LMFDB - Embedded newform 6525.2.a.bx.1.6

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:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 10x^{5} + 19x^{4} + 24x^{3} - 44x^{2} - 3x + 14 $ x^7 - 2x^6 - 10x^5 + 19x^4 + 24x^3 - 44x^2 - 3x + 14
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2175)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$2.26695$ 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)

$f(q)$	$=$	$q+2.26695 q^{2} +3.13907 q^{4} -0.159887 q^{7} +2.58223 q^{8} +O(q^{10})$	$q+2.26695 q^{2} +3.13907 q^{4} -0.159887 q^{7} +2.58223 q^{8} +3.24930 q^{11} -7.05679 q^{13} -0.362456 q^{14} -0.424361 q^{16} +7.63808 q^{17} -3.99028 q^{19} +7.36601 q^{22} +2.06402 q^{23} -15.9974 q^{26} -0.501897 q^{28} +1.00000 q^{29} +10.2044 q^{31} -6.12646 q^{32} +17.3152 q^{34} +4.27815 q^{37} -9.04576 q^{38} +5.12330 q^{41} +3.94180 q^{43} +10.1998 q^{44} +4.67903 q^{46} +5.61626 q^{47} -6.97444 q^{49} -22.1518 q^{52} +12.0822 q^{53} -0.412864 q^{56} +2.26695 q^{58} +7.06781 q^{59} +11.4310 q^{61} +23.1329 q^{62} -13.0397 q^{64} +14.9512 q^{67} +23.9765 q^{68} -3.43347 q^{71} +12.0772 q^{73} +9.69836 q^{74} -12.5258 q^{76} -0.519521 q^{77} -12.7441 q^{79} +11.6143 q^{82} -2.59792 q^{83} +8.93586 q^{86} +8.39044 q^{88} -4.33165 q^{89} +1.12829 q^{91} +6.47910 q^{92} +12.7318 q^{94} +3.88355 q^{97} -15.8107 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 2 q^{2} + 10 q^{4} + q^{7} + 3 q^{8}+O(q^{10}) $ 7 * q + 2 * q^2 + 10 * q^4 + q^7 + 3 * q^8	$ 7 q + 2 q^{2} + 10 q^{4} + q^{7} + 3 q^{8} - 4 q^{11} + q^{13} - 15 q^{14} + 12 q^{16} + 8 q^{17} + 15 q^{19} - 3 q^{22} + 14 q^{23} - 6 q^{26} + 24 q^{28} + 7 q^{29} + 5 q^{31} + 18 q^{32} + 7 q^{34} + 6 q^{37} - 18 q^{38} - 22 q^{41} + 19 q^{43} - 15 q^{44} - 4 q^{46} + 22 q^{47} + 12 q^{49} - 11 q^{52} + 10 q^{53} - 14 q^{56} + 2 q^{58} - 6 q^{59} + 23 q^{61} + 40 q^{62} + 5 q^{64} + 13 q^{67} - 7 q^{68} - 26 q^{71} + 24 q^{73} + 10 q^{74} + 46 q^{76} + 4 q^{77} + 14 q^{79} - 16 q^{82} + 10 q^{83} - 44 q^{86} + 66 q^{88} - 14 q^{89} + 13 q^{91} + 58 q^{92} - 3 q^{94} + 31 q^{97} - 59 q^{98}+O(q^{100}) $ 7 * q + 2 * q^2 + 10 * q^4 + q^7 + 3 * q^8 - 4 * q^11 + q^13 - 15 * q^14 + 12 * q^16 + 8 * q^17 + 15 * q^19 - 3 * q^22 + 14 * q^23 - 6 * q^26 + 24 * q^28 + 7 * q^29 + 5 * q^31 + 18 * q^32 + 7 * q^34 + 6 * q^37 - 18 * q^38 - 22 * q^41 + 19 * q^43 - 15 * q^44 - 4 * q^46 + 22 * q^47 + 12 * q^49 - 11 * q^52 + 10 * q^53 - 14 * q^56 + 2 * q^58 - 6 * q^59 + 23 * q^61 + 40 * q^62 + 5 * q^64 + 13 * q^67 - 7 * q^68 - 26 * q^71 + 24 * q^73 + 10 * q^74 + 46 * q^76 + 4 * q^77 + 14 * q^79 - 16 * q^82 + 10 * q^83 - 44 * q^86 + 66 * q^88 - 14 * q^89 + 13 * q^91 + 58 * q^92 - 3 * q^94 + 31 * q^97 - 59 * 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 $

$2$	2.26695	1.60298	0.801489	−	0.598010i	$-0.204042\pi$
$2$	2.26695	1.60298	0.801489	+	0.598010i	$0.204042\pi$
$3$	0	0
$3$	0	0
$4$	3.13907	1.56954
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.159887	−0.0604316	−0.0302158	−	0.999543i	$-0.509619\pi$
$7$	−0.159887	−0.0604316	−0.0302158	+	0.999543i	$0.509619\pi$
$8$	2.58223	0.912955
$9$	0	0
$10$	0	0
$11$	3.24930	0.979701	0.489851	−	0.871806i	$-0.337051\pi$
$11$	3.24930	0.979701	0.489851	+	0.871806i	$0.337051\pi$
$12$	0	0
$13$	−7.05679	−1.95720	−0.978600	−	0.205770i	$-0.934030\pi$
$13$	−7.05679	−1.95720	−0.978600	+	0.205770i	$0.934030\pi$
$14$	−0.362456	−0.0968704
$15$	0	0
$16$	−0.424361	−0.106090
$17$	7.63808	1.85251	0.926254	−	0.376901i	$-0.123010\pi$
$17$	7.63808	1.85251	0.926254	+	0.376901i	$0.123010\pi$
$18$	0	0
$19$	−3.99028	−0.915432	−0.457716	−	0.889098i	$-0.651332\pi$
$19$	−3.99028	−0.915432	−0.457716	+	0.889098i	$0.651332\pi$
$20$	0	0
$21$	0	0
$22$	7.36601	1.57044
$23$	2.06402	0.430377	0.215189	−	0.976572i	$-0.430963\pi$
$23$	2.06402	0.430377	0.215189	+	0.976572i	$0.430963\pi$
$24$	0	0
$25$	0	0
$26$	−15.9974	−3.13735
$27$	0	0
$28$	−0.501897	−0.0948496
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	10.2044	1.83277	0.916383	−	0.400303i	$-0.131095\pi$
$31$	10.2044	1.83277	0.916383	+	0.400303i	$0.131095\pi$
$32$	−6.12646	−1.08302
$33$	0	0
$34$	17.3152	2.96953
$35$	0	0
$36$	0	0
$37$	4.27815	0.703323	0.351662	−	0.936127i	$-0.385617\pi$
$37$	4.27815	0.703323	0.351662	+	0.936127i	$0.385617\pi$
$38$	−9.04576	−1.46742
$39$	0	0
$40$	0	0
$41$	5.12330	0.800125	0.400063	−	0.916488i	$-0.368988\pi$
$41$	5.12330	0.800125	0.400063	+	0.916488i	$0.368988\pi$
$42$	0	0
$43$	3.94180	0.601118	0.300559	−	0.953763i	$-0.402827\pi$
$43$	3.94180	0.601118	0.300559	+	0.953763i	$0.402827\pi$
$44$	10.1998	1.53768
$45$	0	0
$46$	4.67903	0.689885
$47$	5.61626	0.819215	0.409608	−	0.912262i	$-0.365666\pi$
$47$	5.61626	0.819215	0.409608	+	0.912262i	$0.365666\pi$
$48$	0	0
$49$	−6.97444	−0.996348
$50$	0	0
$51$	0	0
$52$	−22.1518	−3.07190
$53$	12.0822	1.65962	0.829808	−	0.558049i	$-0.188450\pi$
$53$	12.0822	1.65962	0.829808	+	0.558049i	$0.188450\pi$
$54$	0	0
$55$	0	0
$56$	−0.412864	−0.0551713
$57$	0	0
$58$	2.26695	0.297665
$59$	7.06781	0.920151	0.460075	−	0.887880i	$-0.347822\pi$
$59$	7.06781	0.920151	0.460075	+	0.887880i	$0.347822\pi$
$60$	0	0
$61$	11.4310	1.46359	0.731796	−	0.681524i	$-0.238682\pi$
$61$	11.4310	1.46359	0.731796	+	0.681524i	$0.238682\pi$
$62$	23.1329	2.93788
$63$	0	0
$64$	−13.0397	−1.62996
$65$	0	0
$66$	0	0
$67$	14.9512	1.82658	0.913289	−	0.407311i	$-0.133534\pi$
$67$	14.9512	1.82658	0.913289	+	0.407311i	$0.133534\pi$
$68$	23.9765	2.90758
$69$	0	0
$70$	0	0
$71$	−3.43347	−0.407478	−0.203739	−	0.979025i	$-0.565309\pi$
$71$	−3.43347	−0.407478	−0.203739	+	0.979025i	$0.565309\pi$
$72$	0	0
$73$	12.0772	1.41353	0.706764	−	0.707449i	$-0.250154\pi$
$73$	12.0772	1.41353	0.706764	+	0.707449i	$0.250154\pi$
$74$	9.69836	1.12741
$75$	0	0
$76$	−12.5258	−1.43680
$77$	−0.519521	−0.0592049
$78$	0	0
$79$	−12.7441	−1.43383	−0.716913	−	0.697163i	$-0.754446\pi$
$79$	−12.7441	−1.43383	−0.716913	+	0.697163i	$0.754446\pi$
$80$	0	0
$81$	0	0
$82$	11.6143	1.28258
$83$	−2.59792	−0.285159	−0.142580	−	0.989783i	$-0.545540\pi$
$83$	−2.59792	−0.285159	−0.142580	+	0.989783i	$0.545540\pi$
$84$	0	0
$85$	0	0
$86$	8.93586	0.963579
$87$	0	0
$88$	8.39044	0.894423
$89$	−4.33165	−0.459154	−0.229577	−	0.973290i	$-0.573734\pi$
$89$	−4.33165	−0.459154	−0.229577	+	0.973290i	$0.573734\pi$
$90$	0	0
$91$	1.12829	0.118277
$92$	6.47910	0.675493
$93$	0	0
$94$	12.7318	1.31318
$95$	0	0
$96$	0	0
$97$	3.88355	0.394315	0.197158	−	0.980372i	$-0.436829\pi$
$97$	3.88355	0.394315	0.197158	+	0.980372i	$0.436829\pi$
$98$	−15.8107	−1.59712
$99$	0	0
$100$	0	0
$101$	−15.4045	−1.53280	−0.766400	−	0.642363i	$-0.777954\pi$
$101$	−15.4045	−1.53280	−0.766400	+	0.642363i	$0.777954\pi$
$102$	0	0
$103$	−7.68484	−0.757210	−0.378605	−	0.925558i	$-0.623596\pi$
$103$	−7.68484	−0.757210	−0.378605	+	0.925558i	$0.623596\pi$
$104$	−18.2222	−1.78684
$105$	0	0
$106$	27.3897	2.66033
$107$	−4.60924	−0.445592	−0.222796	−	0.974865i	$-0.571518\pi$
$107$	−4.60924	−0.445592	−0.222796	+	0.974865i	$0.571518\pi$
$108$	0	0
$109$	−14.4509	−1.38415	−0.692073	−	0.721827i	$-0.743302\pi$
$109$	−14.4509	−1.38415	−0.692073	+	0.721827i	$0.743302\pi$
$110$	0	0
$111$	0	0
$112$	0.0678498	0.00641120
$113$	−4.08614	−0.384392	−0.192196	−	0.981357i	$-0.561561\pi$
$113$	−4.08614	−0.384392	−0.192196	+	0.981357i	$0.561561\pi$
$114$	0	0
$115$	0	0
$116$	3.13907	0.291456
$117$	0	0
$118$	16.0224	1.47498
$119$	−1.22123	−0.111950
$120$	0	0
$121$	−0.442038	−0.0401853
$122$	25.9136	2.34610
$123$	0	0
$124$	32.0324	2.87659
$125$	0	0
$126$	0	0
$127$	3.69650	0.328011	0.164006	−	0.986459i	$-0.447559\pi$
$127$	3.69650	0.328011	0.164006	+	0.986459i	$0.447559\pi$
$128$	−17.3074	−1.52977
$129$	0	0
$130$	0	0
$131$	9.92811	0.867423	0.433712	−	0.901052i	$-0.357204\pi$
$131$	9.92811	0.867423	0.433712	+	0.901052i	$0.357204\pi$
$132$	0	0
$133$	0.637993	0.0553210
$134$	33.8936	2.92797
$135$	0	0
$136$	19.7233	1.69126
$137$	3.82810	0.327057	0.163528	−	0.986539i	$-0.447712\pi$
$137$	3.82810	0.327057	0.163528	+	0.986539i	$0.447712\pi$
$138$	0	0
$139$	−2.06987	−0.175564	−0.0877822	−	0.996140i	$-0.527978\pi$
$139$	−2.06987	−0.175564	−0.0877822	+	0.996140i	$0.527978\pi$
$140$	0	0
$141$	0	0
$142$	−7.78351	−0.653177
$143$	−22.9296	−1.91747
$144$	0	0
$145$	0	0
$146$	27.3784	2.26585
$147$	0	0
$148$	13.4294	1.10389
$149$	1.46609	0.120107	0.0600536	−	0.998195i	$-0.480873\pi$
$149$	1.46609	0.120107	0.0600536	+	0.998195i	$0.480873\pi$
$150$	0	0
$151$	−20.1279	−1.63799	−0.818995	−	0.573801i	$-0.805468\pi$
$151$	−20.1279	−1.63799	−0.818995	+	0.573801i	$0.805468\pi$
$152$	−10.3038	−0.835748
$153$	0	0
$154$	−1.17773	−0.0949041
$155$	0	0
$156$	0	0
$157$	−7.09106	−0.565928	−0.282964	−	0.959131i	$-0.591318\pi$
$157$	−7.09106	−0.565928	−0.282964	+	0.959131i	$0.591318\pi$
$158$	−28.8903	−2.29839
$159$	0	0
$160$	0	0
$161$	−0.330009	−0.0260084
$162$	0	0
$163$	6.58715	0.515946	0.257973	−	0.966152i	$-0.416945\pi$
$163$	6.58715	0.515946	0.257973	+	0.966152i	$0.416945\pi$
$164$	16.0824	1.25583
$165$	0	0
$166$	−5.88937	−0.457104
$167$	−10.6638	−0.825192	−0.412596	−	0.910914i	$-0.635378\pi$
$167$	−10.6638	−0.825192	−0.412596	+	0.910914i	$0.635378\pi$
$168$	0	0
$169$	36.7983	2.83063
$170$	0	0
$171$	0	0
$172$	12.3736	0.943477
$173$	−13.8799	−1.05527	−0.527633	−	0.849472i	$-0.676920\pi$
$173$	−13.8799	−1.05527	−0.527633	+	0.849472i	$0.676920\pi$
$174$	0	0
$175$	0	0
$176$	−1.37888	−0.103937
$177$	0	0
$178$	−9.81965	−0.736014
$179$	11.1105	0.830439	0.415219	−	0.909721i	$-0.363705\pi$
$179$	11.1105	0.830439	0.415219	+	0.909721i	$0.363705\pi$
$180$	0	0
$181$	10.9453	0.813557	0.406778	−	0.913527i	$-0.366652\pi$
$181$	10.9453	0.813557	0.406778	+	0.913527i	$0.366652\pi$
$182$	2.55777	0.189595
$183$	0	0
$184$	5.32976	0.392915
$185$	0	0
$186$	0	0
$187$	24.8184	1.81490
$188$	17.6298	1.28579
$189$	0	0
$190$	0	0
$191$	−18.1472	−1.31309	−0.656544	−	0.754287i	$-0.727983\pi$
$191$	−18.1472	−1.31309	−0.656544	+	0.754287i	$0.727983\pi$
$192$	0	0
$193$	17.3305	1.24747	0.623737	−	0.781634i	$-0.285614\pi$
$193$	17.3305	1.24747	0.623737	+	0.781634i	$0.285614\pi$
$194$	8.80383	0.632078
$195$	0	0
$196$	−21.8933	−1.56381
$197$	−10.4273	−0.742915	−0.371458	−	0.928450i	$-0.621142\pi$
$197$	−10.4273	−0.742915	−0.371458	+	0.928450i	$0.621142\pi$
$198$	0	0
$199$	11.5570	0.819256	0.409628	−	0.912253i	$-0.365659\pi$
$199$	11.5570	0.819256	0.409628	+	0.912253i	$0.365659\pi$
$200$	0	0
$201$	0	0
$202$	−34.9212	−2.45704
$203$	−0.159887	−0.0112219
$204$	0	0
$205$	0	0
$206$	−17.4212	−1.21379
$207$	0	0
$208$	2.99463	0.207640
$209$	−12.9656	−0.896850
$210$	0	0
$211$	−21.1616	−1.45682	−0.728412	−	0.685140i	$-0.759741\pi$
$211$	−21.1616	−1.45682	−0.728412	+	0.685140i	$0.759741\pi$
$212$	37.9269	2.60483
$213$	0	0
$214$	−10.4489	−0.714274
$215$	0	0
$216$	0	0
$217$	−1.63155	−0.110757
$218$	−32.7595	−2.21876
$219$	0	0
$220$	0	0
$221$	−53.9003	−3.62573
$222$	0	0
$223$	−9.46925	−0.634108	−0.317054	−	0.948408i	$-0.602694\pi$
$223$	−9.46925	−0.634108	−0.317054	+	0.948408i	$0.602694\pi$
$224$	0.979541	0.0654483
$225$	0	0
$226$	−9.26310	−0.616172
$227$	22.4030	1.48694	0.743471	−	0.668768i	$-0.233178\pi$
$227$	22.4030	1.48694	0.743471	+	0.668768i	$0.233178\pi$
$228$	0	0
$229$	2.03117	0.134223	0.0671117	−	0.997745i	$-0.478622\pi$
$229$	2.03117	0.134223	0.0671117	+	0.997745i	$0.478622\pi$
$230$	0	0
$231$	0	0
$232$	2.58223	0.169532
$233$	11.0704	0.725249	0.362624	−	0.931935i	$-0.381881\pi$
$233$	11.0704	0.725249	0.362624	+	0.931935i	$0.381881\pi$
$234$	0	0
$235$	0	0
$236$	22.1864	1.44421
$237$	0	0
$238$	−2.76847	−0.179453
$239$	2.24098	0.144957	0.0724786	−	0.997370i	$-0.476909\pi$
$239$	2.24098	0.144957	0.0724786	+	0.997370i	$0.476909\pi$
$240$	0	0
$241$	14.4838	0.932986	0.466493	−	0.884525i	$-0.345517\pi$
$241$	14.4838	0.932986	0.466493	+	0.884525i	$0.345517\pi$
$242$	−1.00208	−0.0644161
$243$	0	0
$244$	35.8828	2.29716
$245$	0	0
$246$	0	0
$247$	28.1585	1.79168
$248$	26.3501	1.67323
$249$	0	0
$250$	0	0
$251$	16.2044	1.02281	0.511407	−	0.859339i	$-0.329125\pi$
$251$	16.2044	1.02281	0.511407	+	0.859339i	$0.329125\pi$
$252$	0	0
$253$	6.70661	0.421641
$254$	8.37978	0.525794
$255$	0	0
$256$	−13.1557	−0.822232
$257$	−3.36942	−0.210178	−0.105089	−	0.994463i	$-0.533513\pi$
$257$	−3.36942	−0.210178	−0.105089	+	0.994463i	$0.533513\pi$
$258$	0	0
$259$	−0.684020	−0.0425029
$260$	0	0
$261$	0	0
$262$	22.5066	1.39046
$263$	1.79671	0.110790	0.0553950	−	0.998465i	$-0.482358\pi$
$263$	1.79671	0.110790	0.0553950	+	0.998465i	$0.482358\pi$
$264$	0	0
$265$	0	0
$266$	1.44630	0.0886783
$267$	0	0
$268$	46.9329	2.86688
$269$	0.117055	0.00713699	0.00356849	−	0.999994i	$-0.498864\pi$
$269$	0.117055	0.00713699	0.00356849	+	0.999994i	$0.498864\pi$
$270$	0	0
$271$	−4.06294	−0.246806	−0.123403	−	0.992357i	$-0.539381\pi$
$271$	−4.06294	−0.246806	−0.123403	+	0.992357i	$0.539381\pi$
$272$	−3.24131	−0.196533
$273$	0	0
$274$	8.67813	0.524265
$275$	0	0
$276$	0	0
$277$	12.5436	0.753674	0.376837	−	0.926279i	$-0.377012\pi$
$277$	12.5436	0.753674	0.376837	+	0.926279i	$0.377012\pi$
$278$	−4.69231	−0.281426
$279$	0	0
$280$	0	0
$281$	2.83163	0.168921	0.0844605	−	0.996427i	$-0.473083\pi$
$281$	2.83163	0.168921	0.0844605	+	0.996427i	$0.473083\pi$
$282$	0	0
$283$	−28.2249	−1.67780	−0.838898	−	0.544288i	$-0.816800\pi$
$283$	−28.2249	−1.67780	−0.838898	+	0.544288i	$0.816800\pi$
$284$	−10.7779	−0.639551
$285$	0	0
$286$	−51.9804	−3.07366
$287$	−0.819148	−0.0483528
$288$	0	0
$289$	41.3403	2.43178
$290$	0	0
$291$	0	0
$292$	37.9112	2.21859
$293$	21.9337	1.28138	0.640691	−	0.767799i	$-0.278648\pi$
$293$	21.9337	1.28138	0.640691	+	0.767799i	$0.278648\pi$
$294$	0	0
$295$	0	0
$296$	11.0472	0.642103
$297$	0	0
$298$	3.32357	0.192529
$299$	−14.5653	−0.842335
$300$	0	0
$301$	−0.630241	−0.0363265
$302$	−45.6291	−2.62566
$303$	0	0
$304$	1.69332	0.0971185
$305$	0	0
$306$	0	0
$307$	0.461574	0.0263434	0.0131717	−	0.999913i	$-0.495807\pi$
$307$	0.461574	0.0263434	0.0131717	+	0.999913i	$0.495807\pi$
$308$	−1.63081	−0.0929243
$309$	0	0
$310$	0	0
$311$	−27.5507	−1.56226	−0.781128	−	0.624371i	$-0.785355\pi$
$311$	−27.5507	−1.56226	−0.781128	+	0.624371i	$0.785355\pi$
$312$	0	0
$313$	−14.9121	−0.842882	−0.421441	−	0.906856i	$-0.638476\pi$
$313$	−14.9121	−0.842882	−0.421441	+	0.906856i	$0.638476\pi$
$314$	−16.0751	−0.907170
$315$	0	0
$316$	−40.0048	−2.25044
$317$	−22.9266	−1.28769	−0.643843	−	0.765157i	$-0.722661\pi$
$317$	−22.9266	−1.28769	−0.643843	+	0.765157i	$0.722661\pi$
$318$	0	0
$319$	3.24930	0.181926
$320$	0	0
$321$	0	0
$322$	−0.748115	−0.0416908
$323$	−30.4781	−1.69584
$324$	0	0
$325$	0	0
$326$	14.9328	0.827049
$327$	0	0
$328$	13.2295	0.730478
$329$	−0.897966	−0.0495065
$330$	0	0
$331$	−3.23147	−0.177618	−0.0888088	−	0.996049i	$-0.528306\pi$
$331$	−3.23147	−0.177618	−0.0888088	+	0.996049i	$0.528306\pi$
$332$	−8.15507	−0.447568
$333$	0	0
$334$	−24.1744	−1.32277
$335$	0	0
$336$	0	0
$337$	−28.8026	−1.56898	−0.784489	−	0.620143i	$-0.787075\pi$
$337$	−28.8026	−1.56898	−0.784489	+	0.620143i	$0.787075\pi$
$338$	83.4199	4.53744
$339$	0	0
$340$	0	0
$341$	33.1572	1.79556
$342$	0	0
$343$	2.23433	0.120642
$344$	10.1786	0.548794
$345$	0	0
$346$	−31.4650	−1.69157
$347$	−16.4175	−0.881337	−0.440669	−	0.897670i	$-0.645259\pi$
$347$	−16.4175	−0.881337	−0.440669	+	0.897670i	$0.645259\pi$
$348$	0	0
$349$	−18.2107	−0.974796	−0.487398	−	0.873180i	$-0.662054\pi$
$349$	−18.2107	−0.974796	−0.487398	+	0.873180i	$0.662054\pi$
$350$	0	0
$351$	0	0
$352$	−19.9067	−1.06103
$353$	−18.8790	−1.00483	−0.502414	−	0.864627i	$-0.667555\pi$
$353$	−18.8790	−1.00483	−0.502414	+	0.864627i	$0.667555\pi$
$354$	0	0
$355$	0	0
$356$	−13.5974	−0.720660
$357$	0	0
$358$	25.1870	1.33117
$359$	−14.4500	−0.762640	−0.381320	−	0.924443i	$-0.624530\pi$
$359$	−14.4500	−0.762640	−0.381320	+	0.924443i	$0.624530\pi$
$360$	0	0
$361$	−3.07770	−0.161984
$362$	24.8125	1.30411
$363$	0	0
$364$	3.54178	0.185640
$365$	0	0
$366$	0	0
$367$	21.1171	1.10231	0.551153	−	0.834404i	$-0.314188\pi$
$367$	21.1171	1.10231	0.551153	+	0.834404i	$0.314188\pi$
$368$	−0.875889	−0.0456589
$369$	0	0
$370$	0	0
$371$	−1.93178	−0.100293
$372$	0	0
$373$	17.1185	0.886361	0.443181	−	0.896432i	$-0.353850\pi$
$373$	17.1185	0.886361	0.443181	+	0.896432i	$0.353850\pi$
$374$	56.2622	2.90925
$375$	0	0
$376$	14.5025	0.747907
$377$	−7.05679	−0.363443
$378$	0	0
$379$	−0.206688	−0.0106169	−0.00530844	−	0.999986i	$-0.501690\pi$
$379$	−0.206688	−0.0106169	−0.00530844	+	0.999986i	$0.501690\pi$
$380$	0	0
$381$	0	0
$382$	−41.1389	−2.10485
$383$	16.3701	0.836474	0.418237	−	0.908338i	$-0.362648\pi$
$383$	16.3701	0.836474	0.418237	+	0.908338i	$0.362648\pi$
$384$	0	0
$385$	0	0
$386$	39.2873	1.99967
$387$	0	0
$388$	12.1908	0.618892
$389$	−11.7627	−0.596393	−0.298197	−	0.954504i	$-0.596385\pi$
$389$	−11.7627	−0.596393	−0.298197	+	0.954504i	$0.596385\pi$
$390$	0	0
$391$	15.7651	0.797277
$392$	−18.0096	−0.909621
$393$	0	0
$394$	−23.6382	−1.19088
$395$	0	0
$396$	0	0
$397$	4.96134	0.249002	0.124501	−	0.992219i	$-0.460267\pi$
$397$	4.96134	0.249002	0.124501	+	0.992219i	$0.460267\pi$
$398$	26.1992	1.31325
$399$	0	0
$400$	0	0
$401$	−36.1967	−1.80758	−0.903789	−	0.427979i	$-0.859226\pi$
$401$	−36.1967	−1.80758	−0.903789	+	0.427979i	$0.859226\pi$
$402$	0	0
$403$	−72.0103	−3.58709
$404$	−48.3557	−2.40579
$405$	0	0
$406$	−0.362456	−0.0179884
$407$	13.9010	0.689047
$408$	0	0
$409$	−16.1607	−0.799094	−0.399547	−	0.916713i	$-0.630832\pi$
$409$	−16.1607	−0.799094	−0.399547	+	0.916713i	$0.630832\pi$
$410$	0	0
$411$	0	0
$412$	−24.1233	−1.18847
$413$	−1.13005	−0.0556061
$414$	0	0
$415$	0	0
$416$	43.2331	2.11968
$417$	0	0
$418$	−29.3924	−1.43763
$419$	33.5544	1.63924	0.819621	−	0.572906i	$-0.194184\pi$
$419$	33.5544	1.63924	0.819621	+	0.572906i	$0.194184\pi$
$420$	0	0
$421$	26.9847	1.31515	0.657577	−	0.753388i	$-0.271582\pi$
$421$	26.9847	1.31515	0.657577	+	0.753388i	$0.271582\pi$
$422$	−47.9723	−2.33526
$423$	0	0
$424$	31.1990	1.51515
$425$	0	0
$426$	0	0
$427$	−1.82767	−0.0884471
$428$	−14.4687	−0.699373
$429$	0	0
$430$	0	0
$431$	−0.832366	−0.0400937	−0.0200468	−	0.999799i	$-0.506382\pi$
$431$	−0.832366	−0.0400937	−0.0200468	+	0.999799i	$0.506382\pi$
$432$	0	0
$433$	22.5647	1.08439	0.542195	−	0.840253i	$-0.317593\pi$
$433$	22.5647	1.08439	0.542195	+	0.840253i	$0.317593\pi$
$434$	−3.69865	−0.177541
$435$	0	0
$436$	−45.3625	−2.17247
$437$	−8.23600	−0.393981
$438$	0	0
$439$	23.9583	1.14347	0.571734	−	0.820439i	$-0.306271\pi$
$439$	23.9583	1.14347	0.571734	+	0.820439i	$0.306271\pi$
$440$	0	0
$441$	0	0
$442$	−122.190	−5.81196
$443$	17.4925	0.831092	0.415546	−	0.909572i	$-0.363591\pi$
$443$	17.4925	0.831092	0.415546	+	0.909572i	$0.363591\pi$
$444$	0	0
$445$	0	0
$446$	−21.4663	−1.01646
$447$	0	0
$448$	2.08487	0.0985010
$449$	−4.22200	−0.199249	−0.0996243	−	0.995025i	$-0.531764\pi$
$449$	−4.22200	−0.199249	−0.0996243	+	0.995025i	$0.531764\pi$
$450$	0	0
$451$	16.6471	0.783884
$452$	−12.8267	−0.603318
$453$	0	0
$454$	50.7866	2.38353
$455$	0	0
$456$	0	0
$457$	1.28562	0.0601389	0.0300694	−	0.999548i	$-0.490427\pi$
$457$	1.28562	0.0601389	0.0300694	+	0.999548i	$0.490427\pi$
$458$	4.60456	0.215157
$459$	0	0
$460$	0	0
$461$	23.8534	1.11097	0.555483	−	0.831528i	$-0.312534\pi$
$461$	23.8534	1.11097	0.555483	+	0.831528i	$0.312534\pi$
$462$	0	0
$463$	−14.1402	−0.657152	−0.328576	−	0.944478i	$-0.606569\pi$
$463$	−14.1402	−0.657152	−0.328576	+	0.944478i	$0.606569\pi$
$464$	−0.424361	−0.0197005
$465$	0	0
$466$	25.0962	1.16256
$467$	7.06487	0.326923	0.163462	−	0.986550i	$-0.447734\pi$
$467$	7.06487	0.326923	0.163462	+	0.986550i	$0.447734\pi$
$468$	0	0
$469$	−2.39050	−0.110383
$470$	0	0
$471$	0	0
$472$	18.2507	0.840056
$473$	12.8081	0.588916
$474$	0	0
$475$	0	0
$476$	−3.83353	−0.175710
$477$	0	0
$478$	5.08020	0.232363
$479$	−22.9561	−1.04889	−0.524444	−	0.851445i	$-0.675727\pi$
$479$	−22.9561	−1.04889	−0.524444	+	0.851445i	$0.675727\pi$
$480$	0	0
$481$	−30.1900	−1.37654
$482$	32.8342	1.49556
$483$	0	0
$484$	−1.38759	−0.0630723
$485$	0	0
$486$	0	0
$487$	27.0319	1.22493	0.612467	−	0.790496i	$-0.290177\pi$
$487$	27.0319	1.22493	0.612467	+	0.790496i	$0.290177\pi$
$488$	29.5175	1.33619
$489$	0	0
$490$	0	0
$491$	11.5498	0.521235	0.260617	−	0.965442i	$-0.416074\pi$
$491$	11.5498	0.521235	0.260617	+	0.965442i	$0.416074\pi$
$492$	0	0
$493$	7.63808	0.344002
$494$	63.8340	2.87203
$495$	0	0
$496$	−4.33036	−0.194439
$497$	0.548966	0.0246245
$498$	0	0
$499$	3.77942	0.169190	0.0845950	−	0.996415i	$-0.473040\pi$
$499$	3.77942	0.169190	0.0845950	+	0.996415i	$0.473040\pi$
$500$	0	0
$501$	0	0
$502$	36.7346	1.63955
$503$	−14.7684	−0.658489	−0.329244	−	0.944245i	$-0.606794\pi$
$503$	−14.7684	−0.658489	−0.329244	+	0.944245i	$0.606794\pi$
$504$	0	0
$505$	0	0
$506$	15.2036	0.675881
$507$	0	0
$508$	11.6036	0.514826
$509$	42.4182	1.88015	0.940077	−	0.340962i	$-0.110753\pi$
$509$	42.4182	1.88015	0.940077	+	0.340962i	$0.110753\pi$
$510$	0	0
$511$	−1.93098	−0.0854217
$512$	4.79143	0.211753
$513$	0	0
$514$	−7.63831	−0.336911
$515$	0	0
$516$	0	0
$517$	18.2489	0.802586
$518$	−1.55064	−0.0681312
$519$	0	0
$520$	0	0
$521$	−20.3409	−0.891152	−0.445576	−	0.895244i	$-0.647001\pi$
$521$	−20.3409	−0.891152	−0.445576	+	0.895244i	$0.647001\pi$
$522$	0	0
$523$	−1.52190	−0.0665481	−0.0332740	−	0.999446i	$-0.510593\pi$
$523$	−1.52190	−0.0665481	−0.0332740	+	0.999446i	$0.510593\pi$
$524$	31.1651	1.36145
$525$	0	0
$526$	4.07306	0.177594
$527$	77.9421	3.39521
$528$	0	0
$529$	−18.7398	−0.814775
$530$	0	0
$531$	0	0
$532$	2.00271	0.0868283
$533$	−36.1540	−1.56601
$534$	0	0
$535$	0	0
$536$	38.6074	1.66758
$537$	0	0
$538$	0.265359	0.0114404
$539$	−22.6620	−0.976123
$540$	0	0
$541$	−9.40679	−0.404430	−0.202215	−	0.979341i	$-0.564814\pi$
$541$	−9.40679	−0.404430	−0.202215	+	0.979341i	$0.564814\pi$
$542$	−9.21049	−0.395624
$543$	0	0
$544$	−46.7944	−2.00629
$545$	0	0
$546$	0	0
$547$	−19.7232	−0.843304	−0.421652	−	0.906758i	$-0.638550\pi$
$547$	−19.7232	−0.843304	−0.421652	+	0.906758i	$0.638550\pi$
$548$	12.0167	0.513328
$549$	0	0
$550$	0	0
$551$	−3.99028	−0.169991
$552$	0	0
$553$	2.03762	0.0866483
$554$	28.4358	1.20812
$555$	0	0
$556$	−6.49749	−0.275555
$557$	9.01986	0.382184	0.191092	−	0.981572i	$-0.438797\pi$
$557$	9.01986	0.382184	0.191092	+	0.981572i	$0.438797\pi$
$558$	0	0
$559$	−27.8164	−1.17651
$560$	0	0
$561$	0	0
$562$	6.41918	0.270777
$563$	17.5283	0.738728	0.369364	−	0.929285i	$-0.379576\pi$
$563$	17.5283	0.738728	0.369364	+	0.929285i	$0.379576\pi$
$564$	0	0
$565$	0	0
$566$	−63.9845	−2.68947
$567$	0	0
$568$	−8.86599	−0.372009
$569$	−32.3981	−1.35820	−0.679099	−	0.734046i	$-0.737629\pi$
$569$	−32.3981	−1.35820	−0.679099	+	0.734046i	$0.737629\pi$
$570$	0	0
$571$	−11.8393	−0.495458	−0.247729	−	0.968829i	$-0.579684\pi$
$571$	−11.8393	−0.495458	−0.247729	+	0.968829i	$0.579684\pi$
$572$	−71.9778	−3.00954
$573$	0	0
$574$	−1.85697	−0.0775085
$575$	0	0
$576$	0	0
$577$	0.192778	0.00802545	0.00401273	−	0.999992i	$-0.498723\pi$
$577$	0.192778	0.00802545	0.00401273	+	0.999992i	$0.498723\pi$
$578$	93.7165	3.89809
$579$	0	0
$580$	0	0
$581$	0.415374	0.0172326
$582$	0	0
$583$	39.2587	1.62593
$584$	31.1861	1.29049
$585$	0	0
$586$	49.7227	2.05403
$587$	34.5089	1.42433	0.712167	−	0.702010i	$-0.247714\pi$
$587$	34.5089	1.42433	0.712167	+	0.702010i	$0.247714\pi$
$588$	0	0
$589$	−40.7184	−1.67777
$590$	0	0
$591$	0	0
$592$	−1.81548	−0.0746158
$593$	2.42748	0.0996846	0.0498423	−	0.998757i	$-0.484128\pi$
$593$	2.42748	0.0996846	0.0498423	+	0.998757i	$0.484128\pi$
$594$	0	0
$595$	0	0
$596$	4.60218	0.188513
$597$	0	0
$598$	−33.0189	−1.35024
$599$	−29.3312	−1.19844	−0.599220	−	0.800585i	$-0.704522\pi$
$599$	−29.3312	−1.19844	−0.599220	+	0.800585i	$0.704522\pi$
$600$	0	0
$601$	29.5631	1.20590	0.602952	−	0.797778i	$-0.293991\pi$
$601$	29.5631	1.20590	0.602952	+	0.797778i	$0.293991\pi$
$602$	−1.42873	−0.0582306
$603$	0	0
$604$	−63.1831	−2.57089
$605$	0	0
$606$	0	0
$607$	−33.1756	−1.34656	−0.673278	−	0.739390i	$-0.735114\pi$
$607$	−33.1756	−1.34656	−0.673278	+	0.739390i	$0.735114\pi$
$608$	24.4463	0.991427
$609$	0	0
$610$	0	0
$611$	−39.6327	−1.60337
$612$	0	0
$613$	13.9979	0.565371	0.282686	−	0.959213i	$-0.408775\pi$
$613$	13.9979	0.565371	0.282686	+	0.959213i	$0.408775\pi$
$614$	1.04637	0.0422279
$615$	0	0
$616$	−1.34152	−0.0540514
$617$	−29.9344	−1.20511	−0.602556	−	0.798077i	$-0.705851\pi$
$617$	−29.9344	−1.20511	−0.602556	+	0.798077i	$0.705851\pi$
$618$	0	0
$619$	−14.9032	−0.599009	−0.299505	−	0.954095i	$-0.596821\pi$
$619$	−14.9032	−0.599009	−0.299505	+	0.954095i	$0.596821\pi$
$620$	0	0
$621$	0	0
$622$	−62.4561	−2.50426
$623$	0.692574	0.0277474
$624$	0	0
$625$	0	0
$626$	−33.8050	−1.35112
$627$	0	0
$628$	−22.2594	−0.888245
$629$	32.6769	1.30291
$630$	0	0
$631$	−28.8643	−1.14907	−0.574534	−	0.818480i	$-0.694817\pi$
$631$	−28.8643	−1.14907	−0.574534	+	0.818480i	$0.694817\pi$
$632$	−32.9082	−1.30902
$633$	0	0
$634$	−51.9735	−2.06413
$635$	0	0
$636$	0	0
$637$	49.2171	1.95005
$638$	7.36601	0.291623
$639$	0	0
$640$	0	0
$641$	−44.0167	−1.73856	−0.869278	−	0.494324i	$-0.835415\pi$
$641$	−44.0167	−1.73856	−0.869278	+	0.494324i	$0.835415\pi$
$642$	0	0
$643$	29.6239	1.16825	0.584125	−	0.811663i	$-0.301438\pi$
$643$	29.6239	1.16825	0.584125	+	0.811663i	$0.301438\pi$
$644$	−1.03592	−0.0408211
$645$	0	0
$646$	−69.0923	−2.71840
$647$	−39.3943	−1.54875	−0.774375	−	0.632727i	$-0.781936\pi$
$647$	−39.3943	−1.54875	−0.774375	+	0.632727i	$0.781936\pi$
$648$	0	0
$649$	22.9654	0.901473
$650$	0	0
$651$	0	0
$652$	20.6776	0.809796
$653$	22.3518	0.874692	0.437346	−	0.899293i	$-0.355918\pi$
$653$	22.3518	0.874692	0.437346	+	0.899293i	$0.355918\pi$
$654$	0	0
$655$	0	0
$656$	−2.17413	−0.0848855
$657$	0	0
$658$	−2.03565	−0.0793577
$659$	−42.1979	−1.64380	−0.821898	−	0.569634i	$-0.807085\pi$
$659$	−42.1979	−1.64380	−0.821898	+	0.569634i	$0.807085\pi$
$660$	0	0
$661$	8.38035	0.325958	0.162979	−	0.986630i	$-0.447890\pi$
$661$	8.38035	0.325958	0.162979	+	0.986630i	$0.447890\pi$
$662$	−7.32559	−0.284717
$663$	0	0
$664$	−6.70843	−0.260337
$665$	0	0
$666$	0	0
$667$	2.06402	0.0799191
$668$	−33.4746	−1.29517
$669$	0	0
$670$	0	0
$671$	37.1428	1.43388
$672$	0	0
$673$	3.13725	0.120932	0.0604660	−	0.998170i	$-0.480741\pi$
$673$	3.13725	0.120932	0.0604660	+	0.998170i	$0.480741\pi$
$674$	−65.2941	−2.51504
$675$	0	0
$676$	115.512	4.44279
$677$	16.2776	0.625600	0.312800	−	0.949819i	$-0.398733\pi$
$677$	16.2776	0.625600	0.312800	+	0.949819i	$0.398733\pi$
$678$	0	0
$679$	−0.620929	−0.0238291
$680$	0	0
$681$	0	0
$682$	75.1658	2.87825
$683$	−1.78249	−0.0682050	−0.0341025	−	0.999418i	$-0.510857\pi$
$683$	−1.78249	−0.0682050	−0.0341025	+	0.999418i	$0.510857\pi$
$684$	0	0
$685$	0	0
$686$	5.06512	0.193387
$687$	0	0
$688$	−1.67275	−0.0637728
$689$	−85.2614	−3.24820
$690$	0	0
$691$	−20.8504	−0.793187	−0.396594	−	0.917994i	$-0.629808\pi$
$691$	−20.8504	−0.793187	−0.396594	+	0.917994i	$0.629808\pi$
$692$	−43.5699	−1.65628
$693$	0	0
$694$	−37.2177	−1.41276
$695$	0	0
$696$	0	0
$697$	39.1322	1.48224
$698$	−41.2828	−1.56258
$699$	0	0
$700$	0	0
$701$	−32.5504	−1.22941	−0.614707	−	0.788756i	$-0.710726\pi$
$701$	−32.5504	−1.22941	−0.614707	+	0.788756i	$0.710726\pi$
$702$	0	0
$703$	−17.0710	−0.643845
$704$	−42.3698	−1.59687
$705$	0	0
$706$	−42.7978	−1.61072
$707$	2.46297	0.0926295
$708$	0	0
$709$	5.93195	0.222779	0.111390	−	0.993777i	$-0.464470\pi$
$709$	5.93195	0.222779	0.111390	+	0.993777i	$0.464470\pi$
$710$	0	0
$711$	0	0
$712$	−11.1853	−0.419187
$713$	21.0621	0.788781
$714$	0	0
$715$	0	0
$716$	34.8767	1.30340
$717$	0	0
$718$	−32.7574	−1.22250
$719$	13.5279	0.504505	0.252252	−	0.967661i	$-0.418829\pi$
$719$	13.5279	0.504505	0.252252	+	0.967661i	$0.418829\pi$
$720$	0	0
$721$	1.22871	0.0457594
$722$	−6.97701	−0.259657
$723$	0	0
$724$	34.3581	1.27691
$725$	0	0
$726$	0	0
$727$	12.7407	0.472525	0.236263	−	0.971689i	$-0.424077\pi$
$727$	12.7407	0.472525	0.236263	+	0.971689i	$0.424077\pi$
$728$	2.91350	0.107981
$729$	0	0
$730$	0	0
$731$	30.1078	1.11358
$732$	0	0
$733$	−30.9422	−1.14288	−0.571438	−	0.820645i	$-0.693614\pi$
$733$	−30.9422	−1.14288	−0.571438	+	0.820645i	$0.693614\pi$
$734$	47.8716	1.76697
$735$	0	0
$736$	−12.6451	−0.466105
$737$	48.5809	1.78950
$738$	0	0
$739$	35.0047	1.28767	0.643834	−	0.765166i	$-0.277343\pi$
$739$	35.0047	1.28767	0.643834	+	0.765166i	$0.277343\pi$
$740$	0	0
$741$	0	0
$742$	−4.37926	−0.160768
$743$	−12.5515	−0.460468	−0.230234	−	0.973135i	$-0.573949\pi$
$743$	−12.5515	−0.460468	−0.230234	+	0.973135i	$0.573949\pi$
$744$	0	0
$745$	0	0
$746$	38.8068	1.42082
$747$	0	0
$748$	77.9069	2.84856
$749$	0.736956	0.0269278
$750$	0	0
$751$	−33.8278	−1.23439	−0.617197	−	0.786809i	$-0.711732\pi$
$751$	−33.8278	−1.23439	−0.617197	+	0.786809i	$0.711732\pi$
$752$	−2.38332	−0.0869108
$753$	0	0
$754$	−15.9974	−0.582591
$755$	0	0
$756$	0	0
$757$	6.60568	0.240088	0.120044	−	0.992769i	$-0.461697\pi$
$757$	6.60568	0.240088	0.120044	+	0.992769i	$0.461697\pi$
$758$	−0.468553	−0.0170186
$759$	0	0
$760$	0	0
$761$	22.8638	0.828814	0.414407	−	0.910092i	$-0.363989\pi$
$761$	22.8638	0.828814	0.414407	+	0.910092i	$0.363989\pi$
$762$	0	0
$763$	2.31051	0.0836461
$764$	−56.9656	−2.06094
$765$	0	0
$766$	37.1103	1.34085
$767$	−49.8760	−1.80092
$768$	0	0
$769$	−35.0078	−1.26241	−0.631206	−	0.775615i	$-0.717440\pi$
$769$	−35.0078	−1.26241	−0.631206	+	0.775615i	$0.717440\pi$
$770$	0	0
$771$	0	0
$772$	54.4016	1.95796
$773$	17.7019	0.636691	0.318346	−	0.947975i	$-0.396873\pi$
$773$	17.7019	0.636691	0.318346	+	0.947975i	$0.396873\pi$
$774$	0	0
$775$	0	0
$776$	10.0282	0.359992
$777$	0	0
$778$	−26.6655	−0.956005
$779$	−20.4434	−0.732460
$780$	0	0
$781$	−11.1564	−0.399206
$782$	35.7388	1.27802
$783$	0	0
$784$	2.95968	0.105703
$785$	0	0
$786$	0	0
$787$	−5.13176	−0.182928	−0.0914638	−	0.995808i	$-0.529155\pi$
$787$	−5.13176	−0.182928	−0.0914638	+	0.995808i	$0.529155\pi$
$788$	−32.7321	−1.16603
$789$	0	0
$790$	0	0
$791$	0.653321	0.0232294
$792$	0	0
$793$	−80.6662	−2.86454
$794$	11.2471	0.399145
$795$	0	0
$796$	36.2783	1.28585
$797$	27.5492	0.975842	0.487921	−	0.872888i	$-0.337755\pi$
$797$	27.5492	0.975842	0.487921	+	0.872888i	$0.337755\pi$
$798$	0	0
$799$	42.8974	1.51760
$800$	0	0
$801$	0	0
$802$	−82.0562	−2.89751
$803$	39.2424	1.38484
$804$	0	0
$805$	0	0
$806$	−163.244	−5.75002
$807$	0	0
$808$	−39.7778	−1.39938
$809$	5.51555	0.193917	0.0969583	−	0.995288i	$-0.469089\pi$
$809$	5.51555	0.193917	0.0969583	+	0.995288i	$0.469089\pi$
$810$	0	0
$811$	−17.1711	−0.602959	−0.301479	−	0.953473i	$-0.597480\pi$
$811$	−17.1711	−0.602959	−0.301479	+	0.953473i	$0.597480\pi$
$812$	−0.501897	−0.0176131
$813$	0	0
$814$	31.5129	1.10453
$815$	0	0
$816$	0	0
$817$	−15.7288	−0.550283
$818$	−36.6355	−1.28093
$819$	0	0
$820$	0	0
$821$	−6.53740	−0.228157	−0.114078	−	0.993472i	$-0.536391\pi$
$821$	−6.53740	−0.228157	−0.114078	+	0.993472i	$0.536391\pi$
$822$	0	0
$823$	35.8588	1.24996	0.624979	−	0.780641i	$-0.285107\pi$
$823$	35.8588	1.24996	0.624979	+	0.780641i	$0.285107\pi$
$824$	−19.8440	−0.691299
$825$	0	0
$826$	−2.56177	−0.0891354
$827$	−2.36177	−0.0821269	−0.0410635	−	0.999157i	$-0.513075\pi$
$827$	−2.36177	−0.0821269	−0.0410635	+	0.999157i	$0.513075\pi$
$828$	0	0
$829$	54.0332	1.87665	0.938325	−	0.345754i	$-0.112377\pi$
$829$	54.0332	1.87665	0.938325	+	0.345754i	$0.112377\pi$
$830$	0	0
$831$	0	0
$832$	92.0182	3.19016
$833$	−53.2713	−1.84574
$834$	0	0
$835$	0	0
$836$	−40.7000	−1.40764
$837$	0	0
$838$	76.0663	2.62767
$839$	−27.6639	−0.955063	−0.477532	−	0.878615i	$-0.658468\pi$
$839$	−27.6639	−0.955063	−0.477532	+	0.878615i	$0.658468\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	61.1730	2.10816
$843$	0	0
$844$	−66.4278	−2.28654
$845$	0	0
$846$	0	0
$847$	0.0706761	0.00242846
$848$	−5.12721	−0.176069
$849$	0	0
$850$	0	0
$851$	8.83017	0.302694
$852$	0	0
$853$	−16.7180	−0.572413	−0.286207	−	0.958168i	$-0.592394\pi$
$853$	−16.7180	−0.572413	−0.286207	+	0.958168i	$0.592394\pi$
$854$	−4.14324	−0.141779
$855$	0	0
$856$	−11.9021	−0.406805
$857$	−50.6304	−1.72950	−0.864750	−	0.502202i	$-0.832523\pi$
$857$	−50.6304	−1.72950	−0.864750	+	0.502202i	$0.832523\pi$
$858$	0	0
$859$	−23.0473	−0.786363	−0.393182	−	0.919461i	$-0.628626\pi$
$859$	−23.0473	−0.786363	−0.393182	+	0.919461i	$0.628626\pi$
$860$	0	0
$861$	0	0
$862$	−1.88693	−0.0642692
$863$	45.4971	1.54874	0.774370	−	0.632733i	$-0.218067\pi$
$863$	45.4971	1.54874	0.774370	+	0.632733i	$0.218067\pi$
$864$	0	0
$865$	0	0
$866$	51.1531	1.73825
$867$	0	0
$868$	−5.12156	−0.173837
$869$	−41.4095	−1.40472
$870$	0	0
$871$	−105.507	−3.57498
$872$	−37.3155	−1.26366
$873$	0	0
$874$	−18.6706	−0.631543
$875$	0	0
$876$	0	0
$877$	−6.44199	−0.217531	−0.108765	−	0.994067i	$-0.534690\pi$
$877$	−6.44199	−0.217531	−0.108765	+	0.994067i	$0.534690\pi$
$878$	54.3124	1.83295
$879$	0	0
$880$	0	0
$881$	43.3346	1.45998	0.729990	−	0.683458i	$-0.239525\pi$
$881$	43.3346	1.45998	0.729990	+	0.683458i	$0.239525\pi$
$882$	0	0
$883$	24.6736	0.830333	0.415166	−	0.909745i	$-0.363723\pi$
$883$	24.6736	0.830333	0.415166	+	0.909745i	$0.363723\pi$
$884$	−169.197	−5.69072
$885$	0	0
$886$	39.6546	1.33222
$887$	37.1527	1.24747	0.623733	−	0.781637i	$-0.285615\pi$
$887$	37.1527	1.24747	0.623733	+	0.781637i	$0.285615\pi$
$888$	0	0
$889$	−0.591021	−0.0198222
$890$	0	0
$891$	0	0
$892$	−29.7247	−0.995255
$893$	−22.4104	−0.749936
$894$	0	0
$895$	0	0
$896$	2.76723	0.0924466
$897$	0	0
$898$	−9.57108	−0.319391
$899$	10.2044	0.340336
$900$	0	0
$901$	92.2847	3.07445
$902$	37.7383	1.25655
$903$	0	0
$904$	−10.5514	−0.350933
$905$	0	0
$906$	0	0
$907$	−7.49457	−0.248853	−0.124427	−	0.992229i	$-0.539709\pi$
$907$	−7.49457	−0.248853	−0.124427	+	0.992229i	$0.539709\pi$
$908$	70.3248	2.33381
$909$	0	0
$910$	0	0
$911$	−48.6818	−1.61290	−0.806449	−	0.591303i	$-0.798614\pi$
$911$	−48.6818	−1.61290	−0.806449	+	0.591303i	$0.798614\pi$
$912$	0	0
$913$	−8.44143	−0.279371
$914$	2.91444	0.0964012
$915$	0	0
$916$	6.37598	0.210668
$917$	−1.58737	−0.0524197
$918$	0	0
$919$	−26.1563	−0.862816	−0.431408	−	0.902157i	$-0.641983\pi$
$919$	−26.1563	−0.862816	−0.431408	+	0.902157i	$0.641983\pi$
$920$	0	0
$921$	0	0
$922$	54.0746	1.78085
$923$	24.2293	0.797515
$924$	0	0
$925$	0	0
$926$	−32.0552	−1.05340
$927$	0	0
$928$	−6.12646	−0.201111
$929$	8.15366	0.267513	0.133756	−	0.991014i	$-0.457296\pi$
$929$	8.15366	0.267513	0.133756	+	0.991014i	$0.457296\pi$
$930$	0	0
$931$	27.8299	0.912089
$932$	34.7509	1.13830
$933$	0	0
$934$	16.0157	0.524051
$935$	0	0
$936$	0	0
$937$	−41.9169	−1.36936	−0.684682	−	0.728842i	$-0.740059\pi$
$937$	−41.9169	−1.36936	−0.684682	+	0.728842i	$0.740059\pi$
$938$	−5.41915	−0.176941
$939$	0	0
$940$	0	0
$941$	−14.4789	−0.471998	−0.235999	−	0.971753i	$-0.575836\pi$
$941$	−14.4789	−0.471998	−0.235999	+	0.971753i	$0.575836\pi$
$942$	0	0
$943$	10.5746	0.344356
$944$	−2.99931	−0.0976191
$945$	0	0
$946$	29.0353	0.944020
$947$	1.83838	0.0597392	0.0298696	−	0.999554i	$-0.490491\pi$
$947$	1.83838	0.0597392	0.0298696	+	0.999554i	$0.490491\pi$
$948$	0	0
$949$	−85.2262	−2.76656
$950$	0	0
$951$	0	0
$952$	−3.15349	−0.102205
$953$	−49.5140	−1.60391	−0.801957	−	0.597382i	$-0.796208\pi$
$953$	−49.5140	−1.60391	−0.801957	+	0.597382i	$0.796208\pi$
$954$	0	0
$955$	0	0
$956$	7.03461	0.227516
$957$	0	0
$958$	−52.0403	−1.68135
$959$	−0.612063	−0.0197646
$960$	0	0
$961$	73.1299	2.35903
$962$	−68.4393	−2.20657
$963$	0	0
$964$	45.4658	1.46436
$965$	0	0
$966$	0	0
$967$	−5.90299	−0.189827	−0.0949137	−	0.995486i	$-0.530258\pi$
$967$	−5.90299	−0.189827	−0.0949137	+	0.995486i	$0.530258\pi$
$968$	−1.14144	−0.0366874
$969$	0	0
$970$	0	0
$971$	35.7964	1.14876	0.574380	−	0.818588i	$-0.305243\pi$
$971$	35.7964	1.14876	0.574380	+	0.818588i	$0.305243\pi$
$972$	0	0
$973$	0.330946	0.0106096
$974$	61.2801	1.96354
$975$	0	0
$976$	−4.85088	−0.155273
$977$	−1.79673	−0.0574825	−0.0287412	−	0.999587i	$-0.509150\pi$
$977$	−1.79673	−0.0574825	−0.0287412	+	0.999587i	$0.509150\pi$
$978$	0	0
$979$	−14.0748	−0.449834
$980$	0	0
$981$	0	0
$982$	26.1828	0.835528
$983$	6.24151	0.199073	0.0995365	−	0.995034i	$-0.468264\pi$
$983$	6.24151	0.199073	0.0995365	+	0.995034i	$0.468264\pi$
$984$	0	0
$985$	0	0
$986$	17.3152	0.551427
$987$	0	0
$988$	88.3917	2.81211
$989$	8.13593	0.258708
$990$	0	0
$991$	41.0410	1.30371	0.651855	−	0.758344i	$-0.273991\pi$
$991$	41.0410	1.30371	0.651855	+	0.758344i	$0.273991\pi$
$992$	−62.5169	−1.98491
$993$	0	0
$994$	1.24448	0.0394725
$995$	0	0
$996$	0	0
$997$	−29.8910	−0.946657	−0.473329	−	0.880886i	$-0.656948\pi$
$997$	−29.8910	−0.946657	−0.473329	+	0.880886i	$0.656948\pi$
$998$	8.56776	0.271208
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6525.2.a.bx.1.6		7
3.2	odd	2		2175.2.a.ba.1.2	✓	7
5.4	even	2		6525.2.a.bu.1.2		7
15.2	even	4		2175.2.c.o.349.3		14
15.8	even	4		2175.2.c.o.349.12		14
15.14	odd	2		2175.2.a.bb.1.6	yes	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2175.2.a.ba.1.2	✓	7	3.2	odd	2
2175.2.a.bb.1.6	yes	7	15.14	odd	2
2175.2.c.o.349.3		14	15.2	even	4
2175.2.c.o.349.12		14	15.8	even	4
6525.2.a.bu.1.2		7	5.4	even	2
6525.2.a.bx.1.6		7	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+2.26695 q^{2} +3.13907 q^{4} -0.159887 q^{7} +2.58223 q^{8} +O(q^{10})\)	\(q+2.26695 q^{2} +3.13907 q^{4} -0.159887 q^{7} +2.58223 q^{8} +3.24930 q^{11} -7.05679 q^{13} -0.362456 q^{14} -0.424361 q^{16} +7.63808 q^{17} -3.99028 q^{19} +7.36601 q^{22} +2.06402 q^{23} -15.9974 q^{26} -0.501897 q^{28} +1.00000 q^{29} +10.2044 q^{31} -6.12646 q^{32} +17.3152 q^{34} +4.27815 q^{37} -9.04576 q^{38} +5.12330 q^{41} +3.94180 q^{43} +10.1998 q^{44} +4.67903 q^{46} +5.61626 q^{47} -6.97444 q^{49} -22.1518 q^{52} +12.0822 q^{53} -0.412864 q^{56} +2.26695 q^{58} +7.06781 q^{59} +11.4310 q^{61} +23.1329 q^{62} -13.0397 q^{64} +14.9512 q^{67} +23.9765 q^{68} -3.43347 q^{71} +12.0772 q^{73} +9.69836 q^{74} -12.5258 q^{76} -0.519521 q^{77} -12.7441 q^{79} +11.6143 q^{82} -2.59792 q^{83} +8.93586 q^{86} +8.39044 q^{88} -4.33165 q^{89} +1.12829 q^{91} +6.47910 q^{92} +12.7318 q^{94} +3.88355 q^{97} -15.8107 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 2 q^{2} + 10 q^{4} + q^{7} + 3 q^{8}+O(q^{10}) \) 7 * q + 2 * q^2 + 10 * q^4 + q^7 + 3 * q^8	\( 7 q + 2 q^{2} + 10 q^{4} + q^{7} + 3 q^{8} - 4 q^{11} + q^{13} - 15 q^{14} + 12 q^{16} + 8 q^{17} + 15 q^{19} - 3 q^{22} + 14 q^{23} - 6 q^{26} + 24 q^{28} + 7 q^{29} + 5 q^{31} + 18 q^{32} + 7 q^{34} + 6 q^{37} - 18 q^{38} - 22 q^{41} + 19 q^{43} - 15 q^{44} - 4 q^{46} + 22 q^{47} + 12 q^{49} - 11 q^{52} + 10 q^{53} - 14 q^{56} + 2 q^{58} - 6 q^{59} + 23 q^{61} + 40 q^{62} + 5 q^{64} + 13 q^{67} - 7 q^{68} - 26 q^{71} + 24 q^{73} + 10 q^{74} + 46 q^{76} + 4 q^{77} + 14 q^{79} - 16 q^{82} + 10 q^{83} - 44 q^{86} + 66 q^{88} - 14 q^{89} + 13 q^{91} + 58 q^{92} - 3 q^{94} + 31 q^{97} - 59 q^{98}+O(q^{100}) \) 7 * q + 2 * q^2 + 10 * q^4 + q^7 + 3 * q^8 - 4 * q^11 + q^13 - 15 * q^14 + 12 * q^16 + 8 * q^17 + 15 * q^19 - 3 * q^22 + 14 * q^23 - 6 * q^26 + 24 * q^28 + 7 * q^29 + 5 * q^31 + 18 * q^32 + 7 * q^34 + 6 * q^37 - 18 * q^38 - 22 * q^41 + 19 * q^43 - 15 * q^44 - 4 * q^46 + 22 * q^47 + 12 * q^49 - 11 * q^52 + 10 * q^53 - 14 * q^56 + 2 * q^58 - 6 * q^59 + 23 * q^61 + 40 * q^62 + 5 * q^64 + 13 * q^67 - 7 * q^68 - 26 * q^71 + 24 * q^73 + 10 * q^74 + 46 * q^76 + 4 * q^77 + 14 * q^79 - 16 * q^82 + 10 * q^83 - 44 * q^86 + 66 * q^88 - 14 * q^89 + 13 * q^91 + 58 * q^92 - 3 * q^94 + 31 * q^97 - 59 * q^98

Introduction

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