−6.52892	−0.374459
$305$	0	0
$306$	0	0
$307$	−8.59952	−0.490801	−0.245400	−	0.969422i	$-0.578919\pi$
$307$	−8.59952	−0.490801	−0.245400	+	0.969422i	$0.578919\pi$
$308$	24.5340	1.39795
$309$	0	0
$310$	0	0
$311$	12.2313	0.693576	0.346788	−	0.937944i	$-0.387272\pi$
$311$	12.2313	0.693576	0.346788	+	0.937944i	$0.387272\pi$
$312$	0	0
$313$	7.51615	0.424838	0.212419	−	0.977179i	$-0.431866\pi$
$313$	7.51615	0.424838	0.212419	+	0.977179i	$0.431866\pi$
$314$	−34.0094	−1.91926
$315$	0	0
$316$	−60.5383	−3.40555
$317$	−26.5817	−1.49298	−0.746489	−	0.665398i	$-0.768262\pi$
$317$	−26.5817	−1.49298	−0.746489	+	0.665398i	$0.768262\pi$
$318$	0	0
$319$	5.66746	0.317317
$320$	0	0
$321$	0	0
$322$	−23.9644	−1.33548
$323$	6.79085	0.377853
$324$	0	0
$325$	0	0
$326$	54.5588	3.02173
$327$	0	0
$328$	23.4227	1.29330
$329$	21.0121	1.15843
$330$	0	0
$331$	19.3648	1.06439	0.532194	−	0.846623i	$-0.321368\pi$
$331$	19.3648	1.06439	0.532194	+	0.846623i	$0.321368\pi$
$332$	3.80867	0.209028
$333$	0	0
$334$	22.9516	1.25586
$335$	0	0
$336$	0	0
$337$	17.8487	0.972280	0.486140	−	0.873881i	$-0.338405\pi$
$337$	17.8487	0.972280	0.486140	+	0.873881i	$0.338405\pi$
$338$	−7.58675	−0.412665
$339$	0	0
$340$	0	0
$341$	5.81877	0.315104
$342$	0	0
$343$	50.4704	2.72515
$344$	24.2313	1.30647
$345$	0	0
$346$	−16.3521	−0.879092
$347$	19.0578	1.02308	0.511539	−	0.859260i	$-0.329076\pi$
$347$	19.0578	1.02308	0.511539	+	0.859260i	$0.329076\pi$
$348$	0	0
$349$	22.9644	1.22925	0.614627	−	0.788818i	$-0.289307\pi$
$349$	22.9644	1.22925	0.614627	+	0.788818i	$0.289307\pi$
$350$	0	0
$351$	0	0
$352$	−4.98218	−0.265551
$353$	−25.5817	−1.36158	−0.680788	−	0.732480i	$-0.738363\pi$
$353$	−25.5817	−1.36158	−0.680788	+	0.732480i	$0.738363\pi$
$354$	0	0
$355$	0	0
$356$	47.6846	2.52728
$357$	0	0
$358$	−28.5945	−1.51126
$359$	27.3248	1.44215	0.721074	−	0.692858i	$-0.243649\pi$
$359$	27.3248	1.44215	0.721074	+	0.692858i	$0.243649\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	57.2791	3.01052
$363$	0	0
$364$	−86.5784	−4.53794
$365$	0	0
$366$	0	0
$367$	20.3827	1.06397	0.531983	−	0.846755i	$-0.321447\pi$
$367$	20.3827	1.06397	0.531983	+	0.846755i	$0.321447\pi$
$368$	12.5639	0.654938
$369$	0	0
$370$	0	0
$371$	−64.5861	−3.35314
$372$	0	0
$373$	−2.52387	−0.130681	−0.0653405	−	0.997863i	$-0.520813\pi$
$373$	−2.52387	−0.130681	−0.0653405	+	0.997863i	$0.520813\pi$
$374$	19.4660	1.00656
$375$	0	0
$376$	−25.8487	−1.33304
$377$	−20.0000	−1.03005
$378$	0	0
$379$	−9.44049	−0.484925	−0.242463	−	0.970161i	$-0.577955\pi$
$379$	−9.44049	−0.484925	−0.242463	+	0.970161i	$0.577955\pi$
$380$	0	0
$381$	0	0
$382$	61.4449	3.14379
$383$	11.1557	0.570029	0.285015	−	0.958523i	$-0.408002\pi$
$383$	11.1557	0.570029	0.285015	+	0.958523i	$0.408002\pi$
$384$	0	0
$385$	0	0
$386$	16.8810	0.859219
$387$	0	0
$388$	40.9866	2.08078
$389$	−28.3827	−1.43906	−0.719529	−	0.694463i	$-0.755642\pi$
$389$	−28.3827	−1.43906	−0.719529	+	0.694463i	$0.755642\pi$
$390$	0	0
$391$	−13.0679	−0.660874
$392$	−104.493	−5.27767
$393$	0	0
$394$	−58.8964	−2.96716
$395$	0	0
$396$	0	0
$397$	1.40048	0.0702879	0.0351439	−	0.999382i	$-0.488811\pi$
$397$	1.40048	0.0702879	0.0351439	+	0.999382i	$0.488811\pi$
$398$	−63.3241	−3.17415
$399$	0	0
$400$	0	0
$401$	−3.51615	−0.175588	−0.0877940	−	0.996139i	$-0.527982\pi$
$401$	−3.51615	−0.175588	−0.0877940	+	0.996139i	$0.527982\pi$
$402$	0	0
$403$	−20.5340	−1.02287
$404$	12.2670	0.610305
$405$	0	0
$406$	−62.2663	−3.09023
$407$	−10.2670	−0.508915
$408$	0	0
$409$	−31.1379	−1.53967	−0.769834	−	0.638244i	$-0.779661\pi$
$409$	−31.1379	−1.53967	−0.769834	+	0.638244i	$0.779661\pi$
$410$	0	0
$411$	0	0
$412$	−66.3615	−3.26940
$413$	−45.2613	−2.22716
$414$	0	0
$415$	0	0
$416$	17.5817	0.862014
$417$	0	0
$418$	2.86651	0.140205
$419$	−20.4983	−1.00141	−0.500704	−	0.865618i	$-0.666926\pi$
$419$	−20.4983	−1.00141	−0.500704	+	0.865618i	$0.666926\pi$
$420$	0	0
$421$	−3.06794	−0.149522	−0.0747610	−	0.997201i	$-0.523819\pi$
$421$	−3.06794	−0.149522	−0.0747610	+	0.997201i	$0.523819\pi$
$422$	29.7730	1.44933
$423$	0	0
$424$	79.4526	3.85856
$425$	0	0
$426$	0	0
$427$	3.60962	0.174682
$428$	57.3164	2.77049
$429$	0	0
$430$	0	0
$431$	−5.34264	−0.257346	−0.128673	−	0.991687i	$-0.541072\pi$
$431$	−5.34264	−0.257346	−0.128673	+	0.991687i	$0.541072\pi$
$432$	0	0
$433$	6.00000	0.288342	0.144171	−	0.989553i	$-0.453949\pi$
$433$	6.00000	0.288342	0.144171	+	0.989553i	$0.453949\pi$
$434$	−63.9287	−3.06868
$435$	0	0
$436$	46.7653	2.23965
$437$	−1.92434	−0.0920539
$438$	0	0
$439$	3.27470	0.156293	0.0781466	−	0.996942i	$-0.475100\pi$
$439$	3.27470	0.156293	0.0781466	+	0.996942i	$0.475100\pi$
$440$	0	0
$441$	0	0
$442$	−68.6940	−3.26744
$443$	−35.8988	−1.70560	−0.852802	−	0.522235i	$-0.825098\pi$
$443$	−35.8988	−1.70560	−0.852802	+	0.522235i	$0.825098\pi$
$444$	0	0
$445$	0	0
$446$	−9.92434	−0.469931
$447$	0	0
$448$	−9.56388	−0.451851
$449$	−1.26698	−0.0597927	−0.0298963	−	0.999553i	$-0.509518\pi$
$449$	−1.26698	−0.0597927	−0.0298963	+	0.999553i	$0.509518\pi$
$450$	0	0
$451$	−4.38266	−0.206371
$452$	−1.46166	−0.0687505
$453$	0	0
$454$	4.07061	0.191043
$455$	0	0
$456$	0	0
$457$	−1.48385	−0.0694117	−0.0347058	−	0.999398i	$-0.511049\pi$
$457$	−1.48385	−0.0694117	−0.0347058	+	0.999398i	$0.511049\pi$
$458$	4.89205	0.228590
$459$	0	0
$460$	0	0
$461$	4.90652	0.228520	0.114260	−	0.993451i	$-0.463550\pi$
$461$	4.90652	0.228520	0.114260	+	0.993451i	$0.463550\pi$
$462$	0	0
$463$	13.8843	0.645259	0.322630	−	0.946525i	$-0.395433\pi$
$463$	13.8843	0.645259	0.322630	+	0.946525i	$0.395433\pi$
$464$	32.6446	1.51549
$465$	0	0
$466$	4.67518	0.216574
$467$	−29.5060	−1.36538	−0.682689	−	0.730709i	$-0.739189\pi$
$467$	−29.5060	−1.36538	−0.682689	+	0.730709i	$0.739189\pi$
$468$	0	0
$469$	−23.9644	−1.10657
$470$	0	0
$471$	0	0
$472$	55.6796	2.56286
$473$	−4.53397	−0.208472
$474$	0	0
$475$	0	0
$476$	−146.985	−6.73706
$477$	0	0
$478$	5.70748	0.261054
$479$	−10.8766	−0.496965	−0.248482	−	0.968636i	$-0.579932\pi$
$479$	−10.8766	−0.496965	−0.248482	+	0.968636i	$0.579932\pi$
$480$	0	0
$481$	36.2313	1.65201
$482$	48.1957	2.19525
$483$	0	0
$484$	−42.7024	−1.94102
$485$	0	0
$486$	0	0
$487$	33.2791	1.50802	0.754010	−	0.656863i	$-0.228117\pi$
$487$	33.2791	1.50802	0.754010	+	0.656863i	$0.228117\pi$
$488$	−4.44049	−0.201012
$489$	0	0
$490$	0	0
$491$	4.90652	0.221428	0.110714	−	0.993852i	$-0.464686\pi$
$491$	4.90652	0.221428	0.110714	+	0.993852i	$0.464686\pi$
$492$	0	0
$493$	−33.9543	−1.52922
$494$	−10.1157	−0.455126
$495$	0	0
$496$	33.5161	1.50492
$497$	−55.1099	−2.47202
$498$	0	0
$499$	2.00772	0.0898779	0.0449390	−	0.998990i	$-0.485691\pi$
$499$	2.00772	0.0898779	0.0449390	+	0.998990i	$0.485691\pi$
$500$	0	0
$501$	0	0
$502$	67.7519	3.02391
$503$	−7.69738	−0.343209	−0.171605	−	0.985166i	$-0.554895\pi$
$503$	−7.69738	−0.343209	−0.171605	+	0.985166i	$0.554895\pi$
$504$	0	0
$505$	0	0
$506$	−5.51615	−0.245223
$507$	0	0
$508$	−37.1779	−1.64950
$509$	−27.1157	−1.20188	−0.600941	−	0.799294i	$-0.705207\pi$
$509$	−27.1157	−1.20188	−0.600941	+	0.799294i	$0.705207\pi$
$510$	0	0
$511$	−54.7374	−2.42144
$512$	50.4049	2.22760
$513$	0	0
$514$	−41.2586	−1.81984
$515$	0	0
$516$	0	0
$517$	4.83659	0.212713
$518$	112.800	4.95613
$519$	0	0
$520$	0	0
$521$	−17.0800	−0.748290	−0.374145	−	0.927370i	$-0.622064\pi$
$521$	−17.0800	−0.748290	−0.374145	+	0.927370i	$0.622064\pi$
$522$	0	0
$523$	−9.70748	−0.424478	−0.212239	−	0.977218i	$-0.568076\pi$
$523$	−9.70748	−0.424478	−0.212239	+	0.977218i	$0.568076\pi$
$524$	−48.2714	−2.10874
$525$	0	0
$526$	−4.86651	−0.212190
$527$	−34.8608	−1.51856
$528$	0	0
$529$	−19.2969	−0.838996
$530$	0	0
$531$	0	0
$532$	−21.6446	−0.938412
$533$	15.4660	0.669908
$534$	0	0
$535$	0	0
$536$	29.4805	1.27336
$537$	0	0
$538$	49.5205	2.13498
$539$	19.5518	0.842155
$540$	0	0
$541$	27.7475	1.19296	0.596479	−	0.802629i	$-0.296566\pi$
$541$	27.7475	1.19296	0.596479	+	0.802629i	$0.296566\pi$
$542$	−28.3019	−1.21567
$543$	0	0
$544$	29.8487	1.27975
$545$	0	0
$546$	0	0
$547$	2.04002	0.0872248	0.0436124	−	0.999049i	$-0.486113\pi$
$547$	2.04002	0.0872248	0.0436124	+	0.999049i	$0.486113\pi$
$548$	21.5663	0.921265
$549$	0	0
$550$	0	0
$551$	−5.00000	−0.213007
$552$	0	0
$553$	−67.8231	−2.88413
$554$	41.2586	1.75291
$555$	0	0
$556$	12.8537	0.545120
$557$	−24.5340	−1.03954	−0.519769	−	0.854307i	$-0.673982\pi$
$557$	−24.5340	−1.03954	−0.519769	+	0.854307i	$0.673982\pi$
$558$	0	0
$559$	16.0000	0.676728
$560$	0	0
$561$	0	0
$562$	−6.95664	−0.293448
$563$	−38.0878	−1.60521	−0.802604	−	0.596513i	$-0.796553\pi$
$563$	−38.0878	−1.60521	−0.802604	+	0.596513i	$0.796553\pi$
$564$	0	0
$565$	0	0
$566$	−51.1634	−2.15056
$567$	0	0
$568$	67.7952	2.84462
$569$	24.4005	1.02292	0.511461	−	0.859307i	$-0.329105\pi$
$569$	24.4005	1.02292	0.511461	+	0.859307i	$0.329105\pi$
$570$	0	0
$571$	−13.0400	−0.545708	−0.272854	−	0.962055i	$-0.587968\pi$
$571$	−13.0400	−0.545708	−0.272854	+	0.962055i	$0.587968\pi$
$572$	−19.9287	−0.833262
$573$	0	0
$574$	48.1506	2.00977
$575$	0	0
$576$	0	0
$577$	13.0979	0.545271	0.272635	−	0.962117i	$-0.412105\pi$
$577$	13.0979	0.545271	0.272635	+	0.962117i	$0.412105\pi$
$578$	−73.6311	−3.06265
$579$	0	0
$580$	0	0
$581$	4.26698	0.177024
$582$	0	0
$583$	−14.8665	−0.615708
$584$	67.3369	2.78642
$585$	0	0
$586$	−58.1200	−2.40092
$587$	−30.4227	−1.25568	−0.627839	−	0.778343i	$-0.716060\pi$
$587$	−30.4227	−1.25568	−0.627839	+	0.778343i	$0.716060\pi$
$588$	0	0
$589$	−5.13349	−0.211522
$590$	0	0
$591$	0	0
$592$	−59.1379	−2.43055
$593$	30.3470	1.24620	0.623101	−	0.782141i	$-0.285872\pi$
$593$	30.3470	1.24620	0.623101	+	0.782141i	$0.285872\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	19.4660	0.796025
$599$	43.3948	1.77306	0.886531	−	0.462670i	$-0.153108\pi$
$599$	43.3948	1.77306	0.886531	+	0.462670i	$0.153108\pi$
$600$	0	0
$601$	7.84869	0.320155	0.160077	−	0.987104i	$-0.448826\pi$
$601$	7.84869	0.320155	0.160077	+	0.987104i	$0.448826\pi$
$602$	49.8130	2.03023
$603$	0	0
$604$	20.5494	0.836144
$605$	0	0
$606$	0	0
$607$	−13.8087	−0.560477	−0.280238	−	0.959930i	$-0.590414\pi$
$607$	−13.8087	−0.560477	−0.280238	+	0.959930i	$0.590414\pi$
$608$	4.39543	0.178258
$609$	0	0
$610$	0	0
$611$	−17.0679	−0.690495
$612$	0	0
$613$	9.98218	0.403176	0.201588	−	0.979470i	$-0.435390\pi$
$613$	9.98218	0.403176	0.201588	+	0.979470i	$0.435390\pi$
$614$	21.7475	0.877657
$615$	0	0
$616$	−33.8130	−1.36237
$617$	0.115672	0.00465677	0.00232839	−	0.999997i	$-0.499259\pi$
$617$	0.115672	0.00465677	0.00232839	+	0.999997i	$0.499259\pi$
$618$	0	0
$619$	−5.45831	−0.219388	−0.109694	−	0.993965i	$-0.534987\pi$
$619$	−5.45831	−0.219388	−0.109694	+	0.993965i	$0.534987\pi$
$620$	0	0
$621$	0	0
$622$	−30.9321	−1.24026
$623$	53.4227	2.14033
$624$	0	0
$625$	0	0
$626$	−19.0077	−0.759701
$627$	0	0
$628$	59.1106	2.35877
$629$	61.5104	2.45258
$630$	0	0
$631$	45.8130	1.82379	0.911894	−	0.410425i	$-0.134620\pi$
$631$	45.8130	1.82379	0.911894	+	0.410425i	$0.134620\pi$
$632$	83.4348	3.31886
$633$	0	0
$634$	67.2229	2.66976
$635$	0	0
$636$	0	0
$637$	−68.9967	−2.73375
$638$	−14.3325	−0.567431
$639$	0	0
$640$	0	0
$641$	37.1634	1.46787	0.733933	−	0.679222i	$-0.237683\pi$
$641$	37.1634	1.46787	0.733933	+	0.679222i	$0.237683\pi$
$642$	0	0
$643$	11.4583	0.451872	0.225936	−	0.974142i	$-0.427456\pi$
$643$	11.4583	0.451872	0.225936	+	0.974142i	$0.427456\pi$
$644$	41.6516	1.64130
$645$	0	0
$646$	−17.1735	−0.675683
$647$	−34.7952	−1.36794	−0.683971	−	0.729509i	$-0.739748\pi$
$647$	−34.7952	−1.36794	−0.683971	+	0.729509i	$0.739748\pi$
$648$	0	0
$649$	−10.4183	−0.408954
$650$	0	0
$651$	0	0
$652$	−94.8268	−3.71371
$653$	25.2791	0.989247	0.494623	−	0.869107i	$-0.335306\pi$
$653$	25.2791	0.989247	0.494623	+	0.869107i	$0.335306\pi$
$654$	0	0
$655$	0	0
$656$	−25.2441	−0.985617
$657$	0	0
$658$	−53.1379	−2.07153
$659$	−47.6261	−1.85525	−0.927625	−	0.373514i	$-0.878153\pi$
$659$	−47.6261	−1.85525	−0.927625	+	0.373514i	$0.878153\pi$
$660$	0	0
$661$	−14.4882	−0.563527	−0.281763	−	0.959484i	$-0.590919\pi$
$661$	−14.4882	−0.563527	−0.281763	+	0.959484i	$0.590919\pi$
$662$	−48.9721	−1.90335
$663$	0	0
$664$	−5.24916	−0.203707
$665$	0	0
$666$	0	0
$667$	9.62172	0.372554
$668$	−39.8914	−1.54344
$669$	0	0
$670$	0	0
$671$	0.830868	0.0320753
$672$	0	0
$673$	−15.6974	−0.605089	−0.302545	−	0.953135i	$-0.597836\pi$
$673$	−15.6974	−0.605089	−0.302545	+	0.953135i	$0.597836\pi$
$674$	−45.1379	−1.73865
$675$	0	0
$676$	13.1863	0.507165
$677$	26.8130	1.03051	0.515255	−	0.857037i	$-0.327697\pi$
$677$	26.8130	1.03051	0.515255	+	0.857037i	$0.327697\pi$
$678$	0	0
$679$	45.9186	1.76219
$680$	0	0
$681$	0	0
$682$	−14.7152	−0.563474
$683$	−36.4704	−1.39550	−0.697751	−	0.716341i	$-0.745816\pi$
$683$	−36.4704	−1.39550	−0.697751	+	0.716341i	$0.745816\pi$
$684$	0	0
$685$	0	0
$686$	−127.636	−4.87315
$687$	0	0
$688$	−26.1157	−0.995651
$689$	52.4627	1.99867
$690$	0	0
$691$	13.1990	0.502115	0.251058	−	0.967972i	$-0.419222\pi$
$691$	13.1990	0.502115	0.251058	+	0.967972i	$0.419222\pi$
$692$	28.4210	1.08040
$693$	0	0
$694$	−48.1957	−1.82948
$695$	0	0
$696$	0	0
$697$	26.2569	0.994550
$698$	−58.0750	−2.19817
$699$	0	0
$700$	0	0
$701$	−37.7075	−1.42419	−0.712096	−	0.702082i	$-0.752254\pi$
$701$	−37.7075	−1.42419	−0.712096	+	0.702082i	$0.752254\pi$
$702$	0	0
$703$	9.05784	0.341623
$704$	−2.20143	−0.0829694
$705$	0	0
$706$	64.6940	2.43479
$707$	13.7431	0.516863
$708$	0	0
$709$	15.9166	0.597761	0.298881	−	0.954290i	$-0.403387\pi$
$709$	15.9166	0.597761	0.298881	+	0.954290i	$0.403387\pi$
$710$	0	0
$711$	0	0
$712$	−65.7196	−2.46295
$713$	9.87860	0.369957
$714$	0	0
$715$	0	0
$716$	49.6991	1.85734
$717$	0	0
$718$	−69.1022	−2.57887
$719$	−1.43612	−0.0535581	−0.0267790	−	0.999641i	$-0.508525\pi$
$719$	−1.43612	−0.0535581	−0.0267790	+	0.999641i	$0.508525\pi$
$720$	0	0
$721$	−74.3470	−2.76883
$722$	−2.52892	−0.0941166
$723$	0	0
$724$	−99.5548	−3.69993
$725$	0	0
$726$	0	0
$727$	−34.3191	−1.27282	−0.636412	−	0.771349i	$-0.719582\pi$
$727$	−34.3191	−1.27282	−0.636412	+	0.771349i	$0.719582\pi$
$728$	119.323	4.42242
$729$	0	0
$730$	0	0
$731$	27.1634	1.00467
$732$	0	0
$733$	−43.3114	−1.59974	−0.799871	−	0.600172i	$-0.795099\pi$
$733$	−43.3114	−1.59974	−0.799871	+	0.600172i	$0.795099\pi$
$734$	−51.5461	−1.90260
$735$	0	0
$736$	−8.45831	−0.311778
$737$	−5.51615	−0.203190
$738$	0	0
$739$	25.4583	0.936499	0.468250	−	0.883596i	$-0.344885\pi$
$739$	25.4583	0.936499	0.468250	+	0.883596i	$0.344885\pi$
$740$	0	0
$741$	0	0
$742$	163.333	5.99614
$743$	−22.2391	−0.815872	−0.407936	−	0.913010i	$-0.633751\pi$
$743$	−22.2391	−0.815872	−0.407936	+	0.913010i	$0.633751\pi$
$744$	0	0
$745$	0	0
$746$	6.38266	0.233686
$747$	0	0
$748$	−33.8332	−1.23707
$749$	64.2135	2.34631
$750$	0	0
$751$	25.0323	0.913441	0.456721	−	0.889610i	$-0.349024\pi$
$751$	25.0323	0.913441	0.456721	+	0.889610i	$0.349024\pi$
$752$	27.8588	1.01591
$753$	0	0
$754$	50.5784	1.84196
$755$	0	0
$756$	0	0
$757$	−36.7330	−1.33508	−0.667542	−	0.744572i	$-0.732654\pi$
$757$	−36.7330	−1.33508	−0.667542	+	0.744572i	$0.732654\pi$
$758$	23.8742	0.867151
$759$	0	0
$760$	0	0
$761$	22.0901	0.800767	0.400383	−	0.916348i	$-0.368877\pi$
$761$	22.0901	0.800767	0.400383	+	0.916348i	$0.368877\pi$
$762$	0	0
$763$	52.3928	1.89675
$764$	−106.795	−3.86372
$765$	0	0
$766$	−28.2118	−1.01933
$767$	36.7653	1.32752
$768$	0	0
$769$	−34.5817	−1.24705	−0.623524	−	0.781804i	$-0.714300\pi$
$769$	−34.5817	−1.24705	−0.623524	+	0.781804i	$0.714300\pi$
$770$	0	0
$771$	0	0
$772$	−29.3403	−1.05598
$773$	28.0622	1.00933	0.504664	−	0.863316i	$-0.331616\pi$
$773$	28.0622	1.00933	0.504664	+	0.863316i	$0.331616\pi$
$774$	0	0
$775$	0	0
$776$	−56.4882	−2.02781
$777$	0	0
$778$	71.7774	2.57334
$779$	3.86651	0.138532
$780$	0	0
$781$	−12.6853	−0.453915
$782$	33.0477	1.18178
$783$	0	0
$784$	112.618	4.02208
$785$	0	0
$786$	0	0
$787$	−7.54169	−0.268832	−0.134416	−	0.990925i	$-0.542916\pi$
$787$	−7.54169	−0.268832	−0.134416	+	0.990925i	$0.542916\pi$
$788$	102.366	3.64663
$789$	0	0
$790$	0	0
$791$	−1.63754	−0.0582243
$792$	0	0
$793$	−2.93206	−0.104121
$794$	−3.54169	−0.125690
$795$	0	0
$796$	110.062	3.90103
$797$	17.4126	0.616785	0.308392	−	0.951259i	$-0.400209\pi$
$797$	17.4126	0.616785	0.308392	+	0.951259i	$0.400209\pi$
$798$	0	0
$799$	−28.9765	−1.02511
$800$	0	0
$801$	0	0
$802$	8.89205	0.313989
$803$	−12.5995	−0.444628
$804$	0	0
$805$	0	0
$806$	51.9287	1.82911
$807$	0	0
$808$	−16.9065	−0.594769
$809$	5.84869	0.205629	0.102814	−	0.994701i	$-0.467215\pi$
$809$	5.84869	0.205629	0.102814	+	0.994701i	$0.467215\pi$
$810$	0	0
$811$	−4.02992	−0.141510	−0.0707548	−	0.997494i	$-0.522541\pi$
$811$	−4.02992	−0.141510	−0.0707548	+	0.997494i	$0.522541\pi$
$812$	108.223	3.79788
$813$	0	0
$814$	25.9644	0.910050
$815$	0	0
$816$	0	0
$817$	4.00000	0.139942
$818$	78.7451	2.75326
$819$	0	0
$820$	0	0
$821$	−3.97446	−0.138710	−0.0693548	−	0.997592i	$-0.522094\pi$
$821$	−3.97446	−0.138710	−0.0693548	+	0.997592i	$0.522094\pi$
$822$	0	0
$823$	−1.07566	−0.0374950	−0.0187475	−	0.999824i	$-0.505968\pi$
$823$	−1.07566	−0.0374950	−0.0187475	+	0.999824i	$0.505968\pi$
$824$	91.4603	3.18617
$825$	0	0
$826$	114.462	3.98264
$827$	12.2313	0.425325	0.212663	−	0.977126i	$-0.431786\pi$
$827$	12.2313	0.425325	0.212663	+	0.977126i	$0.431786\pi$
$828$	0	0
$829$	−48.0444	−1.66865	−0.834325	−	0.551272i	$-0.814143\pi$
$829$	−48.0444	−1.66865	−0.834325	+	0.551272i	$0.814143\pi$
$830$	0	0
$831$	0	0
$832$	7.76866	0.269330
$833$	−117.137	−4.05854
$834$	0	0
$835$	0	0
$836$	−4.98218	−0.172312
$837$	0	0
$838$	51.8386	1.79073
$839$	−41.6695	−1.43859	−0.719295	−	0.694705i	$-0.755535\pi$
$839$	−41.6695	−1.43859	−0.719295	+	0.694705i	$0.755535\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	7.75856	0.267378
$843$	0	0
$844$	−51.7475	−1.78122
$845$	0	0
$846$	0	0
$847$	−47.8410	−1.64384
$848$	−85.6311	−2.94059
$849$	0	0
$850$	0	0
$851$	−17.4304	−0.597506
$852$	0	0
$853$	25.4126	0.870110	0.435055	−	0.900404i	$-0.356729\pi$
$853$	25.4126	0.870110	0.435055	+	0.900404i	$0.356729\pi$
$854$	−9.12844	−0.312369
$855$	0	0
$856$	−78.9943	−2.69997
$857$	−8.09785	−0.276617	−0.138309	−	0.990389i	$-0.544167\pi$
$857$	−8.09785	−0.276617	−0.138309	+	0.990389i	$0.544167\pi$
$858$	0	0
$859$	−45.8208	−1.56338	−0.781692	−	0.623664i	$-0.785643\pi$
$859$	−45.8208	−1.56338	−0.781692	+	0.623664i	$0.785643\pi$
$860$	0	0
$861$	0	0
$862$	13.5111	0.460190
$863$	17.7051	0.602688	0.301344	−	0.953515i	$-0.402565\pi$
$863$	17.7051	0.602688	0.301344	+	0.953515i	$0.402565\pi$
$864$	0	0
$865$	0	0
$866$	−15.1735	−0.515617
$867$	0	0
$868$	111.112	3.77140
$869$	−15.6116	−0.529588
$870$	0	0
$871$	19.4660	0.659581
$872$	−64.4526	−2.18264
$873$	0	0
$874$	4.86651	0.164612
$875$	0	0
$876$	0	0
$877$	12.9166	0.436163	0.218082	−	0.975931i	$-0.430020\pi$
$877$	12.9166	0.436163	0.218082	+	0.975931i	$0.430020\pi$
$878$	−8.28146	−0.279486
$879$	0	0
$880$	0	0
$881$	−14.1157	−0.475569	−0.237785	−	0.971318i	$-0.576421\pi$
$881$	−14.1157	−0.475569	−0.237785	+	0.971318i	$0.576421\pi$
$882$	0	0
$883$	−7.70510	−0.259297	−0.129649	−	0.991560i	$-0.541385\pi$
$883$	−7.70510	−0.259297	−0.129649	+	0.991560i	$0.541385\pi$
$884$	119.395	4.01568
$885$	0	0
$886$	90.7851	3.04999
$887$	34.6294	1.16274	0.581371	−	0.813638i	$-0.302516\pi$
$887$	34.6294	1.16274	0.581371	+	0.813638i	$0.302516\pi$
$888$	0	0
$889$	−41.6516	−1.39695
$890$	0	0
$891$	0	0
$892$	17.2492	0.577545
$893$	−4.26698	−0.142789
$894$	0	0
$895$	0	0
$896$	67.4755	2.25420
$897$	0	0
$898$	3.20410	0.106922
$899$	25.6675	0.856058
$900$	0	0
$901$	89.0666	2.96724
$902$	11.0834	0.369036
$903$	0	0
$904$	2.01448	0.0670004
$905$	0	0
$906$	0	0
$907$	28.0699	0.932047	0.466023	−	0.884772i	$-0.345686\pi$
$907$	28.0699	0.932047	0.466023	+	0.884772i	$0.345686\pi$
$908$	−7.07498	−0.234792
$909$	0	0
$910$	0	0
$911$	6.35474	0.210542	0.105271	−	0.994444i	$-0.466429\pi$
$911$	6.35474	0.210542	0.105271	+	0.994444i	$0.466429\pi$
$912$	0	0
$913$	0.982180	0.0325054
$914$	3.75254	0.124123
$915$	0	0
$916$	−8.50270	−0.280937
$917$	−54.0800	−1.78588
$918$	0	0
$919$	10.8887	0.359185	0.179593	−	0.983741i	$-0.442522\pi$
$919$	10.8887	0.359185	0.179593	+	0.983741i	$0.442522\pi$
$920$	0	0
$921$	0	0
$922$	−12.4082	−0.408642
$923$	44.7653	1.47347
$924$	0	0
$925$	0	0
$926$	−35.1123	−1.15386
$927$	0	0
$928$	−21.9771	−0.721435
$929$	−22.7552	−0.746574	−0.373287	−	0.927716i	$-0.621769\pi$
$929$	−22.7552	−0.746574	−0.373287	+	0.927716i	$0.621769\pi$
$930$	0	0
$931$	−17.2492	−0.565319
$932$	−8.12577	−0.266168
$933$	0	0
$934$	74.6184	2.44159
$935$	0	0
$936$	0	0
$937$	45.9822	1.50217	0.751086	−	0.660204i	$-0.229530\pi$
$937$	45.9822	1.50217	0.751086	+	0.660204i	$0.229530\pi$
$938$	60.6039	1.97879
$939$	0	0
$940$	0	0
$941$	16.5639	0.539967	0.269984	−	0.962865i	$-0.412982\pi$
$941$	16.5639	0.539967	0.269984	+	0.962865i	$0.412982\pi$
$942$	0	0
$943$	−7.44049	−0.242296
$944$	−60.0094	−1.95314
$945$	0	0
$946$	11.4660	0.372793
$947$	−4.53397	−0.147334	−0.0736671	−	0.997283i	$-0.523470\pi$
$947$	−4.53397	−0.147334	−0.0736671	+	0.997283i	$0.523470\pi$
$948$	0	0
$949$	44.4627	1.44332
$950$	0	0
$951$	0	0
$952$	202.577	6.56556
$953$	−36.4304	−1.18010	−0.590048	−	0.807368i	$-0.700891\pi$
$953$	−36.4304	−1.18010	−0.590048	+	0.807368i	$0.700891\pi$
$954$	0	0
$955$	0	0
$956$	−9.91997	−0.320835
$957$	0	0
$958$	27.5060	0.888680
$959$	24.1614	0.780213
$960$	0	0
$961$	−4.64726	−0.149912
$962$	−91.6261	−2.95414
$963$	0	0
$964$	−83.7673	−2.69796
$965$	0	0
$966$	0	0
$967$	−41.0044	−1.31861	−0.659306	−	0.751875i	$-0.729150\pi$
$967$	−41.0044	−1.31861	−0.659306	+	0.751875i	$0.729150\pi$
$968$	58.8531	1.89161
$969$	0	0
$970$	0	0
$971$	−3.58942	−0.115190	−0.0575951	−	0.998340i	$-0.518343\pi$
$971$	−3.58942	−0.115190	−0.0575951	+	0.998340i	$0.518343\pi$
$972$	0	0
$973$	14.4005	0.461658
$974$	−84.1601	−2.69666
$975$	0	0
$976$	4.78580	0.153190
$977$	42.2313	1.35110	0.675550	−	0.737314i	$-0.263906\pi$
$977$	42.2313	1.35110	0.675550	+	0.737314i	$0.263906\pi$
$978$	0	0
$979$	12.2969	0.393011
$980$	0	0
$981$	0	0
$982$	−12.4082	−0.395961
$983$	−36.0800	−1.15077	−0.575387	−	0.817881i	$-0.695149\pi$
$983$	−36.0800	−1.15077	−0.575387	+	0.817881i	$0.695149\pi$
$984$	0	0
$985$	0	0
$986$	85.8675	2.73458
$987$	0	0
$988$	17.5817	0.559349
$989$	−7.69738	−0.244762
$990$	0	0
$991$	19.3648	0.615144	0.307572	−	0.951525i	$-0.400483\pi$
$991$	19.3648	0.615144	0.307572	+	0.951525i	$0.400483\pi$
$992$	−22.5639	−0.716404
$993$	0	0
$994$	139.369	4.42050
$995$	0	0
$996$	0	0
$997$	20.5639	0.651265	0.325632	−	0.945496i	$-0.394423\pi$
$997$	20.5639	0.651265	0.325632	+	0.945496i	$0.394423\pi$
$998$	−5.07736	−0.160721
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4275.2.a.bf.1.1		3
3.2	odd	2		1425.2.a.t.1.3	✓	3
5.4	even	2		4275.2.a.bg.1.3		3
15.2	even	4		1425.2.c.o.799.6		6
15.8	even	4		1425.2.c.o.799.1		6
15.14	odd	2		1425.2.a.w.1.1	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1425.2.a.t.1.3	✓	3	3.2	odd	2
1425.2.a.w.1.1	yes	3	15.14	odd	2
1425.2.c.o.799.1		6	15.8	even	4
1425.2.c.o.799.6		6	15.2	even	4
4275.2.a.bf.1.1		3	1.1	even	1	trivial
4275.2.a.bg.1.3		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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 4275 = 3^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4275.a (trivial)

\(f(q)\)	\(=\)	\(q-2.52892 q^{2} +4.39543 q^{4} +4.92434 q^{7} -6.05784 q^{8} +1.13349 q^{11} -4.00000 q^{13} -12.4533 q^{14} +6.52892 q^{16} -6.79085 q^{17} -1.00000 q^{19} -2.86651 q^{22} +1.92434 q^{23} +10.1157 q^{26} +21.6446 q^{28} +5.00000 q^{29} +5.13349 q^{31} -4.39543 q^{32} +17.1735 q^{34} -9.05784 q^{37} +2.52892 q^{38} -3.86651 q^{41} -4.00000 q^{43} +4.98218 q^{44} -4.86651 q^{46} +4.26698 q^{47} +17.2492 q^{49} -17.5817 q^{52} -13.1157 q^{53} -29.8309 q^{56} -12.6446 q^{58} -9.19133 q^{59} +0.733016 q^{61} -12.9822 q^{62} -1.94216 q^{64} -4.86651 q^{67} -29.8487 q^{68} -11.1913 q^{71} -11.1157 q^{73} +22.9065 q^{74} -4.39543 q^{76} +5.58170 q^{77} -13.7730 q^{79} +9.77808 q^{82} +0.866508 q^{83} +10.1157 q^{86} -6.86651 q^{88} +10.8487 q^{89} -19.6974 q^{91} +8.45831 q^{92} -10.7909 q^{94} +9.32482 q^{97} -43.6217 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 6 q^{4} - 3 q^{8} + 3 q^{11} - 12 q^{13} - 15 q^{14} + 12 q^{16} - 6 q^{17} - 3 q^{19} - 9 q^{22} - 9 q^{23} + 27 q^{28} + 15 q^{29} + 15 q^{31} - 6 q^{32} + 6 q^{34} - 12 q^{37} - 12 q^{41} - 12 q^{43}+ \cdots - 57 q^{98}+O(q^{100}) \) 3 * q + 6 * q^4 - 3 * q^8 + 3 * q^11 - 12 * q^13 - 15 * q^14 + 12 * q^16 - 6 * q^17 - 3 * q^19 - 9 * q^22 - 9 * q^23 + 27 * q^28 + 15 * q^29 + 15 * q^31 - 6 * q^32 + 6 * q^34 - 12 * q^37 - 12 * q^41 - 12 * q^43 - 15 * q^44 - 15 * q^46 + 12 * q^47 + 21 * q^49 - 24 * q^52 - 9 * q^53 - 30 * q^56 - 12 * q^59 + 3 * q^61 - 9 * q^62 - 21 * q^64 - 15 * q^67 - 60 * q^68 - 18 * q^71 - 3 * q^73 + 24 * q^74 - 6 * q^76 - 12 * q^77 + 3 * q^79 - 9 * q^82 + 3 * q^83 - 21 * q^88 + 3 * q^89 + 9 * q^92 - 18 * q^94 + 12 * q^97 - 57 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

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Database

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