LMFDB - Embedded newform 6525.2.a.ca.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6525,2,Mod(1,6525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6525, 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("6525.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6525 = 3^{2} \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6525.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:	$52.1023873189$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 2x^{8} - 12x^{7} + 21x^{6} + 48x^{5} - 68x^{4} - 73x^{3} + 66x^{2} + 40x - 10 $ x^9 - 2x^8 - 12x^7 + 21x^6 + 48x^5 - 68x^4 - 73x^3 + 66x^2 + 40x - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.77820$ of defining polynomial
Character	$\chi$	$=$	6525.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.77820 q^{2} +1.16198 q^{4} +2.84862 q^{7} +1.49016 q^{8} +O(q^{10})$	$q-1.77820 q^{2} +1.16198 q^{4} +2.84862 q^{7} +1.49016 q^{8} -5.35505 q^{11} +2.50643 q^{13} -5.06541 q^{14} -4.97376 q^{16} -6.50704 q^{17} +2.25433 q^{19} +9.52233 q^{22} +0.355939 q^{23} -4.45692 q^{26} +3.31005 q^{28} +1.00000 q^{29} +5.45828 q^{31} +5.86400 q^{32} +11.5708 q^{34} +2.00281 q^{37} -4.00863 q^{38} +5.37258 q^{41} -12.8773 q^{43} -6.22247 q^{44} -0.632929 q^{46} +4.01725 q^{47} +1.11466 q^{49} +2.91242 q^{52} +3.42758 q^{53} +4.24492 q^{56} -1.77820 q^{58} +4.76617 q^{59} -4.29494 q^{61} -9.70588 q^{62} -0.479810 q^{64} +1.09723 q^{67} -7.56105 q^{68} -5.21539 q^{71} +6.78643 q^{73} -3.56138 q^{74} +2.61948 q^{76} -15.2545 q^{77} -1.84943 q^{79} -9.55349 q^{82} -13.7379 q^{83} +22.8984 q^{86} -7.97990 q^{88} +2.56752 q^{89} +7.13987 q^{91} +0.413594 q^{92} -7.14346 q^{94} -2.73855 q^{97} -1.98209 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 2 q^{2} + 10 q^{4} - q^{7} - 9 q^{8}+O(q^{10}) $ 9 * q - 2 * q^2 + 10 * q^4 - q^7 - 9 * q^8	$ 9 q - 2 q^{2} + 10 q^{4} - q^{7} - 9 q^{8} + 2 q^{11} - q^{13} - 3 q^{14} + 4 q^{16} - 12 q^{17} - q^{19} - 3 q^{22} - 16 q^{23} + 6 q^{26} + 4 q^{28} + 9 q^{29} + 5 q^{31} - 20 q^{32} + 3 q^{34} - 30 q^{38} - 10 q^{41} - 3 q^{43} - 13 q^{44} + 4 q^{46} - 26 q^{47} - 8 q^{49} + 9 q^{52} - 22 q^{53} + 22 q^{56} - 2 q^{58} + 4 q^{59} + 7 q^{61} - 28 q^{62} + 9 q^{64} - 5 q^{67} - 39 q^{68} + 10 q^{73} - 34 q^{74} - 2 q^{76} - 34 q^{77} + 10 q^{79} + 8 q^{82} - 46 q^{83} + 28 q^{86} - 2 q^{88} + 4 q^{89} - 21 q^{91} - 20 q^{92} + 5 q^{94} - 7 q^{97} - 51 q^{98}+O(q^{100}) $ 9 * q - 2 * q^2 + 10 * q^4 - q^7 - 9 * q^8 + 2 * q^11 - q^13 - 3 * q^14 + 4 * q^16 - 12 * q^17 - q^19 - 3 * q^22 - 16 * q^23 + 6 * q^26 + 4 * q^28 + 9 * q^29 + 5 * q^31 - 20 * q^32 + 3 * q^34 - 30 * q^38 - 10 * q^41 - 3 * q^43 - 13 * q^44 + 4 * q^46 - 26 * q^47 - 8 * q^49 + 9 * q^52 - 22 * q^53 + 22 * q^56 - 2 * q^58 + 4 * q^59 + 7 * q^61 - 28 * q^62 + 9 * q^64 - 5 * q^67 - 39 * q^68 + 10 * q^73 - 34 * q^74 - 2 * q^76 - 34 * q^77 + 10 * q^79 + 8 * q^82 - 46 * q^83 + 28 * q^86 - 2 * q^88 + 4 * q^89 - 21 * q^91 - 20 * q^92 + 5 * q^94 - 7 * q^97 - 51 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.77820	−1.25737	−0.628687	−	0.777658i	$-0.716407\pi$
$2$	−1.77820	−1.25737	−0.628687	+	0.777658i	$0.716407\pi$
$3$	0	0
$3$	0	0
$4$	1.16198	0.580990
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.84862	1.07668	0.538339	−	0.842728i	$-0.319052\pi$
$7$	2.84862	1.07668	0.538339	+	0.842728i	$0.319052\pi$
$8$	1.49016	0.526852
$9$	0	0
$10$	0	0
$11$	−5.35505	−1.61461	−0.807305	−	0.590135i	$-0.799075\pi$
$11$	−5.35505	−1.61461	−0.807305	+	0.590135i	$0.799075\pi$
$12$	0	0
$13$	2.50643	0.695158	0.347579	−	0.937651i	$-0.387004\pi$
$13$	2.50643	0.695158	0.347579	+	0.937651i	$0.387004\pi$
$14$	−5.06541	−1.35379
$15$	0	0
$16$	−4.97376	−1.24344
$17$	−6.50704	−1.57819	−0.789095	−	0.614271i	$-0.789450\pi$
$17$	−6.50704	−1.57819	−0.789095	+	0.614271i	$0.789450\pi$
$18$	0	0
$19$	2.25433	0.517178	0.258589	−	0.965987i	$-0.416742\pi$
$19$	2.25433	0.517178	0.258589	+	0.965987i	$0.416742\pi$
$20$	0	0
$21$	0	0
$22$	9.52233	2.03017
$23$	0.355939	0.0742184	0.0371092	−	0.999311i	$-0.488185\pi$
$23$	0.355939	0.0742184	0.0371092	+	0.999311i	$0.488185\pi$
$24$	0	0
$25$	0	0
$26$	−4.45692	−0.874074
$27$	0	0
$28$	3.31005	0.625540
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	5.45828	0.980335	0.490168	−	0.871628i	$-0.336936\pi$
$31$	5.45828	0.980335	0.490168	+	0.871628i	$0.336936\pi$
$32$	5.86400	1.03662
$33$	0	0
$34$	11.5708	1.98438
$35$	0	0
$36$	0	0
$37$	2.00281	0.329259	0.164630	−	0.986355i	$-0.447357\pi$
$37$	2.00281	0.329259	0.164630	+	0.986355i	$0.447357\pi$
$38$	−4.00863	−0.650286
$39$	0	0
$40$	0	0
$41$	5.37258	0.839055	0.419528	−	0.907743i	$-0.362196\pi$
$41$	5.37258	0.839055	0.419528	+	0.907743i	$0.362196\pi$
$42$	0	0
$43$	−12.8773	−1.96377	−0.981886	−	0.189472i	$-0.939322\pi$
$43$	−12.8773	−1.96377	−0.981886	+	0.189472i	$0.939322\pi$
$44$	−6.22247	−0.938072
$45$	0	0
$46$	−0.632929	−0.0933203
$47$	4.01725	0.585976	0.292988	−	0.956116i	$-0.405350\pi$
$47$	4.01725	0.585976	0.292988	+	0.956116i	$0.405350\pi$
$48$	0	0
$49$	1.11466	0.159238
$50$	0	0
$51$	0	0
$52$	2.91242	0.403880
$53$	3.42758	0.470814	0.235407	−	0.971897i	$-0.424358\pi$
$53$	3.42758	0.470814	0.235407	+	0.971897i	$0.424358\pi$
$54$	0	0
$55$	0	0
$56$	4.24492	0.567251
$57$	0	0
$58$	−1.77820	−0.233489
$59$	4.76617	0.620503	0.310251	−	0.950655i	$-0.399587\pi$
$59$	4.76617	0.620503	0.310251	+	0.950655i	$0.399587\pi$
$60$	0	0
$61$	−4.29494	−0.549911	−0.274955	−	0.961457i	$-0.588663\pi$
$61$	−4.29494	−0.549911	−0.274955	+	0.961457i	$0.588663\pi$
$62$	−9.70588	−1.23265
$63$	0	0
$64$	−0.479810	−0.0599762
$65$	0	0
$66$	0	0
$67$	1.09723	0.134048	0.0670239	−	0.997751i	$-0.478650\pi$
$67$	1.09723	0.134048	0.0670239	+	0.997751i	$0.478650\pi$
$68$	−7.56105	−0.916913
$69$	0	0
$70$	0	0
$71$	−5.21539	−0.618953	−0.309476	−	0.950907i	$-0.600154\pi$
$71$	−5.21539	−0.618953	−0.309476	+	0.950907i	$0.600154\pi$
$72$	0	0
$73$	6.78643	0.794291	0.397146	−	0.917756i	$-0.370001\pi$
$73$	6.78643	0.794291	0.397146	+	0.917756i	$0.370001\pi$
$74$	−3.56138	−0.414002
$75$	0	0
$76$	2.61948	0.300475
$77$	−15.2545	−1.73842
$78$	0	0
$79$	−1.84943	−0.208077	−0.104038	−	0.994573i	$-0.533176\pi$
$79$	−1.84943	−0.208077	−0.104038	+	0.994573i	$0.533176\pi$
$80$	0	0
$81$	0	0
$82$	−9.55349	−1.05501
$83$	−13.7379	−1.50793	−0.753963	−	0.656917i	$-0.771860\pi$
$83$	−13.7379	−1.50793	−0.753963	+	0.656917i	$0.771860\pi$
$84$	0	0
$85$	0	0
$86$	22.8984	2.46920
$87$	0	0
$88$	−7.97990	−0.850660
$89$	2.56752	0.272157	0.136078	−	0.990698i	$-0.456550\pi$
$89$	2.56752	0.272157	0.136078	+	0.990698i	$0.456550\pi$
$90$	0	0
$91$	7.13987	0.748462
$92$	0.413594	0.0431202
$93$	0	0
$94$	−7.14346	−0.736792
$95$	0	0
$96$	0	0
$97$	−2.73855	−0.278057	−0.139029	−	0.990288i	$-0.544398\pi$
$97$	−2.73855	−0.278057	−0.139029	+	0.990288i	$0.544398\pi$
$98$	−1.98209	−0.200221
$99$	0	0
$100$	0	0
$101$	5.73985	0.571137	0.285568	−	0.958358i	$-0.407818\pi$
$101$	5.73985	0.571137	0.285568	+	0.958358i	$0.407818\pi$
$102$	0	0
$103$	−15.7896	−1.55579	−0.777895	−	0.628394i	$-0.783713\pi$
$103$	−15.7896	−1.55579	−0.777895	+	0.628394i	$0.783713\pi$
$104$	3.73499	0.366246
$105$	0	0
$106$	−6.09491	−0.591990
$107$	−16.5642	−1.60132	−0.800661	−	0.599118i	$-0.795518\pi$
$107$	−16.5642	−1.60132	−0.800661	+	0.599118i	$0.795518\pi$
$108$	0	0
$109$	9.32556	0.893226	0.446613	−	0.894727i	$-0.352630\pi$
$109$	9.32556	0.893226	0.446613	+	0.894727i	$0.352630\pi$
$110$	0	0
$111$	0	0
$112$	−14.1684	−1.33879
$113$	−14.1979	−1.33563	−0.667815	−	0.744327i	$-0.732770\pi$
$113$	−14.1979	−1.33563	−0.667815	+	0.744327i	$0.732770\pi$
$114$	0	0
$115$	0	0
$116$	1.16198	0.107887
$117$	0	0
$118$	−8.47518	−0.780204
$119$	−18.5361	−1.69920
$120$	0	0
$121$	17.6766	1.60696
$122$	7.63724	0.691444
$123$	0	0
$124$	6.34241	0.569565
$125$	0	0
$126$	0	0
$127$	5.36218	0.475816	0.237908	−	0.971288i	$-0.423538\pi$
$127$	5.36218	0.475816	0.237908	+	0.971288i	$0.423538\pi$
$128$	−10.8748	−0.961205
$129$	0	0
$130$	0	0
$131$	−10.8431	−0.947370	−0.473685	−	0.880694i	$-0.657076\pi$
$131$	−10.8431	−0.947370	−0.473685	+	0.880694i	$0.657076\pi$
$132$	0	0
$133$	6.42173	0.556835
$134$	−1.95109	−0.168548
$135$	0	0
$136$	−9.69656	−0.831473
$137$	−7.41637	−0.633623	−0.316812	−	0.948489i	$-0.602612\pi$
$137$	−7.41637	−0.633623	−0.316812	+	0.948489i	$0.602612\pi$
$138$	0	0
$139$	19.6639	1.66787	0.833937	−	0.551859i	$-0.186082\pi$
$139$	19.6639	1.66787	0.833937	+	0.551859i	$0.186082\pi$
$140$	0	0
$141$	0	0
$142$	9.27398	0.778255
$143$	−13.4221	−1.12241
$144$	0	0
$145$	0	0
$146$	−12.0676	−0.998722
$147$	0	0
$148$	2.32722	0.191296
$149$	−14.4485	−1.18367	−0.591834	−	0.806060i	$-0.701596\pi$
$149$	−14.4485	−1.18367	−0.591834	+	0.806060i	$0.701596\pi$
$150$	0	0
$151$	6.69220	0.544604	0.272302	−	0.962212i	$-0.412215\pi$
$151$	6.69220	0.544604	0.272302	+	0.962212i	$0.412215\pi$
$152$	3.35931	0.272476
$153$	0	0
$154$	27.1256	2.18584
$155$	0	0
$156$	0	0
$157$	17.5789	1.40295	0.701473	−	0.712696i	$-0.252526\pi$
$157$	17.5789	1.40295	0.701473	+	0.712696i	$0.252526\pi$
$158$	3.28864	0.261630
$159$	0	0
$160$	0	0
$161$	1.01394	0.0799094
$162$	0	0
$163$	−19.6576	−1.53971	−0.769853	−	0.638222i	$-0.779670\pi$
$163$	−19.6576	−1.53971	−0.769853	+	0.638222i	$0.779670\pi$
$164$	6.24283	0.487483
$165$	0	0
$166$	24.4286	1.89603
$167$	12.2841	0.950569	0.475285	−	0.879832i	$-0.342345\pi$
$167$	12.2841	0.950569	0.475285	+	0.879832i	$0.342345\pi$
$168$	0	0
$169$	−6.71782	−0.516755
$170$	0	0
$171$	0	0
$172$	−14.9632	−1.14093
$173$	12.9440	0.984118	0.492059	−	0.870562i	$-0.336244\pi$
$173$	12.9440	0.984118	0.492059	+	0.870562i	$0.336244\pi$
$174$	0	0
$175$	0	0
$176$	26.6348	2.00767
$177$	0	0
$178$	−4.56555	−0.342203
$179$	6.01888	0.449872	0.224936	−	0.974374i	$-0.427783\pi$
$179$	6.01888	0.449872	0.224936	+	0.974374i	$0.427783\pi$
$180$	0	0
$181$	−19.8734	−1.47718	−0.738591	−	0.674154i	$-0.764508\pi$
$181$	−19.8734	−1.47718	−0.738591	+	0.674154i	$0.764508\pi$
$182$	−12.6961	−0.941097
$183$	0	0
$184$	0.530407	0.0391021
$185$	0	0
$186$	0	0
$187$	34.8456	2.54816
$188$	4.66797	0.340446
$189$	0	0
$190$	0	0
$191$	−3.04577	−0.220384	−0.110192	−	0.993910i	$-0.535147\pi$
$191$	−3.04577	−0.220384	−0.110192	+	0.993910i	$0.535147\pi$
$192$	0	0
$193$	−21.3408	−1.53615	−0.768073	−	0.640362i	$-0.778784\pi$
$193$	−21.3408	−1.53615	−0.768073	+	0.640362i	$0.778784\pi$
$194$	4.86967	0.349622
$195$	0	0
$196$	1.29522	0.0925154
$197$	10.0730	0.717670	0.358835	−	0.933401i	$-0.383174\pi$
$197$	10.0730	0.717670	0.358835	+	0.933401i	$0.383174\pi$
$198$	0	0
$199$	−4.22842	−0.299744	−0.149872	−	0.988705i	$-0.547886\pi$
$199$	−4.22842	−0.299744	−0.149872	+	0.988705i	$0.547886\pi$
$200$	0	0
$201$	0	0
$202$	−10.2066	−0.718133
$203$	2.84862	0.199934
$204$	0	0
$205$	0	0
$206$	28.0769	1.95621
$207$	0	0
$208$	−12.4664	−0.864388
$209$	−12.0720	−0.835040
$210$	0	0
$211$	28.2252	1.94310	0.971550	−	0.236834i	$-0.0761098\pi$
$211$	28.2252	1.94310	0.971550	+	0.236834i	$0.0761098\pi$
$212$	3.98278	0.273539
$213$	0	0
$214$	29.4544	2.01346
$215$	0	0
$216$	0	0
$217$	15.5486	1.05551
$218$	−16.5827	−1.12312
$219$	0	0
$220$	0	0
$221$	−16.3094	−1.09709
$222$	0	0
$223$	8.48452	0.568165	0.284083	−	0.958800i	$-0.408311\pi$
$223$	8.48452	0.568165	0.284083	+	0.958800i	$0.408311\pi$
$224$	16.7043	1.11610
$225$	0	0
$226$	25.2467	1.67939
$227$	14.7872	0.981460	0.490730	−	0.871312i	$-0.336730\pi$
$227$	14.7872	0.981460	0.490730	+	0.871312i	$0.336730\pi$
$228$	0	0
$229$	−21.0993	−1.39428	−0.697141	−	0.716934i	$-0.745545\pi$
$229$	−21.0993	−1.39428	−0.697141	+	0.716934i	$0.745545\pi$
$230$	0	0
$231$	0	0
$232$	1.49016	0.0978340
$233$	−26.0762	−1.70831	−0.854155	−	0.520018i	$-0.825925\pi$
$233$	−26.0762	−1.70831	−0.854155	+	0.520018i	$0.825925\pi$
$234$	0	0
$235$	0	0
$236$	5.53820	0.360506
$237$	0	0
$238$	32.9609	2.13653
$239$	8.17070	0.528519	0.264259	−	0.964452i	$-0.414872\pi$
$239$	8.17070	0.528519	0.264259	+	0.964452i	$0.414872\pi$
$240$	0	0
$241$	−10.5741	−0.681135	−0.340568	−	0.940220i	$-0.610619\pi$
$241$	−10.5741	−0.681135	−0.340568	+	0.940220i	$0.610619\pi$
$242$	−31.4324	−2.02055
$243$	0	0
$244$	−4.99064	−0.319493
$245$	0	0
$246$	0	0
$247$	5.65031	0.359520
$248$	8.13372	0.516492
$249$	0	0
$250$	0	0
$251$	21.3791	1.34944	0.674720	−	0.738074i	$-0.264264\pi$
$251$	21.3791	1.34944	0.674720	+	0.738074i	$0.264264\pi$
$252$	0	0
$253$	−1.90607	−0.119834
$254$	−9.53500	−0.598279
$255$	0	0
$256$	20.2971	1.26857
$257$	6.90806	0.430913	0.215456	−	0.976513i	$-0.430876\pi$
$257$	6.90806	0.430913	0.215456	+	0.976513i	$0.430876\pi$
$258$	0	0
$259$	5.70524	0.354506
$260$	0	0
$261$	0	0
$262$	19.2812	1.19120
$263$	−2.36575	−0.145878	−0.0729391	−	0.997336i	$-0.523238\pi$
$263$	−2.36575	−0.145878	−0.0729391	+	0.997336i	$0.523238\pi$
$264$	0	0
$265$	0	0
$266$	−11.4191	−0.700150
$267$	0	0
$268$	1.27496	0.0778805
$269$	13.0500	0.795670	0.397835	−	0.917457i	$-0.369762\pi$
$269$	13.0500	0.795670	0.397835	+	0.917457i	$0.369762\pi$
$270$	0	0
$271$	−24.2733	−1.47450	−0.737250	−	0.675620i	$-0.763876\pi$
$271$	−24.2733	−1.47450	−0.737250	+	0.675620i	$0.763876\pi$
$272$	32.3645	1.96239
$273$	0	0
$274$	13.1878	0.796701
$275$	0	0
$276$	0	0
$277$	−0.0846766	−0.00508773	−0.00254386	−	0.999997i	$-0.500810\pi$
$277$	−0.0846766	−0.00508773	−0.00254386	+	0.999997i	$0.500810\pi$
$278$	−34.9663	−2.09714
$279$	0	0
$280$	0	0
$281$	8.18399	0.488216	0.244108	−	0.969748i	$-0.421505\pi$
$281$	8.18399	0.488216	0.244108	+	0.969748i	$0.421505\pi$
$282$	0	0
$283$	−10.6021	−0.630227	−0.315114	−	0.949054i	$-0.602043\pi$
$283$	−10.6021	−0.630227	−0.315114	+	0.949054i	$0.602043\pi$
$284$	−6.06018	−0.359605
$285$	0	0
$286$	23.8670	1.41129
$287$	15.3045	0.903393
$288$	0	0
$289$	25.3416	1.49068
$290$	0	0
$291$	0	0
$292$	7.88570	0.461475
$293$	−26.2617	−1.53422	−0.767112	−	0.641513i	$-0.778307\pi$
$293$	−26.2617	−1.53422	−0.767112	+	0.641513i	$0.778307\pi$
$294$	0	0
$295$	0	0
$296$	2.98451	0.173471
$297$	0	0
$298$	25.6923	1.48831
$299$	0.892136	0.0515935
$300$	0	0
$301$	−36.6826	−2.11435
$302$	−11.9000	−0.684771
$303$	0	0
$304$	−11.2125	−0.643080
$305$	0	0
$306$	0	0
$307$	−20.8923	−1.19239	−0.596193	−	0.802841i	$-0.703321\pi$
$307$	−20.8923	−1.19239	−0.596193	+	0.802841i	$0.703321\pi$
$308$	−17.7255	−1.01000
$309$	0	0
$310$	0	0
$311$	−5.25221	−0.297826	−0.148913	−	0.988850i	$-0.547577\pi$
$311$	−5.25221	−0.297826	−0.148913	+	0.988850i	$0.547577\pi$
$312$	0	0
$313$	32.0146	1.80957	0.904787	−	0.425865i	$-0.140030\pi$
$313$	32.0146	1.80957	0.904787	+	0.425865i	$0.140030\pi$
$314$	−31.2587	−1.76403
$315$	0	0
$316$	−2.14900	−0.120891
$317$	−22.1715	−1.24527	−0.622637	−	0.782511i	$-0.713939\pi$
$317$	−22.1715	−1.24527	−0.622637	+	0.782511i	$0.713939\pi$
$318$	0	0
$319$	−5.35505	−0.299825
$320$	0	0
$321$	0	0
$322$	−1.80298	−0.100476
$323$	−14.6690	−0.816205
$324$	0	0
$325$	0	0
$326$	34.9551	1.93599
$327$	0	0
$328$	8.00602	0.442058
$329$	11.4436	0.630908
$330$	0	0
$331$	−31.8652	−1.75147	−0.875734	−	0.482794i	$-0.839622\pi$
$331$	−31.8652	−1.75147	−0.875734	+	0.482794i	$0.839622\pi$
$332$	−15.9631	−0.876090
$333$	0	0
$334$	−21.8435	−1.19522
$335$	0	0
$336$	0	0
$337$	4.68105	0.254993	0.127496	−	0.991839i	$-0.459306\pi$
$337$	4.68105	0.254993	0.127496	+	0.991839i	$0.459306\pi$
$338$	11.9456	0.649755
$339$	0	0
$340$	0	0
$341$	−29.2294	−1.58286
$342$	0	0
$343$	−16.7651	−0.905231
$344$	−19.1893	−1.03462
$345$	0	0
$346$	−23.0170	−1.23740
$347$	10.0153	0.537650	0.268825	−	0.963189i	$-0.413365\pi$
$347$	10.0153	0.537650	0.268825	+	0.963189i	$0.413365\pi$
$348$	0	0
$349$	14.9709	0.801376	0.400688	−	0.916215i	$-0.368771\pi$
$349$	14.9709	0.801376	0.400688	+	0.916215i	$0.368771\pi$
$350$	0	0
$351$	0	0
$352$	−31.4020	−1.67373
$353$	12.0450	0.641092	0.320546	−	0.947233i	$-0.396134\pi$
$353$	12.0450	0.641092	0.320546	+	0.947233i	$0.396134\pi$
$354$	0	0
$355$	0	0
$356$	2.98341	0.158120
$357$	0	0
$358$	−10.7027	−0.565658
$359$	−4.72615	−0.249436	−0.124718	−	0.992192i	$-0.539803\pi$
$359$	−4.72615	−0.249436	−0.124718	+	0.992192i	$0.539803\pi$
$360$	0	0
$361$	−13.9180	−0.732527
$362$	35.3389	1.85737
$363$	0	0
$364$	8.29639	0.434849
$365$	0	0
$366$	0	0
$367$	11.9598	0.624295	0.312148	−	0.950034i	$-0.398952\pi$
$367$	11.9598	0.624295	0.312148	+	0.950034i	$0.398952\pi$
$368$	−1.77036	−0.0922862
$369$	0	0
$370$	0	0
$371$	9.76389	0.506916
$372$	0	0
$373$	8.31719	0.430648	0.215324	−	0.976543i	$-0.430919\pi$
$373$	8.31719	0.430648	0.215324	+	0.976543i	$0.430919\pi$
$374$	−61.9622	−3.20399
$375$	0	0
$376$	5.98636	0.308723
$377$	2.50643	0.129088
$378$	0	0
$379$	−24.5715	−1.26215	−0.631075	−	0.775721i	$-0.717386\pi$
$379$	−24.5715	−1.26215	−0.631075	+	0.775721i	$0.717386\pi$
$380$	0	0
$381$	0	0
$382$	5.41598	0.277106
$383$	−34.0260	−1.73865	−0.869325	−	0.494241i	$-0.835446\pi$
$383$	−34.0260	−1.73865	−0.869325	+	0.494241i	$0.835446\pi$
$384$	0	0
$385$	0	0
$386$	37.9482	1.93151
$387$	0	0
$388$	−3.18214	−0.161549
$389$	−25.3691	−1.28626	−0.643131	−	0.765756i	$-0.722365\pi$
$389$	−25.3691	−1.28626	−0.643131	+	0.765756i	$0.722365\pi$
$390$	0	0
$391$	−2.31611	−0.117131
$392$	1.66103	0.0838947
$393$	0	0
$394$	−17.9117	−0.902379
$395$	0	0
$396$	0	0
$397$	−16.7222	−0.839265	−0.419632	−	0.907694i	$-0.637841\pi$
$397$	−16.7222	−0.839265	−0.419632	+	0.907694i	$0.637841\pi$
$398$	7.51895	0.376891
$399$	0	0
$400$	0	0
$401$	−8.01416	−0.400208	−0.200104	−	0.979775i	$-0.564128\pi$
$401$	−8.01416	−0.400208	−0.200104	+	0.979775i	$0.564128\pi$
$402$	0	0
$403$	13.6808	0.681488
$404$	6.66960	0.331825
$405$	0	0
$406$	−5.06541	−0.251392
$407$	−10.7251	−0.531625
$408$	0	0
$409$	−19.4335	−0.960924	−0.480462	−	0.877016i	$-0.659531\pi$
$409$	−19.4335	−0.960924	−0.480462	+	0.877016i	$0.659531\pi$
$410$	0	0
$411$	0	0
$412$	−18.3471	−0.903899
$413$	13.5770	0.668082
$414$	0	0
$415$	0	0
$416$	14.6977	0.720613
$417$	0	0
$418$	21.4664	1.04996
$419$	6.92370	0.338245	0.169122	−	0.985595i	$-0.445907\pi$
$419$	6.92370	0.338245	0.169122	+	0.985595i	$0.445907\pi$
$420$	0	0
$421$	−12.9872	−0.632956	−0.316478	−	0.948600i	$-0.602500\pi$
$421$	−12.9872	−0.632956	−0.316478	+	0.948600i	$0.602500\pi$
$422$	−50.1899	−2.44320
$423$	0	0
$424$	5.10766	0.248050
$425$	0	0
$426$	0	0
$427$	−12.2347	−0.592077
$428$	−19.2473	−0.930352
$429$	0	0
$430$	0	0
$431$	39.6752	1.91109	0.955543	−	0.294851i	$-0.0952701\pi$
$431$	39.6752	1.91109	0.955543	+	0.294851i	$0.0952701\pi$
$432$	0	0
$433$	−4.93087	−0.236962	−0.118481	−	0.992956i	$-0.537803\pi$
$433$	−4.93087	−0.236962	−0.118481	+	0.992956i	$0.537803\pi$
$434$	−27.6484	−1.32717
$435$	0	0
$436$	10.8361	0.518956
$437$	0.802403	0.0383841
$438$	0	0
$439$	22.1458	1.05696	0.528481	−	0.848945i	$-0.322762\pi$
$439$	22.1458	1.05696	0.528481	+	0.848945i	$0.322762\pi$
$440$	0	0
$441$	0	0
$442$	29.0014	1.37945
$443$	−1.21156	−0.0575631	−0.0287815	−	0.999586i	$-0.509163\pi$
$443$	−1.21156	−0.0575631	−0.0287815	+	0.999586i	$0.509163\pi$
$444$	0	0
$445$	0	0
$446$	−15.0871	−0.714396
$447$	0	0
$448$	−1.36680	−0.0645751
$449$	−5.22771	−0.246711	−0.123355	−	0.992363i	$-0.539365\pi$
$449$	−5.22771	−0.246711	−0.123355	+	0.992363i	$0.539365\pi$
$450$	0	0
$451$	−28.7704	−1.35475
$452$	−16.4977	−0.775988
$453$	0	0
$454$	−26.2945	−1.23406
$455$	0	0
$456$	0	0
$457$	16.9064	0.790849	0.395425	−	0.918498i	$-0.370597\pi$
$457$	16.9064	0.790849	0.395425	+	0.918498i	$0.370597\pi$
$458$	37.5187	1.75313
$459$	0	0
$460$	0	0
$461$	7.79634	0.363112	0.181556	−	0.983381i	$-0.441887\pi$
$461$	7.79634	0.363112	0.181556	+	0.983381i	$0.441887\pi$
$462$	0	0
$463$	−22.3232	−1.03745	−0.518724	−	0.854942i	$-0.673593\pi$
$463$	−22.3232	−1.03745	−0.518724	+	0.854942i	$0.673593\pi$
$464$	−4.97376	−0.230901
$465$	0	0
$466$	46.3686	2.14799
$467$	−8.19178	−0.379070	−0.189535	−	0.981874i	$-0.560698\pi$
$467$	−8.19178	−0.379070	−0.189535	+	0.981874i	$0.560698\pi$
$468$	0	0
$469$	3.12559	0.144327
$470$	0	0
$471$	0	0
$472$	7.10237	0.326913
$473$	68.9587	3.17072
$474$	0	0
$475$	0	0
$476$	−21.5386	−0.987220
$477$	0	0
$478$	−14.5291	−0.664546
$479$	−17.4830	−0.798817	−0.399409	−	0.916773i	$-0.630785\pi$
$479$	−17.4830	−0.798817	−0.399409	+	0.916773i	$0.630785\pi$
$480$	0	0
$481$	5.01989	0.228887
$482$	18.8028	0.856442
$483$	0	0
$484$	20.5398	0.933630
$485$	0	0
$486$	0	0
$487$	−23.2620	−1.05410	−0.527051	−	0.849833i	$-0.676702\pi$
$487$	−23.2620	−1.05410	−0.527051	+	0.849833i	$0.676702\pi$
$488$	−6.40016	−0.289722
$489$	0	0
$490$	0	0
$491$	−34.4941	−1.55670	−0.778348	−	0.627833i	$-0.783942\pi$
$491$	−34.4941	−1.55670	−0.778348	+	0.627833i	$0.783942\pi$
$492$	0	0
$493$	−6.50704	−0.293062
$494$	−10.0474	−0.452052
$495$	0	0
$496$	−27.1482	−1.21899
$497$	−14.8567	−0.666413
$498$	0	0
$499$	12.8752	0.576372	0.288186	−	0.957574i	$-0.406948\pi$
$499$	12.8752	0.576372	0.288186	+	0.957574i	$0.406948\pi$
$500$	0	0
$501$	0	0
$502$	−38.0163	−1.69675
$503$	−6.19856	−0.276380	−0.138190	−	0.990406i	$-0.544128\pi$
$503$	−6.19856	−0.276380	−0.138190	+	0.990406i	$0.544128\pi$
$504$	0	0
$505$	0	0
$506$	3.38937	0.150676
$507$	0	0
$508$	6.23074	0.276444
$509$	29.6558	1.31447	0.657235	−	0.753686i	$-0.271726\pi$
$509$	29.6558	1.31447	0.657235	+	0.753686i	$0.271726\pi$
$510$	0	0
$511$	19.3320	0.855197
$512$	−14.3427	−0.633863
$513$	0	0
$514$	−12.2839	−0.541819
$515$	0	0
$516$	0	0
$517$	−21.5126	−0.946123
$518$	−10.1450	−0.445747
$519$	0	0
$520$	0	0
$521$	31.2676	1.36986	0.684929	−	0.728610i	$-0.259833\pi$
$521$	31.2676	1.36986	0.684929	+	0.728610i	$0.259833\pi$
$522$	0	0
$523$	−18.7657	−0.820567	−0.410284	−	0.911958i	$-0.634570\pi$
$523$	−18.7657	−0.820567	−0.410284	+	0.911958i	$0.634570\pi$
$524$	−12.5995	−0.550413
$525$	0	0
$526$	4.20676	0.183423
$527$	−35.5172	−1.54716
$528$	0	0
$529$	−22.8733	−0.994492
$530$	0	0
$531$	0	0
$532$	7.46192	0.323515
$533$	13.4660	0.583276
$534$	0	0
$535$	0	0
$536$	1.63505	0.0706234
$537$	0	0
$538$	−23.2054	−1.00045
$539$	−5.96908	−0.257106
$540$	0	0
$541$	−32.7471	−1.40791	−0.703955	−	0.710245i	$-0.748584\pi$
$541$	−32.7471	−1.40791	−0.703955	+	0.710245i	$0.748584\pi$
$542$	43.1627	1.85400
$543$	0	0
$544$	−38.1573	−1.63598
$545$	0	0
$546$	0	0
$547$	−41.0014	−1.75309	−0.876547	−	0.481316i	$-0.840159\pi$
$547$	−41.0014	−1.75309	−0.876547	+	0.481316i	$0.840159\pi$
$548$	−8.61767	−0.368129
$549$	0	0
$550$	0	0
$551$	2.25433	0.0960375
$552$	0	0
$553$	−5.26832	−0.224032
$554$	0.150572	0.00639718
$555$	0	0
$556$	22.8491	0.969018
$557$	−32.8805	−1.39319	−0.696596	−	0.717463i	$-0.745303\pi$
$557$	−32.8805	−1.39319	−0.696596	+	0.717463i	$0.745303\pi$
$558$	0	0
$559$	−32.2761	−1.36513
$560$	0	0
$561$	0	0
$562$	−14.5527	−0.613870
$563$	−19.8934	−0.838407	−0.419204	−	0.907892i	$-0.637691\pi$
$563$	−19.8934	−0.838407	−0.419204	+	0.907892i	$0.637691\pi$
$564$	0	0
$565$	0	0
$566$	18.8525	0.792432
$567$	0	0
$568$	−7.77178	−0.326097
$569$	1.72158	0.0721724	0.0360862	−	0.999349i	$-0.488511\pi$
$569$	1.72158	0.0721724	0.0360862	+	0.999349i	$0.488511\pi$
$570$	0	0
$571$	26.4954	1.10880	0.554399	−	0.832251i	$-0.312948\pi$
$571$	26.4954	1.10880	0.554399	+	0.832251i	$0.312948\pi$
$572$	−15.5962	−0.652108
$573$	0	0
$574$	−27.2143	−1.13590
$575$	0	0
$576$	0	0
$577$	40.8791	1.70182	0.850909	−	0.525314i	$-0.176052\pi$
$577$	40.8791	1.70182	0.850909	+	0.525314i	$0.176052\pi$
$578$	−45.0623	−1.87435
$579$	0	0
$580$	0	0
$581$	−39.1340	−1.62355
$582$	0	0
$583$	−18.3549	−0.760181
$584$	10.1129	0.418474
$585$	0	0
$586$	46.6984	1.92909
$587$	−40.0943	−1.65487	−0.827434	−	0.561563i	$-0.810200\pi$
$587$	−40.0943	−1.65487	−0.827434	+	0.561563i	$0.810200\pi$
$588$	0	0
$589$	12.3047	0.507008
$590$	0	0
$591$	0	0
$592$	−9.96148	−0.409414
$593$	−7.94061	−0.326082	−0.163041	−	0.986619i	$-0.552130\pi$
$593$	−7.94061	−0.326082	−0.163041	+	0.986619i	$0.552130\pi$
$594$	0	0
$595$	0	0
$596$	−16.7889	−0.687700
$597$	0	0
$598$	−1.58639	−0.0648724
$599$	37.2143	1.52054	0.760268	−	0.649609i	$-0.225068\pi$
$599$	37.2143	1.52054	0.760268	+	0.649609i	$0.225068\pi$
$600$	0	0
$601$	−9.87927	−0.402984	−0.201492	−	0.979490i	$-0.564579\pi$
$601$	−9.87927	−0.402984	−0.201492	+	0.979490i	$0.564579\pi$
$602$	65.2289	2.65853
$603$	0	0
$604$	7.77621	0.316409
$605$	0	0
$606$	0	0
$607$	−3.95850	−0.160670	−0.0803352	−	0.996768i	$-0.525599\pi$
$607$	−3.95850	−0.160670	−0.0803352	+	0.996768i	$0.525599\pi$
$608$	13.2194	0.536116
$609$	0	0
$610$	0	0
$611$	10.0690	0.407346
$612$	0	0
$613$	19.8179	0.800436	0.400218	−	0.916420i	$-0.368934\pi$
$613$	19.8179	0.800436	0.400218	+	0.916420i	$0.368934\pi$
$614$	37.1506	1.49927
$615$	0	0
$616$	−22.7317	−0.915888
$617$	8.70648	0.350510	0.175255	−	0.984523i	$-0.443925\pi$
$617$	8.70648	0.350510	0.175255	+	0.984523i	$0.443925\pi$
$618$	0	0
$619$	−18.3012	−0.735589	−0.367794	−	0.929907i	$-0.619887\pi$
$619$	−18.3012	−0.735589	−0.367794	+	0.929907i	$0.619887\pi$
$620$	0	0
$621$	0	0
$622$	9.33946	0.374478
$623$	7.31390	0.293025
$624$	0	0
$625$	0	0
$626$	−56.9283	−2.27531
$627$	0	0
$628$	20.4263	0.815098
$629$	−13.0323	−0.519633
$630$	0	0
$631$	−48.9315	−1.94793	−0.973966	−	0.226693i	$-0.927209\pi$
$631$	−48.9315	−1.94793	−0.973966	+	0.226693i	$0.927209\pi$
$632$	−2.75595	−0.109626
$633$	0	0
$634$	39.4252	1.56578
$635$	0	0
$636$	0	0
$637$	2.79382	0.110695
$638$	9.52233	0.376993
$639$	0	0
$640$	0	0
$641$	5.22083	0.206210	0.103105	−	0.994670i	$-0.467122\pi$
$641$	5.22083	0.206210	0.103105	+	0.994670i	$0.467122\pi$
$642$	0	0
$643$	25.8165	1.01810	0.509051	−	0.860736i	$-0.329996\pi$
$643$	25.8165	1.01810	0.509051	+	0.860736i	$0.329996\pi$
$644$	1.17817	0.0464266
$645$	0	0
$646$	26.0844	1.02628
$647$	−1.68358	−0.0661885	−0.0330943	−	0.999452i	$-0.510536\pi$
$647$	−1.68358	−0.0661885	−0.0330943	+	0.999452i	$0.510536\pi$
$648$	0	0
$649$	−25.5231	−1.00187
$650$	0	0
$651$	0	0
$652$	−22.8418	−0.894553
$653$	−26.5807	−1.04018	−0.520091	−	0.854111i	$-0.674102\pi$
$653$	−26.5807	−1.04018	−0.520091	+	0.854111i	$0.674102\pi$
$654$	0	0
$655$	0	0
$656$	−26.7219	−1.04332
$657$	0	0
$658$	−20.3490	−0.793288
$659$	5.88644	0.229303	0.114652	−	0.993406i	$-0.463425\pi$
$659$	5.88644	0.229303	0.114652	+	0.993406i	$0.463425\pi$
$660$	0	0
$661$	21.3474	0.830319	0.415160	−	0.909749i	$-0.363726\pi$
$661$	21.3474	0.830319	0.415160	+	0.909749i	$0.363726\pi$
$662$	56.6625	2.20225
$663$	0	0
$664$	−20.4717	−0.794454
$665$	0	0
$666$	0	0
$667$	0.355939	0.0137820
$668$	14.2738	0.552271
$669$	0	0
$670$	0	0
$671$	22.9996	0.887891
$672$	0	0
$673$	20.3306	0.783688	0.391844	−	0.920032i	$-0.371837\pi$
$673$	20.3306	0.783688	0.391844	+	0.920032i	$0.371837\pi$
$674$	−8.32382	−0.320622
$675$	0	0
$676$	−7.80597	−0.300230
$677$	−27.0573	−1.03990	−0.519949	−	0.854197i	$-0.674049\pi$
$677$	−27.0573	−1.03990	−0.519949	+	0.854197i	$0.674049\pi$
$678$	0	0
$679$	−7.80110	−0.299379
$680$	0	0
$681$	0	0
$682$	51.9755	1.99025
$683$	−14.8635	−0.568737	−0.284369	−	0.958715i	$-0.591784\pi$
$683$	−14.8635	−0.568737	−0.284369	+	0.958715i	$0.591784\pi$
$684$	0	0
$685$	0	0
$686$	29.8117	1.13821
$687$	0	0
$688$	64.0487	2.44183
$689$	8.59099	0.327291
$690$	0	0
$691$	−3.66787	−0.139532	−0.0697662	−	0.997563i	$-0.522225\pi$
$691$	−3.66787	−0.139532	−0.0697662	+	0.997563i	$0.522225\pi$
$692$	15.0407	0.571763
$693$	0	0
$694$	−17.8092	−0.676028
$695$	0	0
$696$	0	0
$697$	−34.9596	−1.32419
$698$	−26.6212	−1.00763
$699$	0	0
$700$	0	0
$701$	−39.0313	−1.47419	−0.737097	−	0.675787i	$-0.763804\pi$
$701$	−39.0313	−1.47419	−0.737097	+	0.675787i	$0.763804\pi$
$702$	0	0
$703$	4.51498	0.170286
$704$	2.56941	0.0968382
$705$	0	0
$706$	−21.4184	−0.806092
$707$	16.3507	0.614931
$708$	0	0
$709$	−36.9135	−1.38632	−0.693158	−	0.720785i	$-0.743781\pi$
$709$	−36.9135	−1.38632	−0.693158	+	0.720785i	$0.743781\pi$
$710$	0	0
$711$	0	0
$712$	3.82602	0.143386
$713$	1.94281	0.0727589
$714$	0	0
$715$	0	0
$716$	6.99382	0.261371
$717$	0	0
$718$	8.40401	0.313635
$719$	41.9646	1.56502	0.782508	−	0.622640i	$-0.213940\pi$
$719$	41.9646	1.56502	0.782508	+	0.622640i	$0.213940\pi$
$720$	0	0
$721$	−44.9785	−1.67509
$722$	24.7489	0.921060
$723$	0	0
$724$	−23.0926	−0.858228
$725$	0	0
$726$	0	0
$727$	37.0774	1.37513	0.687563	−	0.726125i	$-0.258680\pi$
$727$	37.0774	1.37513	0.687563	+	0.726125i	$0.258680\pi$
$728$	10.6396	0.394329
$729$	0	0
$730$	0	0
$731$	83.7932	3.09920
$732$	0	0
$733$	−6.29483	−0.232505	−0.116252	−	0.993220i	$-0.537088\pi$
$733$	−6.29483	−0.232505	−0.116252	+	0.993220i	$0.537088\pi$
$734$	−21.2668	−0.784973
$735$	0	0
$736$	2.08723	0.0769361
$737$	−5.87572	−0.216435
$738$	0	0
$739$	−34.6989	−1.27642	−0.638210	−	0.769862i	$-0.720325\pi$
$739$	−34.6989	−1.27642	−0.638210	+	0.769862i	$0.720325\pi$
$740$	0	0
$741$	0	0
$742$	−17.3621	−0.637383
$743$	−10.8648	−0.398590	−0.199295	−	0.979940i	$-0.563865\pi$
$743$	−10.8648	−0.398590	−0.199295	+	0.979940i	$0.563865\pi$
$744$	0	0
$745$	0	0
$746$	−14.7896	−0.541485
$747$	0	0
$748$	40.4898	1.48046
$749$	−47.1852	−1.72411
$750$	0	0
$751$	6.02497	0.219854	0.109927	−	0.993940i	$-0.464938\pi$
$751$	6.02497	0.219854	0.109927	+	0.993940i	$0.464938\pi$
$752$	−19.9809	−0.728627
$753$	0	0
$754$	−4.45692	−0.162311
$755$	0	0
$756$	0	0
$757$	−34.1212	−1.24016	−0.620078	−	0.784540i	$-0.712899\pi$
$757$	−34.1212	−1.24016	−0.620078	+	0.784540i	$0.712899\pi$
$758$	43.6929	1.58700
$759$	0	0
$760$	0	0
$761$	14.0126	0.507956	0.253978	−	0.967210i	$-0.418261\pi$
$761$	14.0126	0.507956	0.253978	+	0.967210i	$0.418261\pi$
$762$	0	0
$763$	26.5650	0.961718
$764$	−3.53913	−0.128041
$765$	0	0
$766$	60.5050	2.18613
$767$	11.9461	0.431347
$768$	0	0
$769$	34.0766	1.22883	0.614417	−	0.788982i	$-0.289391\pi$
$769$	34.0766	1.22883	0.614417	+	0.788982i	$0.289391\pi$
$770$	0	0
$771$	0	0
$772$	−24.7976	−0.892486
$773$	14.3678	0.516773	0.258386	−	0.966042i	$-0.416809\pi$
$773$	14.3678	0.516773	0.258386	+	0.966042i	$0.416809\pi$
$774$	0	0
$775$	0	0
$776$	−4.08088	−0.146495
$777$	0	0
$778$	45.1112	1.61731
$779$	12.1115	0.433941
$780$	0	0
$781$	27.9287	0.999367
$782$	4.11850	0.147277
$783$	0	0
$784$	−5.54407	−0.198002
$785$	0	0
$786$	0	0
$787$	1.65976	0.0591642	0.0295821	−	0.999562i	$-0.490582\pi$
$787$	1.65976	0.0591642	0.0295821	+	0.999562i	$0.490582\pi$
$788$	11.7046	0.416959
$789$	0	0
$790$	0	0
$791$	−40.4446	−1.43804
$792$	0	0
$793$	−10.7650	−0.382275
$794$	29.7354	1.05527
$795$	0	0
$796$	−4.91334	−0.174149
$797$	8.57389	0.303703	0.151851	−	0.988403i	$-0.451477\pi$
$797$	8.57389	0.303703	0.151851	+	0.988403i	$0.451477\pi$
$798$	0	0
$799$	−26.1404	−0.924782
$800$	0	0
$801$	0	0
$802$	14.2507	0.503211
$803$	−36.3417	−1.28247
$804$	0	0
$805$	0	0
$806$	−24.3271	−0.856886
$807$	0	0
$808$	8.55332	0.300905
$809$	44.3986	1.56097	0.780485	−	0.625175i	$-0.214972\pi$
$809$	44.3986	1.56097	0.780485	+	0.625175i	$0.214972\pi$
$810$	0	0
$811$	−13.2677	−0.465892	−0.232946	−	0.972490i	$-0.574837\pi$
$811$	−13.2677	−0.465892	−0.232946	+	0.972490i	$0.574837\pi$
$812$	3.31005	0.116160
$813$	0	0
$814$	19.0714	0.668451
$815$	0	0
$816$	0	0
$817$	−29.0297	−1.01562
$818$	34.5565	1.20824
$819$	0	0
$820$	0	0
$821$	18.5405	0.647069	0.323534	−	0.946216i	$-0.395129\pi$
$821$	18.5405	0.647069	0.323534	+	0.946216i	$0.395129\pi$
$822$	0	0
$823$	−4.42801	−0.154351	−0.0771753	−	0.997018i	$-0.524590\pi$
$823$	−4.42801	−0.154351	−0.0771753	+	0.997018i	$0.524590\pi$
$824$	−23.5290	−0.819672
$825$	0	0
$826$	−24.1426	−0.840029
$827$	−26.1067	−0.907819	−0.453910	−	0.891048i	$-0.649971\pi$
$827$	−26.1067	−0.907819	−0.453910	+	0.891048i	$0.649971\pi$
$828$	0	0
$829$	−25.1458	−0.873350	−0.436675	−	0.899619i	$-0.643844\pi$
$829$	−25.1458	−0.873350	−0.436675	+	0.899619i	$0.643844\pi$
$830$	0	0
$831$	0	0
$832$	−1.20261	−0.0416930
$833$	−7.25316	−0.251307
$834$	0	0
$835$	0	0
$836$	−14.0275	−0.485150
$837$	0	0
$838$	−12.3117	−0.425300
$839$	31.3784	1.08330	0.541651	−	0.840603i	$-0.317799\pi$
$839$	31.3784	1.08330	0.541651	+	0.840603i	$0.317799\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	23.0937	0.795863
$843$	0	0
$844$	32.7971	1.12892
$845$	0	0
$846$	0	0
$847$	50.3540	1.73018
$848$	−17.0480	−0.585430
$849$	0	0
$850$	0	0
$851$	0.712877	0.0244371
$852$	0	0
$853$	−36.1547	−1.23791	−0.618956	−	0.785425i	$-0.712444\pi$
$853$	−36.1547	−1.23791	−0.618956	+	0.785425i	$0.712444\pi$
$854$	21.7556	0.744463
$855$	0	0
$856$	−24.6834	−0.843660
$857$	−20.2270	−0.690943	−0.345471	−	0.938429i	$-0.612281\pi$
$857$	−20.2270	−0.690943	−0.345471	+	0.938429i	$0.612281\pi$
$858$	0	0
$859$	29.8005	1.01678	0.508390	−	0.861127i	$-0.330241\pi$
$859$	29.8005	1.01678	0.508390	+	0.861127i	$0.330241\pi$
$860$	0	0
$861$	0	0
$862$	−70.5502	−2.40295
$863$	35.7369	1.21650	0.608248	−	0.793747i	$-0.291872\pi$
$863$	35.7369	1.21650	0.608248	+	0.793747i	$0.291872\pi$
$864$	0	0
$865$	0	0
$866$	8.76805	0.297951
$867$	0	0
$868$	18.0671	0.613239
$869$	9.90378	0.335963
$870$	0	0
$871$	2.75013	0.0931845
$872$	13.8966	0.470598
$873$	0	0
$874$	−1.42683	−0.0482632
$875$	0	0
$876$	0	0
$877$	−0.626314	−0.0211491	−0.0105746	−	0.999944i	$-0.503366\pi$
$877$	−0.626314	−0.0211491	−0.0105746	+	0.999944i	$0.503366\pi$
$878$	−39.3796	−1.32900
$879$	0	0
$880$	0	0
$881$	−6.65299	−0.224145	−0.112072	−	0.993700i	$-0.535749\pi$
$881$	−6.65299	−0.224145	−0.112072	+	0.993700i	$0.535749\pi$
$882$	0	0
$883$	1.03215	0.0347345	0.0173673	−	0.999849i	$-0.494472\pi$
$883$	1.03215	0.0347345	0.0173673	+	0.999849i	$0.494472\pi$
$884$	−18.9512	−0.637399
$885$	0	0
$886$	2.15440	0.0723783
$887$	−55.8755	−1.87611	−0.938057	−	0.346480i	$-0.887377\pi$
$887$	−55.8755	−1.87611	−0.938057	+	0.346480i	$0.887377\pi$
$888$	0	0
$889$	15.2748	0.512301
$890$	0	0
$891$	0	0
$892$	9.85884	0.330098
$893$	9.05620	0.303054
$894$	0	0
$895$	0	0
$896$	−30.9782	−1.03491
$897$	0	0
$898$	9.29588	0.310208
$899$	5.45828	0.182044
$900$	0	0
$901$	−22.3034	−0.743035
$902$	51.1595	1.70342
$903$	0	0
$904$	−21.1572	−0.703679
$905$	0	0
$906$	0	0
$907$	−23.0687	−0.765982	−0.382991	−	0.923752i	$-0.625106\pi$
$907$	−23.0687	−0.765982	−0.382991	+	0.923752i	$0.625106\pi$
$908$	17.1824	0.570219
$909$	0	0
$910$	0	0
$911$	−52.8451	−1.75084	−0.875418	−	0.483367i	$-0.839414\pi$
$911$	−52.8451	−1.75084	−0.875418	+	0.483367i	$0.839414\pi$
$912$	0	0
$913$	73.5670	2.43471
$914$	−30.0629	−0.994393
$915$	0	0
$916$	−24.5170	−0.810064
$917$	−30.8881	−1.02001
$918$	0	0
$919$	−49.6417	−1.63753	−0.818765	−	0.574129i	$-0.805341\pi$
$919$	−49.6417	−1.63753	−0.818765	+	0.574129i	$0.805341\pi$
$920$	0	0
$921$	0	0
$922$	−13.8634	−0.456567
$923$	−13.0720	−0.430270
$924$	0	0
$925$	0	0
$926$	39.6951	1.30446
$927$	0	0
$928$	5.86400	0.192495
$929$	−10.7335	−0.352153	−0.176077	−	0.984376i	$-0.556341\pi$
$929$	−10.7335	−0.352153	−0.176077	+	0.984376i	$0.556341\pi$
$930$	0	0
$931$	2.51281	0.0823542
$932$	−30.3001	−0.992512
$933$	0	0
$934$	14.5666	0.476633
$935$	0	0
$936$	0	0
$937$	11.4028	0.372513	0.186256	−	0.982501i	$-0.440365\pi$
$937$	11.4028	0.372513	0.186256	+	0.982501i	$0.440365\pi$
$938$	−5.55792	−0.181472
$939$	0	0
$940$	0	0
$941$	−26.5670	−0.866060	−0.433030	−	0.901380i	$-0.642556\pi$
$941$	−26.5670	−0.866060	−0.433030	+	0.901380i	$0.642556\pi$
$942$	0	0
$943$	1.91231	0.0622734
$944$	−23.7058	−0.771558
$945$	0	0
$946$	−122.622	−3.98679
$947$	−15.3658	−0.499322	−0.249661	−	0.968333i	$-0.580319\pi$
$947$	−15.3658	−0.499322	−0.249661	+	0.968333i	$0.580319\pi$
$948$	0	0
$949$	17.0097	0.552158
$950$	0	0
$951$	0	0
$952$	−27.6218	−0.895229
$953$	−12.7062	−0.411596	−0.205798	−	0.978595i	$-0.565979\pi$
$953$	−12.7062	−0.411596	−0.205798	+	0.978595i	$0.565979\pi$
$954$	0	0
$955$	0	0
$956$	9.49419	0.307064
$957$	0	0
$958$	31.0881	1.00441
$959$	−21.1264	−0.682209
$960$	0	0
$961$	−1.20722	−0.0389426
$962$	−8.92634	−0.287797
$963$	0	0
$964$	−12.2869	−0.395733
$965$	0	0
$966$	0	0
$967$	−15.8218	−0.508795	−0.254398	−	0.967100i	$-0.581877\pi$
$967$	−15.8218	−0.508795	−0.254398	+	0.967100i	$0.581877\pi$
$968$	26.3410	0.846632
$969$	0	0
$970$	0	0
$971$	−47.8287	−1.53490	−0.767448	−	0.641111i	$-0.778474\pi$
$971$	−47.8287	−1.53490	−0.767448	+	0.641111i	$0.778474\pi$
$972$	0	0
$973$	56.0152	1.79577
$974$	41.3644	1.32540
$975$	0	0
$976$	21.3620	0.683781
$977$	−46.2147	−1.47854	−0.739270	−	0.673409i	$-0.764829\pi$
$977$	−46.2147	−1.47854	−0.739270	+	0.673409i	$0.764829\pi$
$978$	0	0
$979$	−13.7492	−0.439426
$980$	0	0
$981$	0	0
$982$	61.3372	1.95735
$983$	−40.2132	−1.28260	−0.641301	−	0.767289i	$-0.721605\pi$
$983$	−40.2132	−1.28260	−0.641301	+	0.767289i	$0.721605\pi$
$984$	0	0
$985$	0	0
$986$	11.5708	0.368489
$987$	0	0
$988$	6.56555	0.208878
$989$	−4.58354	−0.145748
$990$	0	0
$991$	−8.54051	−0.271298	−0.135649	−	0.990757i	$-0.543312\pi$
$991$	−8.54051	−0.271298	−0.135649	+	0.990757i	$0.543312\pi$
$992$	32.0073	1.01623
$993$	0	0
$994$	26.4181	0.837931
$995$	0	0
$996$	0	0
$997$	−26.7017	−0.845651	−0.422825	−	0.906211i	$-0.638962\pi$
$997$	−26.7017	−0.845651	−0.422825	+	0.906211i	$0.638962\pi$
$998$	−22.8946	−0.724715
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6525.2.a.ca.1.3	✓	9
3.2	odd	2		6525.2.a.cc.1.7	yes	9
5.4	even	2		6525.2.a.cd.1.7	yes	9
15.14	odd	2		6525.2.a.cb.1.3	yes	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6525.2.a.ca.1.3	✓	9	1.1	even	1	trivial
6525.2.a.cb.1.3	yes	9	15.14	odd	2
6525.2.a.cc.1.7	yes	9	3.2	odd	2
6525.2.a.cd.1.7	yes	9	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:	\( N \)	\(=\)	\( 6525 = 3^{2} \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6525.a (trivial)

\(f(q)\)	\(=\)	\(q-1.77820 q^{2} +1.16198 q^{4} +2.84862 q^{7} +1.49016 q^{8} +O(q^{10})\)	\(q-1.77820 q^{2} +1.16198 q^{4} +2.84862 q^{7} +1.49016 q^{8} -5.35505 q^{11} +2.50643 q^{13} -5.06541 q^{14} -4.97376 q^{16} -6.50704 q^{17} +2.25433 q^{19} +9.52233 q^{22} +0.355939 q^{23} -4.45692 q^{26} +3.31005 q^{28} +1.00000 q^{29} +5.45828 q^{31} +5.86400 q^{32} +11.5708 q^{34} +2.00281 q^{37} -4.00863 q^{38} +5.37258 q^{41} -12.8773 q^{43} -6.22247 q^{44} -0.632929 q^{46} +4.01725 q^{47} +1.11466 q^{49} +2.91242 q^{52} +3.42758 q^{53} +4.24492 q^{56} -1.77820 q^{58} +4.76617 q^{59} -4.29494 q^{61} -9.70588 q^{62} -0.479810 q^{64} +1.09723 q^{67} -7.56105 q^{68} -5.21539 q^{71} +6.78643 q^{73} -3.56138 q^{74} +2.61948 q^{76} -15.2545 q^{77} -1.84943 q^{79} -9.55349 q^{82} -13.7379 q^{83} +22.8984 q^{86} -7.97990 q^{88} +2.56752 q^{89} +7.13987 q^{91} +0.413594 q^{92} -7.14346 q^{94} -2.73855 q^{97} -1.98209 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 2 q^{2} + 10 q^{4} - q^{7} - 9 q^{8}+O(q^{10}) \) 9 * q - 2 * q^2 + 10 * q^4 - q^7 - 9 * q^8	\( 9 q - 2 q^{2} + 10 q^{4} - q^{7} - 9 q^{8} + 2 q^{11} - q^{13} - 3 q^{14} + 4 q^{16} - 12 q^{17} - q^{19} - 3 q^{22} - 16 q^{23} + 6 q^{26} + 4 q^{28} + 9 q^{29} + 5 q^{31} - 20 q^{32} + 3 q^{34} - 30 q^{38} - 10 q^{41} - 3 q^{43} - 13 q^{44} + 4 q^{46} - 26 q^{47} - 8 q^{49} + 9 q^{52} - 22 q^{53} + 22 q^{56} - 2 q^{58} + 4 q^{59} + 7 q^{61} - 28 q^{62} + 9 q^{64} - 5 q^{67} - 39 q^{68} + 10 q^{73} - 34 q^{74} - 2 q^{76} - 34 q^{77} + 10 q^{79} + 8 q^{82} - 46 q^{83} + 28 q^{86} - 2 q^{88} + 4 q^{89} - 21 q^{91} - 20 q^{92} + 5 q^{94} - 7 q^{97} - 51 q^{98}+O(q^{100}) \) 9 * q - 2 * q^2 + 10 * q^4 - q^7 - 9 * q^8 + 2 * q^11 - q^13 - 3 * q^14 + 4 * q^16 - 12 * q^17 - q^19 - 3 * q^22 - 16 * q^23 + 6 * q^26 + 4 * q^28 + 9 * q^29 + 5 * q^31 - 20 * q^32 + 3 * q^34 - 30 * q^38 - 10 * q^41 - 3 * q^43 - 13 * q^44 + 4 * q^46 - 26 * q^47 - 8 * q^49 + 9 * q^52 - 22 * q^53 + 22 * q^56 - 2 * q^58 + 4 * q^59 + 7 * q^61 - 28 * q^62 + 9 * q^64 - 5 * q^67 - 39 * q^68 + 10 * q^73 - 34 * q^74 - 2 * q^76 - 34 * q^77 + 10 * q^79 + 8 * q^82 - 46 * q^83 + 28 * q^86 - 2 * q^88 + 4 * q^89 - 21 * q^91 - 20 * q^92 + 5 * q^94 - 7 * q^97 - 51 * q^98

Introduction

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