LMFDB - Embedded newform 2925.2.a.bm.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2925,2,Mod(1,2925)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2925, 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("2925.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2925 = 3^{2} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2925.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:	$23.3562425912$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.1821184.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 8x^{3} + 13x - 4 $ x^5 - 8x^3 + 13x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 195)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.77159$ of defining polynomial
Character	$\chi$	$=$	2925.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.77159 q^{2} +1.13853 q^{4} -0.437634 q^{7} +1.52618 q^{8} +O(q^{10})$	$q-1.77159 q^{2} +1.13853 q^{4} -0.437634 q^{7} +1.52618 q^{8} -5.73540 q^{11} +1.00000 q^{13} +0.775308 q^{14} -4.98081 q^{16} -3.98081 q^{17} +4.77531 q^{19} +10.1608 q^{22} -0.337673 q^{23} -1.77159 q^{26} -0.498258 q^{28} -1.72295 q^{29} -7.86166 q^{31} +5.77159 q^{32} +7.05236 q^{34} +1.66233 q^{37} -8.45988 q^{38} -2.68304 q^{41} +2.27705 q^{43} -6.52990 q^{44} +0.598218 q^{46} +11.7162 q^{47} -6.80848 q^{49} +1.13853 q^{52} +14.2993 q^{53} -0.667908 q^{56} +3.05236 q^{58} -11.4392 q^{59} -2.25786 q^{61} +13.9276 q^{62} -0.263257 q^{64} +2.17977 q^{67} -4.53225 q^{68} +13.6970 q^{71} -2.55410 q^{73} -2.94496 q^{74} +5.43681 q^{76} +2.51001 q^{77} +2.55144 q^{79} +4.75325 q^{82} +5.41158 q^{83} -4.03400 q^{86} -8.75325 q^{88} +4.72448 q^{89} -0.437634 q^{91} -0.384450 q^{92} -20.7563 q^{94} -14.7901 q^{97} +12.0618 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 6 q^{4} + 5 q^{7}+O(q^{10}) $ 5 * q + 6 * q^4 + 5 * q^7	$ 5 q + 6 q^{4} + 5 q^{7} - 5 q^{11} + 5 q^{13} - 12 q^{14} + 5 q^{17} + 8 q^{19} - 8 q^{22} + 7 q^{23} + 14 q^{28} - 8 q^{29} + 12 q^{31} + 20 q^{32} + 20 q^{34} + 17 q^{37} + 24 q^{38} - 5 q^{41} + 12 q^{43} - 18 q^{44} - 12 q^{46} + 10 q^{47} + 22 q^{49} + 6 q^{52} + 13 q^{53} - 8 q^{59} + 13 q^{61} + 40 q^{62} - 16 q^{64} + 28 q^{67} + 14 q^{68} - 5 q^{71} - 14 q^{73} - 12 q^{74} + 35 q^{77} + q^{79} + 16 q^{82} + 6 q^{83} - 36 q^{88} - 19 q^{89} + 5 q^{91} + 10 q^{92} - 12 q^{94} - 13 q^{97} - 28 q^{98}+O(q^{100}) $ 5 * q + 6 * q^4 + 5 * q^7 - 5 * q^11 + 5 * q^13 - 12 * q^14 + 5 * q^17 + 8 * q^19 - 8 * q^22 + 7 * q^23 + 14 * q^28 - 8 * q^29 + 12 * q^31 + 20 * q^32 + 20 * q^34 + 17 * q^37 + 24 * q^38 - 5 * q^41 + 12 * q^43 - 18 * q^44 - 12 * q^46 + 10 * q^47 + 22 * q^49 + 6 * q^52 + 13 * q^53 - 8 * q^59 + 13 * q^61 + 40 * q^62 - 16 * q^64 + 28 * q^67 + 14 * q^68 - 5 * q^71 - 14 * q^73 - 12 * q^74 + 35 * q^77 + q^79 + 16 * q^82 + 6 * q^83 - 36 * q^88 - 19 * q^89 + 5 * q^91 + 10 * q^92 - 12 * q^94 - 13 * q^97 - 28 * 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 $

$2$	−1.77159	−1.25270	−0.626351	−	0.779541i	$-0.715452\pi$
$2$	−1.77159	−1.25270	−0.626351	+	0.779541i	$0.715452\pi$
$3$	0	0
$3$	0	0
$4$	1.13853	0.569263
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.437634	−0.165410	−0.0827051	−	0.996574i	$-0.526356\pi$
$7$	−0.437634	−0.165410	−0.0827051	+	0.996574i	$0.526356\pi$
$8$	1.52618	0.539586
$9$	0	0
$10$	0	0
$11$	−5.73540	−1.72929	−0.864644	−	0.502385i	$-0.832456\pi$
$11$	−5.73540	−1.72929	−0.864644	+	0.502385i	$0.832456\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0.775308	0.207210
$15$	0	0
$16$	−4.98081	−1.24520
$17$	−3.98081	−0.965488	−0.482744	−	0.875761i	$-0.660360\pi$
$17$	−3.98081	−0.965488	−0.482744	+	0.875761i	$0.660360\pi$
$18$	0	0
$19$	4.77531	1.09553	0.547765	−	0.836632i	$-0.315479\pi$
$19$	4.77531	1.09553	0.547765	+	0.836632i	$0.315479\pi$
$20$	0	0
$21$	0	0
$22$	10.1608	2.16628
$23$	−0.337673	−0.0704098	−0.0352049	−	0.999380i	$-0.511208\pi$
$23$	−0.337673	−0.0704098	−0.0352049	+	0.999380i	$0.511208\pi$
$24$	0	0
$25$	0	0
$26$	−1.77159	−0.347437
$27$	0	0
$28$	−0.498258	−0.0941618
$29$	−1.72295	−0.319944	−0.159972	−	0.987122i	$-0.551140\pi$
$29$	−1.72295	−0.319944	−0.159972	+	0.987122i	$0.551140\pi$
$30$	0	0
$31$	−7.86166	−1.41200	−0.705998	−	0.708214i	$-0.749501\pi$
$31$	−7.86166	−1.41200	−0.705998	+	0.708214i	$0.749501\pi$
$32$	5.77159	1.02028
$33$	0	0
$34$	7.05236	1.20947
$35$	0	0
$36$	0	0
$37$	1.66233	0.273285	0.136642	−	0.990620i	$-0.456369\pi$
$37$	1.66233	0.273285	0.136642	+	0.990620i	$0.456369\pi$
$38$	−8.45988	−1.37237
$39$	0	0
$40$	0	0
$41$	−2.68304	−0.419021	−0.209511	−	0.977806i	$-0.567187\pi$
$41$	−2.68304	−0.419021	−0.209511	+	0.977806i	$0.567187\pi$
$42$	0	0
$43$	2.27705	0.347247	0.173623	−	0.984812i	$-0.444452\pi$
$43$	2.27705	0.347247	0.173623	+	0.984812i	$0.444452\pi$
$44$	−6.52990	−0.984419
$45$	0	0
$46$	0.598218	0.0882025
$47$	11.7162	1.70899	0.854493	−	0.519464i	$-0.173868\pi$
$47$	11.7162	1.70899	0.854493	+	0.519464i	$0.173868\pi$
$48$	0	0
$49$	−6.80848	−0.972639
$50$	0	0
$51$	0	0
$52$	1.13853	0.157885
$53$	14.2993	1.96416	0.982080	−	0.188467i	$-0.0603517\pi$
$53$	14.2993	1.96416	0.982080	+	0.188467i	$0.0603517\pi$
$54$	0	0
$55$	0	0
$56$	−0.667908	−0.0892530
$57$	0	0
$58$	3.05236	0.400794
$59$	−11.4392	−1.48925	−0.744626	−	0.667482i	$-0.767372\pi$
$59$	−11.4392	−1.48925	−0.744626	+	0.667482i	$0.767372\pi$
$60$	0	0
$61$	−2.25786	−0.289089	−0.144545	−	0.989498i	$-0.546172\pi$
$61$	−2.25786	−0.289089	−0.144545	+	0.989498i	$0.546172\pi$
$62$	13.9276	1.76881
$63$	0	0
$64$	−0.263257	−0.0329071
$65$	0	0
$66$	0	0
$67$	2.17977	0.266302	0.133151	−	0.991096i	$-0.457491\pi$
$67$	2.17977	0.266302	0.133151	+	0.991096i	$0.457491\pi$
$68$	−4.53225	−0.549616
$69$	0	0
$70$	0	0
$71$	13.6970	1.62554	0.812769	−	0.582586i	$-0.197959\pi$
$71$	13.6970	1.62554	0.812769	+	0.582586i	$0.197959\pi$
$72$	0	0
$73$	−2.55410	−0.298935	−0.149467	−	0.988767i	$-0.547756\pi$
$73$	−2.55410	−0.298935	−0.149467	+	0.988767i	$0.547756\pi$
$74$	−2.94496	−0.342344
$75$	0	0
$76$	5.43681	0.623645
$77$	2.51001	0.286042
$78$	0	0
$79$	2.55144	0.287060	0.143530	−	0.989646i	$-0.454155\pi$
$79$	2.55144	0.287060	0.143530	+	0.989646i	$0.454155\pi$
$80$	0	0
$81$	0	0
$82$	4.75325	0.524909
$83$	5.41158	0.593998	0.296999	−	0.954878i	$-0.404014\pi$
$83$	5.41158	0.593998	0.296999	+	0.954878i	$0.404014\pi$
$84$	0	0
$85$	0	0
$86$	−4.03400	−0.434997
$87$	0	0
$88$	−8.75325	−0.933099
$89$	4.72448	0.500794	0.250397	−	0.968143i	$-0.419439\pi$
$89$	4.72448	0.500794	0.250397	+	0.968143i	$0.419439\pi$
$90$	0	0
$91$	−0.437634	−0.0458765
$92$	−0.384450	−0.0400817
$93$	0	0
$94$	−20.7563	−2.14085
$95$	0	0
$96$	0	0
$97$	−14.7901	−1.50171	−0.750854	−	0.660468i	$-0.770358\pi$
$97$	−14.7901	−1.50171	−0.750854	+	0.660468i	$0.770358\pi$
$98$	12.0618	1.21843
$99$	0	0
$100$	0	0
$101$	−12.5955	−1.25330	−0.626651	−	0.779300i	$-0.715575\pi$
$101$	−12.5955	−1.25330	−0.626651	+	0.779300i	$0.715575\pi$
$102$	0	0
$103$	−6.23867	−0.614715	−0.307357	−	0.951594i	$-0.599445\pi$
$103$	−6.23867	−0.614715	−0.307357	+	0.951594i	$0.599445\pi$
$104$	1.52618	0.149654
$105$	0	0
$106$	−25.3325	−2.46051
$107$	1.10554	0.106877	0.0534384	−	0.998571i	$-0.482982\pi$
$107$	1.10554	0.106877	0.0534384	+	0.998571i	$0.482982\pi$
$108$	0	0
$109$	7.92019	0.758616	0.379308	−	0.925270i	$-0.376162\pi$
$109$	7.92019	0.758616	0.379308	+	0.925270i	$0.376162\pi$
$110$	0	0
$111$	0	0
$112$	2.17977	0.205969
$113$	11.4432	1.07649	0.538244	−	0.842789i	$-0.319088\pi$
$113$	11.4432	1.07649	0.538244	+	0.842789i	$0.319088\pi$
$114$	0	0
$115$	0	0
$116$	−1.96162	−0.182132
$117$	0	0
$118$	20.2655	1.86559
$119$	1.74214	0.159702
$120$	0	0
$121$	21.8948	1.99044
$122$	4.00000	0.362143
$123$	0	0
$124$	−8.95070	−0.803796
$125$	0	0
$126$	0	0
$127$	1.42937	0.126836	0.0634180	−	0.997987i	$-0.479800\pi$
$127$	1.42937	0.126836	0.0634180	+	0.997987i	$0.479800\pi$
$128$	−11.0768	−0.979060
$129$	0	0
$130$	0	0
$131$	−17.6819	−1.54487	−0.772437	−	0.635092i	$-0.780962\pi$
$131$	−17.6819	−1.54487	−0.772437	+	0.635092i	$0.780962\pi$
$132$	0	0
$133$	−2.08984	−0.181212
$134$	−3.86166	−0.333597
$135$	0	0
$136$	−6.07543	−0.520964
$137$	−10.7686	−0.920021	−0.460011	−	0.887913i	$-0.652154\pi$
$137$	−10.7686	−0.920021	−0.460011	+	0.887913i	$0.652154\pi$
$138$	0	0
$139$	3.80848	0.323031	0.161515	−	0.986870i	$-0.448362\pi$
$139$	3.80848	0.323031	0.161515	+	0.986870i	$0.448362\pi$
$140$	0	0
$141$	0	0
$142$	−24.2655	−2.03631
$143$	−5.73540	−0.479618
$144$	0	0
$145$	0	0
$146$	4.52481	0.374476
$147$	0	0
$148$	1.89260	0.155571
$149$	16.3649	1.34067	0.670334	−	0.742060i	$-0.266151\pi$
$149$	16.3649	1.34067	0.670334	+	0.742060i	$0.266151\pi$
$150$	0	0
$151$	20.3774	1.65829	0.829144	−	0.559035i	$-0.188828\pi$
$151$	20.3774	1.65829	0.829144	+	0.559035i	$0.188828\pi$
$152$	7.28797	0.591133
$153$	0	0
$154$	−4.44670	−0.358325
$155$	0	0
$156$	0	0
$157$	−13.1911	−1.05276	−0.526381	−	0.850249i	$-0.676451\pi$
$157$	−13.1911	−1.05276	−0.526381	+	0.850249i	$0.676451\pi$
$158$	−4.52010	−0.359600
$159$	0	0
$160$	0	0
$161$	0.147777	0.0116465
$162$	0	0
$163$	10.3578	0.811287	0.405644	−	0.914031i	$-0.367047\pi$
$163$	10.3578	0.811287	0.405644	+	0.914031i	$0.367047\pi$
$164$	−3.05471	−0.238533
$165$	0	0
$166$	−9.58708	−0.744102
$167$	10.6713	0.825769	0.412885	−	0.910783i	$-0.364521\pi$
$167$	10.6713	0.825769	0.412885	+	0.910783i	$0.364521\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	2.59248	0.197675
$173$	−5.64045	−0.428836	−0.214418	−	0.976742i	$-0.568785\pi$
$173$	−5.64045	−0.428836	−0.214418	+	0.976742i	$0.568785\pi$
$174$	0	0
$175$	0	0
$176$	28.5669	2.15331
$177$	0	0
$178$	−8.36983	−0.627345
$179$	11.9616	0.894054	0.447027	−	0.894521i	$-0.352483\pi$
$179$	11.9616	0.894054	0.447027	+	0.894521i	$0.352483\pi$
$180$	0	0
$181$	11.3442	0.843209	0.421604	−	0.906780i	$-0.361467\pi$
$181$	11.3442	0.843209	0.421604	+	0.906780i	$0.361467\pi$
$182$	0.775308	0.0574696
$183$	0	0
$184$	−0.515350	−0.0379921
$185$	0	0
$186$	0	0
$187$	22.8315	1.66961
$188$	13.3392	0.972861
$189$	0	0
$190$	0	0
$191$	19.8691	1.43768	0.718839	−	0.695177i	$-0.244674\pi$
$191$	19.8691	1.43768	0.718839	+	0.695177i	$0.244674\pi$
$192$	0	0
$193$	1.10823	0.0797719	0.0398859	−	0.999204i	$-0.487301\pi$
$193$	1.10823	0.0797719	0.0398859	+	0.999204i	$0.487301\pi$
$194$	26.2020	1.88119
$195$	0	0
$196$	−7.75162	−0.553687
$197$	−5.37758	−0.383137	−0.191568	−	0.981479i	$-0.561357\pi$
$197$	−5.37758	−0.383137	−0.191568	+	0.981479i	$0.561357\pi$
$198$	0	0
$199$	−11.2736	−0.799162	−0.399581	−	0.916698i	$-0.630844\pi$
$199$	−11.2736	−0.799162	−0.399581	+	0.916698i	$0.630844\pi$
$200$	0	0
$201$	0	0
$202$	22.3141	1.57001
$203$	0.754022	0.0529220
$204$	0	0
$205$	0	0
$206$	11.0524	0.770054
$207$	0	0
$208$	−4.98081	−0.345357
$209$	−27.3883	−1.89449
$210$	0	0
$211$	−16.4498	−1.13245	−0.566224	−	0.824251i	$-0.691596\pi$
$211$	−16.4498	−1.13245	−0.566224	+	0.824251i	$0.691596\pi$
$212$	16.2801	1.11812
$213$	0	0
$214$	−1.95857	−0.133885
$215$	0	0
$216$	0	0
$217$	3.44053	0.233559
$218$	−14.0313	−0.950320
$219$	0	0
$220$	0	0
$221$	−3.98081	−0.267778
$222$	0	0
$223$	25.0527	1.67765	0.838827	−	0.544397i	$-0.183242\pi$
$223$	25.0527	1.67765	0.838827	+	0.544397i	$0.183242\pi$
$224$	−2.52584	−0.168765
$225$	0	0
$226$	−20.2727	−1.34852
$227$	8.11414	0.538554	0.269277	−	0.963063i	$-0.413215\pi$
$227$	8.11414	0.538554	0.269277	+	0.963063i	$0.413215\pi$
$228$	0	0
$229$	9.57717	0.632877	0.316439	−	0.948613i	$-0.397513\pi$
$229$	9.57717	0.632877	0.316439	+	0.948613i	$0.397513\pi$
$230$	0	0
$231$	0	0
$232$	−2.62953	−0.172637
$233$	16.3791	1.07303	0.536516	−	0.843890i	$-0.319740\pi$
$233$	16.3791	1.07303	0.536516	+	0.843890i	$0.319740\pi$
$234$	0	0
$235$	0	0
$236$	−13.0238	−0.847775
$237$	0	0
$238$	−3.08635	−0.200059
$239$	22.2511	1.43931	0.719653	−	0.694334i	$-0.244301\pi$
$239$	22.2511	1.43931	0.719653	+	0.694334i	$0.244301\pi$
$240$	0	0
$241$	23.8031	1.53329	0.766647	−	0.642068i	$-0.221924\pi$
$241$	23.8031	1.53329	0.766647	+	0.642068i	$0.221924\pi$
$242$	−38.7886	−2.49343
$243$	0	0
$244$	−2.57063	−0.164568
$245$	0	0
$246$	0	0
$247$	4.77531	0.303846
$248$	−11.9983	−0.761893
$249$	0	0
$250$	0	0
$251$	4.35992	0.275196	0.137598	−	0.990488i	$-0.456062\pi$
$251$	4.35992	0.275196	0.137598	+	0.990488i	$0.456062\pi$
$252$	0	0
$253$	1.93669	0.121759
$254$	−2.53225	−0.158888
$255$	0	0
$256$	20.1500	1.25938
$257$	14.9965	0.935457	0.467728	−	0.883872i	$-0.345073\pi$
$257$	14.9965	0.935457	0.467728	+	0.883872i	$0.345073\pi$
$258$	0	0
$259$	−0.727491	−0.0452041
$260$	0	0
$261$	0	0
$262$	31.3250	1.93527
$263$	10.7029	0.659971	0.329986	−	0.943986i	$-0.392956\pi$
$263$	10.7029	0.659971	0.329986	+	0.943986i	$0.392956\pi$
$264$	0	0
$265$	0	0
$266$	3.70233	0.227005
$267$	0	0
$268$	2.48173	0.151596
$269$	−15.0076	−0.915028	−0.457514	−	0.889202i	$-0.651260\pi$
$269$	−15.0076	−0.915028	−0.457514	+	0.889202i	$0.651260\pi$
$270$	0	0
$271$	−1.71386	−0.104109	−0.0520547	−	0.998644i	$-0.516577\pi$
$271$	−1.71386	−0.104109	−0.0520547	+	0.998644i	$0.516577\pi$
$272$	19.8277	1.20223
$273$	0	0
$274$	19.0775	1.15251
$275$	0	0
$276$	0	0
$277$	21.1527	1.27094	0.635471	−	0.772125i	$-0.280806\pi$
$277$	21.1527	1.27094	0.635471	+	0.772125i	$0.280806\pi$
$278$	−6.74705	−0.404661
$279$	0	0
$280$	0	0
$281$	−18.7440	−1.11818	−0.559088	−	0.829108i	$-0.688849\pi$
$281$	−18.7440	−1.11818	−0.559088	+	0.829108i	$0.688849\pi$
$282$	0	0
$283$	−9.29744	−0.552675	−0.276338	−	0.961061i	$-0.589121\pi$
$283$	−9.29744	−0.552675	−0.276338	+	0.961061i	$0.589121\pi$
$284$	15.5944	0.925358
$285$	0	0
$286$	10.1608	0.600819
$287$	1.17419	0.0693103
$288$	0	0
$289$	−1.15315	−0.0678321
$290$	0	0
$291$	0	0
$292$	−2.90791	−0.170172
$293$	−17.9762	−1.05018	−0.525090	−	0.851047i	$-0.675968\pi$
$293$	−17.9762	−1.05018	−0.525090	+	0.851047i	$0.675968\pi$
$294$	0	0
$295$	0	0
$296$	2.53701	0.147461
$297$	0	0
$298$	−28.9919	−1.67946
$299$	−0.337673	−0.0195282
$300$	0	0
$301$	−0.996515	−0.0574382
$302$	−36.1003	−2.07734
$303$	0	0
$304$	−23.7849	−1.36416
$305$	0	0
$306$	0	0
$307$	13.3129	0.759807	0.379904	−	0.925026i	$-0.375957\pi$
$307$	13.3129	0.759807	0.379904	+	0.925026i	$0.375957\pi$
$308$	2.85771	0.162833
$309$	0	0
$310$	0	0
$311$	−2.64776	−0.150141	−0.0750703	−	0.997178i	$-0.523918\pi$
$311$	−2.64776	−0.150141	−0.0750703	+	0.997178i	$0.523918\pi$
$312$	0	0
$313$	−15.8718	−0.897126	−0.448563	−	0.893751i	$-0.648064\pi$
$313$	−15.8718	−0.897126	−0.448563	+	0.893751i	$0.648064\pi$
$314$	23.3691	1.31880
$315$	0	0
$316$	2.90488	0.163412
$317$	3.85187	0.216342	0.108171	−	0.994132i	$-0.465501\pi$
$317$	3.85187	0.216342	0.108171	+	0.994132i	$0.465501\pi$
$318$	0	0
$319$	9.88181	0.553275
$320$	0	0
$321$	0	0
$322$	−0.261801	−0.0145896
$323$	−19.0096	−1.05772
$324$	0	0
$325$	0	0
$326$	−18.3498	−1.01630
$327$	0	0
$328$	−4.09480	−0.226098
$329$	−5.12742	−0.282684
$330$	0	0
$331$	−8.15256	−0.448105	−0.224053	−	0.974577i	$-0.571929\pi$
$331$	−8.15256	−0.448105	−0.224053	+	0.974577i	$0.571929\pi$
$332$	6.16121	0.338141
$333$	0	0
$334$	−18.9051	−1.03444
$335$	0	0
$336$	0	0
$337$	−0.343016	−0.0186852	−0.00934262	−	0.999956i	$-0.502974\pi$
$337$	−0.343016	−0.0186852	−0.00934262	+	0.999956i	$0.502974\pi$
$338$	−1.77159	−0.0963617
$339$	0	0
$340$	0	0
$341$	45.0898	2.44175
$342$	0	0
$343$	6.04306	0.326295
$344$	3.47519	0.187369
$345$	0	0
$346$	9.99256	0.537203
$347$	32.2306	1.73023	0.865116	−	0.501572i	$-0.167245\pi$
$347$	32.2306	1.73023	0.865116	+	0.501572i	$0.167245\pi$
$348$	0	0
$349$	23.6186	1.26427	0.632137	−	0.774856i	$-0.282178\pi$
$349$	23.6186	1.26427	0.632137	+	0.774856i	$0.282178\pi$
$350$	0	0
$351$	0	0
$352$	−33.1024	−1.76436
$353$	30.5552	1.62629	0.813144	−	0.582063i	$-0.197754\pi$
$353$	30.5552	1.62629	0.813144	+	0.582063i	$0.197754\pi$
$354$	0	0
$355$	0	0
$356$	5.37894	0.285083
$357$	0	0
$358$	−21.1911	−1.11998
$359$	30.2293	1.59544	0.797719	−	0.603029i	$-0.206040\pi$
$359$	30.2293	1.59544	0.797719	+	0.603029i	$0.206040\pi$
$360$	0	0
$361$	3.80356	0.200188
$362$	−20.0973	−1.05629
$363$	0	0
$364$	−0.498258	−0.0261158
$365$	0	0
$366$	0	0
$367$	10.0441	0.524299	0.262149	−	0.965027i	$-0.415569\pi$
$367$	10.0441	0.524299	0.262149	+	0.965027i	$0.415569\pi$
$368$	1.68189	0.0876744
$369$	0	0
$370$	0	0
$371$	−6.25786	−0.324892
$372$	0	0
$373$	28.8097	1.49171	0.745854	−	0.666109i	$-0.232042\pi$
$373$	28.8097	1.49171	0.745854	+	0.666109i	$0.232042\pi$
$374$	−40.4481	−2.09152
$375$	0	0
$376$	17.8810	0.922144
$377$	−1.72295	−0.0887364
$378$	0	0
$379$	17.0581	0.876216	0.438108	−	0.898922i	$-0.355649\pi$
$379$	17.0581	0.876216	0.438108	+	0.898922i	$0.355649\pi$
$380$	0	0
$381$	0	0
$382$	−35.1999	−1.80098
$383$	6.58842	0.336653	0.168326	−	0.985731i	$-0.446164\pi$
$383$	6.58842	0.336653	0.168326	+	0.985731i	$0.446164\pi$
$384$	0	0
$385$	0	0
$386$	−1.96332	−0.0999304
$387$	0	0
$388$	−16.8389	−0.854866
$389$	−30.5741	−1.55017	−0.775083	−	0.631859i	$-0.782292\pi$
$389$	−30.5741	−1.55017	−0.775083	+	0.631859i	$0.782292\pi$
$390$	0	0
$391$	1.34421	0.0679798
$392$	−10.3910	−0.524822
$393$	0	0
$394$	9.52686	0.479956
$395$	0	0
$396$	0	0
$397$	−21.5069	−1.07940	−0.539700	−	0.841857i	$-0.681462\pi$
$397$	−21.5069	−1.07940	−0.539700	+	0.841857i	$0.681462\pi$
$398$	19.9721	1.00111
$399$	0	0
$400$	0	0
$401$	17.8990	0.893835	0.446918	−	0.894575i	$-0.352522\pi$
$401$	17.8990	0.893835	0.446918	+	0.894575i	$0.352522\pi$
$402$	0	0
$403$	−7.86166	−0.391617
$404$	−14.3403	−0.713458
$405$	0	0
$406$	−1.33582	−0.0662954
$407$	−9.53411	−0.472588
$408$	0	0
$409$	22.3844	1.10684	0.553420	−	0.832902i	$-0.313322\pi$
$409$	22.3844	1.10684	0.553420	+	0.832902i	$0.313322\pi$
$410$	0	0
$411$	0	0
$412$	−7.10288	−0.349934
$413$	5.00617	0.246337
$414$	0	0
$415$	0	0
$416$	5.77159	0.282975
$417$	0	0
$418$	48.5208	2.37323
$419$	27.5153	1.34421	0.672105	−	0.740456i	$-0.265390\pi$
$419$	27.5153	1.34421	0.672105	+	0.740456i	$0.265390\pi$
$420$	0	0
$421$	6.34607	0.309289	0.154644	−	0.987970i	$-0.450577\pi$
$421$	6.34607	0.309289	0.154644	+	0.987970i	$0.450577\pi$
$422$	29.1422	1.41862
$423$	0	0
$424$	21.8233	1.05983
$425$	0	0
$426$	0	0
$427$	0.988117	0.0478183
$428$	1.25869	0.0608410
$429$	0	0
$430$	0	0
$431$	3.78383	0.182261	0.0911304	−	0.995839i	$-0.470952\pi$
$431$	3.78383	0.182261	0.0911304	+	0.995839i	$0.470952\pi$
$432$	0	0
$433$	−2.55410	−0.122742	−0.0613711	−	0.998115i	$-0.519547\pi$
$433$	−2.55410	−0.122742	−0.0613711	+	0.998115i	$0.519547\pi$
$434$	−6.09521	−0.292579
$435$	0	0
$436$	9.01733	0.431852
$437$	−1.61249	−0.0771361
$438$	0	0
$439$	2.34073	0.111717	0.0558585	−	0.998439i	$-0.482210\pi$
$439$	2.34073	0.111717	0.0558585	+	0.998439i	$0.482210\pi$
$440$	0	0
$441$	0	0
$442$	7.05236	0.335446
$443$	0.468574	0.0222626	0.0111313	−	0.999938i	$-0.496457\pi$
$443$	0.468574	0.0222626	0.0111313	+	0.999938i	$0.496457\pi$
$444$	0	0
$445$	0	0
$446$	−44.3831	−2.10160
$447$	0	0
$448$	0.115210	0.00544317
$449$	17.5863	0.829947	0.414974	−	0.909833i	$-0.363791\pi$
$449$	17.5863	0.829947	0.414974	+	0.909833i	$0.363791\pi$
$450$	0	0
$451$	15.3883	0.724608
$452$	13.0284	0.612804
$453$	0	0
$454$	−14.3749	−0.674648
$455$	0	0
$456$	0	0
$457$	−11.9132	−0.557276	−0.278638	−	0.960396i	$-0.589883\pi$
$457$	−11.9132	−0.557276	−0.278638	+	0.960396i	$0.589883\pi$
$458$	−16.9668	−0.792807
$459$	0	0
$460$	0	0
$461$	17.7540	0.826886	0.413443	−	0.910530i	$-0.364326\pi$
$461$	17.7540	0.826886	0.413443	+	0.910530i	$0.364326\pi$
$462$	0	0
$463$	39.4856	1.83505	0.917526	−	0.397676i	$-0.130183\pi$
$463$	39.4856	1.83505	0.917526	+	0.397676i	$0.130183\pi$
$464$	8.58169	0.398395
$465$	0	0
$466$	−29.0170	−1.34419
$467$	−19.2793	−0.892139	−0.446069	−	0.894998i	$-0.647177\pi$
$467$	−19.2793	−0.892139	−0.446069	+	0.894998i	$0.647177\pi$
$468$	0	0
$469$	−0.953943	−0.0440490
$470$	0	0
$471$	0	0
$472$	−17.4582	−0.803579
$473$	−13.0598	−0.600490
$474$	0	0
$475$	0	0
$476$	1.98347	0.0909122
$477$	0	0
$478$	−39.4198	−1.80302
$479$	−38.8194	−1.77371	−0.886853	−	0.462052i	$-0.847113\pi$
$479$	−38.8194	−1.77371	−0.886853	+	0.462052i	$0.847113\pi$
$480$	0	0
$481$	1.66233	0.0757956
$482$	−42.1694	−1.92076
$483$	0	0
$484$	24.9278	1.13308
$485$	0	0
$486$	0	0
$487$	−29.8866	−1.35429	−0.677145	−	0.735849i	$-0.736783\pi$
$487$	−29.8866	−1.35429	−0.677145	+	0.735849i	$0.736783\pi$
$488$	−3.44590	−0.155989
$489$	0	0
$490$	0	0
$491$	−26.7474	−1.20709	−0.603547	−	0.797327i	$-0.706246\pi$
$491$	−26.7474	−1.20709	−0.603547	+	0.797327i	$0.706246\pi$
$492$	0	0
$493$	6.85874	0.308902
$494$	−8.45988	−0.380628
$495$	0	0
$496$	39.1574	1.75822
$497$	−5.99429	−0.268880
$498$	0	0
$499$	31.9694	1.43115	0.715574	−	0.698537i	$-0.246165\pi$
$499$	31.9694	1.43115	0.715574	+	0.698537i	$0.246165\pi$
$500$	0	0
$501$	0	0
$502$	−7.72398	−0.344738
$503$	9.37993	0.418231	0.209115	−	0.977891i	$-0.432942\pi$
$503$	9.37993	0.418231	0.209115	+	0.977891i	$0.432942\pi$
$504$	0	0
$505$	0	0
$506$	−3.43102	−0.152528
$507$	0	0
$508$	1.62737	0.0722030
$509$	−11.0042	−0.487753	−0.243877	−	0.969806i	$-0.578419\pi$
$509$	−11.0042	−0.487753	−0.243877	+	0.969806i	$0.578419\pi$
$510$	0	0
$511$	1.11776	0.0494469
$512$	−13.5440	−0.598565
$513$	0	0
$514$	−26.5677	−1.17185
$515$	0	0
$516$	0	0
$517$	−67.1972	−2.95533
$518$	1.28881	0.0566273
$519$	0	0
$520$	0	0
$521$	−29.1305	−1.27623	−0.638115	−	0.769941i	$-0.720285\pi$
$521$	−29.1305	−1.27623	−0.638115	+	0.769941i	$0.720285\pi$
$522$	0	0
$523$	−13.5675	−0.593266	−0.296633	−	0.954992i	$-0.595864\pi$
$523$	−13.5675	−0.593266	−0.296633	+	0.954992i	$0.595864\pi$
$524$	−20.1313	−0.879439
$525$	0	0
$526$	−18.9612	−0.826747
$527$	31.2958	1.36327
$528$	0	0
$529$	−22.8860	−0.995042
$530$	0	0
$531$	0	0
$532$	−2.37933	−0.103157
$533$	−2.68304	−0.116216
$534$	0	0
$535$	0	0
$536$	3.32672	0.143693
$537$	0	0
$538$	26.5872	1.14626
$539$	39.0493	1.68197
$540$	0	0
$541$	−22.6672	−0.974541	−0.487270	−	0.873251i	$-0.662007\pi$
$541$	−22.6672	−0.974541	−0.487270	+	0.873251i	$0.662007\pi$
$542$	3.03625	0.130418
$543$	0	0
$544$	−22.9756	−0.985071
$545$	0	0
$546$	0	0
$547$	33.5014	1.43242	0.716209	−	0.697886i	$-0.245876\pi$
$547$	33.5014	1.43242	0.716209	+	0.697886i	$0.245876\pi$
$548$	−12.2603	−0.523734
$549$	0	0
$550$	0	0
$551$	−8.22762	−0.350508
$552$	0	0
$553$	−1.11660	−0.0474826
$554$	−37.4739	−1.59211
$555$	0	0
$556$	4.33605	0.183889
$557$	−42.6185	−1.80580	−0.902902	−	0.429846i	$-0.858568\pi$
$557$	−42.6185	−1.80580	−0.902902	+	0.429846i	$0.858568\pi$
$558$	0	0
$559$	2.27705	0.0963090
$560$	0	0
$561$	0	0
$562$	33.2067	1.40074
$563$	17.1688	0.723580	0.361790	−	0.932260i	$-0.382166\pi$
$563$	17.1688	0.723580	0.361790	+	0.932260i	$0.382166\pi$
$564$	0	0
$565$	0	0
$566$	16.4712	0.692338
$567$	0	0
$568$	20.9041	0.877117
$569$	−16.9018	−0.708561	−0.354281	−	0.935139i	$-0.615274\pi$
$569$	−16.9018	−0.708561	−0.354281	+	0.935139i	$0.615274\pi$
$570$	0	0
$571$	−29.9408	−1.25298	−0.626491	−	0.779428i	$-0.715510\pi$
$571$	−29.9408	−1.25298	−0.626491	+	0.779428i	$0.715510\pi$
$572$	−6.52990	−0.273029
$573$	0	0
$574$	−2.08018	−0.0868252
$575$	0	0
$576$	0	0
$577$	44.8430	1.86684	0.933419	−	0.358789i	$-0.116810\pi$
$577$	44.8430	1.86684	0.933419	+	0.358789i	$0.116810\pi$
$578$	2.04290	0.0849734
$579$	0	0
$580$	0	0
$581$	−2.36829	−0.0982532
$582$	0	0
$583$	−82.0122	−3.39660
$584$	−3.89801	−0.161301
$585$	0	0
$586$	31.8464	1.31556
$587$	−24.2703	−1.00174	−0.500872	−	0.865522i	$-0.666987\pi$
$587$	−24.2703	−1.00174	−0.500872	+	0.865522i	$0.666987\pi$
$588$	0	0
$589$	−37.5418	−1.54688
$590$	0	0
$591$	0	0
$592$	−8.27973	−0.340295
$593$	−44.2659	−1.81778	−0.908892	−	0.417032i	$-0.863070\pi$
$593$	−44.2659	−1.81778	−0.908892	+	0.417032i	$0.863070\pi$
$594$	0	0
$595$	0	0
$596$	18.6319	0.763192
$597$	0	0
$598$	0.598218	0.0244630
$599$	18.9996	0.776301	0.388151	−	0.921596i	$-0.373114\pi$
$599$	18.9996	0.776301	0.388151	+	0.921596i	$0.373114\pi$
$600$	0	0
$601$	0.722123	0.0294560	0.0147280	−	0.999892i	$-0.495312\pi$
$601$	0.722123	0.0294560	0.0147280	+	0.999892i	$0.495312\pi$
$602$	1.76541	0.0719529
$603$	0	0
$604$	23.2002	0.944001
$605$	0	0
$606$	0	0
$607$	29.9236	1.21456	0.607281	−	0.794487i	$-0.292260\pi$
$607$	29.9236	1.21456	0.607281	+	0.794487i	$0.292260\pi$
$608$	27.5611	1.11775
$609$	0	0
$610$	0	0
$611$	11.7162	0.473987
$612$	0	0
$613$	34.6980	1.40144	0.700719	−	0.713438i	$-0.252863\pi$
$613$	34.6980	1.40144	0.700719	+	0.713438i	$0.252863\pi$
$614$	−23.5850	−0.951812
$615$	0	0
$616$	3.83072	0.154344
$617$	−25.8247	−1.03966	−0.519831	−	0.854269i	$-0.674005\pi$
$617$	−25.8247	−1.03966	−0.519831	+	0.854269i	$0.674005\pi$
$618$	0	0
$619$	−29.3106	−1.17809	−0.589047	−	0.808099i	$-0.700497\pi$
$619$	−29.3106	−1.17809	−0.589047	+	0.808099i	$0.700497\pi$
$620$	0	0
$621$	0	0
$622$	4.69074	0.188082
$623$	−2.06759	−0.0828364
$624$	0	0
$625$	0	0
$626$	28.1183	1.12383
$627$	0	0
$628$	−15.0184	−0.599298
$629$	−6.61741	−0.263853
$630$	0	0
$631$	−22.9480	−0.913546	−0.456773	−	0.889583i	$-0.650995\pi$
$631$	−22.9480	−0.913546	−0.456773	+	0.889583i	$0.650995\pi$
$632$	3.89396	0.154893
$633$	0	0
$634$	−6.82392	−0.271013
$635$	0	0
$636$	0	0
$637$	−6.80848	−0.269762
$638$	−17.5065	−0.693089
$639$	0	0
$640$	0	0
$641$	−24.3185	−0.960522	−0.480261	−	0.877126i	$-0.659458\pi$
$641$	−24.3185	−0.960522	−0.480261	+	0.877126i	$0.659458\pi$
$642$	0	0
$643$	11.2911	0.445276	0.222638	−	0.974901i	$-0.428533\pi$
$643$	11.2911	0.445276	0.222638	+	0.974901i	$0.428533\pi$
$644$	0.168248	0.00662991
$645$	0	0
$646$	33.6772	1.32501
$647$	−7.55633	−0.297070	−0.148535	−	0.988907i	$-0.547456\pi$
$647$	−7.55633	−0.297070	−0.148535	+	0.988907i	$0.547456\pi$
$648$	0	0
$649$	65.6082	2.57535
$650$	0	0
$651$	0	0
$652$	11.7926	0.461835
$653$	−33.6711	−1.31765	−0.658826	−	0.752295i	$-0.728947\pi$
$653$	−33.6711	−1.31765	−0.658826	+	0.752295i	$0.728947\pi$
$654$	0	0
$655$	0	0
$656$	13.3637	0.521766
$657$	0	0
$658$	9.08367	0.354118
$659$	17.1039	0.666274	0.333137	−	0.942878i	$-0.391893\pi$
$659$	17.1039	0.666274	0.333137	+	0.942878i	$0.391893\pi$
$660$	0	0
$661$	19.1964	0.746655	0.373327	−	0.927700i	$-0.378217\pi$
$661$	19.1964	0.746655	0.373327	+	0.927700i	$0.378217\pi$
$662$	14.4430	0.561342
$663$	0	0
$664$	8.25903	0.320513
$665$	0	0
$666$	0	0
$667$	0.581794	0.0225272
$668$	12.1495	0.470080
$669$	0	0
$670$	0	0
$671$	12.9497	0.499919
$672$	0	0
$673$	−27.9616	−1.07784	−0.538921	−	0.842357i	$-0.681168\pi$
$673$	−27.9616	−1.07784	−0.538921	+	0.842357i	$0.681168\pi$
$674$	0.607682	0.0234070
$675$	0	0
$676$	1.13853	0.0437894
$677$	6.98533	0.268468	0.134234	−	0.990950i	$-0.457143\pi$
$677$	6.98533	0.268468	0.134234	+	0.990950i	$0.457143\pi$
$678$	0	0
$679$	6.47266	0.248398
$680$	0	0
$681$	0	0
$682$	−79.8805	−3.05878
$683$	18.0969	0.692460	0.346230	−	0.938150i	$-0.387462\pi$
$683$	18.0969	0.692460	0.346230	+	0.938150i	$0.387462\pi$
$684$	0	0
$685$	0	0
$686$	−10.7058	−0.408750
$687$	0	0
$688$	−11.3416	−0.432393
$689$	14.2993	0.544760
$690$	0	0
$691$	13.9818	0.531892	0.265946	−	0.963988i	$-0.414316\pi$
$691$	13.9818	0.531892	0.265946	+	0.963988i	$0.414316\pi$
$692$	−6.42180	−0.244120
$693$	0	0
$694$	−57.0994	−2.16746
$695$	0	0
$696$	0	0
$697$	10.6807	0.404560
$698$	−41.8424	−1.58376
$699$	0	0
$700$	0	0
$701$	2.64470	0.0998891	0.0499445	−	0.998752i	$-0.484096\pi$
$701$	2.64470	0.0998891	0.0499445	+	0.998752i	$0.484096\pi$
$702$	0	0
$703$	7.93812	0.299392
$704$	1.50988	0.0569058
$705$	0	0
$706$	−54.1312	−2.03725
$707$	5.51224	0.207309
$708$	0	0
$709$	9.89834	0.371740	0.185870	−	0.982574i	$-0.440490\pi$
$709$	9.89834	0.371740	0.185870	+	0.982574i	$0.440490\pi$
$710$	0	0
$711$	0	0
$712$	7.21040	0.270221
$713$	2.65467	0.0994183
$714$	0	0
$715$	0	0
$716$	13.6186	0.508951
$717$	0	0
$718$	−53.5538	−1.99861
$719$	−21.1288	−0.787972	−0.393986	−	0.919116i	$-0.628904\pi$
$719$	−21.1288	−0.787972	−0.393986	+	0.919116i	$0.628904\pi$
$720$	0	0
$721$	2.73026	0.101680
$722$	−6.73835	−0.250775
$723$	0	0
$724$	12.9157	0.480007
$725$	0	0
$726$	0	0
$727$	−18.7471	−0.695290	−0.347645	−	0.937626i	$-0.613019\pi$
$727$	−18.7471	−0.695290	−0.347645	+	0.937626i	$0.613019\pi$
$728$	−0.667908	−0.0247543
$729$	0	0
$730$	0	0
$731$	−9.06451	−0.335263
$732$	0	0
$733$	47.2675	1.74586	0.872932	−	0.487842i	$-0.162216\pi$
$733$	47.2675	1.74586	0.872932	+	0.487842i	$0.162216\pi$
$734$	−17.7940	−0.656790
$735$	0	0
$736$	−1.94891	−0.0718379
$737$	−12.5019	−0.460512
$738$	0	0
$739$	−40.6501	−1.49534	−0.747670	−	0.664071i	$-0.768827\pi$
$739$	−40.6501	−1.49534	−0.747670	+	0.664071i	$0.768827\pi$
$740$	0	0
$741$	0	0
$742$	11.0864	0.406993
$743$	1.06271	0.0389871	0.0194936	−	0.999810i	$-0.493795\pi$
$743$	1.06271	0.0389871	0.0194936	+	0.999810i	$0.493795\pi$
$744$	0	0
$745$	0	0
$746$	−51.0389	−1.86867
$747$	0	0
$748$	25.9943	0.950445
$749$	−0.483823	−0.0176785
$750$	0	0
$751$	−10.8780	−0.396945	−0.198473	−	0.980106i	$-0.563598\pi$
$751$	−10.8780	−0.396945	−0.198473	+	0.980106i	$0.563598\pi$
$752$	−58.3562	−2.12803
$753$	0	0
$754$	3.05236	0.111160
$755$	0	0
$756$	0	0
$757$	−16.2279	−0.589812	−0.294906	−	0.955526i	$-0.595288\pi$
$757$	−16.2279	−0.589812	−0.294906	+	0.955526i	$0.595288\pi$
$758$	−30.2199	−1.09764
$759$	0	0
$760$	0	0
$761$	−37.3880	−1.35531	−0.677657	−	0.735379i	$-0.737004\pi$
$761$	−37.3880	−1.35531	−0.677657	+	0.735379i	$0.737004\pi$
$762$	0	0
$763$	−3.46615	−0.125483
$764$	22.6215	0.818416
$765$	0	0
$766$	−11.6720	−0.421726
$767$	−11.4392	−0.413044
$768$	0	0
$769$	−12.5874	−0.453914	−0.226957	−	0.973905i	$-0.572878\pi$
$769$	−12.5874	−0.453914	−0.226957	+	0.973905i	$0.572878\pi$
$770$	0	0
$771$	0	0
$772$	1.26174	0.0454111
$773$	0.746723	0.0268577	0.0134289	−	0.999910i	$-0.495725\pi$
$773$	0.746723	0.0268577	0.0134289	+	0.999910i	$0.495725\pi$
$774$	0	0
$775$	0	0
$776$	−22.5724	−0.810300
$777$	0	0
$778$	54.1646	1.94190
$779$	−12.8124	−0.459050
$780$	0	0
$781$	−78.5579	−2.81102
$782$	−2.38139	−0.0851585
$783$	0	0
$784$	33.9117	1.21113
$785$	0	0
$786$	0	0
$787$	33.3585	1.18910	0.594551	−	0.804058i	$-0.297330\pi$
$787$	33.3585	1.18910	0.594551	+	0.804058i	$0.297330\pi$
$788$	−6.12251	−0.218105
$789$	0	0
$790$	0	0
$791$	−5.00794	−0.178062
$792$	0	0
$793$	−2.25786	−0.0801790
$794$	38.1014	1.35217
$795$	0	0
$796$	−12.8352	−0.454933
$797$	−0.233988	−0.00828829	−0.00414415	−	0.999991i	$-0.501319\pi$
$797$	−0.233988	−0.00828829	−0.00414415	+	0.999991i	$0.501319\pi$
$798$	0	0
$799$	−46.6400	−1.65001
$800$	0	0
$801$	0	0
$802$	−31.7097	−1.11971
$803$	14.6488	0.516945
$804$	0	0
$805$	0	0
$806$	13.9276	0.490580
$807$	0	0
$808$	−19.2230	−0.676264
$809$	22.2003	0.780521	0.390260	−	0.920705i	$-0.372385\pi$
$809$	22.2003	0.780521	0.390260	+	0.920705i	$0.372385\pi$
$810$	0	0
$811$	−43.4456	−1.52558	−0.762791	−	0.646645i	$-0.776171\pi$
$811$	−43.4456	−1.52558	−0.762791	+	0.646645i	$0.776171\pi$
$812$	0.858473	0.0301265
$813$	0	0
$814$	16.8905	0.592012
$815$	0	0
$816$	0	0
$817$	10.8736	0.380420
$818$	−39.6560	−1.38654
$819$	0	0
$820$	0	0
$821$	6.76172	0.235986	0.117993	−	0.993014i	$-0.462354\pi$
$821$	6.76172	0.235986	0.117993	+	0.993014i	$0.462354\pi$
$822$	0	0
$823$	−28.6208	−0.997659	−0.498829	−	0.866700i	$-0.666237\pi$
$823$	−28.6208	−0.997659	−0.498829	+	0.866700i	$0.666237\pi$
$824$	−9.52133	−0.331691
$825$	0	0
$826$	−8.86887	−0.308587
$827$	−8.19215	−0.284869	−0.142435	−	0.989804i	$-0.545493\pi$
$827$	−8.19215	−0.284869	−0.142435	+	0.989804i	$0.545493\pi$
$828$	0	0
$829$	−3.60208	−0.125105	−0.0625526	−	0.998042i	$-0.519924\pi$
$829$	−3.60208	−0.125105	−0.0625526	+	0.998042i	$0.519924\pi$
$830$	0	0
$831$	0	0
$832$	−0.263257	−0.00912678
$833$	27.1033	0.939072
$834$	0	0
$835$	0	0
$836$	−31.1823	−1.07846
$837$	0	0
$838$	−48.7458	−1.68389
$839$	−1.50713	−0.0520318	−0.0260159	−	0.999662i	$-0.508282\pi$
$839$	−1.50713	−0.0520318	−0.0260159	+	0.999662i	$0.508282\pi$
$840$	0	0
$841$	−26.0314	−0.897636
$842$	−11.2426	−0.387446
$843$	0	0
$844$	−18.7285	−0.644660
$845$	0	0
$846$	0	0
$847$	−9.58193	−0.329239
$848$	−71.2221	−2.44578
$849$	0	0
$850$	0	0
$851$	−0.561324	−0.0192419
$852$	0	0
$853$	8.35952	0.286225	0.143112	−	0.989706i	$-0.454289\pi$
$853$	8.35952	0.286225	0.143112	+	0.989706i	$0.454289\pi$
$854$	−1.75054	−0.0599021
$855$	0	0
$856$	1.68726	0.0576692
$857$	12.8565	0.439168	0.219584	−	0.975594i	$-0.429530\pi$
$857$	12.8565	0.439168	0.219584	+	0.975594i	$0.429530\pi$
$858$	0	0
$859$	−16.3734	−0.558652	−0.279326	−	0.960196i	$-0.590111\pi$
$859$	−16.3734	−0.558652	−0.279326	+	0.960196i	$0.590111\pi$
$860$	0	0
$861$	0	0
$862$	−6.70339	−0.228318
$863$	50.0600	1.70406	0.852031	−	0.523492i	$-0.175371\pi$
$863$	50.0600	1.70406	0.852031	+	0.523492i	$0.175371\pi$
$864$	0	0
$865$	0	0
$866$	4.52481	0.153759
$867$	0	0
$868$	3.91713	0.132956
$869$	−14.6335	−0.496409
$870$	0	0
$871$	2.17977	0.0738588
$872$	12.0876	0.409339
$873$	0	0
$874$	2.85668	0.0966285
$875$	0	0
$876$	0	0
$877$	48.1027	1.62431	0.812157	−	0.583439i	$-0.198293\pi$
$877$	48.1027	1.62431	0.812157	+	0.583439i	$0.198293\pi$
$878$	−4.14681	−0.139948
$879$	0	0
$880$	0	0
$881$	41.2109	1.38843	0.694216	−	0.719767i	$-0.255751\pi$
$881$	41.2109	1.38843	0.694216	+	0.719767i	$0.255751\pi$
$882$	0	0
$883$	−13.9344	−0.468930	−0.234465	−	0.972125i	$-0.575334\pi$
$883$	−13.9344	−0.468930	−0.234465	+	0.972125i	$0.575334\pi$
$884$	−4.53225	−0.152436
$885$	0	0
$886$	−0.830120	−0.0278884
$887$	−47.8092	−1.60528	−0.802638	−	0.596466i	$-0.796571\pi$
$887$	−47.8092	−1.60528	−0.802638	+	0.596466i	$0.796571\pi$
$888$	0	0
$889$	−0.625541	−0.0209800
$890$	0	0
$891$	0	0
$892$	28.5232	0.955026
$893$	55.9485	1.87225
$894$	0	0
$895$	0	0
$896$	4.84758	0.161946
$897$	0	0
$898$	−31.1556	−1.03968
$899$	13.5452	0.451759
$900$	0	0
$901$	−56.9228	−1.89637
$902$	−27.2618	−0.907718
$903$	0	0
$904$	17.4644	0.580857
$905$	0	0
$906$	0	0
$907$	−8.74334	−0.290318	−0.145159	−	0.989408i	$-0.546369\pi$
$907$	−8.74334	−0.290318	−0.145159	+	0.989408i	$0.546369\pi$
$908$	9.23815	0.306579
$909$	0	0
$910$	0	0
$911$	2.67266	0.0885493	0.0442746	−	0.999019i	$-0.485902\pi$
$911$	2.67266	0.0885493	0.0442746	+	0.999019i	$0.485902\pi$
$912$	0	0
$913$	−31.0376	−1.02719
$914$	21.1053	0.698100
$915$	0	0
$916$	10.9039	0.360273
$917$	7.73820	0.255538
$918$	0	0
$919$	−15.4118	−0.508389	−0.254195	−	0.967153i	$-0.581810\pi$
$919$	−15.4118	−0.508389	−0.254195	+	0.967153i	$0.581810\pi$
$920$	0	0
$921$	0	0
$922$	−31.4528	−1.03584
$923$	13.6970	0.450843
$924$	0	0
$925$	0	0
$926$	−69.9522	−2.29877
$927$	0	0
$928$	−9.94416	−0.326433
$929$	−16.7575	−0.549795	−0.274898	−	0.961473i	$-0.588644\pi$
$929$	−16.7575	−0.549795	−0.274898	+	0.961473i	$0.588644\pi$
$930$	0	0
$931$	−32.5126	−1.06556
$932$	18.6480	0.610836
$933$	0	0
$934$	34.1549	1.11758
$935$	0	0
$936$	0	0
$937$	−34.3945	−1.12362	−0.561809	−	0.827267i	$-0.689895\pi$
$937$	−34.3945	−1.12362	−0.561809	+	0.827267i	$0.689895\pi$
$938$	1.68999	0.0551803
$939$	0	0
$940$	0	0
$941$	−3.90298	−0.127234	−0.0636168	−	0.997974i	$-0.520264\pi$
$941$	−3.90298	−0.127234	−0.0636168	+	0.997974i	$0.520264\pi$
$942$	0	0
$943$	0.905993	0.0295032
$944$	56.9763	1.85442
$945$	0	0
$946$	23.1366	0.752235
$947$	32.7611	1.06459	0.532297	−	0.846558i	$-0.321329\pi$
$947$	32.7611	1.06459	0.532297	+	0.846558i	$0.321329\pi$
$948$	0	0
$949$	−2.55410	−0.0829096
$950$	0	0
$951$	0	0
$952$	2.65882	0.0861727
$953$	−3.05611	−0.0989970	−0.0494985	−	0.998774i	$-0.515762\pi$
$953$	−3.05611	−0.0989970	−0.0494985	+	0.998774i	$0.515762\pi$
$954$	0	0
$955$	0	0
$956$	25.3335	0.819343
$957$	0	0
$958$	68.7721	2.22192
$959$	4.71269	0.152181
$960$	0	0
$961$	30.8057	0.993733
$962$	−2.94496	−0.0949493
$963$	0	0
$964$	27.1005	0.872847
$965$	0	0
$966$	0	0
$967$	−20.5538	−0.660966	−0.330483	−	0.943812i	$-0.607212\pi$
$967$	−20.5538	−0.660966	−0.330483	+	0.943812i	$0.607212\pi$
$968$	33.4154	1.07401
$969$	0	0
$970$	0	0
$971$	49.3281	1.58301	0.791507	−	0.611160i	$-0.209297\pi$
$971$	49.3281	1.58301	0.791507	+	0.611160i	$0.209297\pi$
$972$	0	0
$973$	−1.66672	−0.0534326
$974$	52.9467	1.69652
$975$	0	0
$976$	11.2460	0.359975
$977$	−19.0164	−0.608390	−0.304195	−	0.952610i	$-0.598387\pi$
$977$	−19.0164	−0.608390	−0.304195	+	0.952610i	$0.598387\pi$
$978$	0	0
$979$	−27.0968	−0.866017
$980$	0	0
$981$	0	0
$982$	47.3854	1.51213
$983$	−20.8696	−0.665636	−0.332818	−	0.942991i	$-0.607999\pi$
$983$	−20.8696	−0.665636	−0.332818	+	0.942991i	$0.607999\pi$
$984$	0	0
$985$	0	0
$986$	−12.1509	−0.386962
$987$	0	0
$988$	5.43681	0.172968
$989$	−0.768899	−0.0244496
$990$	0	0
$991$	15.6435	0.496932	0.248466	−	0.968641i	$-0.420074\pi$
$991$	15.6435	0.496932	0.248466	+	0.968641i	$0.420074\pi$
$992$	−45.3743	−1.44063
$993$	0	0
$994$	10.6194	0.336827
$995$	0	0
$996$	0	0
$997$	6.21108	0.196707	0.0983535	−	0.995152i	$-0.468642\pi$
$997$	6.21108	0.196707	0.0983535	+	0.995152i	$0.468642\pi$
$998$	−56.6367	−1.79280
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2925.2.a.bm.1.2		5
3.2	odd	2		975.2.a.s.1.4		5
5.2	odd	4		585.2.c.c.469.3		10
5.3	odd	4		585.2.c.c.469.8		10
5.4	even	2		2925.2.a.bl.1.4		5
15.2	even	4		195.2.c.b.79.8	yes	10
15.8	even	4		195.2.c.b.79.3	✓	10
15.14	odd	2		975.2.a.r.1.2		5
60.23	odd	4		3120.2.l.p.1249.4		10
60.47	odd	4		3120.2.l.p.1249.9		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
195.2.c.b.79.3	✓	10	15.8	even	4
195.2.c.b.79.8	yes	10	15.2	even	4
585.2.c.c.469.3		10	5.2	odd	4
585.2.c.c.469.8		10	5.3	odd	4
975.2.a.r.1.2		5	15.14	odd	2
975.2.a.s.1.4		5	3.2	odd	2
2925.2.a.bl.1.4		5	5.4	even	2
2925.2.a.bm.1.2		5	1.1	even	1	trivial
3120.2.l.p.1249.4		10	60.23	odd	4
3120.2.l.p.1249.9		10	60.47	odd	4

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:	\( N \)	\(=\)	\( 2925 = 3^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2925.a (trivial)

\(f(q)\)	\(=\)	\(q-1.77159 q^{2} +1.13853 q^{4} -0.437634 q^{7} +1.52618 q^{8} +O(q^{10})\)	\(q-1.77159 q^{2} +1.13853 q^{4} -0.437634 q^{7} +1.52618 q^{8} -5.73540 q^{11} +1.00000 q^{13} +0.775308 q^{14} -4.98081 q^{16} -3.98081 q^{17} +4.77531 q^{19} +10.1608 q^{22} -0.337673 q^{23} -1.77159 q^{26} -0.498258 q^{28} -1.72295 q^{29} -7.86166 q^{31} +5.77159 q^{32} +7.05236 q^{34} +1.66233 q^{37} -8.45988 q^{38} -2.68304 q^{41} +2.27705 q^{43} -6.52990 q^{44} +0.598218 q^{46} +11.7162 q^{47} -6.80848 q^{49} +1.13853 q^{52} +14.2993 q^{53} -0.667908 q^{56} +3.05236 q^{58} -11.4392 q^{59} -2.25786 q^{61} +13.9276 q^{62} -0.263257 q^{64} +2.17977 q^{67} -4.53225 q^{68} +13.6970 q^{71} -2.55410 q^{73} -2.94496 q^{74} +5.43681 q^{76} +2.51001 q^{77} +2.55144 q^{79} +4.75325 q^{82} +5.41158 q^{83} -4.03400 q^{86} -8.75325 q^{88} +4.72448 q^{89} -0.437634 q^{91} -0.384450 q^{92} -20.7563 q^{94} -14.7901 q^{97} +12.0618 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 6 q^{4} + 5 q^{7}+O(q^{10}) \) 5 * q + 6 * q^4 + 5 * q^7	\( 5 q + 6 q^{4} + 5 q^{7} - 5 q^{11} + 5 q^{13} - 12 q^{14} + 5 q^{17} + 8 q^{19} - 8 q^{22} + 7 q^{23} + 14 q^{28} - 8 q^{29} + 12 q^{31} + 20 q^{32} + 20 q^{34} + 17 q^{37} + 24 q^{38} - 5 q^{41} + 12 q^{43} - 18 q^{44} - 12 q^{46} + 10 q^{47} + 22 q^{49} + 6 q^{52} + 13 q^{53} - 8 q^{59} + 13 q^{61} + 40 q^{62} - 16 q^{64} + 28 q^{67} + 14 q^{68} - 5 q^{71} - 14 q^{73} - 12 q^{74} + 35 q^{77} + q^{79} + 16 q^{82} + 6 q^{83} - 36 q^{88} - 19 q^{89} + 5 q^{91} + 10 q^{92} - 12 q^{94} - 13 q^{97} - 28 q^{98}+O(q^{100}) \) 5 * q + 6 * q^4 + 5 * q^7 - 5 * q^11 + 5 * q^13 - 12 * q^14 + 5 * q^17 + 8 * q^19 - 8 * q^22 + 7 * q^23 + 14 * q^28 - 8 * q^29 + 12 * q^31 + 20 * q^32 + 20 * q^34 + 17 * q^37 + 24 * q^38 - 5 * q^41 + 12 * q^43 - 18 * q^44 - 12 * q^46 + 10 * q^47 + 22 * q^49 + 6 * q^52 + 13 * q^53 - 8 * q^59 + 13 * q^61 + 40 * q^62 - 16 * q^64 + 28 * q^67 + 14 * q^68 - 5 * q^71 - 14 * q^73 - 12 * q^74 + 35 * q^77 + q^79 + 16 * q^82 + 6 * q^83 - 36 * q^88 - 19 * q^89 + 5 * q^91 + 10 * q^92 - 12 * q^94 - 13 * q^97 - 28 * q^98

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