LMFDB - Embedded newform 2475.4.a.bu.1.4

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:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 4 x^{9} - 49 x^{8} + 180 x^{7} + 753 x^{6} - 2420 x^{5} - 4059 x^{4} + 9796 x^{3} + 7226 x^{2} + \cdots - 2112 $ x^10 - 4x^9 - 49x^8 + 180x^7 + 753x^6 - 2420x^5 - 4059x^4 + 9796x^3 + 7226x^2 - 9120*x - 2112
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.12679$ 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)

$f(q)$	$=$	$q-2.12679 q^{2} -3.47675 q^{4} -3.88078 q^{7} +24.4087 q^{8} +O(q^{10})$	$q-2.12679 q^{2} -3.47675 q^{4} -3.88078 q^{7} +24.4087 q^{8} +11.0000 q^{11} -85.4842 q^{13} +8.25362 q^{14} -24.0982 q^{16} +10.9966 q^{17} -85.7726 q^{19} -23.3947 q^{22} +86.2973 q^{23} +181.807 q^{26} +13.4925 q^{28} +253.485 q^{29} +65.4544 q^{31} -144.018 q^{32} -23.3876 q^{34} +180.436 q^{37} +182.421 q^{38} -57.8120 q^{41} +202.238 q^{43} -38.2443 q^{44} -183.537 q^{46} +130.827 q^{47} -327.940 q^{49} +297.207 q^{52} -261.450 q^{53} -94.7247 q^{56} -539.110 q^{58} -469.494 q^{59} +282.657 q^{61} -139.208 q^{62} +499.081 q^{64} -25.4984 q^{67} -38.2326 q^{68} +468.080 q^{71} -840.768 q^{73} -383.750 q^{74} +298.210 q^{76} -42.6886 q^{77} -476.318 q^{79} +122.954 q^{82} -1015.07 q^{83} -430.119 q^{86} +268.495 q^{88} -171.263 q^{89} +331.746 q^{91} -300.034 q^{92} -278.242 q^{94} -153.908 q^{97} +697.460 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 4 q^{2} + 34 q^{4} - 2 q^{7} - 48 q^{8}+O(q^{10}) $ 10 * q - 4 * q^2 + 34 * q^4 - 2 * q^7 - 48 * q^8	$ 10 q - 4 q^{2} + 34 q^{4} - 2 q^{7} - 48 q^{8} + 110 q^{11} - 26 q^{13} - 72 q^{14} + 206 q^{16} - 148 q^{17} - 114 q^{19} - 44 q^{22} + 34 q^{23} - 100 q^{26} + 86 q^{28} + 38 q^{29} + 232 q^{31} - 448 q^{32} - 20 q^{34} - 754 q^{37} - 780 q^{38} - 160 q^{41} + 66 q^{43} + 374 q^{44} + 682 q^{46} - 450 q^{47} + 590 q^{49} - 200 q^{52} - 1068 q^{53} - 268 q^{56} + 138 q^{58} + 838 q^{59} - 566 q^{61} - 1230 q^{62} + 462 q^{64} - 430 q^{67} - 2234 q^{68} - 518 q^{71} + 184 q^{73} - 402 q^{74} + 386 q^{76} - 22 q^{77} + 956 q^{79} + 2180 q^{82} - 2094 q^{83} - 892 q^{86} - 528 q^{88} + 512 q^{89} - 858 q^{91} - 4476 q^{92} - 294 q^{94} + 1006 q^{97} - 2226 q^{98}+O(q^{100}) $ 10 * q - 4 * q^2 + 34 * q^4 - 2 * q^7 - 48 * q^8 + 110 * q^11 - 26 * q^13 - 72 * q^14 + 206 * q^16 - 148 * q^17 - 114 * q^19 - 44 * q^22 + 34 * q^23 - 100 * q^26 + 86 * q^28 + 38 * q^29 + 232 * q^31 - 448 * q^32 - 20 * q^34 - 754 * q^37 - 780 * q^38 - 160 * q^41 + 66 * q^43 + 374 * q^44 + 682 * q^46 - 450 * q^47 + 590 * q^49 - 200 * q^52 - 1068 * q^53 - 268 * q^56 + 138 * q^58 + 838 * q^59 - 566 * q^61 - 1230 * q^62 + 462 * q^64 - 430 * q^67 - 2234 * q^68 - 518 * q^71 + 184 * q^73 - 402 * q^74 + 386 * q^76 - 22 * q^77 + 956 * q^79 + 2180 * q^82 - 2094 * q^83 - 892 * q^86 - 528 * q^88 + 512 * q^89 - 858 * q^91 - 4476 * q^92 - 294 * q^94 + 1006 * q^97 - 2226 * 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 $

$2$	−2.12679	−0.751935	−0.375967	−	0.926633i	$-0.622690\pi$
$2$	−2.12679	−0.751935	−0.375967	+	0.926633i	$0.622690\pi$
$3$	0	0
$3$	0	0
$4$	−3.47675	−0.434594
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.88078	−0.209542	−0.104771	−	0.994496i	$-0.533411\pi$
$7$	−3.88078	−0.209542	−0.104771	+	0.994496i	$0.533411\pi$
$8$	24.4087	1.07872
$9$	0	0
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	−85.4842	−1.82377	−0.911887	−	0.410442i	$-0.865374\pi$
$13$	−85.4842	−1.82377	−0.911887	+	0.410442i	$0.865374\pi$
$14$	8.25362	0.157562
$15$	0	0
$16$	−24.0982	−0.376534
$17$	10.9966	0.156887	0.0784435	−	0.996919i	$-0.475005\pi$
$17$	10.9966	0.156887	0.0784435	+	0.996919i	$0.475005\pi$
$18$	0	0
$19$	−85.7726	−1.03566	−0.517831	−	0.855483i	$-0.673261\pi$
$19$	−85.7726	−1.03566	−0.517831	+	0.855483i	$0.673261\pi$
$20$	0	0
$21$	0	0
$22$	−23.3947	−0.226717
$23$	86.2973	0.782358	0.391179	−	0.920315i	$-0.372067\pi$
$23$	86.2973	0.782358	0.391179	+	0.920315i	$0.372067\pi$
$24$	0	0
$25$	0	0
$26$	181.807	1.37136
$27$	0	0
$28$	13.4925	0.0910659
$29$	253.485	1.62314	0.811569	−	0.584257i	$-0.198614\pi$
$29$	253.485	1.62314	0.811569	+	0.584257i	$0.198614\pi$
$30$	0	0
$31$	65.4544	0.379225	0.189612	−	0.981859i	$-0.439277\pi$
$31$	65.4544	0.379225	0.189612	+	0.981859i	$0.439277\pi$
$32$	−144.018	−0.795592
$33$	0	0
$34$	−23.3876	−0.117969
$35$	0	0
$36$	0	0
$37$	180.436	0.801716	0.400858	−	0.916140i	$-0.368712\pi$
$37$	180.436	0.801716	0.400858	+	0.916140i	$0.368712\pi$
$38$	182.421	0.778751
$39$	0	0
$40$	0	0
$41$	−57.8120	−0.220213	−0.110106	−	0.993920i	$-0.535119\pi$
$41$	−57.8120	−0.220213	−0.110106	+	0.993920i	$0.535119\pi$
$42$	0	0
$43$	202.238	0.717234	0.358617	−	0.933485i	$-0.383248\pi$
$43$	202.238	0.717234	0.358617	+	0.933485i	$0.383248\pi$
$44$	−38.2443	−0.131035
$45$	0	0
$46$	−183.537	−0.588282
$47$	130.827	0.406023	0.203011	−	0.979176i	$-0.434927\pi$
$47$	130.827	0.406023	0.203011	+	0.979176i	$0.434927\pi$
$48$	0	0
$49$	−327.940	−0.956092
$50$	0	0
$51$	0	0
$52$	297.207	0.792601
$53$	−261.450	−0.677601	−0.338801	−	0.940858i	$-0.610021\pi$
$53$	−261.450	−0.677601	−0.338801	+	0.940858i	$0.610021\pi$
$54$	0	0
$55$	0	0
$56$	−94.7247	−0.226038
$57$	0	0
$58$	−539.110	−1.22049
$59$	−469.494	−1.03598	−0.517990	−	0.855386i	$-0.673320\pi$
$59$	−469.494	−1.03598	−0.517990	+	0.855386i	$0.673320\pi$
$60$	0	0
$61$	282.657	0.593288	0.296644	−	0.954988i	$-0.404133\pi$
$61$	282.657	0.593288	0.296644	+	0.954988i	$0.404133\pi$
$62$	−139.208	−0.285152
$63$	0	0
$64$	499.081	0.974768
$65$	0	0
$66$	0	0
$67$	−25.4984	−0.0464945	−0.0232472	−	0.999730i	$-0.507400\pi$
$67$	−25.4984	−0.0464945	−0.0232472	+	0.999730i	$0.507400\pi$
$68$	−38.2326	−0.0681821
$69$	0	0
$70$	0	0
$71$	468.080	0.782406	0.391203	−	0.920304i	$-0.372059\pi$
$71$	468.080	0.782406	0.391203	+	0.920304i	$0.372059\pi$
$72$	0	0
$73$	−840.768	−1.34801	−0.674003	−	0.738729i	$-0.735427\pi$
$73$	−840.768	−1.34801	−0.674003	+	0.738729i	$0.735427\pi$
$74$	−383.750	−0.602838
$75$	0	0
$76$	298.210	0.450093
$77$	−42.6886	−0.0631794
$78$	0	0
$79$	−476.318	−0.678354	−0.339177	−	0.940723i	$-0.610149\pi$
$79$	−476.318	−0.678354	−0.339177	+	0.940723i	$0.610149\pi$
$80$	0	0
$81$	0	0
$82$	122.954	0.165586
$83$	−1015.07	−1.34239	−0.671195	−	0.741281i	$-0.734219\pi$
$83$	−1015.07	−1.34239	−0.671195	+	0.741281i	$0.734219\pi$
$84$	0	0
$85$	0	0
$86$	−430.119	−0.539313
$87$	0	0
$88$	268.495	0.325247
$89$	−171.263	−0.203976	−0.101988	−	0.994786i	$-0.532520\pi$
$89$	−171.263	−0.203976	−0.101988	+	0.994786i	$0.532520\pi$
$90$	0	0
$91$	331.746	0.382158
$92$	−300.034	−0.340008
$93$	0	0
$94$	−278.242	−0.305303
$95$	0	0
$96$	0	0
$97$	−153.908	−0.161103	−0.0805516	−	0.996750i	$-0.525668\pi$
$97$	−153.908	−0.161103	−0.0805516	+	0.996750i	$0.525668\pi$
$98$	697.460	0.718919
$99$	0	0
$100$	0	0
$101$	1664.61	1.63995	0.819973	−	0.572402i	$-0.193988\pi$
$101$	1664.61	1.63995	0.819973	+	0.572402i	$0.193988\pi$
$102$	0	0
$103$	138.810	0.132790	0.0663951	−	0.997793i	$-0.478850\pi$
$103$	138.810	0.132790	0.0663951	+	0.997793i	$0.478850\pi$
$104$	−2086.56	−1.96734
$105$	0	0
$106$	556.049	0.509512
$107$	1850.92	1.67229	0.836147	−	0.548505i	$-0.184803\pi$
$107$	1850.92	1.67229	0.836147	+	0.548505i	$0.184803\pi$
$108$	0	0
$109$	1943.21	1.70757	0.853786	−	0.520624i	$-0.174301\pi$
$109$	1943.21	1.70757	0.853786	+	0.520624i	$0.174301\pi$
$110$	0	0
$111$	0	0
$112$	93.5198	0.0788999
$113$	1274.87	1.06132	0.530661	−	0.847584i	$-0.321944\pi$
$113$	1274.87	1.06132	0.530661	+	0.847584i	$0.321944\pi$
$114$	0	0
$115$	0	0
$116$	−881.305	−0.705406
$117$	0	0
$118$	998.516	0.778990
$119$	−42.6756	−0.0328745
$120$	0	0
$121$	121.000	0.0909091
$122$	−601.154	−0.446114
$123$	0	0
$124$	−227.569	−0.164809
$125$	0	0
$126$	0	0
$127$	1939.17	1.35491	0.677453	−	0.735566i	$-0.263084\pi$
$127$	1939.17	1.35491	0.677453	+	0.735566i	$0.263084\pi$
$128$	90.6984	0.0626303
$129$	0	0
$130$	0	0
$131$	239.855	0.159971	0.0799856	−	0.996796i	$-0.474513\pi$
$131$	239.855	0.159971	0.0799856	+	0.996796i	$0.474513\pi$
$132$	0	0
$133$	332.865	0.217015
$134$	54.2299	0.0349608
$135$	0	0
$136$	268.414	0.169237
$137$	−1884.28	−1.17507	−0.587537	−	0.809197i	$-0.699902\pi$
$137$	−1884.28	−1.17507	−0.587537	+	0.809197i	$0.699902\pi$
$138$	0	0
$139$	862.966	0.526588	0.263294	−	0.964716i	$-0.415191\pi$
$139$	862.966	0.526588	0.263294	+	0.964716i	$0.415191\pi$
$140$	0	0
$141$	0	0
$142$	−995.508	−0.588318
$143$	−940.326	−0.549888
$144$	0	0
$145$	0	0
$146$	1788.14	1.01361
$147$	0	0
$148$	−627.331	−0.348421
$149$	1835.40	1.00914	0.504570	−	0.863371i	$-0.331651\pi$
$149$	1835.40	1.00914	0.504570	+	0.863371i	$0.331651\pi$
$150$	0	0
$151$	2527.85	1.36234	0.681170	−	0.732125i	$-0.261471\pi$
$151$	2527.85	1.36234	0.681170	+	0.732125i	$0.261471\pi$
$152$	−2093.60	−1.11719
$153$	0	0
$154$	90.7898	0.0475068
$155$	0	0
$156$	0	0
$157$	946.180	0.480977	0.240489	−	0.970652i	$-0.422692\pi$
$157$	946.180	0.480977	0.240489	+	0.970652i	$0.422692\pi$
$158$	1013.03	0.510078
$159$	0	0
$160$	0	0
$161$	−334.901	−0.163937
$162$	0	0
$163$	−1659.87	−0.797616	−0.398808	−	0.917034i	$-0.630576\pi$
$163$	−1659.87	−0.797616	−0.398808	+	0.917034i	$0.630576\pi$
$164$	200.998	0.0957031
$165$	0	0
$166$	2158.84	1.00939
$167$	−3276.34	−1.51815	−0.759075	−	0.651004i	$-0.774348\pi$
$167$	−3276.34	−1.51815	−0.759075	+	0.651004i	$0.774348\pi$
$168$	0	0
$169$	5110.55	2.32615
$170$	0	0
$171$	0	0
$172$	−703.132	−0.311705
$173$	371.955	0.163463	0.0817317	−	0.996654i	$-0.473955\pi$
$173$	371.955	0.163463	0.0817317	+	0.996654i	$0.473955\pi$
$174$	0	0
$175$	0	0
$176$	−265.080	−0.113529
$177$	0	0
$178$	364.242	0.153377
$179$	−501.684	−0.209484	−0.104742	−	0.994499i	$-0.533402\pi$
$179$	−501.684	−0.209484	−0.104742	+	0.994499i	$0.533402\pi$
$180$	0	0
$181$	−965.257	−0.396392	−0.198196	−	0.980162i	$-0.563508\pi$
$181$	−965.257	−0.396392	−0.198196	+	0.980162i	$0.563508\pi$
$182$	−705.554	−0.287358
$183$	0	0
$184$	2106.40	0.843946
$185$	0	0
$186$	0	0
$187$	120.963	0.0473032
$188$	−454.853	−0.176455
$189$	0	0
$190$	0	0
$191$	3617.87	1.37057	0.685287	−	0.728273i	$-0.259677\pi$
$191$	3617.87	1.37057	0.685287	+	0.728273i	$0.259677\pi$
$192$	0	0
$193$	−4013.18	−1.49676	−0.748380	−	0.663270i	$-0.769168\pi$
$193$	−4013.18	−1.49676	−0.748380	+	0.663270i	$0.769168\pi$
$194$	327.331	0.121139
$195$	0	0
$196$	1140.16	0.415512
$197$	−3033.97	−1.09727	−0.548634	−	0.836063i	$-0.684852\pi$
$197$	−3033.97	−1.09727	−0.548634	+	0.836063i	$0.684852\pi$
$198$	0	0
$199$	−4224.14	−1.50473	−0.752365	−	0.658746i	$-0.771087\pi$
$199$	−4224.14	−1.50473	−0.752365	+	0.658746i	$0.771087\pi$
$200$	0	0
$201$	0	0
$202$	−3540.27	−1.23313
$203$	−983.720	−0.340116
$204$	0	0
$205$	0	0
$206$	−295.221	−0.0998496
$207$	0	0
$208$	2060.01	0.686713
$209$	−943.499	−0.312264
$210$	0	0
$211$	3552.99	1.15923	0.579616	−	0.814890i	$-0.303203\pi$
$211$	3552.99	1.15923	0.579616	+	0.814890i	$0.303203\pi$
$212$	908.995	0.294481
$213$	0	0
$214$	−3936.53	−1.25746
$215$	0	0
$216$	0	0
$217$	−254.014	−0.0794637
$218$	−4132.80	−1.28398
$219$	0	0
$220$	0	0
$221$	−940.040	−0.286126
$222$	0	0
$223$	−4158.51	−1.24877	−0.624383	−	0.781119i	$-0.714649\pi$
$223$	−4158.51	−1.24877	−0.624383	+	0.781119i	$0.714649\pi$
$224$	558.901	0.166710
$225$	0	0
$226$	−2711.38	−0.798045
$227$	−2198.15	−0.642714	−0.321357	−	0.946958i	$-0.604139\pi$
$227$	−2198.15	−0.642714	−0.321357	+	0.946958i	$0.604139\pi$
$228$	0	0
$229$	−1456.62	−0.420332	−0.210166	−	0.977666i	$-0.567400\pi$
$229$	−1456.62	−0.420332	−0.210166	+	0.977666i	$0.567400\pi$
$230$	0	0
$231$	0	0
$232$	6187.24	1.75091
$233$	−4469.97	−1.25681	−0.628407	−	0.777885i	$-0.716293\pi$
$233$	−4469.97	−1.25681	−0.628407	+	0.777885i	$0.716293\pi$
$234$	0	0
$235$	0	0
$236$	1632.31	0.450231
$237$	0	0
$238$	90.7621	0.0247195
$239$	−6647.69	−1.79918	−0.899589	−	0.436738i	$-0.856134\pi$
$239$	−6647.69	−1.79918	−0.899589	+	0.436738i	$0.856134\pi$
$240$	0	0
$241$	4889.71	1.30694	0.653472	−	0.756950i	$-0.273312\pi$
$241$	4889.71	1.30694	0.653472	+	0.756950i	$0.273312\pi$
$242$	−257.342	−0.0683577
$243$	0	0
$244$	−982.729	−0.257839
$245$	0	0
$246$	0	0
$247$	7332.21	1.88881
$248$	1597.66	0.409078
$249$	0	0
$250$	0	0
$251$	1508.15	0.379256	0.189628	−	0.981856i	$-0.439272\pi$
$251$	1508.15	0.379256	0.189628	+	0.981856i	$0.439272\pi$
$252$	0	0
$253$	949.271	0.235890
$254$	−4124.20	−1.01880
$255$	0	0
$256$	−4185.54	−1.02186
$257$	−5681.62	−1.37903	−0.689513	−	0.724273i	$-0.742176\pi$
$257$	−5681.62	−1.37903	−0.689513	+	0.724273i	$0.742176\pi$
$258$	0	0
$259$	−700.233	−0.167994
$260$	0	0
$261$	0	0
$262$	−510.122	−0.120288
$263$	−3309.65	−0.775976	−0.387988	−	0.921664i	$-0.626830\pi$
$263$	−3309.65	−0.775976	−0.387988	+	0.921664i	$0.626830\pi$
$264$	0	0
$265$	0	0
$266$	−707.934	−0.163181
$267$	0	0
$268$	88.6518	0.0202062
$269$	−636.345	−0.144233	−0.0721165	−	0.997396i	$-0.522975\pi$
$269$	−636.345	−0.144233	−0.0721165	+	0.997396i	$0.522975\pi$
$270$	0	0
$271$	741.562	0.166224	0.0831120	−	0.996540i	$-0.473514\pi$
$271$	741.562	0.166224	0.0831120	+	0.996540i	$0.473514\pi$
$272$	−264.999	−0.0590733
$273$	0	0
$274$	4007.48	0.883579
$275$	0	0
$276$	0	0
$277$	1517.93	0.329255	0.164628	−	0.986356i	$-0.447358\pi$
$277$	1517.93	0.329255	0.164628	+	0.986356i	$0.447358\pi$
$278$	−1835.35	−0.395960
$279$	0	0
$280$	0	0
$281$	2737.36	0.581129	0.290564	−	0.956855i	$-0.406157\pi$
$281$	2737.36	0.581129	0.290564	+	0.956855i	$0.406157\pi$
$282$	0	0
$283$	4197.10	0.881596	0.440798	−	0.897606i	$-0.354695\pi$
$283$	4197.10	0.881596	0.440798	+	0.897606i	$0.354695\pi$
$284$	−1627.40	−0.340029
$285$	0	0
$286$	1999.88	0.413480
$287$	224.356	0.0461439
$288$	0	0
$289$	−4792.07	−0.975386
$290$	0	0
$291$	0	0
$292$	2923.14	0.585835
$293$	1388.33	0.276817	0.138408	−	0.990375i	$-0.455801\pi$
$293$	1388.33	0.276817	0.138408	+	0.990375i	$0.455801\pi$
$294$	0	0
$295$	0	0
$296$	4404.20	0.864828
$297$	0	0
$298$	−3903.51	−0.758807
$299$	−7377.06	−1.42684
$300$	0	0
$301$	−784.842	−0.150291
$302$	−5376.21	−1.02439
$303$	0	0
$304$	2066.96	0.389962
$305$	0	0
$306$	0	0
$307$	−5083.07	−0.944972	−0.472486	−	0.881338i	$-0.656643\pi$
$307$	−5083.07	−0.944972	−0.472486	+	0.881338i	$0.656643\pi$
$308$	148.418	0.0274574
$309$	0	0
$310$	0	0
$311$	−1770.65	−0.322844	−0.161422	−	0.986885i	$-0.551608\pi$
$311$	−1770.65	−0.322844	−0.161422	+	0.986885i	$0.551608\pi$
$312$	0	0
$313$	−8347.62	−1.50746	−0.753730	−	0.657184i	$-0.771748\pi$
$313$	−8347.62	−1.50746	−0.753730	+	0.657184i	$0.771748\pi$
$314$	−2012.33	−0.361663
$315$	0	0
$316$	1656.04	0.294809
$317$	−55.1725	−0.00977538	−0.00488769	−	0.999988i	$-0.501556\pi$
$317$	−55.1725	−0.00977538	−0.00488769	+	0.999988i	$0.501556\pi$
$318$	0	0
$319$	2788.34	0.489394
$320$	0	0
$321$	0	0
$322$	712.265	0.123270
$323$	−943.211	−0.162482
$324$	0	0
$325$	0	0
$326$	3530.21	0.599755
$327$	0	0
$328$	−1411.11	−0.237548
$329$	−507.711	−0.0850790
$330$	0	0
$331$	−7328.81	−1.21700	−0.608501	−	0.793553i	$-0.708229\pi$
$331$	−7328.81	−1.21700	−0.608501	+	0.793553i	$0.708229\pi$
$332$	3529.15	0.583395
$333$	0	0
$334$	6968.10	1.14155
$335$	0	0
$336$	0	0
$337$	−1760.01	−0.284492	−0.142246	−	0.989831i	$-0.545432\pi$
$337$	−1760.01	−0.284492	−0.142246	+	0.989831i	$0.545432\pi$
$338$	−10869.1	−1.74911
$339$	0	0
$340$	0	0
$341$	719.999	0.114341
$342$	0	0
$343$	2603.77	0.409884
$344$	4936.37	0.773695
$345$	0	0
$346$	−791.070	−0.122914
$347$	−10928.7	−1.69073	−0.845366	−	0.534188i	$-0.820618\pi$
$347$	−10928.7	−1.69073	−0.845366	+	0.534188i	$0.820618\pi$
$348$	0	0
$349$	−2078.09	−0.318733	−0.159366	−	0.987220i	$-0.550945\pi$
$349$	−2078.09	−0.318733	−0.159366	+	0.987220i	$0.550945\pi$
$350$	0	0
$351$	0	0
$352$	−1584.19	−0.239880
$353$	−9404.80	−1.41804	−0.709019	−	0.705190i	$-0.750862\pi$
$353$	−9404.80	−1.41804	−0.709019	+	0.705190i	$0.750862\pi$
$354$	0	0
$355$	0	0
$356$	595.440	0.0886468
$357$	0	0
$358$	1066.98	0.157518
$359$	−914.674	−0.134470	−0.0672349	−	0.997737i	$-0.521418\pi$
$359$	−914.674	−0.134470	−0.0672349	+	0.997737i	$0.521418\pi$
$360$	0	0
$361$	497.942	0.0725969
$362$	2052.90	0.298061
$363$	0	0
$364$	−1153.40	−0.166084
$365$	0	0
$366$	0	0
$367$	208.708	0.0296852	0.0148426	−	0.999890i	$-0.495275\pi$
$367$	208.708	0.0296852	0.0148426	+	0.999890i	$0.495275\pi$
$368$	−2079.61	−0.294585
$369$	0	0
$370$	0	0
$371$	1014.63	0.141986
$372$	0	0
$373$	9381.11	1.30224	0.651120	−	0.758975i	$-0.274300\pi$
$373$	9381.11	1.30224	0.651120	+	0.758975i	$0.274300\pi$
$374$	−257.263	−0.0355689
$375$	0	0
$376$	3193.31	0.437985
$377$	−21669.0	−2.96024
$378$	0	0
$379$	−12833.6	−1.73936	−0.869678	−	0.493620i	$-0.835673\pi$
$379$	−12833.6	−1.73936	−0.869678	+	0.493620i	$0.835673\pi$
$380$	0	0
$381$	0	0
$382$	−7694.45	−1.03058
$383$	10330.9	1.37828	0.689141	−	0.724627i	$-0.257988\pi$
$383$	10330.9	1.37828	0.689141	+	0.724627i	$0.257988\pi$
$384$	0	0
$385$	0	0
$386$	8535.19	1.12547
$387$	0	0
$388$	535.101	0.0700145
$389$	14335.6	1.86850	0.934249	−	0.356621i	$-0.116071\pi$
$389$	14335.6	1.86850	0.934249	+	0.356621i	$0.116071\pi$
$390$	0	0
$391$	948.981	0.122742
$392$	−8004.57	−1.03136
$393$	0	0
$394$	6452.64	0.825074
$395$	0	0
$396$	0	0
$397$	1287.56	0.162773	0.0813867	−	0.996683i	$-0.474065\pi$
$397$	1287.56	0.162773	0.0813867	+	0.996683i	$0.474065\pi$
$398$	8983.87	1.13146
$399$	0	0
$400$	0	0
$401$	−3729.86	−0.464490	−0.232245	−	0.972657i	$-0.574607\pi$
$401$	−3729.86	−0.464490	−0.232245	+	0.972657i	$0.574607\pi$
$402$	0	0
$403$	−5595.32	−0.691620
$404$	−5787.43	−0.712711
$405$	0	0
$406$	2092.17	0.255745
$407$	1984.80	0.241727
$408$	0	0
$409$	−3237.70	−0.391427	−0.195714	−	0.980661i	$-0.562702\pi$
$409$	−3237.70	−0.391427	−0.195714	+	0.980661i	$0.562702\pi$
$410$	0	0
$411$	0	0
$412$	−482.609	−0.0577098
$413$	1822.00	0.217082
$414$	0	0
$415$	0	0
$416$	12311.2	1.45098
$417$	0	0
$418$	2006.63	0.234802
$419$	−9034.62	−1.05339	−0.526694	−	0.850055i	$-0.676569\pi$
$419$	−9034.62	−1.05339	−0.526694	+	0.850055i	$0.676569\pi$
$420$	0	0
$421$	−2978.78	−0.344838	−0.172419	−	0.985024i	$-0.555158\pi$
$421$	−2978.78	−0.344838	−0.172419	+	0.985024i	$0.555158\pi$
$422$	−7556.48	−0.871667
$423$	0	0
$424$	−6381.64	−0.730943
$425$	0	0
$426$	0	0
$427$	−1096.93	−0.124319
$428$	−6435.20	−0.726769
$429$	0	0
$430$	0	0
$431$	−2598.18	−0.290371	−0.145186	−	0.989404i	$-0.546378\pi$
$431$	−2598.18	−0.290371	−0.145186	+	0.989404i	$0.546378\pi$
$432$	0	0
$433$	11555.2	1.28246	0.641231	−	0.767348i	$-0.278424\pi$
$433$	11555.2	1.28246	0.641231	+	0.767348i	$0.278424\pi$
$434$	540.236	0.0597515
$435$	0	0
$436$	−6756.04	−0.742101
$437$	−7401.95	−0.810259
$438$	0	0
$439$	16419.7	1.78513	0.892564	−	0.450920i	$-0.148904\pi$
$439$	16419.7	1.78513	0.892564	+	0.450920i	$0.148904\pi$
$440$	0	0
$441$	0	0
$442$	1999.27	0.215148
$443$	−4861.44	−0.521386	−0.260693	−	0.965422i	$-0.583951\pi$
$443$	−4861.44	−0.521386	−0.260693	+	0.965422i	$0.583951\pi$
$444$	0	0
$445$	0	0
$446$	8844.30	0.938990
$447$	0	0
$448$	−1936.82	−0.204255
$449$	5551.79	0.583530	0.291765	−	0.956490i	$-0.405757\pi$
$449$	5551.79	0.583530	0.291765	+	0.956490i	$0.405757\pi$
$450$	0	0
$451$	−635.932	−0.0663966
$452$	−4432.39	−0.461244
$453$	0	0
$454$	4675.00	0.483279
$455$	0	0
$456$	0	0
$457$	991.538	0.101493	0.0507464	−	0.998712i	$-0.483840\pi$
$457$	991.538	0.101493	0.0507464	+	0.998712i	$0.483840\pi$
$458$	3097.92	0.316062
$459$	0	0
$460$	0	0
$461$	−2049.47	−0.207057	−0.103528	−	0.994626i	$-0.533013\pi$
$461$	−2049.47	−0.207057	−0.103528	+	0.994626i	$0.533013\pi$
$462$	0	0
$463$	3504.87	0.351804	0.175902	−	0.984408i	$-0.443716\pi$
$463$	3504.87	0.351804	0.175902	+	0.984408i	$0.443716\pi$
$464$	−6108.53	−0.611167
$465$	0	0
$466$	9506.71	0.945042
$467$	9599.12	0.951166	0.475583	−	0.879671i	$-0.342237\pi$
$467$	9599.12	0.951166	0.475583	+	0.879671i	$0.342237\pi$
$468$	0	0
$469$	98.9539	0.00974257
$470$	0	0
$471$	0	0
$472$	−11459.7	−1.11753
$473$	2224.62	0.216254
$474$	0	0
$475$	0	0
$476$	148.372	0.0142871
$477$	0	0
$478$	14138.3	1.35286
$479$	−18201.4	−1.73621	−0.868104	−	0.496382i	$-0.834661\pi$
$479$	−18201.4	−1.73621	−0.868104	+	0.496382i	$0.834661\pi$
$480$	0	0
$481$	−15424.4	−1.46215
$482$	−10399.4	−0.982737
$483$	0	0
$484$	−420.687	−0.0395085
$485$	0	0
$486$	0	0
$487$	−4205.87	−0.391348	−0.195674	−	0.980669i	$-0.562689\pi$
$487$	−4205.87	−0.391348	−0.195674	+	0.980669i	$0.562689\pi$
$488$	6899.29	0.639992
$489$	0	0
$490$	0	0
$491$	16430.7	1.51019	0.755097	−	0.655614i	$-0.227590\pi$
$491$	16430.7	1.51019	0.755097	+	0.655614i	$0.227590\pi$
$492$	0	0
$493$	2787.49	0.254649
$494$	−15594.1	−1.42027
$495$	0	0
$496$	−1577.33	−0.142791
$497$	−1816.51	−0.163947
$498$	0	0
$499$	−1454.87	−0.130519	−0.0652597	−	0.997868i	$-0.520788\pi$
$499$	−1454.87	−0.130519	−0.0652597	+	0.997868i	$0.520788\pi$
$500$	0	0
$501$	0	0
$502$	−3207.51	−0.285176
$503$	−14199.0	−1.25865	−0.629326	−	0.777141i	$-0.716669\pi$
$503$	−14199.0	−1.25865	−0.629326	+	0.777141i	$0.716669\pi$
$504$	0	0
$505$	0	0
$506$	−2018.90	−0.177374
$507$	0	0
$508$	−6742.00	−0.588834
$509$	−555.954	−0.0484130	−0.0242065	−	0.999707i	$-0.507706\pi$
$509$	−555.954	−0.0484130	−0.0242065	+	0.999707i	$0.507706\pi$
$510$	0	0
$511$	3262.84	0.282465
$512$	8176.20	0.705743
$513$	0	0
$514$	12083.6	1.03694
$515$	0	0
$516$	0	0
$517$	1439.10	0.122420
$518$	1489.25	0.126320
$519$	0	0
$520$	0	0
$521$	15651.5	1.31613	0.658064	−	0.752962i	$-0.271376\pi$
$521$	15651.5	1.31613	0.658064	+	0.752962i	$0.271376\pi$
$522$	0	0
$523$	−7860.01	−0.657159	−0.328580	−	0.944476i	$-0.606570\pi$
$523$	−7860.01	−0.657159	−0.328580	+	0.944476i	$0.606570\pi$
$524$	−833.916	−0.0695225
$525$	0	0
$526$	7038.94	0.583484
$527$	719.779	0.0594954
$528$	0	0
$529$	−4719.77	−0.387916
$530$	0	0
$531$	0	0
$532$	−1157.29	−0.0943135
$533$	4942.01	0.401618
$534$	0	0
$535$	0	0
$536$	−622.383	−0.0501546
$537$	0	0
$538$	1353.37	0.108454
$539$	−3607.33	−0.288273
$540$	0	0
$541$	−23426.5	−1.86171	−0.930854	−	0.365392i	$-0.880935\pi$
$541$	−23426.5	−1.86171	−0.930854	+	0.365392i	$0.880935\pi$
$542$	−1577.15	−0.124990
$543$	0	0
$544$	−1583.71	−0.124818
$545$	0	0
$546$	0	0
$547$	−8493.19	−0.663880	−0.331940	−	0.943300i	$-0.607703\pi$
$547$	−8493.19	−0.663880	−0.331940	+	0.943300i	$0.607703\pi$
$548$	6551.18	0.510680
$549$	0	0
$550$	0	0
$551$	−21742.1	−1.68102
$552$	0	0
$553$	1848.49	0.142144
$554$	−3228.33	−0.247579
$555$	0	0
$556$	−3000.32	−0.228852
$557$	−10057.4	−0.765072	−0.382536	−	0.923941i	$-0.624949\pi$
$557$	−10057.4	−0.765072	−0.382536	+	0.923941i	$0.624949\pi$
$558$	0	0
$559$	−17288.2	−1.30807
$560$	0	0
$561$	0	0
$562$	−5821.80	−0.436971
$563$	−18854.9	−1.41144	−0.705720	−	0.708491i	$-0.749376\pi$
$563$	−18854.9	−1.41144	−0.705720	+	0.708491i	$0.749376\pi$
$564$	0	0
$565$	0	0
$566$	−8926.37	−0.662903
$567$	0	0
$568$	11425.2	0.843998
$569$	−17550.8	−1.29309	−0.646545	−	0.762876i	$-0.723787\pi$
$569$	−17550.8	−1.29309	−0.646545	+	0.762876i	$0.723787\pi$
$570$	0	0
$571$	−14843.0	−1.08785	−0.543924	−	0.839135i	$-0.683062\pi$
$571$	−14843.0	−1.08785	−0.543924	+	0.839135i	$0.683062\pi$
$572$	3269.28	0.238978
$573$	0	0
$574$	−477.158	−0.0346972
$575$	0	0
$576$	0	0
$577$	−10410.8	−0.751137	−0.375568	−	0.926795i	$-0.622552\pi$
$577$	−10410.8	−0.751137	−0.375568	+	0.926795i	$0.622552\pi$
$578$	10191.7	0.733427
$579$	0	0
$580$	0	0
$581$	3939.27	0.281288
$582$	0	0
$583$	−2875.94	−0.204304
$584$	−20522.0	−1.45412
$585$	0	0
$586$	−2952.70	−0.208148
$587$	6592.84	0.463570	0.231785	−	0.972767i	$-0.425543\pi$
$587$	6592.84	0.463570	0.231785	+	0.972767i	$0.425543\pi$
$588$	0	0
$589$	−5614.20	−0.392749
$590$	0	0
$591$	0	0
$592$	−4348.18	−0.301874
$593$	−12467.3	−0.863354	−0.431677	−	0.902028i	$-0.642078\pi$
$593$	−12467.3	−0.863354	−0.431677	+	0.902028i	$0.642078\pi$
$594$	0	0
$595$	0	0
$596$	−6381.23	−0.438566
$597$	0	0
$598$	15689.5	1.07289
$599$	7470.81	0.509598	0.254799	−	0.966994i	$-0.417991\pi$
$599$	7470.81	0.509598	0.254799	+	0.966994i	$0.417991\pi$
$600$	0	0
$601$	13892.6	0.942916	0.471458	−	0.881888i	$-0.343728\pi$
$601$	13892.6	0.942916	0.471458	+	0.881888i	$0.343728\pi$
$602$	1669.20	0.113009
$603$	0	0
$604$	−8788.70	−0.592065
$605$	0	0
$606$	0	0
$607$	−2950.93	−0.197322	−0.0986610	−	0.995121i	$-0.531456\pi$
$607$	−2950.93	−0.197322	−0.0986610	+	0.995121i	$0.531456\pi$
$608$	12352.8	0.823965
$609$	0	0
$610$	0	0
$611$	−11183.6	−0.740494
$612$	0	0
$613$	−18514.3	−1.21988	−0.609940	−	0.792448i	$-0.708806\pi$
$613$	−18514.3	−1.21988	−0.609940	+	0.792448i	$0.708806\pi$
$614$	10810.6	0.710557
$615$	0	0
$616$	−1041.97	−0.0681530
$617$	15254.7	0.995350	0.497675	−	0.867364i	$-0.334187\pi$
$617$	15254.7	0.995350	0.497675	+	0.867364i	$0.334187\pi$
$618$	0	0
$619$	−18333.5	−1.19044	−0.595222	−	0.803561i	$-0.702936\pi$
$619$	−18333.5	−1.19044	−0.595222	+	0.803561i	$0.702936\pi$
$620$	0	0
$621$	0	0
$622$	3765.81	0.242758
$623$	664.636	0.0427417
$624$	0	0
$625$	0	0
$626$	17753.7	1.13351
$627$	0	0
$628$	−3289.63	−0.209030
$629$	1984.19	0.125779
$630$	0	0
$631$	−14697.9	−0.927278	−0.463639	−	0.886024i	$-0.653457\pi$
$631$	−14697.9	−0.927278	−0.463639	+	0.886024i	$0.653457\pi$
$632$	−11626.3	−0.731755
$633$	0	0
$634$	117.340	0.00735045
$635$	0	0
$636$	0	0
$637$	28033.7	1.74370
$638$	−5930.21	−0.367993
$639$	0	0
$640$	0	0
$641$	10617.2	0.654221	0.327110	−	0.944986i	$-0.393925\pi$
$641$	10617.2	0.654221	0.327110	+	0.944986i	$0.393925\pi$
$642$	0	0
$643$	2668.41	0.163657	0.0818287	−	0.996646i	$-0.473924\pi$
$643$	2668.41	0.163657	0.0818287	+	0.996646i	$0.473924\pi$
$644$	1164.37	0.0712461
$645$	0	0
$646$	2006.01	0.122176
$647$	−13366.2	−0.812180	−0.406090	−	0.913833i	$-0.633108\pi$
$647$	−13366.2	−0.812180	−0.406090	+	0.913833i	$0.633108\pi$
$648$	0	0
$649$	−5164.43	−0.312360
$650$	0	0
$651$	0	0
$652$	5770.97	0.346639
$653$	−9762.08	−0.585022	−0.292511	−	0.956262i	$-0.594491\pi$
$653$	−9762.08	−0.585022	−0.292511	+	0.956262i	$0.594491\pi$
$654$	0	0
$655$	0	0
$656$	1393.16	0.0829176
$657$	0	0
$658$	1079.80	0.0639739
$659$	16304.9	0.963808	0.481904	−	0.876224i	$-0.339945\pi$
$659$	16304.9	0.963808	0.481904	+	0.876224i	$0.339945\pi$
$660$	0	0
$661$	−3424.69	−0.201520	−0.100760	−	0.994911i	$-0.532127\pi$
$661$	−3424.69	−0.201520	−0.100760	+	0.994911i	$0.532127\pi$
$662$	15586.9	0.915107
$663$	0	0
$664$	−24776.5	−1.44807
$665$	0	0
$666$	0	0
$667$	21875.1	1.26988
$668$	11391.0	0.659778
$669$	0	0
$670$	0	0
$671$	3109.23	0.178883
$672$	0	0
$673$	19841.6	1.13646	0.568229	−	0.822871i	$-0.307629\pi$
$673$	19841.6	1.13646	0.568229	+	0.822871i	$0.307629\pi$
$674$	3743.17	0.213919
$675$	0	0
$676$	−17768.1	−1.01093
$677$	−17070.8	−0.969106	−0.484553	−	0.874762i	$-0.661018\pi$
$677$	−17070.8	−0.969106	−0.484553	+	0.874762i	$0.661018\pi$
$678$	0	0
$679$	597.284	0.0337580
$680$	0	0
$681$	0	0
$682$	−1531.29	−0.0859766
$683$	−8354.04	−0.468021	−0.234011	−	0.972234i	$-0.575185\pi$
$683$	−8354.04	−0.468021	−0.234011	+	0.972234i	$0.575185\pi$
$684$	0	0
$685$	0	0
$686$	−5537.68	−0.308206
$687$	0	0
$688$	−4873.58	−0.270063
$689$	22349.8	1.23579
$690$	0	0
$691$	−12032.4	−0.662423	−0.331211	−	0.943557i	$-0.607457\pi$
$691$	−12032.4	−0.662423	−0.331211	+	0.943557i	$0.607457\pi$
$692$	−1293.19	−0.0710402
$693$	0	0
$694$	23243.1	1.27132
$695$	0	0
$696$	0	0
$697$	−635.738	−0.0345485
$698$	4419.67	0.239666
$699$	0	0
$700$	0	0
$701$	17125.3	0.922704	0.461352	−	0.887217i	$-0.347365\pi$
$701$	17125.3	0.922704	0.461352	+	0.887217i	$0.347365\pi$
$702$	0	0
$703$	−15476.5	−0.830308
$704$	5489.89	0.293903
$705$	0	0
$706$	20002.1	1.06627
$707$	−6459.98	−0.343638
$708$	0	0
$709$	5672.11	0.300452	0.150226	−	0.988652i	$-0.452000\pi$
$709$	5672.11	0.300452	0.150226	+	0.988652i	$0.452000\pi$
$710$	0	0
$711$	0	0
$712$	−4180.31	−0.220033
$713$	5648.54	0.296689
$714$	0	0
$715$	0	0
$716$	1744.23	0.0910404
$717$	0	0
$718$	1945.32	0.101113
$719$	13811.8	0.716404	0.358202	−	0.933644i	$-0.383390\pi$
$719$	13811.8	0.716404	0.358202	+	0.933644i	$0.383390\pi$
$720$	0	0
$721$	−538.693	−0.0278252
$722$	−1059.02	−0.0545882
$723$	0	0
$724$	3355.96	0.172270
$725$	0	0
$726$	0	0
$727$	−15346.0	−0.782877	−0.391439	−	0.920204i	$-0.628023\pi$
$727$	−15346.0	−0.782877	−0.391439	+	0.920204i	$0.628023\pi$
$728$	8097.47	0.412242
$729$	0	0
$730$	0	0
$731$	2223.94	0.112525
$732$	0	0
$733$	318.563	0.0160524	0.00802618	−	0.999968i	$-0.497445\pi$
$733$	318.563	0.0160524	0.00802618	+	0.999968i	$0.497445\pi$
$734$	−443.878	−0.0223213
$735$	0	0
$736$	−12428.3	−0.622438
$737$	−280.483	−0.0140186
$738$	0	0
$739$	−2124.08	−0.105731	−0.0528656	−	0.998602i	$-0.516836\pi$
$739$	−2124.08	−0.105731	−0.0528656	+	0.998602i	$0.516836\pi$
$740$	0	0
$741$	0	0
$742$	−2157.90	−0.106764
$743$	−28988.6	−1.43134	−0.715672	−	0.698436i	$-0.753880\pi$
$743$	−28988.6	−1.43134	−0.715672	+	0.698436i	$0.753880\pi$
$744$	0	0
$745$	0	0
$746$	−19951.7	−0.979199
$747$	0	0
$748$	−420.559	−0.0205577
$749$	−7183.03	−0.350417
$750$	0	0
$751$	−96.6619	−0.00469673	−0.00234837	−	0.999997i	$-0.500748\pi$
$751$	−96.6619	−0.00469673	−0.00234837	+	0.999997i	$0.500748\pi$
$752$	−3152.69	−0.152881
$753$	0	0
$754$	46085.4	2.22590
$755$	0	0
$756$	0	0
$757$	1980.85	0.0951060	0.0475530	−	0.998869i	$-0.484858\pi$
$757$	1980.85	0.0951060	0.0475530	+	0.998869i	$0.484858\pi$
$758$	27294.3	1.30788
$759$	0	0
$760$	0	0
$761$	12309.4	0.586356	0.293178	−	0.956058i	$-0.405287\pi$
$761$	12309.4	0.586356	0.293178	+	0.956058i	$0.405287\pi$
$762$	0	0
$763$	−7541.16	−0.357809
$764$	−12578.4	−0.595643
$765$	0	0
$766$	−21971.6	−1.03638
$767$	40134.3	1.88939
$768$	0	0
$769$	−27723.1	−1.30003	−0.650014	−	0.759922i	$-0.725237\pi$
$769$	−27723.1	−1.30003	−0.650014	+	0.759922i	$0.725237\pi$
$770$	0	0
$771$	0	0
$772$	13952.8	0.650483
$773$	24457.0	1.13798	0.568988	−	0.822346i	$-0.307335\pi$
$773$	24457.0	1.13798	0.568988	+	0.822346i	$0.307335\pi$
$774$	0	0
$775$	0	0
$776$	−3756.70	−0.173786
$777$	0	0
$778$	−30488.9	−1.40499
$779$	4958.69	0.228066
$780$	0	0
$781$	5148.88	0.235904
$782$	−2018.29	−0.0922938
$783$	0	0
$784$	7902.75	0.360001
$785$	0	0
$786$	0	0
$787$	−19137.6	−0.866811	−0.433405	−	0.901199i	$-0.642688\pi$
$787$	−19137.6	−0.866811	−0.433405	+	0.901199i	$0.642688\pi$
$788$	10548.4	0.476866
$789$	0	0
$790$	0	0
$791$	−4947.48	−0.222392
$792$	0	0
$793$	−24162.7	−1.08202
$794$	−2738.38	−0.122395
$795$	0	0
$796$	14686.3	0.653947
$797$	−26916.9	−1.19629	−0.598147	−	0.801387i	$-0.704096\pi$
$797$	−26916.9	−1.19629	−0.598147	+	0.801387i	$0.704096\pi$
$798$	0	0
$799$	1438.66	0.0636997
$800$	0	0
$801$	0	0
$802$	7932.64	0.349266
$803$	−9248.45	−0.406439
$804$	0	0
$805$	0	0
$806$	11900.1	0.520053
$807$	0	0
$808$	40630.9	1.76905
$809$	39344.3	1.70986	0.854928	−	0.518747i	$-0.173601\pi$
$809$	39344.3	1.70986	0.854928	+	0.518747i	$0.173601\pi$
$810$	0	0
$811$	−7219.28	−0.312581	−0.156290	−	0.987711i	$-0.549954\pi$
$811$	−7219.28	−0.312581	−0.156290	+	0.987711i	$0.549954\pi$
$812$	3420.15	0.147813
$813$	0	0
$814$	−4221.25	−0.181763
$815$	0	0
$816$	0	0
$817$	−17346.5	−0.742812
$818$	6885.91	0.294328
$819$	0	0
$820$	0	0
$821$	−29293.1	−1.24523	−0.622616	−	0.782528i	$-0.713930\pi$
$821$	−29293.1	−1.24523	−0.622616	+	0.782528i	$0.713930\pi$
$822$	0	0
$823$	19271.1	0.816218	0.408109	−	0.912933i	$-0.366188\pi$
$823$	19271.1	0.816218	0.408109	+	0.912933i	$0.366188\pi$
$824$	3388.18	0.143244
$825$	0	0
$826$	−3875.02	−0.163232
$827$	−17490.2	−0.735424	−0.367712	−	0.929940i	$-0.619859\pi$
$827$	−17490.2	−0.735424	−0.367712	+	0.929940i	$0.619859\pi$
$828$	0	0
$829$	−33612.8	−1.40823	−0.704113	−	0.710088i	$-0.748655\pi$
$829$	−33612.8	−1.40823	−0.704113	+	0.710088i	$0.748655\pi$
$830$	0	0
$831$	0	0
$832$	−42663.5	−1.77776
$833$	−3606.23	−0.149998
$834$	0	0
$835$	0	0
$836$	3280.31	0.135708
$837$	0	0
$838$	19214.8	0.792080
$839$	2111.96	0.0869047	0.0434523	−	0.999056i	$-0.486164\pi$
$839$	2111.96	0.0869047	0.0434523	+	0.999056i	$0.486164\pi$
$840$	0	0
$841$	39865.7	1.63458
$842$	6335.25	0.259296
$843$	0	0
$844$	−12352.9	−0.503795
$845$	0	0
$846$	0	0
$847$	−469.575	−0.0190493
$848$	6300.46	0.255140
$849$	0	0
$850$	0	0
$851$	15571.1	0.627229
$852$	0	0
$853$	18992.4	0.762355	0.381178	−	0.924502i	$-0.375519\pi$
$853$	18992.4	0.762355	0.381178	+	0.924502i	$0.375519\pi$
$854$	2332.95	0.0934798
$855$	0	0
$856$	45178.6	1.80394
$857$	45968.8	1.83228	0.916139	−	0.400860i	$-0.131289\pi$
$857$	45968.8	1.83228	0.916139	+	0.400860i	$0.131289\pi$
$858$	0	0
$859$	20490.0	0.813865	0.406932	−	0.913458i	$-0.366598\pi$
$859$	20490.0	0.813865	0.406932	+	0.913458i	$0.366598\pi$
$860$	0	0
$861$	0	0
$862$	5525.79	0.218340
$863$	15810.3	0.623624	0.311812	−	0.950144i	$-0.399064\pi$
$863$	15810.3	0.623624	0.311812	+	0.950144i	$0.399064\pi$
$864$	0	0
$865$	0	0
$866$	−24575.5	−0.964328
$867$	0	0
$868$	883.145	0.0345344
$869$	−5239.50	−0.204531
$870$	0	0
$871$	2179.71	0.0847954
$872$	47431.1	1.84199
$873$	0	0
$874$	15742.4	0.609262
$875$	0	0
$876$	0	0
$877$	9390.46	0.361566	0.180783	−	0.983523i	$-0.442137\pi$
$877$	9390.46	0.361566	0.180783	+	0.983523i	$0.442137\pi$
$878$	−34921.4	−1.34230
$879$	0	0
$880$	0	0
$881$	−41165.1	−1.57422	−0.787110	−	0.616813i	$-0.788423\pi$
$881$	−41165.1	−1.57422	−0.787110	+	0.616813i	$0.788423\pi$
$882$	0	0
$883$	10729.3	0.408912	0.204456	−	0.978876i	$-0.434457\pi$
$883$	10729.3	0.408912	0.204456	+	0.978876i	$0.434457\pi$
$884$	3268.28	0.124349
$885$	0	0
$886$	10339.3	0.392048
$887$	33673.5	1.27468	0.637342	−	0.770581i	$-0.280034\pi$
$887$	33673.5	1.27468	0.637342	+	0.770581i	$0.280034\pi$
$888$	0	0
$889$	−7525.48	−0.283910
$890$	0	0
$891$	0	0
$892$	14458.1	0.542706
$893$	−11221.4	−0.420503
$894$	0	0
$895$	0	0
$896$	−351.981	−0.0131237
$897$	0	0
$898$	−11807.5	−0.438777
$899$	16591.7	0.615534
$900$	0	0
$901$	−2875.07	−0.106307
$902$	1352.50	0.0499259
$903$	0	0
$904$	31117.8	1.14487
$905$	0	0
$906$	0	0
$907$	−12927.4	−0.473260	−0.236630	−	0.971600i	$-0.576043\pi$
$907$	−12927.4	−0.473260	−0.236630	+	0.971600i	$0.576043\pi$
$908$	7642.41	0.279320
$909$	0	0
$910$	0	0
$911$	−13572.8	−0.493619	−0.246810	−	0.969064i	$-0.579382\pi$
$911$	−13572.8	−0.493619	−0.246810	+	0.969064i	$0.579382\pi$
$912$	0	0
$913$	−11165.8	−0.404746
$914$	−2108.80	−0.0763160
$915$	0	0
$916$	5064.30	0.182674
$917$	−930.825	−0.0335208
$918$	0	0
$919$	17600.3	0.631751	0.315876	−	0.948801i	$-0.397702\pi$
$919$	17600.3	0.631751	0.315876	+	0.948801i	$0.397702\pi$
$920$	0	0
$921$	0	0
$922$	4358.79	0.155693
$923$	−40013.4	−1.42693
$924$	0	0
$925$	0	0
$926$	−7454.14	−0.264534
$927$	0	0
$928$	−36506.3	−1.29136
$929$	−43159.1	−1.52422	−0.762112	−	0.647445i	$-0.775837\pi$
$929$	−43159.1	−1.52422	−0.762112	+	0.647445i	$0.775837\pi$
$930$	0	0
$931$	28128.2	0.990189
$932$	15541.0	0.546204
$933$	0	0
$934$	−20415.3	−0.715215
$935$	0	0
$936$	0	0
$937$	4899.29	0.170814	0.0854071	−	0.996346i	$-0.472781\pi$
$937$	4899.29	0.170814	0.0854071	+	0.996346i	$0.472781\pi$
$938$	−210.454	−0.00732578
$939$	0	0
$940$	0	0
$941$	−53441.5	−1.85137	−0.925687	−	0.378290i	$-0.876512\pi$
$941$	−53441.5	−1.85137	−0.925687	+	0.378290i	$0.876512\pi$
$942$	0	0
$943$	−4989.02	−0.172285
$944$	11313.9	0.390082
$945$	0	0
$946$	−4731.31	−0.162609
$947$	−23650.7	−0.811556	−0.405778	−	0.913972i	$-0.632999\pi$
$947$	−23650.7	−0.811556	−0.405778	+	0.913972i	$0.632999\pi$
$948$	0	0
$949$	71872.4	2.45846
$950$	0	0
$951$	0	0
$952$	−1041.65	−0.0354624
$953$	−50070.7	−1.70194	−0.850970	−	0.525215i	$-0.823985\pi$
$953$	−50070.7	−1.70194	−0.850970	+	0.525215i	$0.823985\pi$
$954$	0	0
$955$	0	0
$956$	23112.4	0.781912
$957$	0	0
$958$	38710.6	1.30552
$959$	7312.49	0.246228
$960$	0	0
$961$	−25506.7	−0.856189
$962$	32804.6	1.09944
$963$	0	0
$964$	−17000.3	−0.567990
$965$	0	0
$966$	0	0
$967$	59560.1	1.98068	0.990342	−	0.138643i	$-0.0442742\pi$
$967$	59560.1	1.98068	0.990342	+	0.138643i	$0.0442742\pi$
$968$	2953.45	0.0980656
$969$	0	0
$970$	0	0
$971$	−57479.7	−1.89970	−0.949851	−	0.312702i	$-0.898766\pi$
$971$	−57479.7	−1.89970	−0.949851	+	0.312702i	$0.898766\pi$
$972$	0	0
$973$	−3348.98	−0.110343
$974$	8945.02	0.294268
$975$	0	0
$976$	−6811.53	−0.223393
$977$	29720.5	0.973227	0.486613	−	0.873617i	$-0.338232\pi$
$977$	29720.5	0.973227	0.486613	+	0.873617i	$0.338232\pi$
$978$	0	0
$979$	−1883.90	−0.0615011
$980$	0	0
$981$	0	0
$982$	−34944.6	−1.13557
$983$	9713.31	0.315164	0.157582	−	0.987506i	$-0.449630\pi$
$983$	9713.31	0.315164	0.157582	+	0.987506i	$0.449630\pi$
$984$	0	0
$985$	0	0
$986$	−5928.41	−0.191480
$987$	0	0
$988$	−25492.3	−0.820867
$989$	17452.6	0.561134
$990$	0	0
$991$	16225.9	0.520112	0.260056	−	0.965594i	$-0.416259\pi$
$991$	16225.9	0.520112	0.260056	+	0.965594i	$0.416259\pi$
$992$	−9426.59	−0.301708
$993$	0	0
$994$	3863.35	0.123278
$995$	0	0
$996$	0	0
$997$	50587.7	1.60695	0.803475	−	0.595338i	$-0.202982\pi$
$997$	50587.7	1.60695	0.803475	+	0.595338i	$0.202982\pi$
$998$	3094.22	0.0981420
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.4.a.bu.1.4	✓	10
3.2	odd	2		2475.4.a.bx.1.7	yes	10
5.4	even	2		2475.4.a.by.1.7	yes	10
15.14	odd	2		2475.4.a.bv.1.4	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2475.4.a.bu.1.4	✓	10	1.1	even	1	trivial
2475.4.a.bv.1.4	yes	10	15.14	odd	2
2475.4.a.bx.1.7	yes	10	3.2	odd	2
2475.4.a.by.1.7	yes	10	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:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2475.a (trivial)

\(f(q)\)	\(=\)	\(q-2.12679 q^{2} -3.47675 q^{4} -3.88078 q^{7} +24.4087 q^{8} +O(q^{10})\)	\(q-2.12679 q^{2} -3.47675 q^{4} -3.88078 q^{7} +24.4087 q^{8} +11.0000 q^{11} -85.4842 q^{13} +8.25362 q^{14} -24.0982 q^{16} +10.9966 q^{17} -85.7726 q^{19} -23.3947 q^{22} +86.2973 q^{23} +181.807 q^{26} +13.4925 q^{28} +253.485 q^{29} +65.4544 q^{31} -144.018 q^{32} -23.3876 q^{34} +180.436 q^{37} +182.421 q^{38} -57.8120 q^{41} +202.238 q^{43} -38.2443 q^{44} -183.537 q^{46} +130.827 q^{47} -327.940 q^{49} +297.207 q^{52} -261.450 q^{53} -94.7247 q^{56} -539.110 q^{58} -469.494 q^{59} +282.657 q^{61} -139.208 q^{62} +499.081 q^{64} -25.4984 q^{67} -38.2326 q^{68} +468.080 q^{71} -840.768 q^{73} -383.750 q^{74} +298.210 q^{76} -42.6886 q^{77} -476.318 q^{79} +122.954 q^{82} -1015.07 q^{83} -430.119 q^{86} +268.495 q^{88} -171.263 q^{89} +331.746 q^{91} -300.034 q^{92} -278.242 q^{94} -153.908 q^{97} +697.460 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 4 q^{2} + 34 q^{4} - 2 q^{7} - 48 q^{8}+O(q^{10}) \) 10 * q - 4 * q^2 + 34 * q^4 - 2 * q^7 - 48 * q^8	\( 10 q - 4 q^{2} + 34 q^{4} - 2 q^{7} - 48 q^{8} + 110 q^{11} - 26 q^{13} - 72 q^{14} + 206 q^{16} - 148 q^{17} - 114 q^{19} - 44 q^{22} + 34 q^{23} - 100 q^{26} + 86 q^{28} + 38 q^{29} + 232 q^{31} - 448 q^{32} - 20 q^{34} - 754 q^{37} - 780 q^{38} - 160 q^{41} + 66 q^{43} + 374 q^{44} + 682 q^{46} - 450 q^{47} + 590 q^{49} - 200 q^{52} - 1068 q^{53} - 268 q^{56} + 138 q^{58} + 838 q^{59} - 566 q^{61} - 1230 q^{62} + 462 q^{64} - 430 q^{67} - 2234 q^{68} - 518 q^{71} + 184 q^{73} - 402 q^{74} + 386 q^{76} - 22 q^{77} + 956 q^{79} + 2180 q^{82} - 2094 q^{83} - 892 q^{86} - 528 q^{88} + 512 q^{89} - 858 q^{91} - 4476 q^{92} - 294 q^{94} + 1006 q^{97} - 2226 q^{98}+O(q^{100}) \) 10 * q - 4 * q^2 + 34 * q^4 - 2 * q^7 - 48 * q^8 + 110 * q^11 - 26 * q^13 - 72 * q^14 + 206 * q^16 - 148 * q^17 - 114 * q^19 - 44 * q^22 + 34 * q^23 - 100 * q^26 + 86 * q^28 + 38 * q^29 + 232 * q^31 - 448 * q^32 - 20 * q^34 - 754 * q^37 - 780 * q^38 - 160 * q^41 + 66 * q^43 + 374 * q^44 + 682 * q^46 - 450 * q^47 + 590 * q^49 - 200 * q^52 - 1068 * q^53 - 268 * q^56 + 138 * q^58 + 838 * q^59 - 566 * q^61 - 1230 * q^62 + 462 * q^64 - 430 * q^67 - 2234 * q^68 - 518 * q^71 + 184 * q^73 - 402 * q^74 + 386 * q^76 - 22 * q^77 + 956 * q^79 + 2180 * q^82 - 2094 * q^83 - 892 * q^86 - 528 * q^88 + 512 * q^89 - 858 * q^91 - 4476 * q^92 - 294 * q^94 + 1006 * q^97 - 2226 * q^98

Introduction

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