LMFDB - Embedded newform 5625.2.a.u.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5625,2,Mod(1,5625)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5625, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("5625.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5625 = 3^{2} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5625.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:	$44.9158511370$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.13366265625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 12x^{6} + 10x^{5} + 41x^{4} - 20x^{3} - 48x^{2} + 8x + 16 $ x^8 - x^7 - 12x^6 + 10x^5 + 41x^4 - 20x^3 - 48x^2 + 8x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1875)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-2.69767$ of defining polynomial
Character	$\chi$	$=$	5625.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.69767 q^{2} +5.27745 q^{4} -3.56649 q^{7} +8.84149 q^{8} +O(q^{10})$	$q+2.69767 q^{2} +5.27745 q^{4} -3.56649 q^{7} +8.84149 q^{8} +0.0695627 q^{11} +4.85751 q^{13} -9.62122 q^{14} +13.2966 q^{16} -3.03430 q^{17} +5.05259 q^{19} +0.187658 q^{22} +7.43231 q^{23} +13.1040 q^{26} -18.8220 q^{28} -1.95440 q^{29} -5.63589 q^{31} +18.1868 q^{32} -8.18554 q^{34} +6.21419 q^{37} +13.6302 q^{38} +5.63577 q^{41} +0.244040 q^{43} +0.367114 q^{44} +20.0499 q^{46} +3.23073 q^{47} +5.71984 q^{49} +25.6353 q^{52} -8.37482 q^{53} -31.5331 q^{56} -5.27233 q^{58} +1.60530 q^{59} +3.12503 q^{61} -15.2038 q^{62} +22.4690 q^{64} -2.94777 q^{67} -16.0133 q^{68} -7.25925 q^{71} +3.69960 q^{73} +16.7638 q^{74} +26.6648 q^{76} -0.248095 q^{77} +4.70868 q^{79} +15.2035 q^{82} +8.80783 q^{83} +0.658341 q^{86} +0.615038 q^{88} +3.55155 q^{89} -17.3243 q^{91} +39.2236 q^{92} +8.71545 q^{94} +8.80534 q^{97} +15.4303 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - q^{2} + 9 q^{4} + 12 q^{7} - 3 q^{8}+O(q^{10}) $ 8 * q - q^2 + 9 * q^4 + 12 * q^7 - 3 * q^8	$ 8 q - q^{2} + 9 q^{4} + 12 q^{7} - 3 q^{8} - 12 q^{11} + 14 q^{13} - 16 q^{14} + 15 q^{16} + q^{17} + 16 q^{19} + 18 q^{22} + 4 q^{23} + 34 q^{26} - 21 q^{28} - 2 q^{29} + 13 q^{31} + 18 q^{32} - 37 q^{34} - 8 q^{37} + 24 q^{38} + 12 q^{41} + 20 q^{43} - 47 q^{44} + 33 q^{46} + 15 q^{47} + 30 q^{49} - q^{52} + 4 q^{53} - 60 q^{56} + 2 q^{58} - 14 q^{59} + 10 q^{61} - 4 q^{62} + 41 q^{64} + 19 q^{67} + 33 q^{68} - 21 q^{71} - 19 q^{73} + 9 q^{74} - q^{76} + 11 q^{77} + 10 q^{79} + 24 q^{82} + 27 q^{83} - 42 q^{86} + 53 q^{88} + 9 q^{89} - 12 q^{91} + 63 q^{92} + 14 q^{94} + 24 q^{97} + 24 q^{98}+O(q^{100}) $ 8 * q - q^2 + 9 * q^4 + 12 * q^7 - 3 * q^8 - 12 * q^11 + 14 * q^13 - 16 * q^14 + 15 * q^16 + q^17 + 16 * q^19 + 18 * q^22 + 4 * q^23 + 34 * q^26 - 21 * q^28 - 2 * q^29 + 13 * q^31 + 18 * q^32 - 37 * q^34 - 8 * q^37 + 24 * q^38 + 12 * q^41 + 20 * q^43 - 47 * q^44 + 33 * q^46 + 15 * q^47 + 30 * q^49 - q^52 + 4 * q^53 - 60 * q^56 + 2 * q^58 - 14 * q^59 + 10 * q^61 - 4 * q^62 + 41 * q^64 + 19 * q^67 + 33 * q^68 - 21 * q^71 - 19 * q^73 + 9 * q^74 - q^76 + 11 * q^77 + 10 * q^79 + 24 * q^82 + 27 * q^83 - 42 * q^86 + 53 * q^88 + 9 * q^89 - 12 * q^91 + 63 * q^92 + 14 * q^94 + 24 * q^97 + 24 * 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 $

$2$	2.69767	1.90754	0.953772	−	0.300531i	$-0.0971640\pi$
$2$	2.69767	1.90754	0.953772	+	0.300531i	$0.0971640\pi$
$3$	0	0
$3$	0	0
$4$	5.27745	2.63872
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.56649	−1.34801	−0.674003	−	0.738729i	$-0.735427\pi$
$7$	−3.56649	−1.34801	−0.674003	+	0.738729i	$0.735427\pi$
$8$	8.84149	3.12594
$9$	0	0
$10$	0	0
$11$	0.0695627	0.0209740	0.0104870	−	0.999945i	$-0.496662\pi$
$11$	0.0695627	0.0209740	0.0104870	+	0.999945i	$0.496662\pi$
$12$	0	0
$13$	4.85751	1.34723	0.673616	−	0.739082i	$-0.264740\pi$
$13$	4.85751	1.34723	0.673616	+	0.739082i	$0.264740\pi$
$14$	−9.62122	−2.57138
$15$	0	0
$16$	13.2966	3.32414
$17$	−3.03430	−0.735925	−0.367963	−	0.929841i	$-0.619945\pi$
$17$	−3.03430	−0.735925	−0.367963	+	0.929841i	$0.619945\pi$
$18$	0	0
$19$	5.05259	1.15914	0.579571	−	0.814922i	$-0.303220\pi$
$19$	5.05259	1.15914	0.579571	+	0.814922i	$0.303220\pi$
$20$	0	0
$21$	0	0
$22$	0.187658	0.0400087
$23$	7.43231	1.54974	0.774872	−	0.632119i	$-0.217815\pi$
$23$	7.43231	1.54974	0.774872	+	0.632119i	$0.217815\pi$
$24$	0	0
$25$	0	0
$26$	13.1040	2.56990
$27$	0	0
$28$	−18.8220	−3.55702
$29$	−1.95440	−0.362923	−0.181461	−	0.983398i	$-0.558083\pi$
$29$	−1.95440	−0.362923	−0.181461	+	0.983398i	$0.558083\pi$
$30$	0	0
$31$	−5.63589	−1.01224	−0.506118	−	0.862464i	$-0.668920\pi$
$31$	−5.63589	−1.01224	−0.506118	+	0.862464i	$0.668920\pi$
$32$	18.1868	3.21500
$33$	0	0
$34$	−8.18554	−1.40381
$35$	0	0
$36$	0	0
$37$	6.21419	1.02161	0.510803	−	0.859698i	$-0.329348\pi$
$37$	6.21419	1.02161	0.510803	+	0.859698i	$0.329348\pi$
$38$	13.6302	2.21112
$39$	0	0
$40$	0	0
$41$	5.63577	0.880159	0.440079	−	0.897959i	$-0.354950\pi$
$41$	5.63577	0.880159	0.440079	+	0.897959i	$0.354950\pi$
$42$	0	0
$43$	0.244040	0.0372158	0.0186079	−	0.999827i	$-0.494077\pi$
$43$	0.244040	0.0372158	0.0186079	+	0.999827i	$0.494077\pi$
$44$	0.367114	0.0553445
$45$	0	0
$46$	20.0499	2.95620
$47$	3.23073	0.471250	0.235625	−	0.971844i	$-0.424286\pi$
$47$	3.23073	0.471250	0.235625	+	0.971844i	$0.424286\pi$
$48$	0	0
$49$	5.71984	0.817120
$50$	0	0
$51$	0	0
$52$	25.6353	3.55497
$53$	−8.37482	−1.15037	−0.575185	−	0.818023i	$-0.695070\pi$
$53$	−8.37482	−1.15037	−0.575185	+	0.818023i	$0.695070\pi$
$54$	0	0
$55$	0	0
$56$	−31.5331	−4.21378
$57$	0	0
$58$	−5.27233	−0.692291
$59$	1.60530	0.208993	0.104496	−	0.994525i	$-0.466677\pi$
$59$	1.60530	0.208993	0.104496	+	0.994525i	$0.466677\pi$
$60$	0	0
$61$	3.12503	0.400119	0.200060	−	0.979784i	$-0.435886\pi$
$61$	3.12503	0.400119	0.200060	+	0.979784i	$0.435886\pi$
$62$	−15.2038	−1.93088
$63$	0	0
$64$	22.4690	2.80862
$65$	0	0
$66$	0	0
$67$	−2.94777	−0.360127	−0.180063	−	0.983655i	$-0.557630\pi$
$67$	−2.94777	−0.360127	−0.180063	+	0.983655i	$0.557630\pi$
$68$	−16.0133	−1.94190
$69$	0	0
$70$	0	0
$71$	−7.25925	−0.861514	−0.430757	−	0.902468i	$-0.641753\pi$
$71$	−7.25925	−0.861514	−0.430757	+	0.902468i	$0.641753\pi$
$72$	0	0
$73$	3.69960	0.433005	0.216503	−	0.976282i	$-0.430535\pi$
$73$	3.69960	0.433005	0.216503	+	0.976282i	$0.430535\pi$
$74$	16.7638	1.94876
$75$	0	0
$76$	26.6648	3.05866
$77$	−0.248095	−0.0282730
$78$	0	0
$79$	4.70868	0.529768	0.264884	−	0.964280i	$-0.414666\pi$
$79$	4.70868	0.529768	0.264884	+	0.964280i	$0.414666\pi$
$80$	0	0
$81$	0	0
$82$	15.2035	1.67894
$83$	8.80783	0.966785	0.483393	−	0.875404i	$-0.339404\pi$
$83$	8.80783	0.966785	0.483393	+	0.875404i	$0.339404\pi$
$84$	0	0
$85$	0	0
$86$	0.658341	0.0709908
$87$	0	0
$88$	0.615038	0.0655633
$89$	3.55155	0.376463	0.188232	−	0.982125i	$-0.439724\pi$
$89$	3.55155	0.376463	0.188232	+	0.982125i	$0.439724\pi$
$90$	0	0
$91$	−17.3243	−1.81608
$92$	39.2236	4.08934
$93$	0	0
$94$	8.71545	0.898930
$95$	0	0
$96$	0	0
$97$	8.80534	0.894047	0.447023	−	0.894522i	$-0.352484\pi$
$97$	8.80534	0.894047	0.447023	+	0.894522i	$0.352484\pi$
$98$	15.4303	1.55869
$99$	0	0
$100$	0	0
$101$	−13.0891	−1.30241	−0.651207	−	0.758900i	$-0.725737\pi$
$101$	−13.0891	−1.30241	−0.651207	+	0.758900i	$0.725737\pi$
$102$	0	0
$103$	18.8222	1.85461	0.927303	−	0.374312i	$-0.122121\pi$
$103$	18.8222	1.85461	0.927303	+	0.374312i	$0.122121\pi$
$104$	42.9476	4.21136
$105$	0	0
$106$	−22.5925	−2.19438
$107$	6.15954	0.595465	0.297733	−	0.954649i	$-0.403770\pi$
$107$	6.15954	0.595465	0.297733	+	0.954649i	$0.403770\pi$
$108$	0	0
$109$	6.18006	0.591943	0.295971	−	0.955197i	$-0.404357\pi$
$109$	6.18006	0.591943	0.295971	+	0.955197i	$0.404357\pi$
$110$	0	0
$111$	0	0
$112$	−47.4220	−4.48096
$113$	−15.9751	−1.50281	−0.751407	−	0.659839i	$-0.770624\pi$
$113$	−15.9751	−1.50281	−0.751407	+	0.659839i	$0.770624\pi$
$114$	0	0
$115$	0	0
$116$	−10.3142	−0.957653
$117$	0	0
$118$	4.33058	0.398662
$119$	10.8218	0.992031
$120$	0	0
$121$	−10.9952	−0.999560
$122$	8.43031	0.763245
$123$	0	0
$124$	−29.7431	−2.67101
$125$	0	0
$126$	0	0
$127$	19.9990	1.77463	0.887314	−	0.461166i	$-0.152569\pi$
$127$	19.9990	1.77463	0.887314	+	0.461166i	$0.152569\pi$
$128$	24.2404	2.14257
$129$	0	0
$130$	0	0
$131$	−9.50363	−0.830336	−0.415168	−	0.909745i	$-0.636277\pi$
$131$	−9.50363	−0.830336	−0.415168	+	0.909745i	$0.636277\pi$
$132$	0	0
$133$	−18.0200	−1.56253
$134$	−7.95211	−0.686958
$135$	0	0
$136$	−26.8277	−2.30046
$137$	12.5608	1.07314	0.536571	−	0.843855i	$-0.319719\pi$
$137$	12.5608	1.07314	0.536571	+	0.843855i	$0.319719\pi$
$138$	0	0
$139$	2.55389	0.216618	0.108309	−	0.994117i	$-0.465456\pi$
$139$	2.55389	0.216618	0.108309	+	0.994117i	$0.465456\pi$
$140$	0	0
$141$	0	0
$142$	−19.5831	−1.64338
$143$	0.337902	0.0282568
$144$	0	0
$145$	0	0
$146$	9.98031	0.825976
$147$	0	0
$148$	32.7950	2.69574
$149$	−16.3046	−1.33572	−0.667861	−	0.744286i	$-0.732790\pi$
$149$	−16.3046	−1.33572	−0.667861	+	0.744286i	$0.732790\pi$
$150$	0	0
$151$	−5.24543	−0.426867	−0.213434	−	0.976958i	$-0.568465\pi$
$151$	−5.24543	−0.426867	−0.213434	+	0.976958i	$0.568465\pi$
$152$	44.6724	3.62341
$153$	0	0
$154$	−0.669279	−0.0539320
$155$	0	0
$156$	0	0
$157$	8.73858	0.697415	0.348708	−	0.937232i	$-0.386621\pi$
$157$	8.73858	0.697415	0.348708	+	0.937232i	$0.386621\pi$
$158$	12.7025	1.01056
$159$	0	0
$160$	0	0
$161$	−26.5072	−2.08906
$162$	0	0
$163$	−20.7968	−1.62893	−0.814467	−	0.580210i	$-0.802971\pi$
$163$	−20.7968	−1.62893	−0.814467	+	0.580210i	$0.802971\pi$
$164$	29.7425	2.32250
$165$	0	0
$166$	23.7607	1.84419
$167$	−24.9576	−1.93128	−0.965639	−	0.259889i	$-0.916314\pi$
$167$	−24.9576	−1.93128	−0.965639	+	0.259889i	$0.916314\pi$
$168$	0	0
$169$	10.5954	0.815032
$170$	0	0
$171$	0	0
$172$	1.28791	0.0982022
$173$	6.78592	0.515924	0.257962	−	0.966155i	$-0.416949\pi$
$173$	6.78592	0.515924	0.257962	+	0.966155i	$0.416949\pi$
$174$	0	0
$175$	0	0
$176$	0.924945	0.0697204
$177$	0	0
$178$	9.58091	0.718120
$179$	−9.73723	−0.727795	−0.363897	−	0.931439i	$-0.618554\pi$
$179$	−9.73723	−0.727795	−0.363897	+	0.931439i	$0.618554\pi$
$180$	0	0
$181$	−7.24814	−0.538750	−0.269375	−	0.963035i	$-0.586817\pi$
$181$	−7.24814	−0.538750	−0.269375	+	0.963035i	$0.586817\pi$
$182$	−46.7352	−3.46424
$183$	0	0
$184$	65.7126	4.84440
$185$	0	0
$186$	0	0
$187$	−0.211074	−0.0154353
$188$	17.0500	1.24350
$189$	0	0
$190$	0	0
$191$	21.3284	1.54327	0.771635	−	0.636065i	$-0.219439\pi$
$191$	21.3284	1.54327	0.771635	+	0.636065i	$0.219439\pi$
$192$	0	0
$193$	24.9564	1.79640	0.898201	−	0.439586i	$-0.144875\pi$
$193$	24.9564	1.79640	0.898201	+	0.439586i	$0.144875\pi$
$194$	23.7539	1.70543
$195$	0	0
$196$	30.1862	2.15615
$197$	−18.0515	−1.28611	−0.643056	−	0.765819i	$-0.722334\pi$
$197$	−18.0515	−1.28611	−0.643056	+	0.765819i	$0.722334\pi$
$198$	0	0
$199$	10.9432	0.775744	0.387872	−	0.921713i	$-0.373210\pi$
$199$	10.9432	0.775744	0.387872	+	0.921713i	$0.373210\pi$
$200$	0	0
$201$	0	0
$202$	−35.3101	−2.48441
$203$	6.97034	0.489222
$204$	0	0
$205$	0	0
$206$	50.7762	3.53774
$207$	0	0
$208$	64.5882	4.47838
$209$	0.351472	0.0243118
$210$	0	0
$211$	−8.21731	−0.565703	−0.282852	−	0.959164i	$-0.591280\pi$
$211$	−8.21731	−0.565703	−0.282852	+	0.959164i	$0.591280\pi$
$212$	−44.1977	−3.03551
$213$	0	0
$214$	16.6164	1.13588
$215$	0	0
$216$	0	0
$217$	20.1003	1.36450
$218$	16.6718	1.12916
$219$	0	0
$220$	0	0
$221$	−14.7391	−0.991461
$222$	0	0
$223$	−1.92221	−0.128721	−0.0643604	−	0.997927i	$-0.520501\pi$
$223$	−1.92221	−0.128721	−0.0643604	+	0.997927i	$0.520501\pi$
$224$	−64.8631	−4.33385
$225$	0	0
$226$	−43.0957	−2.86668
$227$	−7.29101	−0.483922	−0.241961	−	0.970286i	$-0.577791\pi$
$227$	−7.29101	−0.483922	−0.241961	+	0.970286i	$0.577791\pi$
$228$	0	0
$229$	−17.0476	−1.12653	−0.563267	−	0.826275i	$-0.690456\pi$
$229$	−17.0476	−1.12653	−0.563267	+	0.826275i	$0.690456\pi$
$230$	0	0
$231$	0	0
$232$	−17.2798	−1.13447
$233$	−26.4383	−1.73203	−0.866015	−	0.500018i	$-0.833327\pi$
$233$	−26.4383	−1.73203	−0.866015	+	0.500018i	$0.833327\pi$
$234$	0	0
$235$	0	0
$236$	8.47190	0.551474
$237$	0	0
$238$	29.1936	1.89234
$239$	−11.7151	−0.757784	−0.378892	−	0.925441i	$-0.623695\pi$
$239$	−11.7151	−0.757784	−0.378892	+	0.925441i	$0.623695\pi$
$240$	0	0
$241$	−8.71335	−0.561276	−0.280638	−	0.959814i	$-0.590546\pi$
$241$	−8.71335	−0.561276	−0.280638	+	0.959814i	$0.590546\pi$
$242$	−29.6614	−1.90670
$243$	0	0
$244$	16.4922	1.05580
$245$	0	0
$246$	0	0
$247$	24.5430	1.56163
$248$	−49.8297	−3.16419
$249$	0	0
$250$	0	0
$251$	21.3939	1.35037	0.675186	−	0.737648i	$-0.264063\pi$
$251$	21.3939	1.35037	0.675186	+	0.737648i	$0.264063\pi$
$252$	0	0
$253$	0.517012	0.0325042
$254$	53.9509	3.38518
$255$	0	0
$256$	20.4547	1.27842
$257$	0.433099	0.0270160	0.0135080	−	0.999909i	$-0.495700\pi$
$257$	0.433099	0.0270160	0.0135080	+	0.999909i	$0.495700\pi$
$258$	0	0
$259$	−22.1628	−1.37713
$260$	0	0
$261$	0	0
$262$	−25.6377	−1.58390
$263$	−21.9239	−1.35189	−0.675943	−	0.736954i	$-0.736263\pi$
$263$	−21.9239	−1.35189	−0.675943	+	0.736954i	$0.736263\pi$
$264$	0	0
$265$	0	0
$266$	−48.6121	−2.98060
$267$	0	0
$268$	−15.5567	−0.950276
$269$	−25.9465	−1.58199	−0.790993	−	0.611825i	$-0.790436\pi$
$269$	−25.9465	−1.58199	−0.790993	+	0.611825i	$0.790436\pi$
$270$	0	0
$271$	−4.79789	−0.291451	−0.145725	−	0.989325i	$-0.546552\pi$
$271$	−4.79789	−0.291451	−0.145725	+	0.989325i	$0.546552\pi$
$272$	−40.3457	−2.44632
$273$	0	0
$274$	33.8850	2.04707
$275$	0	0
$276$	0	0
$277$	−2.42344	−0.145611	−0.0728053	−	0.997346i	$-0.523195\pi$
$277$	−2.42344	−0.145611	−0.0728053	+	0.997346i	$0.523195\pi$
$278$	6.88955	0.413208
$279$	0	0
$280$	0	0
$281$	7.91275	0.472035	0.236017	−	0.971749i	$-0.424158\pi$
$281$	7.91275	0.472035	0.236017	+	0.971749i	$0.424158\pi$
$282$	0	0
$283$	−17.0311	−1.01239	−0.506195	−	0.862419i	$-0.668949\pi$
$283$	−17.0311	−1.01239	−0.506195	+	0.862419i	$0.668949\pi$
$284$	−38.3103	−2.27330
$285$	0	0
$286$	0.911549	0.0539010
$287$	−20.0999	−1.18646
$288$	0	0
$289$	−7.79304	−0.458414
$290$	0	0
$291$	0	0
$292$	19.5244	1.14258
$293$	6.65379	0.388718	0.194359	−	0.980930i	$-0.437737\pi$
$293$	6.65379	0.388718	0.194359	+	0.980930i	$0.437737\pi$
$294$	0	0
$295$	0	0
$296$	54.9426	3.19348
$297$	0	0
$298$	−43.9844	−2.54795
$299$	36.1025	2.08786
$300$	0	0
$301$	−0.870367	−0.0501671
$302$	−14.1505	−0.814268
$303$	0	0
$304$	67.1820	3.85315
$305$	0	0
$306$	0	0
$307$	−5.48149	−0.312845	−0.156423	−	0.987690i	$-0.549996\pi$
$307$	−5.48149	−0.312845	−0.156423	+	0.987690i	$0.549996\pi$
$308$	−1.30931	−0.0746047
$309$	0	0
$310$	0	0
$311$	−31.9826	−1.81357	−0.906783	−	0.421597i	$-0.861470\pi$
$311$	−31.9826	−1.81357	−0.906783	+	0.421597i	$0.861470\pi$
$312$	0	0
$313$	0.728573	0.0411814	0.0205907	−	0.999788i	$-0.493445\pi$
$313$	0.728573	0.0411814	0.0205907	+	0.999788i	$0.493445\pi$
$314$	23.5739	1.33035
$315$	0	0
$316$	24.8498	1.39791
$317$	29.0469	1.63144	0.815718	−	0.578450i	$-0.196342\pi$
$317$	29.0469	1.63144	0.815718	+	0.578450i	$0.196342\pi$
$318$	0	0
$319$	−0.135953	−0.00761193
$320$	0	0
$321$	0	0
$322$	−71.5079	−3.98498
$323$	−15.3310	−0.853042
$324$	0	0
$325$	0	0
$326$	−56.1031	−3.10726
$327$	0	0
$328$	49.8286	2.75132
$329$	−11.5223	−0.635248
$330$	0	0
$331$	2.95713	0.162539	0.0812693	−	0.996692i	$-0.474103\pi$
$331$	2.95713	0.162539	0.0812693	+	0.996692i	$0.474103\pi$
$332$	46.4829	2.55108
$333$	0	0
$334$	−67.3275	−3.68400
$335$	0	0
$336$	0	0
$337$	−26.6236	−1.45028	−0.725139	−	0.688602i	$-0.758225\pi$
$337$	−26.6236	−1.45028	−0.725139	+	0.688602i	$0.758225\pi$
$338$	28.5830	1.55471
$339$	0	0
$340$	0	0
$341$	−0.392048	−0.0212306
$342$	0	0
$343$	4.56568	0.246523
$344$	2.15768	0.116334
$345$	0	0
$346$	18.3062	0.984148
$347$	−12.0229	−0.645421	−0.322710	−	0.946498i	$-0.604594\pi$
$347$	−12.0229	−0.645421	−0.322710	+	0.946498i	$0.604594\pi$
$348$	0	0
$349$	32.1053	1.71856	0.859279	−	0.511507i	$-0.170913\pi$
$349$	32.1053	1.71856	0.859279	+	0.511507i	$0.170913\pi$
$350$	0	0
$351$	0	0
$352$	1.26512	0.0674314
$353$	−10.8691	−0.578504	−0.289252	−	0.957253i	$-0.593407\pi$
$353$	−10.8691	−0.578504	−0.289252	+	0.957253i	$0.593407\pi$
$354$	0	0
$355$	0	0
$356$	18.7431	0.993382
$357$	0	0
$358$	−26.2679	−1.38830
$359$	−28.8562	−1.52297	−0.761486	−	0.648182i	$-0.775530\pi$
$359$	−28.8562	−1.52297	−0.761486	+	0.648182i	$0.775530\pi$
$360$	0	0
$361$	6.52862	0.343611
$362$	−19.5531	−1.02769
$363$	0	0
$364$	−91.4279	−4.79212
$365$	0	0
$366$	0	0
$367$	6.78347	0.354095	0.177047	−	0.984202i	$-0.443345\pi$
$367$	6.78347	0.354095	0.177047	+	0.984202i	$0.443345\pi$
$368$	98.8241	5.15156
$369$	0	0
$370$	0	0
$371$	29.8687	1.55071
$372$	0	0
$373$	−5.09871	−0.264001	−0.132001	−	0.991250i	$-0.542140\pi$
$373$	−5.09871	−0.264001	−0.132001	+	0.991250i	$0.542140\pi$
$374$	−0.569409	−0.0294434
$375$	0	0
$376$	28.5644	1.47310
$377$	−9.49351	−0.488941
$378$	0	0
$379$	−0.991089	−0.0509088	−0.0254544	−	0.999676i	$-0.508103\pi$
$379$	−0.991089	−0.0509088	−0.0254544	+	0.999676i	$0.508103\pi$
$380$	0	0
$381$	0	0
$382$	57.5372	2.94386
$383$	4.25182	0.217258	0.108629	−	0.994082i	$-0.465354\pi$
$383$	4.25182	0.217258	0.108629	+	0.994082i	$0.465354\pi$
$384$	0	0
$385$	0	0
$386$	67.3243	3.42671
$387$	0	0
$388$	46.4697	2.35914
$389$	−9.50264	−0.481803	−0.240902	−	0.970550i	$-0.577443\pi$
$389$	−9.50264	−0.481803	−0.240902	+	0.970550i	$0.577443\pi$
$390$	0	0
$391$	−22.5518	−1.14049
$392$	50.5719	2.55427
$393$	0	0
$394$	−48.6970	−2.45332
$395$	0	0
$396$	0	0
$397$	0.796576	0.0399790	0.0199895	−	0.999800i	$-0.493637\pi$
$397$	0.796576	0.0399790	0.0199895	+	0.999800i	$0.493637\pi$
$398$	29.5212	1.47977
$399$	0	0
$400$	0	0
$401$	−29.0334	−1.44986	−0.724928	−	0.688824i	$-0.758127\pi$
$401$	−29.0334	−1.44986	−0.724928	+	0.688824i	$0.758127\pi$
$402$	0	0
$403$	−27.3764	−1.36372
$404$	−69.0770	−3.43671
$405$	0	0
$406$	18.8037	0.933212
$407$	0.432276	0.0214271
$408$	0	0
$409$	−21.1715	−1.04686	−0.523432	−	0.852067i	$-0.675349\pi$
$409$	−21.1715	−1.04686	−0.523432	+	0.852067i	$0.675349\pi$
$410$	0	0
$411$	0	0
$412$	99.3331	4.89379
$413$	−5.72529	−0.281723
$414$	0	0
$415$	0	0
$416$	88.3426	4.33135
$417$	0	0
$418$	0.948156	0.0463758
$419$	5.61833	0.274473	0.137237	−	0.990538i	$-0.456178\pi$
$419$	5.61833	0.274473	0.137237	+	0.990538i	$0.456178\pi$
$420$	0	0
$421$	−8.10655	−0.395089	−0.197545	−	0.980294i	$-0.563297\pi$
$421$	−8.10655	−0.395089	−0.197545	+	0.980294i	$0.563297\pi$
$422$	−22.1676	−1.07910
$423$	0	0
$424$	−74.0459	−3.59599
$425$	0	0
$426$	0	0
$427$	−11.1454	−0.539363
$428$	32.5066	1.57127
$429$	0	0
$430$	0	0
$431$	−13.8047	−0.664947	−0.332473	−	0.943113i	$-0.607883\pi$
$431$	−13.8047	−0.664947	−0.332473	+	0.943113i	$0.607883\pi$
$432$	0	0
$433$	−39.7463	−1.91009	−0.955043	−	0.296466i	$-0.904192\pi$
$433$	−39.7463	−1.91009	−0.955043	+	0.296466i	$0.904192\pi$
$434$	54.2242	2.60284
$435$	0	0
$436$	32.6150	1.56197
$437$	37.5524	1.79637
$438$	0	0
$439$	17.2705	0.824277	0.412139	−	0.911121i	$-0.364782\pi$
$439$	17.2705	0.824277	0.412139	+	0.911121i	$0.364782\pi$
$440$	0	0
$441$	0	0
$442$	−39.7614	−1.89126
$443$	26.9182	1.27892	0.639461	−	0.768824i	$-0.279158\pi$
$443$	26.9182	1.27892	0.639461	+	0.768824i	$0.279158\pi$
$444$	0	0
$445$	0	0
$446$	−5.18550	−0.245541
$447$	0	0
$448$	−80.1354	−3.78604
$449$	0.0865077	0.00408255	0.00204128	−	0.999998i	$-0.499350\pi$
$449$	0.0865077	0.00408255	0.00204128	+	0.999998i	$0.499350\pi$
$450$	0	0
$451$	0.392039	0.0184604
$452$	−84.3079	−3.96551
$453$	0	0
$454$	−19.6688	−0.923102
$455$	0	0
$456$	0	0
$457$	−29.1547	−1.36380	−0.681899	−	0.731446i	$-0.738846\pi$
$457$	−29.1547	−1.36380	−0.681899	+	0.731446i	$0.738846\pi$
$458$	−45.9888	−2.14891
$459$	0	0
$460$	0	0
$461$	9.80698	0.456757	0.228378	−	0.973572i	$-0.426658\pi$
$461$	9.80698	0.456757	0.228378	+	0.973572i	$0.426658\pi$
$462$	0	0
$463$	−26.8716	−1.24883	−0.624414	−	0.781093i	$-0.714662\pi$
$463$	−26.8716	−1.24883	−0.624414	+	0.781093i	$0.714662\pi$
$464$	−25.9868	−1.20641
$465$	0	0
$466$	−71.3219	−3.30392
$467$	−4.06672	−0.188185	−0.0940926	−	0.995563i	$-0.529995\pi$
$467$	−4.06672	−0.188185	−0.0940926	+	0.995563i	$0.529995\pi$
$468$	0	0
$469$	10.5132	0.485453
$470$	0	0
$471$	0	0
$472$	14.1933	0.653298
$473$	0.0169761	0.000780562 0
$474$	0	0
$475$	0	0
$476$	57.1114	2.61770
$477$	0	0
$478$	−31.6034	−1.44551
$479$	0.636070	0.0290628	0.0145314	−	0.999894i	$-0.495374\pi$
$479$	0.636070	0.0290628	0.0145314	+	0.999894i	$0.495374\pi$
$480$	0	0
$481$	30.1855	1.37634
$482$	−23.5058	−1.07066
$483$	0	0
$484$	−58.0264	−2.63756
$485$	0	0
$486$	0	0
$487$	0.591183	0.0267890	0.0133945	−	0.999910i	$-0.495736\pi$
$487$	0.591183	0.0267890	0.0133945	+	0.999910i	$0.495736\pi$
$488$	27.6299	1.25075
$489$	0	0
$490$	0	0
$491$	17.4485	0.787438	0.393719	−	0.919231i	$-0.371188\pi$
$491$	17.4485	0.787438	0.393719	+	0.919231i	$0.371188\pi$
$492$	0	0
$493$	5.93023	0.267084
$494$	66.2090	2.97888
$495$	0	0
$496$	−74.9380	−3.36481
$497$	25.8900	1.16133
$498$	0	0
$499$	13.5297	0.605673	0.302836	−	0.953043i	$-0.402066\pi$
$499$	13.5297	0.605673	0.302836	+	0.953043i	$0.402066\pi$
$500$	0	0
$501$	0	0
$502$	57.7138	2.57589
$503$	3.92944	0.175205	0.0876025	−	0.996156i	$-0.472079\pi$
$503$	3.92944	0.175205	0.0876025	+	0.996156i	$0.472079\pi$
$504$	0	0
$505$	0	0
$506$	1.39473	0.0620033
$507$	0	0
$508$	105.544	4.68275
$509$	−1.93748	−0.0858773	−0.0429386	−	0.999078i	$-0.513672\pi$
$509$	−1.93748	−0.0858773	−0.0429386	+	0.999078i	$0.513672\pi$
$510$	0	0
$511$	−13.1946	−0.583693
$512$	6.69932	0.296071
$513$	0	0
$514$	1.16836	0.0515342
$515$	0	0
$516$	0	0
$517$	0.224738	0.00988397
$518$	−59.7881	−2.62694
$519$	0	0
$520$	0	0
$521$	−31.0365	−1.35973	−0.679866	−	0.733336i	$-0.737962\pi$
$521$	−31.0365	−1.35973	−0.679866	+	0.733336i	$0.737962\pi$
$522$	0	0
$523$	45.0692	1.97074	0.985369	−	0.170435i	$-0.0545173\pi$
$523$	45.0692	1.97074	0.985369	+	0.170435i	$0.0545173\pi$
$524$	−50.1549	−2.19103
$525$	0	0
$526$	−59.1435	−2.57878
$527$	17.1010	0.744930
$528$	0	0
$529$	32.2392	1.40170
$530$	0	0
$531$	0	0
$532$	−95.0995	−4.12309
$533$	27.3758	1.18578
$534$	0	0
$535$	0	0
$536$	−26.0626	−1.12573
$537$	0	0
$538$	−69.9953	−3.01771
$539$	0.397888	0.0171382
$540$	0	0
$541$	15.7996	0.679276	0.339638	−	0.940556i	$-0.389695\pi$
$541$	15.7996	0.679276	0.339638	+	0.940556i	$0.389695\pi$
$542$	−12.9431	−0.555956
$543$	0	0
$544$	−55.1842	−2.36600
$545$	0	0
$546$	0	0
$547$	−4.25467	−0.181917	−0.0909583	−	0.995855i	$-0.528993\pi$
$547$	−4.25467	−0.181917	−0.0909583	+	0.995855i	$0.528993\pi$
$548$	66.2890	2.83173
$549$	0	0
$550$	0	0
$551$	−9.87477	−0.420679
$552$	0	0
$553$	−16.7934	−0.714130
$554$	−6.53766	−0.277759
$555$	0	0
$556$	13.4780	0.571594
$557$	12.2716	0.519963	0.259982	−	0.965614i	$-0.416283\pi$
$557$	12.2716	0.519963	0.259982	+	0.965614i	$0.416283\pi$
$558$	0	0
$559$	1.18543	0.0501383
$560$	0	0
$561$	0	0
$562$	21.3460	0.900427
$563$	28.9344	1.21944	0.609721	−	0.792616i	$-0.291282\pi$
$563$	28.9344	1.21944	0.609721	+	0.792616i	$0.291282\pi$
$564$	0	0
$565$	0	0
$566$	−45.9442	−1.93118
$567$	0	0
$568$	−64.1825	−2.69304
$569$	20.8189	0.872774	0.436387	−	0.899759i	$-0.356258\pi$
$569$	20.8189	0.872774	0.436387	+	0.899759i	$0.356258\pi$
$570$	0	0
$571$	−26.7522	−1.11954	−0.559771	−	0.828647i	$-0.689111\pi$
$571$	−26.7522	−1.11954	−0.559771	+	0.828647i	$0.689111\pi$
$572$	1.78326	0.0745618
$573$	0	0
$574$	−54.2230	−2.26322
$575$	0	0
$576$	0	0
$577$	−28.7636	−1.19744	−0.598722	−	0.800957i	$-0.704325\pi$
$577$	−28.7636	−1.19744	−0.598722	+	0.800957i	$0.704325\pi$
$578$	−21.0231	−0.874446
$579$	0	0
$580$	0	0
$581$	−31.4130	−1.30323
$582$	0	0
$583$	−0.582576	−0.0241278
$584$	32.7099	1.35355
$585$	0	0
$586$	17.9498	0.741498
$587$	19.6227	0.809916	0.404958	−	0.914335i	$-0.367286\pi$
$587$	19.6227	0.809916	0.404958	+	0.914335i	$0.367286\pi$
$588$	0	0
$589$	−28.4758	−1.17333
$590$	0	0
$591$	0	0
$592$	82.6273	3.39596
$593$	20.8486	0.856150	0.428075	−	0.903743i	$-0.359192\pi$
$593$	20.8486	0.856150	0.428075	+	0.903743i	$0.359192\pi$
$594$	0	0
$595$	0	0
$596$	−86.0465	−3.52460
$597$	0	0
$598$	97.3928	3.98269
$599$	21.6873	0.886118	0.443059	−	0.896492i	$-0.353893\pi$
$599$	21.6873	0.886118	0.443059	+	0.896492i	$0.353893\pi$
$600$	0	0
$601$	−5.34431	−0.217999	−0.108999	−	0.994042i	$-0.534765\pi$
$601$	−5.34431	−0.217999	−0.108999	+	0.994042i	$0.534765\pi$
$602$	−2.34797	−0.0956960
$603$	0	0
$604$	−27.6825	−1.12639
$605$	0	0
$606$	0	0
$607$	33.9628	1.37851	0.689254	−	0.724520i	$-0.257939\pi$
$607$	33.9628	1.37851	0.689254	+	0.724520i	$0.257939\pi$
$608$	91.8904	3.72665
$609$	0	0
$610$	0	0
$611$	15.6933	0.634882
$612$	0	0
$613$	9.86615	0.398490	0.199245	−	0.979950i	$-0.436151\pi$
$613$	9.86615	0.398490	0.199245	+	0.979950i	$0.436151\pi$
$614$	−14.7873	−0.596766
$615$	0	0
$616$	−2.19353	−0.0883797
$617$	11.1935	0.450633	0.225316	−	0.974286i	$-0.427658\pi$
$617$	11.1935	0.450633	0.225316	+	0.974286i	$0.427658\pi$
$618$	0	0
$619$	37.0266	1.48822	0.744112	−	0.668054i	$-0.232873\pi$
$619$	37.0266	1.48822	0.744112	+	0.668054i	$0.232873\pi$
$620$	0	0
$621$	0	0
$622$	−86.2786	−3.45946
$623$	−12.6665	−0.507474
$624$	0	0
$625$	0	0
$626$	1.96545	0.0785553
$627$	0	0
$628$	46.1174	1.84029
$629$	−18.8557	−0.751825
$630$	0	0
$631$	−29.7805	−1.18554	−0.592771	−	0.805371i	$-0.701966\pi$
$631$	−29.7805	−1.18554	−0.592771	+	0.805371i	$0.701966\pi$
$632$	41.6317	1.65602
$633$	0	0
$634$	78.3591	3.11204
$635$	0	0
$636$	0	0
$637$	27.7842	1.10085
$638$	−0.366758	−0.0145201
$639$	0	0
$640$	0	0
$641$	−41.5447	−1.64092	−0.820459	−	0.571705i	$-0.806282\pi$
$641$	−41.5447	−1.64092	−0.820459	+	0.571705i	$0.806282\pi$
$642$	0	0
$643$	16.6650	0.657202	0.328601	−	0.944469i	$-0.393423\pi$
$643$	16.6650	0.657202	0.328601	+	0.944469i	$0.393423\pi$
$644$	−139.891	−5.51246
$645$	0	0
$646$	−41.3582	−1.62722
$647$	36.8673	1.44940	0.724702	−	0.689062i	$-0.241977\pi$
$647$	36.8673	1.44940	0.724702	+	0.689062i	$0.241977\pi$
$648$	0	0
$649$	0.111669	0.00438340
$650$	0	0
$651$	0	0
$652$	−109.754	−4.29831
$653$	28.3184	1.10819	0.554093	−	0.832455i	$-0.313065\pi$
$653$	28.3184	1.10819	0.554093	+	0.832455i	$0.313065\pi$
$654$	0	0
$655$	0	0
$656$	74.9363	2.92577
$657$	0	0
$658$	−31.0835	−1.21176
$659$	−35.8536	−1.39666	−0.698329	−	0.715777i	$-0.746073\pi$
$659$	−35.8536	−1.39666	−0.698329	+	0.715777i	$0.746073\pi$
$660$	0	0
$661$	−34.8735	−1.35642	−0.678210	−	0.734868i	$-0.737244\pi$
$661$	−34.8735	−1.35642	−0.678210	+	0.734868i	$0.737244\pi$
$662$	7.97738	0.310050
$663$	0	0
$664$	77.8743	3.02211
$665$	0	0
$666$	0	0
$667$	−14.5257	−0.562437
$668$	−131.712	−5.09611
$669$	0	0
$670$	0	0
$671$	0.217386	0.00839208
$672$	0	0
$673$	−7.43183	−0.286476	−0.143238	−	0.989688i	$-0.545751\pi$
$673$	−7.43183	−0.286476	−0.143238	+	0.989688i	$0.545751\pi$
$674$	−71.8217	−2.76647
$675$	0	0
$676$	55.9167	2.15064
$677$	33.1734	1.27496	0.637478	−	0.770468i	$-0.279978\pi$
$677$	33.1734	1.27496	0.637478	+	0.770468i	$0.279978\pi$
$678$	0	0
$679$	−31.4041	−1.20518
$680$	0	0
$681$	0	0
$682$	−1.05762	−0.0404983
$683$	−46.0779	−1.76312	−0.881561	−	0.472070i	$-0.843507\pi$
$683$	−46.0779	−1.76312	−0.881561	+	0.472070i	$0.843507\pi$
$684$	0	0
$685$	0	0
$686$	12.3167	0.470254
$687$	0	0
$688$	3.24490	0.123710
$689$	−40.6808	−1.54981
$690$	0	0
$691$	40.0004	1.52169	0.760843	−	0.648936i	$-0.224786\pi$
$691$	40.0004	1.52169	0.760843	+	0.648936i	$0.224786\pi$
$692$	35.8123	1.36138
$693$	0	0
$694$	−32.4338	−1.23117
$695$	0	0
$696$	0	0
$697$	−17.1006	−0.647731
$698$	86.6097	3.27823
$699$	0	0
$700$	0	0
$701$	28.1775	1.06425	0.532124	−	0.846666i	$-0.321394\pi$
$701$	28.1775	1.06425	0.532124	+	0.846666i	$0.321394\pi$
$702$	0	0
$703$	31.3977	1.18419
$704$	1.56300	0.0589079
$705$	0	0
$706$	−29.3213	−1.10352
$707$	46.6821	1.75566
$708$	0	0
$709$	−38.6459	−1.45138	−0.725689	−	0.688023i	$-0.758479\pi$
$709$	−38.6459	−1.45138	−0.725689	+	0.688023i	$0.758479\pi$
$710$	0	0
$711$	0	0
$712$	31.4009	1.17680
$713$	−41.8877	−1.56871
$714$	0	0
$715$	0	0
$716$	−51.3877	−1.92045
$717$	0	0
$718$	−77.8446	−2.90514
$719$	−14.1179	−0.526508	−0.263254	−	0.964727i	$-0.584796\pi$
$719$	−14.1179	−0.526508	−0.263254	+	0.964727i	$0.584796\pi$
$720$	0	0
$721$	−67.1291	−2.50002
$722$	17.6121	0.655454
$723$	0	0
$724$	−38.2517	−1.42161
$725$	0	0
$726$	0	0
$727$	−6.18443	−0.229368	−0.114684	−	0.993402i	$-0.536586\pi$
$727$	−6.18443	−0.229368	−0.114684	+	0.993402i	$0.536586\pi$
$728$	−153.172	−5.67694
$729$	0	0
$730$	0	0
$731$	−0.740491	−0.0273880
$732$	0	0
$733$	33.8511	1.25032	0.625159	−	0.780498i	$-0.285034\pi$
$733$	33.8511	1.25032	0.625159	+	0.780498i	$0.285034\pi$
$734$	18.2996	0.675451
$735$	0	0
$736$	135.170	4.98243
$737$	−0.205055	−0.00755329
$738$	0	0
$739$	28.1239	1.03455	0.517277	−	0.855818i	$-0.326946\pi$
$739$	28.1239	1.03455	0.517277	+	0.855818i	$0.326946\pi$
$740$	0	0
$741$	0	0
$742$	80.5760	2.95804
$743$	22.1429	0.812346	0.406173	−	0.913796i	$-0.366863\pi$
$743$	22.1429	0.812346	0.406173	+	0.913796i	$0.366863\pi$
$744$	0	0
$745$	0	0
$746$	−13.7547	−0.503594
$747$	0	0
$748$	−1.11393	−0.0407294
$749$	−21.9679	−0.802690
$750$	0	0
$751$	−5.75539	−0.210017	−0.105009	−	0.994471i	$-0.533487\pi$
$751$	−5.75539	−0.210017	−0.105009	+	0.994471i	$0.533487\pi$
$752$	42.9575	1.56650
$753$	0	0
$754$	−25.6104	−0.932676
$755$	0	0
$756$	0	0
$757$	−18.4324	−0.669937	−0.334968	−	0.942229i	$-0.608726\pi$
$757$	−18.4324	−0.669937	−0.334968	+	0.942229i	$0.608726\pi$
$758$	−2.67364	−0.0971108
$759$	0	0
$760$	0	0
$761$	25.3526	0.919031	0.459515	−	0.888170i	$-0.348023\pi$
$761$	25.3526	0.919031	0.459515	+	0.888170i	$0.348023\pi$
$762$	0	0
$763$	−22.0411	−0.797942
$764$	112.560	4.07227
$765$	0	0
$766$	11.4700	0.414429
$767$	7.79777	0.281561
$768$	0	0
$769$	30.3124	1.09309	0.546546	−	0.837429i	$-0.315942\pi$
$769$	30.3124	1.09309	0.546546	+	0.837429i	$0.315942\pi$
$770$	0	0
$771$	0	0
$772$	131.706	4.74021
$773$	−3.26668	−0.117494	−0.0587471	−	0.998273i	$-0.518711\pi$
$773$	−3.26668	−0.117494	−0.0587471	+	0.998273i	$0.518711\pi$
$774$	0	0
$775$	0	0
$776$	77.8523	2.79473
$777$	0	0
$778$	−25.6350	−0.919060
$779$	28.4752	1.02023
$780$	0	0
$781$	−0.504973	−0.0180694
$782$	−60.8375	−2.17554
$783$	0	0
$784$	76.0542	2.71622
$785$	0	0
$786$	0	0
$787$	−2.44831	−0.0872728	−0.0436364	−	0.999047i	$-0.513894\pi$
$787$	−2.44831	−0.0872728	−0.0436364	+	0.999047i	$0.513894\pi$
$788$	−95.2656	−3.39370
$789$	0	0
$790$	0	0
$791$	56.9751	2.02580
$792$	0	0
$793$	15.1799	0.539053
$794$	2.14890	0.0762617
$795$	0	0
$796$	57.7523	2.04697
$797$	−35.2494	−1.24860	−0.624299	−	0.781186i	$-0.714615\pi$
$797$	−35.2494	−1.24860	−0.624299	+	0.781186i	$0.714615\pi$
$798$	0	0
$799$	−9.80298	−0.346805
$800$	0	0
$801$	0	0
$802$	−78.3226	−2.76567
$803$	0.257354	0.00908183
$804$	0	0
$805$	0	0
$806$	−73.8526	−2.60135
$807$	0	0
$808$	−115.727	−4.07126
$809$	16.2002	0.569570	0.284785	−	0.958591i	$-0.408078\pi$
$809$	16.2002	0.569570	0.284785	+	0.958591i	$0.408078\pi$
$810$	0	0
$811$	−47.4540	−1.66633	−0.833167	−	0.553021i	$-0.813475\pi$
$811$	−47.4540	−1.66633	−0.833167	+	0.553021i	$0.813475\pi$
$812$	36.7856	1.29092
$813$	0	0
$814$	1.16614	0.0408732
$815$	0	0
$816$	0	0
$817$	1.23303	0.0431384
$818$	−57.1139	−1.99694
$819$	0	0
$820$	0	0
$821$	−34.4342	−1.20176	−0.600881	−	0.799338i	$-0.705184\pi$
$821$	−34.4342	−1.20176	−0.600881	+	0.799338i	$0.705184\pi$
$822$	0	0
$823$	33.0865	1.15332	0.576661	−	0.816983i	$-0.304355\pi$
$823$	33.0865	1.15332	0.576661	+	0.816983i	$0.304355\pi$
$824$	166.416	5.79738
$825$	0	0
$826$	−15.4450	−0.537399
$827$	22.4568	0.780900	0.390450	−	0.920624i	$-0.372319\pi$
$827$	22.4568	0.780900	0.390450	+	0.920624i	$0.372319\pi$
$828$	0	0
$829$	4.99720	0.173560	0.0867800	−	0.996227i	$-0.472342\pi$
$829$	4.99720	0.173560	0.0867800	+	0.996227i	$0.472342\pi$
$830$	0	0
$831$	0	0
$832$	109.143	3.78386
$833$	−17.3557	−0.601339
$834$	0	0
$835$	0	0
$836$	1.85487	0.0641521
$837$	0	0
$838$	15.1564	0.523570
$839$	25.8577	0.892705	0.446353	−	0.894857i	$-0.352723\pi$
$839$	25.8577	0.892705	0.446353	+	0.894857i	$0.352723\pi$
$840$	0	0
$841$	−25.1803	−0.868287
$842$	−21.8688	−0.753650
$843$	0	0
$844$	−43.3664	−1.49273
$845$	0	0
$846$	0	0
$847$	39.2141	1.34741
$848$	−111.356	−3.82399
$849$	0	0
$850$	0	0
$851$	46.1857	1.58323
$852$	0	0
$853$	−46.1376	−1.57972	−0.789860	−	0.613287i	$-0.789847\pi$
$853$	−46.1376	−1.57972	−0.789860	+	0.613287i	$0.789847\pi$
$854$	−30.0666	−1.02886
$855$	0	0
$856$	54.4595	1.86139
$857$	17.8220	0.608787	0.304393	−	0.952546i	$-0.401546\pi$
$857$	17.8220	0.608787	0.304393	+	0.952546i	$0.401546\pi$
$858$	0	0
$859$	30.7896	1.05053	0.525263	−	0.850940i	$-0.323967\pi$
$859$	30.7896	1.05053	0.525263	+	0.850940i	$0.323967\pi$
$860$	0	0
$861$	0	0
$862$	−37.2405	−1.26842
$863$	52.1333	1.77464	0.887319	−	0.461155i	$-0.152565\pi$
$863$	52.1333	1.77464	0.887319	+	0.461155i	$0.152565\pi$
$864$	0	0
$865$	0	0
$866$	−107.223	−3.64357
$867$	0	0
$868$	106.079	3.60054
$869$	0.327549	0.0111113
$870$	0	0
$871$	−14.3188	−0.485174
$872$	54.6410	1.85038
$873$	0	0
$874$	101.304	3.42666
$875$	0	0
$876$	0	0
$877$	42.3249	1.42921	0.714605	−	0.699528i	$-0.246606\pi$
$877$	42.3249	1.42921	0.714605	+	0.699528i	$0.246606\pi$
$878$	46.5903	1.57234
$879$	0	0
$880$	0	0
$881$	−6.21376	−0.209347	−0.104673	−	0.994507i	$-0.533380\pi$
$881$	−6.21376	−0.209347	−0.104673	+	0.994507i	$0.533380\pi$
$882$	0	0
$883$	30.9588	1.04185	0.520924	−	0.853603i	$-0.325588\pi$
$883$	30.9588	1.04185	0.520924	+	0.853603i	$0.325588\pi$
$884$	−77.7850	−2.61619
$885$	0	0
$886$	72.6165	2.43960
$887$	39.7363	1.33422	0.667108	−	0.744961i	$-0.267532\pi$
$887$	39.7363	1.33422	0.667108	+	0.744961i	$0.267532\pi$
$888$	0	0
$889$	−71.3263	−2.39221
$890$	0	0
$891$	0	0
$892$	−10.1444	−0.339659
$893$	16.3235	0.546246
$894$	0	0
$895$	0	0
$896$	−86.4530	−2.88819
$897$	0	0
$898$	0.233370	0.00778765
$899$	11.0148	0.367363
$900$	0	0
$901$	25.4117	0.846586
$902$	1.05759	0.0352141
$903$	0	0
$904$	−141.244	−4.69770
$905$	0	0
$906$	0	0
$907$	−17.1801	−0.570454	−0.285227	−	0.958460i	$-0.592069\pi$
$907$	−17.1801	−0.570454	−0.285227	+	0.958460i	$0.592069\pi$
$908$	−38.4779	−1.27694
$909$	0	0
$910$	0	0
$911$	10.0621	0.333371	0.166685	−	0.986010i	$-0.446694\pi$
$911$	10.0621	0.333371	0.166685	+	0.986010i	$0.446694\pi$
$912$	0	0
$913$	0.612697	0.0202773
$914$	−78.6498	−2.60150
$915$	0	0
$916$	−89.9676	−2.97261
$917$	33.8946	1.11930
$918$	0	0
$919$	12.2601	0.404424	0.202212	−	0.979342i	$-0.435187\pi$
$919$	12.2601	0.404424	0.202212	+	0.979342i	$0.435187\pi$
$920$	0	0
$921$	0	0
$922$	26.4560	0.871283
$923$	−35.2619	−1.16066
$924$	0	0
$925$	0	0
$926$	−72.4908	−2.38220
$927$	0	0
$928$	−35.5443	−1.16680
$929$	−2.29196	−0.0751969	−0.0375984	−	0.999293i	$-0.511971\pi$
$929$	−2.29196	−0.0751969	−0.0375984	+	0.999293i	$0.511971\pi$
$930$	0	0
$931$	28.9000	0.947158
$932$	−139.527	−4.57035
$933$	0	0
$934$	−10.9707	−0.358972
$935$	0	0
$936$	0	0
$937$	39.3565	1.28572	0.642861	−	0.765983i	$-0.277747\pi$
$937$	39.3565	1.28572	0.642861	+	0.765983i	$0.277747\pi$
$938$	28.3611	0.926023
$939$	0	0
$940$	0	0
$941$	22.6288	0.737678	0.368839	−	0.929493i	$-0.379755\pi$
$941$	22.6288	0.737678	0.368839	+	0.929493i	$0.379755\pi$
$942$	0	0
$943$	41.8868	1.36402
$944$	21.3450	0.694720
$945$	0	0
$946$	0.0457960	0.00148896
$947$	44.2268	1.43718	0.718589	−	0.695435i	$-0.244788\pi$
$947$	44.2268	1.43718	0.718589	+	0.695435i	$0.244788\pi$
$948$	0	0
$949$	17.9708	0.583358
$950$	0	0
$951$	0	0
$952$	95.6807	3.10103
$953$	32.4366	1.05072	0.525362	−	0.850879i	$-0.323930\pi$
$953$	32.4366	1.05072	0.525362	+	0.850879i	$0.323930\pi$
$954$	0	0
$955$	0	0
$956$	−61.8256	−1.99958
$957$	0	0
$958$	1.71591	0.0554385
$959$	−44.7980	−1.44660
$960$	0	0
$961$	0.763274	0.0246217
$962$	81.4306	2.62543
$963$	0	0
$964$	−45.9842	−1.48105
$965$	0	0
$966$	0	0
$967$	−10.1870	−0.327591	−0.163796	−	0.986494i	$-0.552374\pi$
$967$	−10.1870	−0.327591	−0.163796	+	0.986494i	$0.552374\pi$
$968$	−97.2136	−3.12456
$969$	0	0
$970$	0	0
$971$	−6.22735	−0.199845	−0.0999225	−	0.994995i	$-0.531859\pi$
$971$	−6.22735	−0.199845	−0.0999225	+	0.994995i	$0.531859\pi$
$972$	0	0
$973$	−9.10840	−0.292002
$974$	1.59482	0.0511013
$975$	0	0
$976$	41.5521	1.33005
$977$	8.75116	0.279974	0.139987	−	0.990153i	$-0.455294\pi$
$977$	8.75116	0.279974	0.139987	+	0.990153i	$0.455294\pi$
$978$	0	0
$979$	0.247055	0.00789592
$980$	0	0
$981$	0	0
$982$	47.0703	1.50207
$983$	0.807527	0.0257561	0.0128781	−	0.999917i	$-0.495901\pi$
$983$	0.807527	0.0257561	0.0128781	+	0.999917i	$0.495901\pi$
$984$	0	0
$985$	0	0
$986$	15.9978	0.509474
$987$	0	0
$988$	129.524	4.12072
$989$	1.81378	0.0576749
$990$	0	0
$991$	6.58244	0.209098	0.104549	−	0.994520i	$-0.466660\pi$
$991$	6.58244	0.209098	0.104549	+	0.994520i	$0.466660\pi$
$992$	−102.499	−3.25434
$993$	0	0
$994$	69.8428	2.21528
$995$	0	0
$996$	0	0
$997$	−33.4939	−1.06076	−0.530381	−	0.847760i	$-0.677951\pi$
$997$	−33.4939	−1.06076	−0.530381	+	0.847760i	$0.677951\pi$
$998$	36.4987	1.15535
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.u.1.8		8
3.2	odd	2		1875.2.a.o.1.1	yes	8
5.4	even	2		5625.2.a.bc.1.1		8
15.2	even	4		1875.2.b.g.1249.1		16
15.8	even	4		1875.2.b.g.1249.16		16
15.14	odd	2		1875.2.a.n.1.8	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1875.2.a.n.1.8	✓	8	15.14	odd	2
1875.2.a.o.1.1	yes	8	3.2	odd	2
1875.2.b.g.1249.1		16	15.2	even	4
1875.2.b.g.1249.16		16	15.8	even	4
5625.2.a.u.1.8		8	1.1	even	1	trivial
5625.2.a.bc.1.1		8	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:	\( N \)	\(=\)	\( 5625 = 3^{2} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5625.a (trivial)

\(f(q)\)	\(=\)	\(q+2.69767 q^{2} +5.27745 q^{4} -3.56649 q^{7} +8.84149 q^{8} +O(q^{10})\)	\(q+2.69767 q^{2} +5.27745 q^{4} -3.56649 q^{7} +8.84149 q^{8} +0.0695627 q^{11} +4.85751 q^{13} -9.62122 q^{14} +13.2966 q^{16} -3.03430 q^{17} +5.05259 q^{19} +0.187658 q^{22} +7.43231 q^{23} +13.1040 q^{26} -18.8220 q^{28} -1.95440 q^{29} -5.63589 q^{31} +18.1868 q^{32} -8.18554 q^{34} +6.21419 q^{37} +13.6302 q^{38} +5.63577 q^{41} +0.244040 q^{43} +0.367114 q^{44} +20.0499 q^{46} +3.23073 q^{47} +5.71984 q^{49} +25.6353 q^{52} -8.37482 q^{53} -31.5331 q^{56} -5.27233 q^{58} +1.60530 q^{59} +3.12503 q^{61} -15.2038 q^{62} +22.4690 q^{64} -2.94777 q^{67} -16.0133 q^{68} -7.25925 q^{71} +3.69960 q^{73} +16.7638 q^{74} +26.6648 q^{76} -0.248095 q^{77} +4.70868 q^{79} +15.2035 q^{82} +8.80783 q^{83} +0.658341 q^{86} +0.615038 q^{88} +3.55155 q^{89} -17.3243 q^{91} +39.2236 q^{92} +8.71545 q^{94} +8.80534 q^{97} +15.4303 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{2} + 9 q^{4} + 12 q^{7} - 3 q^{8}+O(q^{10}) \) 8 * q - q^2 + 9 * q^4 + 12 * q^7 - 3 * q^8	\( 8 q - q^{2} + 9 q^{4} + 12 q^{7} - 3 q^{8} - 12 q^{11} + 14 q^{13} - 16 q^{14} + 15 q^{16} + q^{17} + 16 q^{19} + 18 q^{22} + 4 q^{23} + 34 q^{26} - 21 q^{28} - 2 q^{29} + 13 q^{31} + 18 q^{32} - 37 q^{34} - 8 q^{37} + 24 q^{38} + 12 q^{41} + 20 q^{43} - 47 q^{44} + 33 q^{46} + 15 q^{47} + 30 q^{49} - q^{52} + 4 q^{53} - 60 q^{56} + 2 q^{58} - 14 q^{59} + 10 q^{61} - 4 q^{62} + 41 q^{64} + 19 q^{67} + 33 q^{68} - 21 q^{71} - 19 q^{73} + 9 q^{74} - q^{76} + 11 q^{77} + 10 q^{79} + 24 q^{82} + 27 q^{83} - 42 q^{86} + 53 q^{88} + 9 q^{89} - 12 q^{91} + 63 q^{92} + 14 q^{94} + 24 q^{97} + 24 q^{98}+O(q^{100}) \) 8 * q - q^2 + 9 * q^4 + 12 * q^7 - 3 * q^8 - 12 * q^11 + 14 * q^13 - 16 * q^14 + 15 * q^16 + q^17 + 16 * q^19 + 18 * q^22 + 4 * q^23 + 34 * q^26 - 21 * q^28 - 2 * q^29 + 13 * q^31 + 18 * q^32 - 37 * q^34 - 8 * q^37 + 24 * q^38 + 12 * q^41 + 20 * q^43 - 47 * q^44 + 33 * q^46 + 15 * q^47 + 30 * q^49 - q^52 + 4 * q^53 - 60 * q^56 + 2 * q^58 - 14 * q^59 + 10 * q^61 - 4 * q^62 + 41 * q^64 + 19 * q^67 + 33 * q^68 - 21 * q^71 - 19 * q^73 + 9 * q^74 - q^76 + 11 * q^77 + 10 * q^79 + 24 * q^82 + 27 * q^83 - 42 * q^86 + 53 * q^88 + 9 * q^89 - 12 * q^91 + 63 * q^92 + 14 * q^94 + 24 * q^97 + 24 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more