LMFDB - Embedded newform 2475.4.a.br.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,4,Mod(1,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("2475.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2475.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:	$146.029727264$
Analytic rank:	$1$
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} - 3x^{6} - 44x^{5} + 118x^{4} + 515x^{3} - 1279x^{2} - 892x + 1840 $ x^7 - 3x^6 - 44x^5 + 118x^4 + 515x^3 - 1279x^2 - 892x + 1840
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$2.62783$ of defining polynomial
Character	$\chi$	$=$	2475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.62783 q^{2} -1.09451 q^{4} +24.0582 q^{7} -23.8988 q^{8} +O(q^{10})$	$q+2.62783 q^{2} -1.09451 q^{4} +24.0582 q^{7} -23.8988 q^{8} +11.0000 q^{11} -70.7912 q^{13} +63.2207 q^{14} -54.0459 q^{16} -27.9494 q^{17} +50.9926 q^{19} +28.9061 q^{22} +201.498 q^{23} -186.027 q^{26} -26.3320 q^{28} -126.395 q^{29} +30.1877 q^{31} +49.1672 q^{32} -73.4462 q^{34} +7.32183 q^{37} +134.000 q^{38} -366.098 q^{41} +33.4661 q^{43} -12.0397 q^{44} +529.503 q^{46} -14.3267 q^{47} +235.795 q^{49} +77.4820 q^{52} -154.671 q^{53} -574.962 q^{56} -332.145 q^{58} -611.926 q^{59} -83.1222 q^{61} +79.3282 q^{62} +561.570 q^{64} -737.743 q^{67} +30.5910 q^{68} +930.806 q^{71} -491.999 q^{73} +19.2405 q^{74} -55.8121 q^{76} +264.640 q^{77} +687.955 q^{79} -962.043 q^{82} -712.253 q^{83} +87.9431 q^{86} -262.887 q^{88} +1332.36 q^{89} -1703.11 q^{91} -220.543 q^{92} -37.6482 q^{94} -598.852 q^{97} +619.630 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 3 q^{2} + 41 q^{4} - 50 q^{7} + 21 q^{8}+O(q^{10}) $ 7 * q + 3 * q^2 + 41 * q^4 - 50 * q^7 + 21 * q^8	$ 7 q + 3 q^{2} + 41 q^{4} - 50 q^{7} + 21 q^{8} + 77 q^{11} - 24 q^{13} - 142 q^{14} + 181 q^{16} + 38 q^{17} + 26 q^{19} + 33 q^{22} + 228 q^{23} - 476 q^{26} - 840 q^{28} - 572 q^{29} - 140 q^{31} + 991 q^{32} - 806 q^{34} + 104 q^{37} + 498 q^{38} - 896 q^{41} - 614 q^{43} + 451 q^{44} - 344 q^{46} + 520 q^{47} + 295 q^{49} + 26 q^{52} + 380 q^{53} - 1522 q^{56} - 1600 q^{58} - 1316 q^{59} - 386 q^{61} + 440 q^{62} + 869 q^{64} - 348 q^{67} + 332 q^{68} - 804 q^{71} - 468 q^{73} + 748 q^{74} - 1698 q^{76} - 550 q^{77} - 374 q^{79} + 620 q^{82} + 3128 q^{83} + 2534 q^{86} + 231 q^{88} - 694 q^{89} - 3376 q^{91} - 1184 q^{92} - 2920 q^{94} + 8 q^{97} + 4211 q^{98}+O(q^{100}) $ 7 * q + 3 * q^2 + 41 * q^4 - 50 * q^7 + 21 * q^8 + 77 * q^11 - 24 * q^13 - 142 * q^14 + 181 * q^16 + 38 * q^17 + 26 * q^19 + 33 * q^22 + 228 * q^23 - 476 * q^26 - 840 * q^28 - 572 * q^29 - 140 * q^31 + 991 * q^32 - 806 * q^34 + 104 * q^37 + 498 * q^38 - 896 * q^41 - 614 * q^43 + 451 * q^44 - 344 * q^46 + 520 * q^47 + 295 * q^49 + 26 * q^52 + 380 * q^53 - 1522 * q^56 - 1600 * q^58 - 1316 * q^59 - 386 * q^61 + 440 * q^62 + 869 * q^64 - 348 * q^67 + 332 * q^68 - 804 * q^71 - 468 * q^73 + 748 * q^74 - 1698 * q^76 - 550 * q^77 - 374 * q^79 + 620 * q^82 + 3128 * q^83 + 2534 * q^86 + 231 * q^88 - 694 * q^89 - 3376 * q^91 - 1184 * q^92 - 2920 * q^94 + 8 * q^97 + 4211 * 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 $

$2$	2.62783	0.929078	0.464539	−	0.885553i	$-0.346220\pi$
$2$	2.62783	0.929078	0.464539	+	0.885553i	$0.346220\pi$
$3$	0	0
$3$	0	0
$4$	−1.09451	−0.136814
$5$	0	0
$5$	0	0
$6$	0	0
$7$	24.0582	1.29902	0.649509	−	0.760354i	$-0.274974\pi$
$7$	24.0582	1.29902	0.649509	+	0.760354i	$0.274974\pi$
$8$	−23.8988	−1.05619
$9$	0	0
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	−70.7912	−1.51030	−0.755152	−	0.655549i	$-0.772437\pi$
$13$	−70.7912	−1.51030	−0.755152	+	0.655549i	$0.772437\pi$
$14$	63.2207	1.20689
$15$	0	0
$16$	−54.0459	−0.844468
$17$	−27.9494	−0.398749	−0.199374	−	0.979923i	$-0.563891\pi$
$17$	−27.9494	−0.398749	−0.199374	+	0.979923i	$0.563891\pi$
$18$	0	0
$19$	50.9926	0.615711	0.307855	−	0.951433i	$-0.400389\pi$
$19$	50.9926	0.615711	0.307855	+	0.951433i	$0.400389\pi$
$20$	0	0
$21$	0	0
$22$	28.9061	0.280128
$23$	201.498	1.82675	0.913376	−	0.407118i	$-0.133466\pi$
$23$	201.498	1.82675	0.913376	+	0.407118i	$0.133466\pi$
$24$	0	0
$25$	0	0
$26$	−186.027	−1.40319
$27$	0	0
$28$	−26.3320	−0.177724
$29$	−126.395	−0.809345	−0.404672	−	0.914462i	$-0.632614\pi$
$29$	−126.395	−0.809345	−0.404672	+	0.914462i	$0.632614\pi$
$30$	0	0
$31$	30.1877	0.174899	0.0874496	−	0.996169i	$-0.472128\pi$
$31$	30.1877	0.174899	0.0874496	+	0.996169i	$0.472128\pi$
$32$	49.1672	0.271613
$33$	0	0
$34$	−73.4462	−0.370468
$35$	0	0
$36$	0	0
$37$	7.32183	0.0325325	0.0162662	−	0.999868i	$-0.494822\pi$
$37$	7.32183	0.0325325	0.0162662	+	0.999868i	$0.494822\pi$
$38$	134.000	0.572043
$39$	0	0
$40$	0	0
$41$	−366.098	−1.39451	−0.697255	−	0.716824i	$-0.745595\pi$
$41$	−366.098	−1.39451	−0.697255	+	0.716824i	$0.745595\pi$
$42$	0	0
$43$	33.4661	0.118687	0.0593434	−	0.998238i	$-0.481099\pi$
$43$	33.4661	0.118687	0.0593434	+	0.998238i	$0.481099\pi$
$44$	−12.0397	−0.0412511
$45$	0	0
$46$	529.503	1.69719
$47$	−14.3267	−0.0444632	−0.0222316	−	0.999753i	$-0.507077\pi$
$47$	−14.3267	−0.0444632	−0.0222316	+	0.999753i	$0.507077\pi$
$48$	0	0
$49$	235.795	0.687450
$50$	0	0
$51$	0	0
$52$	77.4820	0.206631
$53$	−154.671	−0.400863	−0.200432	−	0.979708i	$-0.564234\pi$
$53$	−154.671	−0.400863	−0.200432	+	0.979708i	$0.564234\pi$
$54$	0	0
$55$	0	0
$56$	−574.962	−1.37201
$57$	0	0
$58$	−332.145	−0.751944
$59$	−611.926	−1.35027	−0.675135	−	0.737694i	$-0.735915\pi$
$59$	−611.926	−1.35027	−0.675135	+	0.737694i	$0.735915\pi$
$60$	0	0
$61$	−83.1222	−0.174471	−0.0872353	−	0.996188i	$-0.527803\pi$
$61$	−83.1222	−0.174471	−0.0872353	+	0.996188i	$0.527803\pi$
$62$	79.3282	0.162495
$63$	0	0
$64$	561.570	1.09682
$65$	0	0
$66$	0	0
$67$	−737.743	−1.34522	−0.672609	−	0.739998i	$-0.734826\pi$
$67$	−737.743	−1.34522	−0.672609	+	0.739998i	$0.734826\pi$
$68$	30.5910	0.0545545
$69$	0	0
$70$	0	0
$71$	930.806	1.55586	0.777932	−	0.628348i	$-0.216269\pi$
$71$	930.806	1.55586	0.777932	+	0.628348i	$0.216269\pi$
$72$	0	0
$73$	−491.999	−0.788824	−0.394412	−	0.918934i	$-0.629052\pi$
$73$	−491.999	−0.788824	−0.394412	+	0.918934i	$0.629052\pi$
$74$	19.2405	0.0302252
$75$	0	0
$76$	−55.8121	−0.0842380
$77$	264.640	0.391669
$78$	0	0
$79$	687.955	0.979759	0.489879	−	0.871790i	$-0.337041\pi$
$79$	687.955	0.979759	0.489879	+	0.871790i	$0.337041\pi$
$80$	0	0
$81$	0	0
$82$	−962.043	−1.29561
$83$	−712.253	−0.941927	−0.470963	−	0.882153i	$-0.656094\pi$
$83$	−712.253	−0.941927	−0.470963	+	0.882153i	$0.656094\pi$
$84$	0	0
$85$	0	0
$86$	87.9431	0.110269
$87$	0	0
$88$	−262.887	−0.318453
$89$	1332.36	1.58685	0.793423	−	0.608670i	$-0.208297\pi$
$89$	1332.36	1.58685	0.793423	+	0.608670i	$0.208297\pi$
$90$	0	0
$91$	−1703.11	−1.96191
$92$	−220.543	−0.249926
$93$	0	0
$94$	−37.6482	−0.0413097
$95$	0	0
$96$	0	0
$97$	−598.852	−0.626847	−0.313424	−	0.949613i	$-0.601476\pi$
$97$	−598.852	−0.626847	−0.313424	+	0.949613i	$0.601476\pi$
$98$	619.630	0.638694
$99$	0	0
$100$	0	0
$101$	−392.723	−0.386905	−0.193452	−	0.981110i	$-0.561968\pi$
$101$	−392.723	−0.386905	−0.193452	+	0.981110i	$0.561968\pi$
$102$	0	0
$103$	−889.816	−0.851225	−0.425613	−	0.904905i	$-0.639941\pi$
$103$	−889.816	−0.851225	−0.425613	+	0.904905i	$0.639941\pi$
$104$	1691.83	1.59517
$105$	0	0
$106$	−406.450	−0.372433
$107$	−911.308	−0.823360	−0.411680	−	0.911328i	$-0.635058\pi$
$107$	−911.308	−0.823360	−0.411680	+	0.911328i	$0.635058\pi$
$108$	0	0
$109$	−1552.29	−1.36406	−0.682029	−	0.731325i	$-0.738902\pi$
$109$	−1552.29	−1.36406	−0.682029	+	0.731325i	$0.738902\pi$
$110$	0	0
$111$	0	0
$112$	−1300.25	−1.09698
$113$	457.867	0.381173	0.190586	−	0.981670i	$-0.438961\pi$
$113$	457.867	0.381173	0.190586	+	0.981670i	$0.438961\pi$
$114$	0	0
$115$	0	0
$116$	138.341	0.110730
$117$	0	0
$118$	−1608.04	−1.25451
$119$	−672.411	−0.517982
$120$	0	0
$121$	121.000	0.0909091
$122$	−218.431	−0.162097
$123$	0	0
$124$	−33.0409	−0.0239287
$125$	0	0
$126$	0	0
$127$	911.406	0.636805	0.318402	−	0.947956i	$-0.396854\pi$
$127$	911.406	0.636805	0.318402	+	0.947956i	$0.396854\pi$
$128$	1082.37	0.747416
$129$	0	0
$130$	0	0
$131$	−2552.41	−1.70233	−0.851165	−	0.524898i	$-0.824103\pi$
$131$	−2552.41	−1.70233	−0.851165	+	0.524898i	$0.824103\pi$
$132$	0	0
$133$	1226.79	0.799820
$134$	−1938.66	−1.24981
$135$	0	0
$136$	667.958	0.421154
$137$	−390.364	−0.243438	−0.121719	−	0.992565i	$-0.538841\pi$
$137$	−390.364	−0.243438	−0.121719	+	0.992565i	$0.538841\pi$
$138$	0	0
$139$	−2657.82	−1.62182	−0.810912	−	0.585168i	$-0.801028\pi$
$139$	−2657.82	−1.62182	−0.810912	+	0.585168i	$0.801028\pi$
$140$	0	0
$141$	0	0
$142$	2446.00	1.44552
$143$	−778.704	−0.455374
$144$	0	0
$145$	0	0
$146$	−1292.89	−0.732879
$147$	0	0
$148$	−8.01385	−0.00445091
$149$	266.568	0.146565	0.0732823	−	0.997311i	$-0.476653\pi$
$149$	266.568	0.146565	0.0732823	+	0.997311i	$0.476653\pi$
$150$	0	0
$151$	2565.20	1.38247	0.691235	−	0.722630i	$-0.257067\pi$
$151$	2565.20	1.38247	0.691235	+	0.722630i	$0.257067\pi$
$152$	−1218.66	−0.650307
$153$	0	0
$154$	695.428	0.363891
$155$	0	0
$156$	0	0
$157$	−483.922	−0.245995	−0.122997	−	0.992407i	$-0.539251\pi$
$157$	−483.922	−0.245995	−0.122997	+	0.992407i	$0.539251\pi$
$158$	1807.83	0.910272
$159$	0	0
$160$	0	0
$161$	4847.68	2.37298
$162$	0	0
$163$	−3701.47	−1.77866	−0.889330	−	0.457266i	$-0.848829\pi$
$163$	−3701.47	−1.77866	−0.889330	+	0.457266i	$0.848829\pi$
$164$	400.699	0.190789
$165$	0	0
$166$	−1871.68	−0.875123
$167$	2361.35	1.09417	0.547086	−	0.837077i	$-0.315737\pi$
$167$	2361.35	1.09417	0.547086	+	0.837077i	$0.315737\pi$
$168$	0	0
$169$	2814.40	1.28102
$170$	0	0
$171$	0	0
$172$	−36.6291	−0.0162380
$173$	3613.18	1.58789	0.793944	−	0.607990i	$-0.208024\pi$
$173$	3613.18	1.58789	0.793944	+	0.607990i	$0.208024\pi$
$174$	0	0
$175$	0	0
$176$	−594.505	−0.254617
$177$	0	0
$178$	3501.20	1.47430
$179$	−2632.47	−1.09922	−0.549608	−	0.835422i	$-0.685223\pi$
$179$	−2632.47	−1.09922	−0.549608	+	0.835422i	$0.685223\pi$
$180$	0	0
$181$	−3956.26	−1.62468	−0.812338	−	0.583187i	$-0.801806\pi$
$181$	−3956.26	−1.62468	−0.812338	+	0.583187i	$0.801806\pi$
$182$	−4475.48	−1.82277
$183$	0	0
$184$	−4815.57	−1.92939
$185$	0	0
$186$	0	0
$187$	−307.443	−0.120227
$188$	15.6808	0.00608320
$189$	0	0
$190$	0	0
$191$	2921.65	1.10682	0.553412	−	0.832908i	$-0.313326\pi$
$191$	2921.65	1.10682	0.553412	+	0.832908i	$0.313326\pi$
$192$	0	0
$193$	−510.977	−0.190575	−0.0952875	−	0.995450i	$-0.530377\pi$
$193$	−510.977	−0.190575	−0.0952875	+	0.995450i	$0.530377\pi$
$194$	−1573.68	−0.582390
$195$	0	0
$196$	−258.081	−0.0940529
$197$	2020.44	0.730711	0.365356	−	0.930868i	$-0.380947\pi$
$197$	2020.44	0.730711	0.365356	+	0.930868i	$0.380947\pi$
$198$	0	0
$199$	−3608.68	−1.28549	−0.642745	−	0.766080i	$-0.722205\pi$
$199$	−3608.68	−1.28549	−0.642745	+	0.766080i	$0.722205\pi$
$200$	0	0
$201$	0	0
$202$	−1032.01	−0.359464
$203$	−3040.84	−1.05135
$204$	0	0
$205$	0	0
$206$	−2338.29	−0.790855
$207$	0	0
$208$	3825.98	1.27540
$209$	560.919	0.185644
$210$	0	0
$211$	2279.36	0.743685	0.371842	−	0.928296i	$-0.378726\pi$
$211$	2279.36	0.743685	0.371842	+	0.928296i	$0.378726\pi$
$212$	169.290	0.0548438
$213$	0	0
$214$	−2394.76	−0.764965
$215$	0	0
$216$	0	0
$217$	726.261	0.227197
$218$	−4079.15	−1.26732
$219$	0	0
$220$	0	0
$221$	1978.57	0.602232
$222$	0	0
$223$	−6504.82	−1.95334	−0.976670	−	0.214744i	$-0.931108\pi$
$223$	−6504.82	−1.95334	−0.976670	+	0.214744i	$0.931108\pi$
$224$	1182.87	0.352830
$225$	0	0
$226$	1203.20	0.354139
$227$	−5152.78	−1.50662	−0.753308	−	0.657668i	$-0.771543\pi$
$227$	−5152.78	−1.50662	−0.753308	+	0.657668i	$0.771543\pi$
$228$	0	0
$229$	−5961.82	−1.72039	−0.860193	−	0.509969i	$-0.829657\pi$
$229$	−5961.82	−1.72039	−0.860193	+	0.509969i	$0.829657\pi$
$230$	0	0
$231$	0	0
$232$	3020.70	0.854821
$233$	−4516.75	−1.26996	−0.634982	−	0.772527i	$-0.718993\pi$
$233$	−4516.75	−1.26996	−0.634982	+	0.772527i	$0.718993\pi$
$234$	0	0
$235$	0	0
$236$	669.762	0.184736
$237$	0	0
$238$	−1766.98	−0.481245
$239$	−3298.29	−0.892672	−0.446336	−	0.894865i	$-0.647271\pi$
$239$	−3298.29	−0.892672	−0.446336	+	0.894865i	$0.647271\pi$
$240$	0	0
$241$	−500.141	−0.133680	−0.0668400	−	0.997764i	$-0.521292\pi$
$241$	−500.141	−0.133680	−0.0668400	+	0.997764i	$0.521292\pi$
$242$	317.967	0.0844616
$243$	0	0
$244$	90.9784	0.0238701
$245$	0	0
$246$	0	0
$247$	−3609.83	−0.929911
$248$	−721.451	−0.184727
$249$	0	0
$250$	0	0
$251$	−1070.12	−0.269106	−0.134553	−	0.990906i	$-0.542960\pi$
$251$	−1070.12	−0.269106	−0.134553	+	0.990906i	$0.542960\pi$
$252$	0	0
$253$	2216.48	0.550786
$254$	2395.02	0.591641
$255$	0	0
$256$	−1648.27	−0.402410
$257$	3868.69	0.938997	0.469499	−	0.882933i	$-0.344435\pi$
$257$	3868.69	0.938997	0.469499	+	0.882933i	$0.344435\pi$
$258$	0	0
$259$	176.150	0.0422603
$260$	0	0
$261$	0	0
$262$	−6707.30	−1.58160
$263$	7328.33	1.71819	0.859096	−	0.511815i	$-0.171026\pi$
$263$	7328.33	1.71819	0.859096	+	0.511815i	$0.171026\pi$
$264$	0	0
$265$	0	0
$266$	3223.79	0.743095
$267$	0	0
$268$	807.470	0.184045
$269$	−2589.62	−0.586960	−0.293480	−	0.955965i	$-0.594813\pi$
$269$	−2589.62	−0.586960	−0.293480	+	0.955965i	$0.594813\pi$
$270$	0	0
$271$	4148.33	0.929863	0.464932	−	0.885347i	$-0.346079\pi$
$271$	4148.33	0.929863	0.464932	+	0.885347i	$0.346079\pi$
$272$	1510.55	0.336730
$273$	0	0
$274$	−1025.81	−0.226173
$275$	0	0
$276$	0	0
$277$	−839.252	−0.182042	−0.0910212	−	0.995849i	$-0.529013\pi$
$277$	−839.252	−0.182042	−0.0910212	+	0.995849i	$0.529013\pi$
$278$	−6984.30	−1.50680
$279$	0	0
$280$	0	0
$281$	−737.536	−0.156575	−0.0782877	−	0.996931i	$-0.524945\pi$
$281$	−737.536	−0.156575	−0.0782877	+	0.996931i	$0.524945\pi$
$282$	0	0
$283$	−2013.44	−0.422921	−0.211460	−	0.977387i	$-0.567822\pi$
$283$	−2013.44	−0.422921	−0.211460	+	0.977387i	$0.567822\pi$
$284$	−1018.78	−0.212864
$285$	0	0
$286$	−2046.30	−0.423078
$287$	−8807.64	−1.81149
$288$	0	0
$289$	−4131.83	−0.841000
$290$	0	0
$291$	0	0
$292$	538.500	0.107922
$293$	4857.22	0.968469	0.484235	−	0.874938i	$-0.339098\pi$
$293$	4857.22	0.968469	0.484235	+	0.874938i	$0.339098\pi$
$294$	0	0
$295$	0	0
$296$	−174.983	−0.0343605
$297$	0	0
$298$	700.496	0.136170
$299$	−14264.3	−2.75895
$300$	0	0
$301$	805.132	0.154176
$302$	6740.91	1.28442
$303$	0	0
$304$	−2755.94	−0.519948
$305$	0	0
$306$	0	0
$307$	−10201.0	−1.89642	−0.948211	−	0.317641i	$-0.897109\pi$
$307$	−10201.0	−1.89642	−0.948211	+	0.317641i	$0.897109\pi$
$308$	−289.652	−0.0535859
$309$	0	0
$310$	0	0
$311$	690.017	0.125811	0.0629056	−	0.998019i	$-0.479963\pi$
$311$	690.017	0.125811	0.0629056	+	0.998019i	$0.479963\pi$
$312$	0	0
$313$	5360.35	0.968003	0.484002	−	0.875067i	$-0.339183\pi$
$313$	5360.35	0.968003	0.484002	+	0.875067i	$0.339183\pi$
$314$	−1271.67	−0.228548
$315$	0	0
$316$	−752.976	−0.134045
$317$	−10817.5	−1.91662	−0.958312	−	0.285724i	$-0.907766\pi$
$317$	−10817.5	−1.91662	−0.958312	+	0.285724i	$0.907766\pi$
$318$	0	0
$319$	−1390.35	−0.244027
$320$	0	0
$321$	0	0
$322$	12738.9	2.20469
$323$	−1425.21	−0.245514
$324$	0	0
$325$	0	0
$326$	−9726.83	−1.65251
$327$	0	0
$328$	8749.31	1.47287
$329$	−344.675	−0.0577585
$330$	0	0
$331$	−10694.7	−1.77593	−0.887966	−	0.459909i	$-0.847882\pi$
$331$	−10694.7	−1.77593	−0.887966	+	0.459909i	$0.847882\pi$
$332$	779.571	0.128869
$333$	0	0
$334$	6205.22	1.01657
$335$	0	0
$336$	0	0
$337$	−656.983	−0.106196	−0.0530981	−	0.998589i	$-0.516910\pi$
$337$	−656.983	−0.106196	−0.0530981	+	0.998589i	$0.516910\pi$
$338$	7395.76	1.19017
$339$	0	0
$340$	0	0
$341$	332.065	0.0527341
$342$	0	0
$343$	−2579.15	−0.406009
$344$	−799.800	−0.125356
$345$	0	0
$346$	9494.81	1.47527
$347$	−887.275	−0.137266	−0.0686331	−	0.997642i	$-0.521864\pi$
$347$	−887.275	−0.137266	−0.0686331	+	0.997642i	$0.521864\pi$
$348$	0	0
$349$	5723.95	0.877925	0.438963	−	0.898505i	$-0.355346\pi$
$349$	5723.95	0.877925	0.438963	+	0.898505i	$0.355346\pi$
$350$	0	0
$351$	0	0
$352$	540.839	0.0818944
$353$	3266.86	0.492571	0.246285	−	0.969197i	$-0.420790\pi$
$353$	3266.86	0.492571	0.246285	+	0.969197i	$0.420790\pi$
$354$	0	0
$355$	0	0
$356$	−1458.28	−0.217103
$357$	0	0
$358$	−6917.67	−1.02126
$359$	11275.9	1.65771	0.828854	−	0.559466i	$-0.188994\pi$
$359$	11275.9	1.65771	0.828854	+	0.559466i	$0.188994\pi$
$360$	0	0
$361$	−4258.76	−0.620900
$362$	−10396.4	−1.50945
$363$	0	0
$364$	1864.08	0.268418
$365$	0	0
$366$	0	0
$367$	−2789.28	−0.396728	−0.198364	−	0.980128i	$-0.563563\pi$
$367$	−2789.28	−0.396728	−0.198364	+	0.980128i	$0.563563\pi$
$368$	−10890.2	−1.54263
$369$	0	0
$370$	0	0
$371$	−3721.11	−0.520729
$372$	0	0
$373$	13032.2	1.80907	0.904533	−	0.426403i	$-0.140220\pi$
$373$	13032.2	1.80907	0.904533	+	0.426403i	$0.140220\pi$
$374$	−807.909	−0.111700
$375$	0	0
$376$	342.392	0.0469615
$377$	8947.67	1.22236
$378$	0	0
$379$	−3913.88	−0.530455	−0.265227	−	0.964186i	$-0.585447\pi$
$379$	−3913.88	−0.530455	−0.265227	+	0.964186i	$0.585447\pi$
$380$	0	0
$381$	0	0
$382$	7677.60	1.02833
$383$	3589.82	0.478932	0.239466	−	0.970905i	$-0.423028\pi$
$383$	3589.82	0.478932	0.239466	+	0.970905i	$0.423028\pi$
$384$	0	0
$385$	0	0
$386$	−1342.76	−0.177059
$387$	0	0
$388$	655.452	0.0857616
$389$	5556.23	0.724196	0.362098	−	0.932140i	$-0.382061\pi$
$389$	5556.23	0.724196	0.362098	+	0.932140i	$0.382061\pi$
$390$	0	0
$391$	−5631.75	−0.728414
$392$	−5635.23	−0.726077
$393$	0	0
$394$	5309.36	0.678887
$395$	0	0
$396$	0	0
$397$	−355.562	−0.0449500	−0.0224750	−	0.999747i	$-0.507155\pi$
$397$	−355.562	−0.0449500	−0.0224750	+	0.999747i	$0.507155\pi$
$398$	−9482.99	−1.19432
$399$	0	0
$400$	0	0
$401$	−4439.53	−0.552867	−0.276434	−	0.961033i	$-0.589153\pi$
$401$	−4439.53	−0.552867	−0.276434	+	0.961033i	$0.589153\pi$
$402$	0	0
$403$	−2137.03	−0.264151
$404$	429.840	0.0529341
$405$	0	0
$406$	−7990.80	−0.976790
$407$	80.5402	0.00980891
$408$	0	0
$409$	−7022.79	−0.849033	−0.424517	−	0.905420i	$-0.639556\pi$
$409$	−7022.79	−0.849033	−0.424517	+	0.905420i	$0.639556\pi$
$410$	0	0
$411$	0	0
$412$	973.917	0.116460
$413$	−14721.8	−1.75403
$414$	0	0
$415$	0	0
$416$	−3480.61	−0.410218
$417$	0	0
$418$	1474.00	0.172478
$419$	−10786.9	−1.25769	−0.628847	−	0.777529i	$-0.716473\pi$
$419$	−10786.9	−1.25769	−0.628847	+	0.777529i	$0.716473\pi$
$420$	0	0
$421$	−683.119	−0.0790812	−0.0395406	−	0.999218i	$-0.512589\pi$
$421$	−683.119	−0.0790812	−0.0395406	+	0.999218i	$0.512589\pi$
$422$	5989.76	0.690941
$423$	0	0
$424$	3696.46	0.423387
$425$	0	0
$426$	0	0
$427$	−1999.77	−0.226641
$428$	997.440	0.112647
$429$	0	0
$430$	0	0
$431$	5515.28	0.616384	0.308192	−	0.951324i	$-0.400276\pi$
$431$	5515.28	0.616384	0.308192	+	0.951324i	$0.400276\pi$
$432$	0	0
$433$	1021.73	0.113398	0.0566989	−	0.998391i	$-0.481942\pi$
$433$	1021.73	0.113398	0.0566989	+	0.998391i	$0.481942\pi$
$434$	1908.49	0.211084
$435$	0	0
$436$	1699.00	0.186623
$437$	10274.9	1.12475
$438$	0	0
$439$	−15902.3	−1.72887	−0.864435	−	0.502745i	$-0.832324\pi$
$439$	−15902.3	−1.72887	−0.864435	+	0.502745i	$0.832324\pi$
$440$	0	0
$441$	0	0
$442$	5199.35	0.559520
$443$	−8290.70	−0.889172	−0.444586	−	0.895736i	$-0.646649\pi$
$443$	−8290.70	−0.889172	−0.444586	+	0.895736i	$0.646649\pi$
$444$	0	0
$445$	0	0
$446$	−17093.6	−1.81481
$447$	0	0
$448$	13510.4	1.42479
$449$	−14375.1	−1.51092	−0.755461	−	0.655194i	$-0.772587\pi$
$449$	−14375.1	−1.51092	−0.755461	+	0.655194i	$0.772587\pi$
$450$	0	0
$451$	−4027.08	−0.420460
$452$	−501.142	−0.0521499
$453$	0	0
$454$	−13540.6	−1.39976
$455$	0	0
$456$	0	0
$457$	2187.59	0.223919	0.111959	−	0.993713i	$-0.464287\pi$
$457$	2187.59	0.223919	0.111959	+	0.993713i	$0.464287\pi$
$458$	−15666.6	−1.59837
$459$	0	0
$460$	0	0
$461$	1016.23	0.102669	0.0513344	−	0.998682i	$-0.483653\pi$
$461$	1016.23	0.102669	0.0513344	+	0.998682i	$0.483653\pi$
$462$	0	0
$463$	9523.44	0.955921	0.477961	−	0.878381i	$-0.341376\pi$
$463$	9523.44	0.955921	0.477961	+	0.878381i	$0.341376\pi$
$464$	6831.15	0.683465
$465$	0	0
$466$	−11869.2	−1.17990
$467$	7251.93	0.718585	0.359293	−	0.933225i	$-0.383018\pi$
$467$	7251.93	0.718585	0.359293	+	0.933225i	$0.383018\pi$
$468$	0	0
$469$	−17748.7	−1.74746
$470$	0	0
$471$	0	0
$472$	14624.3	1.42614
$473$	368.127	0.0357854
$474$	0	0
$475$	0	0
$476$	735.964	0.0708673
$477$	0	0
$478$	−8667.34	−0.829362
$479$	10523.6	1.00383	0.501916	−	0.864916i	$-0.332629\pi$
$479$	10523.6	1.00383	0.501916	+	0.864916i	$0.332629\pi$
$480$	0	0
$481$	−518.322	−0.0491340
$482$	−1314.28	−0.124199
$483$	0	0
$484$	−132.436	−0.0124377
$485$	0	0
$486$	0	0
$487$	−555.512	−0.0516892	−0.0258446	−	0.999666i	$-0.508228\pi$
$487$	−555.512	−0.0516892	−0.0258446	+	0.999666i	$0.508228\pi$
$488$	1986.52	0.184274
$489$	0	0
$490$	0	0
$491$	19782.4	1.81826	0.909132	−	0.416509i	$-0.136747\pi$
$491$	19782.4	1.81826	0.909132	+	0.416509i	$0.136747\pi$
$492$	0	0
$493$	3532.67	0.322725
$494$	−9486.01	−0.863959
$495$	0	0
$496$	−1631.52	−0.147697
$497$	22393.5	2.02110
$498$	0	0
$499$	−2749.43	−0.246657	−0.123328	−	0.992366i	$-0.539357\pi$
$499$	−2749.43	−0.246657	−0.123328	+	0.992366i	$0.539357\pi$
$500$	0	0
$501$	0	0
$502$	−2812.10	−0.250020
$503$	13224.1	1.17223	0.586116	−	0.810227i	$-0.300656\pi$
$503$	13224.1	1.17223	0.586116	+	0.810227i	$0.300656\pi$
$504$	0	0
$505$	0	0
$506$	5824.53	0.511723
$507$	0	0
$508$	−997.547	−0.0871240
$509$	2851.62	0.248322	0.124161	−	0.992262i	$-0.460376\pi$
$509$	2851.62	0.248322	0.124161	+	0.992262i	$0.460376\pi$
$510$	0	0
$511$	−11836.6	−1.02470
$512$	−12990.4	−1.12129
$513$	0	0
$514$	10166.3	0.872402
$515$	0	0
$516$	0	0
$517$	−157.594	−0.0134061
$518$	462.892	0.0392631
$519$	0	0
$520$	0	0
$521$	−18549.3	−1.55980	−0.779902	−	0.625901i	$-0.784731\pi$
$521$	−18549.3	−1.55980	−0.779902	+	0.625901i	$0.784731\pi$
$522$	0	0
$523$	−11766.4	−0.983760	−0.491880	−	0.870663i	$-0.663690\pi$
$523$	−11766.4	−0.983760	−0.491880	+	0.870663i	$0.663690\pi$
$524$	2793.65	0.232903
$525$	0	0
$526$	19257.6	1.59633
$527$	−843.729	−0.0697408
$528$	0	0
$529$	28434.5	2.33702
$530$	0	0
$531$	0	0
$532$	−1342.74	−0.109427
$533$	25916.5	2.10613
$534$	0	0
$535$	0	0
$536$	17631.2	1.42080
$537$	0	0
$538$	−6805.09	−0.545331
$539$	2593.75	0.207274
$540$	0	0
$541$	13561.9	1.07777	0.538884	−	0.842380i	$-0.318846\pi$
$541$	13561.9	1.07777	0.538884	+	0.842380i	$0.318846\pi$
$542$	10901.1	0.863915
$543$	0	0
$544$	−1374.19	−0.108305
$545$	0	0
$546$	0	0
$547$	16092.5	1.25789	0.628944	−	0.777451i	$-0.283487\pi$
$547$	16092.5	1.25789	0.628944	+	0.777451i	$0.283487\pi$
$548$	427.259	0.0333058
$549$	0	0
$550$	0	0
$551$	−6445.22	−0.498322
$552$	0	0
$553$	16550.9	1.27273
$554$	−2205.41	−0.169132
$555$	0	0
$556$	2909.02	0.221889
$557$	19100.1	1.45296	0.726478	−	0.687190i	$-0.241156\pi$
$557$	19100.1	1.45296	0.726478	+	0.687190i	$0.241156\pi$
$558$	0	0
$559$	−2369.11	−0.179253
$560$	0	0
$561$	0	0
$562$	−1938.12	−0.145471
$563$	6898.73	0.516424	0.258212	−	0.966088i	$-0.416867\pi$
$563$	6898.73	0.516424	0.258212	+	0.966088i	$0.416867\pi$
$564$	0	0
$565$	0	0
$566$	−5290.98	−0.392926
$567$	0	0
$568$	−22245.2	−1.64329
$569$	161.850	0.0119246	0.00596231	−	0.999982i	$-0.498102\pi$
$569$	161.850	0.0119246	0.00596231	+	0.999982i	$0.498102\pi$
$570$	0	0
$571$	19826.3	1.45307	0.726537	−	0.687127i	$-0.241128\pi$
$571$	19826.3	1.45307	0.726537	+	0.687127i	$0.241128\pi$
$572$	852.302	0.0623017
$573$	0	0
$574$	−23145.0	−1.68302
$575$	0	0
$576$	0	0
$577$	25367.5	1.83027	0.915134	−	0.403150i	$-0.132085\pi$
$577$	25367.5	1.83027	0.915134	+	0.403150i	$0.132085\pi$
$578$	−10857.7	−0.781354
$579$	0	0
$580$	0	0
$581$	−17135.5	−1.22358
$582$	0	0
$583$	−1701.39	−0.120865
$584$	11758.2	0.833147
$585$	0	0
$586$	12763.9	0.899784
$587$	14760.7	1.03789	0.518943	−	0.854809i	$-0.326326\pi$
$587$	14760.7	1.03789	0.518943	+	0.854809i	$0.326326\pi$
$588$	0	0
$589$	1539.35	0.107687
$590$	0	0
$591$	0	0
$592$	−395.715	−0.0274726
$593$	−10705.6	−0.741357	−0.370679	−	0.928761i	$-0.620875\pi$
$593$	−10705.6	−0.741357	−0.370679	+	0.928761i	$0.620875\pi$
$594$	0	0
$595$	0	0
$596$	−291.763	−0.0200521
$597$	0	0
$598$	−37484.2	−2.56328
$599$	14015.0	0.955986	0.477993	−	0.878364i	$-0.341364\pi$
$599$	14015.0	0.955986	0.477993	+	0.878364i	$0.341364\pi$
$600$	0	0
$601$	−27307.6	−1.85341	−0.926706	−	0.375788i	$-0.877372\pi$
$601$	−27307.6	−1.85341	−0.926706	+	0.375788i	$0.877372\pi$
$602$	2115.75	0.143242
$603$	0	0
$604$	−2807.65	−0.189142
$605$	0	0
$606$	0	0
$607$	−4494.13	−0.300513	−0.150256	−	0.988647i	$-0.548010\pi$
$607$	−4494.13	−0.300513	−0.150256	+	0.988647i	$0.548010\pi$
$608$	2507.16	0.167235
$609$	0	0
$610$	0	0
$611$	1014.21	0.0671529
$612$	0	0
$613$	9965.04	0.656581	0.328291	−	0.944577i	$-0.393528\pi$
$613$	9965.04	0.656581	0.328291	+	0.944577i	$0.393528\pi$
$614$	−26806.5	−1.76192
$615$	0	0
$616$	−6324.58	−0.413676
$617$	20508.4	1.33815	0.669075	−	0.743195i	$-0.266691\pi$
$617$	20508.4	1.33815	0.669075	+	0.743195i	$0.266691\pi$
$618$	0	0
$619$	6213.15	0.403437	0.201718	−	0.979444i	$-0.435347\pi$
$619$	6213.15	0.403437	0.201718	+	0.979444i	$0.435347\pi$
$620$	0	0
$621$	0	0
$622$	1813.25	0.116888
$623$	32054.0	2.06134
$624$	0	0
$625$	0	0
$626$	14086.1	0.899350
$627$	0	0
$628$	529.660	0.0336556
$629$	−204.641	−0.0129723
$630$	0	0
$631$	−26557.9	−1.67552	−0.837762	−	0.546036i	$-0.816136\pi$
$631$	−26557.9	−1.67552	−0.837762	+	0.546036i	$0.816136\pi$
$632$	−16441.3	−1.03481
$633$	0	0
$634$	−28426.5	−1.78069
$635$	0	0
$636$	0	0
$637$	−16692.2	−1.03826
$638$	−3653.59	−0.226720
$639$	0	0
$640$	0	0
$641$	15431.3	0.950859	0.475430	−	0.879754i	$-0.342293\pi$
$641$	15431.3	0.950859	0.475430	+	0.879754i	$0.342293\pi$
$642$	0	0
$643$	19148.5	1.17441	0.587204	−	0.809439i	$-0.300229\pi$
$643$	19148.5	1.17441	0.587204	+	0.809439i	$0.300229\pi$
$644$	−5305.85	−0.324658
$645$	0	0
$646$	−3745.21	−0.228101
$647$	−15497.7	−0.941699	−0.470849	−	0.882214i	$-0.656052\pi$
$647$	−15497.7	−0.941699	−0.470849	+	0.882214i	$0.656052\pi$
$648$	0	0
$649$	−6731.19	−0.407122
$650$	0	0
$651$	0	0
$652$	4051.31	0.243346
$653$	11208.7	0.671716	0.335858	−	0.941913i	$-0.390974\pi$
$653$	11208.7	0.671716	0.335858	+	0.941913i	$0.390974\pi$
$654$	0	0
$655$	0	0
$656$	19786.1	1.17762
$657$	0	0
$658$	−905.747	−0.0536621
$659$	12671.7	0.749045	0.374523	−	0.927218i	$-0.377807\pi$
$659$	12671.7	0.749045	0.374523	+	0.927218i	$0.377807\pi$
$660$	0	0
$661$	29609.9	1.74235	0.871173	−	0.490977i	$-0.163360\pi$
$661$	29609.9	1.74235	0.871173	+	0.490977i	$0.163360\pi$
$662$	−28103.8	−1.64998
$663$	0	0
$664$	17022.0	0.994853
$665$	0	0
$666$	0	0
$667$	−25468.4	−1.47847
$668$	−2584.53	−0.149698
$669$	0	0
$670$	0	0
$671$	−914.344	−0.0526049
$672$	0	0
$673$	−4928.54	−0.282290	−0.141145	−	0.989989i	$-0.545078\pi$
$673$	−4928.54	−0.282290	−0.141145	+	0.989989i	$0.545078\pi$
$674$	−1726.44	−0.0986646
$675$	0	0
$676$	−3080.40	−0.175262
$677$	17239.1	0.978659	0.489330	−	0.872099i	$-0.337241\pi$
$677$	17239.1	0.978659	0.489330	+	0.872099i	$0.337241\pi$
$678$	0	0
$679$	−14407.3	−0.814286
$680$	0	0
$681$	0	0
$682$	872.610	0.0489941
$683$	16810.9	0.941804	0.470902	−	0.882185i	$-0.343929\pi$
$683$	16810.9	0.941804	0.470902	+	0.882185i	$0.343929\pi$
$684$	0	0
$685$	0	0
$686$	−6777.56	−0.377214
$687$	0	0
$688$	−1808.71	−0.100227
$689$	10949.4	0.605425
$690$	0	0
$691$	11163.7	0.614597	0.307298	−	0.951613i	$-0.400575\pi$
$691$	11163.7	0.614597	0.307298	+	0.951613i	$0.400575\pi$
$692$	−3954.67	−0.217246
$693$	0	0
$694$	−2331.61	−0.127531
$695$	0	0
$696$	0	0
$697$	10232.2	0.556059
$698$	15041.6	0.815661
$699$	0	0
$700$	0	0
$701$	−28098.2	−1.51391	−0.756956	−	0.653466i	$-0.773314\pi$
$701$	−28098.2	−1.51391	−0.756956	+	0.653466i	$0.773314\pi$
$702$	0	0
$703$	373.359	0.0200306
$704$	6177.27	0.330703
$705$	0	0
$706$	8584.75	0.457637
$707$	−9448.18	−0.502596
$708$	0	0
$709$	−23232.4	−1.23062	−0.615310	−	0.788285i	$-0.710969\pi$
$709$	−23232.4	−1.23062	−0.615310	+	0.788285i	$0.710969\pi$
$710$	0	0
$711$	0	0
$712$	−31841.7	−1.67601
$713$	6082.77	0.319497
$714$	0	0
$715$	0	0
$716$	2881.27	0.150389
$717$	0	0
$718$	29631.0	1.54014
$719$	−24964.9	−1.29490	−0.647450	−	0.762108i	$-0.724165\pi$
$719$	−24964.9	−1.29490	−0.647450	+	0.762108i	$0.724165\pi$
$720$	0	0
$721$	−21407.3	−1.10576
$722$	−11191.3	−0.576865
$723$	0	0
$724$	4330.18	0.222279
$725$	0	0
$726$	0	0
$727$	12425.8	0.633904	0.316952	−	0.948442i	$-0.397341\pi$
$727$	12425.8	0.633904	0.316952	+	0.948442i	$0.397341\pi$
$728$	40702.3	2.07215
$729$	0	0
$730$	0	0
$731$	−935.357	−0.0473262
$732$	0	0
$733$	3632.29	0.183031	0.0915154	−	0.995804i	$-0.470829\pi$
$733$	3632.29	0.183031	0.0915154	+	0.995804i	$0.470829\pi$
$734$	−7329.74	−0.368591
$735$	0	0
$736$	9907.10	0.496169
$737$	−8115.17	−0.405599
$738$	0	0
$739$	15663.6	0.779693	0.389847	−	0.920880i	$-0.372528\pi$
$739$	15663.6	0.779693	0.389847	+	0.920880i	$0.372528\pi$
$740$	0	0
$741$	0	0
$742$	−9778.44	−0.483798
$743$	20450.2	1.00975	0.504875	−	0.863193i	$-0.331539\pi$
$743$	20450.2	1.00975	0.504875	+	0.863193i	$0.331539\pi$
$744$	0	0
$745$	0	0
$746$	34246.4	1.68076
$747$	0	0
$748$	336.501	0.0164488
$749$	−21924.4	−1.06956
$750$	0	0
$751$	−18814.2	−0.914168	−0.457084	−	0.889424i	$-0.651106\pi$
$751$	−18814.2	−0.914168	−0.457084	+	0.889424i	$0.651106\pi$
$752$	774.302	0.0375477
$753$	0	0
$754$	23513.0	1.13566
$755$	0	0
$756$	0	0
$757$	−1994.60	−0.0957663	−0.0478831	−	0.998853i	$-0.515248\pi$
$757$	−1994.60	−0.0957663	−0.0478831	+	0.998853i	$0.515248\pi$
$758$	−10285.0	−0.492834
$759$	0	0
$760$	0	0
$761$	3002.11	0.143004	0.0715021	−	0.997440i	$-0.477221\pi$
$761$	3002.11	0.143004	0.0715021	+	0.997440i	$0.477221\pi$
$762$	0	0
$763$	−37345.2	−1.77194
$764$	−3197.79	−0.151429
$765$	0	0
$766$	9433.43	0.444966
$767$	43319.0	2.03932
$768$	0	0
$769$	5156.00	0.241781	0.120891	−	0.992666i	$-0.461425\pi$
$769$	5156.00	0.241781	0.120891	+	0.992666i	$0.461425\pi$
$770$	0	0
$771$	0	0
$772$	559.272	0.0260734
$773$	−13932.5	−0.648276	−0.324138	−	0.946010i	$-0.605074\pi$
$773$	−13932.5	−0.648276	−0.324138	+	0.946010i	$0.605074\pi$
$774$	0	0
$775$	0	0
$776$	14311.9	0.662069
$777$	0	0
$778$	14600.8	0.672834
$779$	−18668.3	−0.858614
$780$	0	0
$781$	10238.9	0.469111
$782$	−14799.3	−0.676754
$783$	0	0
$784$	−12743.8	−0.580529
$785$	0	0
$786$	0	0
$787$	−30180.5	−1.36699	−0.683493	−	0.729957i	$-0.739540\pi$
$787$	−30180.5	−1.36699	−0.683493	+	0.729957i	$0.739540\pi$
$788$	−2211.40	−0.0999717
$789$	0	0
$790$	0	0
$791$	11015.4	0.495151
$792$	0	0
$793$	5884.32	0.263504
$794$	−934.357	−0.0417621
$795$	0	0
$796$	3949.75	0.175873
$797$	19931.9	0.885853	0.442927	−	0.896558i	$-0.353940\pi$
$797$	19931.9	0.885853	0.442927	+	0.896558i	$0.353940\pi$
$798$	0	0
$799$	400.424	0.0177296
$800$	0	0
$801$	0	0
$802$	−11666.3	−0.513657
$803$	−5411.99	−0.237839
$804$	0	0
$805$	0	0
$806$	−5615.74	−0.245417
$807$	0	0
$808$	9385.61	0.408644
$809$	10844.1	0.471272	0.235636	−	0.971841i	$-0.424283\pi$
$809$	10844.1	0.471272	0.235636	+	0.971841i	$0.424283\pi$
$810$	0	0
$811$	8058.46	0.348916	0.174458	−	0.984665i	$-0.444183\pi$
$811$	8058.46	0.348916	0.174458	+	0.984665i	$0.444183\pi$
$812$	3328.24	0.143840
$813$	0	0
$814$	211.646	0.00911325
$815$	0	0
$816$	0	0
$817$	1706.52	0.0730767
$818$	−18454.7	−0.788818
$819$	0	0
$820$	0	0
$821$	−19340.1	−0.822136	−0.411068	−	0.911605i	$-0.634844\pi$
$821$	−19340.1	−0.822136	−0.411068	+	0.911605i	$0.634844\pi$
$822$	0	0
$823$	−10347.7	−0.438271	−0.219135	−	0.975694i	$-0.570324\pi$
$823$	−10347.7	−0.438271	−0.219135	+	0.975694i	$0.570324\pi$
$824$	21265.6	0.899055
$825$	0	0
$826$	−38686.4	−1.62963
$827$	4117.22	0.173119	0.0865597	−	0.996247i	$-0.472413\pi$
$827$	4117.22	0.173119	0.0865597	+	0.996247i	$0.472413\pi$
$828$	0	0
$829$	35735.0	1.49714	0.748569	−	0.663057i	$-0.230741\pi$
$829$	35735.0	1.49714	0.748569	+	0.663057i	$0.230741\pi$
$830$	0	0
$831$	0	0
$832$	−39754.3	−1.65653
$833$	−6590.34	−0.274120
$834$	0	0
$835$	0	0
$836$	−613.933	−0.0253987
$837$	0	0
$838$	−28346.1	−1.16850
$839$	−9919.13	−0.408160	−0.204080	−	0.978954i	$-0.565420\pi$
$839$	−9919.13	−0.408160	−0.204080	+	0.978954i	$0.565420\pi$
$840$	0	0
$841$	−8413.25	−0.344961
$842$	−1795.12	−0.0734726
$843$	0	0
$844$	−2494.79	−0.101747
$845$	0	0
$846$	0	0
$847$	2911.04	0.118093
$848$	8359.36	0.338516
$849$	0	0
$850$	0	0
$851$	1475.34	0.0594288
$852$	0	0
$853$	14138.6	0.567523	0.283761	−	0.958895i	$-0.408418\pi$
$853$	14138.6	0.567523	0.283761	+	0.958895i	$0.408418\pi$
$854$	−5255.05	−0.210567
$855$	0	0
$856$	21779.2	0.869624
$857$	−9474.85	−0.377660	−0.188830	−	0.982010i	$-0.560470\pi$
$857$	−9474.85	−0.377660	−0.188830	+	0.982010i	$0.560470\pi$
$858$	0	0
$859$	14024.6	0.557059	0.278529	−	0.960428i	$-0.410153\pi$
$859$	14024.6	0.557059	0.278529	+	0.960428i	$0.410153\pi$
$860$	0	0
$861$	0	0
$862$	14493.2	0.572669
$863$	32886.7	1.29719	0.648596	−	0.761133i	$-0.275357\pi$
$863$	32886.7	1.29719	0.648596	+	0.761133i	$0.275357\pi$
$864$	0	0
$865$	0	0
$866$	2684.93	0.105355
$867$	0	0
$868$	−794.903	−0.0310838
$869$	7567.50	0.295408
$870$	0	0
$871$	52225.7	2.03169
$872$	37097.9	1.44070
$873$	0	0
$874$	27000.7	1.04498
$875$	0	0
$876$	0	0
$877$	−7459.91	−0.287233	−0.143616	−	0.989633i	$-0.545873\pi$
$877$	−7459.91	−0.287233	−0.143616	+	0.989633i	$0.545873\pi$
$878$	−41788.5	−1.60625
$879$	0	0
$880$	0	0
$881$	−6549.49	−0.250463	−0.125231	−	0.992128i	$-0.539967\pi$
$881$	−6549.49	−0.250463	−0.125231	+	0.992128i	$0.539967\pi$
$882$	0	0
$883$	433.667	0.0165278	0.00826391	−	0.999966i	$-0.497369\pi$
$883$	433.667	0.0165278	0.00826391	+	0.999966i	$0.497369\pi$
$884$	−2165.58	−0.0823939
$885$	0	0
$886$	−21786.5	−0.826110
$887$	−8848.55	−0.334955	−0.167478	−	0.985876i	$-0.553562\pi$
$887$	−8848.55	−0.334955	−0.167478	+	0.985876i	$0.553562\pi$
$888$	0	0
$889$	21926.8	0.827222
$890$	0	0
$891$	0	0
$892$	7119.62	0.267245
$893$	−730.557	−0.0273764
$894$	0	0
$895$	0	0
$896$	26039.9	0.970907
$897$	0	0
$898$	−37775.4	−1.40376
$899$	−3815.58	−0.141554
$900$	0	0
$901$	4322.97	0.159844
$902$	−10582.5	−0.390640
$903$	0	0
$904$	−10942.5	−0.402591
$905$	0	0
$906$	0	0
$907$	36540.9	1.33773	0.668865	−	0.743384i	$-0.266781\pi$
$907$	36540.9	1.33773	0.668865	+	0.743384i	$0.266781\pi$
$908$	5639.79	0.206127
$909$	0	0
$910$	0	0
$911$	−13231.9	−0.481220	−0.240610	−	0.970622i	$-0.577347\pi$
$911$	−13231.9	−0.481220	−0.240610	+	0.970622i	$0.577347\pi$
$912$	0	0
$913$	−7834.78	−0.284002
$914$	5748.60	0.208038
$915$	0	0
$916$	6525.30	0.235373
$917$	−61406.3	−2.21136
$918$	0	0
$919$	15568.9	0.558838	0.279419	−	0.960169i	$-0.409858\pi$
$919$	15568.9	0.558838	0.279419	+	0.960169i	$0.409858\pi$
$920$	0	0
$921$	0	0
$922$	2670.47	0.0953874
$923$	−65892.9	−2.34983
$924$	0	0
$925$	0	0
$926$	25026.0	0.888125
$927$	0	0
$928$	−6214.49	−0.219828
$929$	19507.9	0.688950	0.344475	−	0.938795i	$-0.388057\pi$
$929$	19507.9	0.688950	0.344475	+	0.938795i	$0.388057\pi$
$930$	0	0
$931$	12023.8	0.423270
$932$	4943.64	0.173749
$933$	0	0
$934$	19056.8	0.667622
$935$	0	0
$936$	0	0
$937$	9493.57	0.330994	0.165497	−	0.986210i	$-0.447077\pi$
$937$	9493.57	0.330994	0.165497	+	0.986210i	$0.447077\pi$
$938$	−46640.6	−1.62353
$939$	0	0
$940$	0	0
$941$	−39841.1	−1.38022	−0.690108	−	0.723706i	$-0.742437\pi$
$941$	−39841.1	−1.38022	−0.690108	+	0.723706i	$0.742437\pi$
$942$	0	0
$943$	−73768.1	−2.54742
$944$	33072.1	1.14026
$945$	0	0
$946$	967.374	0.0332474
$947$	8119.72	0.278622	0.139311	−	0.990249i	$-0.455511\pi$
$947$	8119.72	0.278622	0.139311	+	0.990249i	$0.455511\pi$
$948$	0	0
$949$	34829.2	1.19136
$950$	0	0
$951$	0	0
$952$	16069.8	0.547087
$953$	21781.7	0.740376	0.370188	−	0.928957i	$-0.379293\pi$
$953$	21781.7	0.740376	0.370188	+	0.928957i	$0.379293\pi$
$954$	0	0
$955$	0	0
$956$	3610.02	0.122130
$957$	0	0
$958$	27654.2	0.932638
$959$	−9391.43	−0.316231
$960$	0	0
$961$	−28879.7	−0.969410
$962$	−1362.06	−0.0456493
$963$	0	0
$964$	547.411	0.0182893
$965$	0	0
$966$	0	0
$967$	−9740.88	−0.323936	−0.161968	−	0.986796i	$-0.551784\pi$
$967$	−9740.88	−0.323936	−0.161968	+	0.986796i	$0.551784\pi$
$968$	−2891.76	−0.0960172
$969$	0	0
$970$	0	0
$971$	9950.95	0.328879	0.164439	−	0.986387i	$-0.447419\pi$
$971$	9950.95	0.328879	0.164439	+	0.986387i	$0.447419\pi$
$972$	0	0
$973$	−63942.3	−2.10678
$974$	−1459.79	−0.0480233
$975$	0	0
$976$	4492.41	0.147335
$977$	−43872.0	−1.43663	−0.718317	−	0.695716i	$-0.755087\pi$
$977$	−43872.0	−1.43663	−0.718317	+	0.695716i	$0.755087\pi$
$978$	0	0
$979$	14655.9	0.478452
$980$	0	0
$981$	0	0
$982$	51984.8	1.68931
$983$	6647.35	0.215684	0.107842	−	0.994168i	$-0.465606\pi$
$983$	6647.35	0.215684	0.107842	+	0.994168i	$0.465606\pi$
$984$	0	0
$985$	0	0
$986$	9283.25	0.299837
$987$	0	0
$988$	3951.01	0.127225
$989$	6743.35	0.216811
$990$	0	0
$991$	−13097.7	−0.419840	−0.209920	−	0.977719i	$-0.567320\pi$
$991$	−13097.7	−0.419840	−0.209920	+	0.977719i	$0.567320\pi$
$992$	1484.24	0.0475049
$993$	0	0
$994$	58846.3	1.87776
$995$	0	0
$996$	0	0
$997$	27231.3	0.865019	0.432510	−	0.901629i	$-0.357628\pi$
$997$	27231.3	0.865019	0.432510	+	0.901629i	$0.357628\pi$
$998$	−7225.05	−0.229163
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.4.a.br.1.5		7
3.2	odd	2		825.4.a.bb.1.3		7
5.2	odd	4		495.4.c.c.199.10		14
5.3	odd	4		495.4.c.c.199.5		14
5.4	even	2		2475.4.a.bq.1.3		7
15.2	even	4		165.4.c.a.34.5	✓	14
15.8	even	4		165.4.c.a.34.10	yes	14
15.14	odd	2		825.4.a.bc.1.5		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.4.c.a.34.5	✓	14	15.2	even	4
165.4.c.a.34.10	yes	14	15.8	even	4
495.4.c.c.199.5		14	5.3	odd	4
495.4.c.c.199.10		14	5.2	odd	4
825.4.a.bb.1.3		7	3.2	odd	2
825.4.a.bc.1.5		7	15.14	odd	2
2475.4.a.bq.1.3		7	5.4	even	2
2475.4.a.br.1.5		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:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2475.a (trivial)

\(f(q)\)	\(=\)	\(q+2.62783 q^{2} -1.09451 q^{4} +24.0582 q^{7} -23.8988 q^{8} +O(q^{10})\)	\(q+2.62783 q^{2} -1.09451 q^{4} +24.0582 q^{7} -23.8988 q^{8} +11.0000 q^{11} -70.7912 q^{13} +63.2207 q^{14} -54.0459 q^{16} -27.9494 q^{17} +50.9926 q^{19} +28.9061 q^{22} +201.498 q^{23} -186.027 q^{26} -26.3320 q^{28} -126.395 q^{29} +30.1877 q^{31} +49.1672 q^{32} -73.4462 q^{34} +7.32183 q^{37} +134.000 q^{38} -366.098 q^{41} +33.4661 q^{43} -12.0397 q^{44} +529.503 q^{46} -14.3267 q^{47} +235.795 q^{49} +77.4820 q^{52} -154.671 q^{53} -574.962 q^{56} -332.145 q^{58} -611.926 q^{59} -83.1222 q^{61} +79.3282 q^{62} +561.570 q^{64} -737.743 q^{67} +30.5910 q^{68} +930.806 q^{71} -491.999 q^{73} +19.2405 q^{74} -55.8121 q^{76} +264.640 q^{77} +687.955 q^{79} -962.043 q^{82} -712.253 q^{83} +87.9431 q^{86} -262.887 q^{88} +1332.36 q^{89} -1703.11 q^{91} -220.543 q^{92} -37.6482 q^{94} -598.852 q^{97} +619.630 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 3 q^{2} + 41 q^{4} - 50 q^{7} + 21 q^{8}+O(q^{10}) \) 7 * q + 3 * q^2 + 41 * q^4 - 50 * q^7 + 21 * q^8	\( 7 q + 3 q^{2} + 41 q^{4} - 50 q^{7} + 21 q^{8} + 77 q^{11} - 24 q^{13} - 142 q^{14} + 181 q^{16} + 38 q^{17} + 26 q^{19} + 33 q^{22} + 228 q^{23} - 476 q^{26} - 840 q^{28} - 572 q^{29} - 140 q^{31} + 991 q^{32} - 806 q^{34} + 104 q^{37} + 498 q^{38} - 896 q^{41} - 614 q^{43} + 451 q^{44} - 344 q^{46} + 520 q^{47} + 295 q^{49} + 26 q^{52} + 380 q^{53} - 1522 q^{56} - 1600 q^{58} - 1316 q^{59} - 386 q^{61} + 440 q^{62} + 869 q^{64} - 348 q^{67} + 332 q^{68} - 804 q^{71} - 468 q^{73} + 748 q^{74} - 1698 q^{76} - 550 q^{77} - 374 q^{79} + 620 q^{82} + 3128 q^{83} + 2534 q^{86} + 231 q^{88} - 694 q^{89} - 3376 q^{91} - 1184 q^{92} - 2920 q^{94} + 8 q^{97} + 4211 q^{98}+O(q^{100}) \) 7 * q + 3 * q^2 + 41 * q^4 - 50 * q^7 + 21 * q^8 + 77 * q^11 - 24 * q^13 - 142 * q^14 + 181 * q^16 + 38 * q^17 + 26 * q^19 + 33 * q^22 + 228 * q^23 - 476 * q^26 - 840 * q^28 - 572 * q^29 - 140 * q^31 + 991 * q^32 - 806 * q^34 + 104 * q^37 + 498 * q^38 - 896 * q^41 - 614 * q^43 + 451 * q^44 - 344 * q^46 + 520 * q^47 + 295 * q^49 + 26 * q^52 + 380 * q^53 - 1522 * q^56 - 1600 * q^58 - 1316 * q^59 - 386 * q^61 + 440 * q^62 + 869 * q^64 - 348 * q^67 + 332 * q^68 - 804 * q^71 - 468 * q^73 + 748 * q^74 - 1698 * q^76 - 550 * q^77 - 374 * q^79 + 620 * q^82 + 3128 * q^83 + 2534 * q^86 + 231 * q^88 - 694 * q^89 - 3376 * q^91 - 1184 * q^92 - 2920 * q^94 + 8 * q^97 + 4211 * q^98

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