LMFDB - Embedded newform 1225.4.a.bl.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1225,4,Mod(1,1225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1225.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1225 = 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1225.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:	$72.2773397570$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 45x^{6} + 26x^{5} + 566x^{4} - 137x^{3} - 2154x^{2} + 376x + 1224 $ x^8 - x^7 - 45x^6 + 26x^5 + 566x^4 - 137x^3 - 2154x^2 + 376x + 1224
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 175)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-2.92095$ of defining polynomial
Character	$\chi$	$=$	1225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.92095 q^{2} +10.0483 q^{3} +0.531934 q^{4} +29.3505 q^{6} -21.8138 q^{8} +73.9678 q^{9} +15.5936 q^{11} +5.34501 q^{12} +63.9123 q^{13} -67.9725 q^{16} -10.5777 q^{17} +216.056 q^{18} -38.7727 q^{19} +45.5481 q^{22} +24.1227 q^{23} -219.191 q^{24} +186.684 q^{26} +471.945 q^{27} +226.731 q^{29} -143.130 q^{31} -24.0335 q^{32} +156.689 q^{33} -30.8968 q^{34} +39.3460 q^{36} +285.253 q^{37} -113.253 q^{38} +642.208 q^{39} +32.0457 q^{41} +235.711 q^{43} +8.29477 q^{44} +70.4612 q^{46} -144.296 q^{47} -683.006 q^{48} -106.287 q^{51} +33.9971 q^{52} -318.581 q^{53} +1378.53 q^{54} -389.599 q^{57} +662.270 q^{58} +448.322 q^{59} +166.695 q^{61} -418.076 q^{62} +473.580 q^{64} +457.680 q^{66} -741.042 q^{67} -5.62662 q^{68} +242.392 q^{69} -373.652 q^{71} -1613.52 q^{72} -237.314 q^{73} +833.209 q^{74} -20.6245 q^{76} +1875.86 q^{78} -465.390 q^{79} +2745.10 q^{81} +93.6037 q^{82} -691.289 q^{83} +688.498 q^{86} +2278.26 q^{87} -340.156 q^{88} +1013.18 q^{89} +12.8317 q^{92} -1438.21 q^{93} -421.482 q^{94} -241.495 q^{96} +1151.67 q^{97} +1153.42 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - q^{2} + 6 q^{3} + 27 q^{4} + 4 q^{6} - 42 q^{8} + 42 q^{9} + 103 q^{12} + 138 q^{13} + 191 q^{16} + 138 q^{17} + 81 q^{18} - 42 q^{19} + 127 q^{22} + 124 q^{23} - 292 q^{24} + 23 q^{26} + 204 q^{27}+ \cdots + 4086 q^{99}+O(q^{100}) $ 8 * q - q^2 + 6 * q^3 + 27 * q^4 + 4 * q^6 - 42 * q^8 + 42 * q^9 + 103 * q^12 + 138 * q^13 + 191 * q^16 + 138 * q^17 + 81 * q^18 - 42 * q^19 + 127 * q^22 + 124 * q^23 - 292 * q^24 + 23 * q^26 + 204 * q^27 + 288 * q^29 + 18 * q^31 - 562 * q^32 + 236 * q^33 - 488 * q^34 - 711 * q^36 + 220 * q^37 + 1104 * q^38 + 22 * q^39 + 168 * q^41 + 60 * q^43 - 880 * q^44 - 98 * q^46 + 1098 * q^47 + 338 * q^48 + 628 * q^51 + 1370 * q^52 - 734 * q^53 + 2163 * q^54 + 914 * q^57 + 924 * q^58 + 558 * q^59 - 1496 * q^61 + 801 * q^62 + 2436 * q^64 - 682 * q^66 - 1166 * q^67 + 2455 * q^68 + 822 * q^69 - 724 * q^71 - 2191 * q^72 + 892 * q^73 + 235 * q^74 - 1509 * q^76 + 2122 * q^78 - 840 * q^79 + 2392 * q^81 + 4154 * q^82 + 1146 * q^83 + 1816 * q^86 + 4286 * q^87 - 788 * q^88 + 2678 * q^89 + 6932 * q^92 - 1986 * q^93 + 3625 * q^94 - 6601 * q^96 + 664 * q^97 + 4086 * q^99

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 $

$2$	2.92095	1.03271	0.516355	−	0.856374i	$-0.327288\pi$
$2$	2.92095	1.03271	0.516355	+	0.856374i	$0.327288\pi$
$3$	10.0483	1.93379	0.966895	−	0.255173i	$-0.0821324\pi$
$3$	10.0483	1.93379	0.966895	+	0.255173i	$0.0821324\pi$
$4$	0.531934	0.0664917
$5$	0	0
$5$	0	0
$6$	29.3505	1.99705
$7$	0	0
$7$	0	0
$8$	−21.8138	−0.964044
$9$	73.9678	2.73955
$10$	0	0
$11$	15.5936	0.427423	0.213711	−	0.976897i	$-0.431445\pi$
$11$	15.5936	0.427423	0.213711	+	0.976897i	$0.431445\pi$
$12$	5.34501	0.128581
$13$	63.9123	1.36354	0.681772	−	0.731565i	$-0.261210\pi$
$13$	63.9123	1.36354	0.681772	+	0.731565i	$0.261210\pi$
$14$	0	0
$15$	0	0
$16$	−67.9725	−1.06207
$17$	−10.5777	−0.150909	−0.0754547	−	0.997149i	$-0.524041\pi$
$17$	−10.5777	−0.150909	−0.0754547	+	0.997149i	$0.524041\pi$
$18$	216.056	2.82916
$19$	−38.7727	−0.468162	−0.234081	−	0.972217i	$-0.575208\pi$
$19$	−38.7727	−0.468162	−0.234081	+	0.972217i	$0.575208\pi$
$20$	0	0
$21$	0	0
$22$	45.5481	0.441404
$23$	24.1227	0.218693	0.109346	−	0.994004i	$-0.465124\pi$
$23$	24.1227	0.218693	0.109346	+	0.994004i	$0.465124\pi$
$24$	−219.191	−1.86426
$25$	0	0
$26$	186.684	1.40815
$27$	471.945	3.36392
$28$	0	0
$29$	226.731	1.45182	0.725912	−	0.687787i	$-0.241418\pi$
$29$	226.731	1.45182	0.725912	+	0.687787i	$0.241418\pi$
$30$	0	0
$31$	−143.130	−0.829256	−0.414628	−	0.909991i	$-0.636088\pi$
$31$	−143.130	−0.829256	−0.414628	+	0.909991i	$0.636088\pi$
$32$	−24.0335	−0.132768
$33$	156.689	0.826546
$34$	−30.8968	−0.155846
$35$	0	0
$36$	39.3460	0.182157
$37$	285.253	1.26744	0.633720	−	0.773562i	$-0.281527\pi$
$37$	285.253	1.26744	0.633720	+	0.773562i	$0.281527\pi$
$38$	−113.253	−0.483476
$39$	642.208	2.63681
$40$	0	0
$41$	32.0457	0.122066	0.0610328	−	0.998136i	$-0.480561\pi$
$41$	32.0457	0.122066	0.0610328	+	0.998136i	$0.480561\pi$
$42$	0	0
$43$	235.711	0.835942	0.417971	−	0.908460i	$-0.362741\pi$
$43$	235.711	0.835942	0.417971	+	0.908460i	$0.362741\pi$
$44$	8.29477	0.0284201
$45$	0	0
$46$	70.4612	0.225846
$47$	−144.296	−0.447825	−0.223912	−	0.974609i	$-0.571883\pi$
$47$	−144.296	−0.447825	−0.223912	+	0.974609i	$0.571883\pi$
$48$	−683.006	−2.05382
$49$	0	0
$50$	0	0
$51$	−106.287	−0.291827
$52$	33.9971	0.0906644
$53$	−318.581	−0.825670	−0.412835	−	0.910806i	$-0.635461\pi$
$53$	−318.581	−0.825670	−0.412835	+	0.910806i	$0.635461\pi$
$54$	1378.53	3.47396
$55$	0	0
$56$	0	0
$57$	−389.599	−0.905327
$58$	662.270	1.49931
$59$	448.322	0.989263	0.494632	−	0.869103i	$-0.335303\pi$
$59$	448.322	0.989263	0.494632	+	0.869103i	$0.335303\pi$
$60$	0	0
$61$	166.695	0.349886	0.174943	−	0.984579i	$-0.444026\pi$
$61$	166.695	0.349886	0.174943	+	0.984579i	$0.444026\pi$
$62$	−418.076	−0.856382
$63$	0	0
$64$	473.580	0.924960
$65$	0	0
$66$	457.680	0.853583
$67$	−741.042	−1.35123	−0.675617	−	0.737253i	$-0.736123\pi$
$67$	−741.042	−1.35123	−0.675617	+	0.737253i	$0.736123\pi$
$68$	−5.62662	−0.0100342
$69$	242.392	0.422906
$70$	0	0
$71$	−373.652	−0.624567	−0.312284	−	0.949989i	$-0.601094\pi$
$71$	−373.652	−0.624567	−0.312284	+	0.949989i	$0.601094\pi$
$72$	−1613.52	−2.64104
$73$	−237.314	−0.380486	−0.190243	−	0.981737i	$-0.560928\pi$
$73$	−237.314	−0.380486	−0.190243	+	0.981737i	$0.560928\pi$
$74$	833.209	1.30890
$75$	0	0
$76$	−20.6245	−0.0311289
$77$	0	0
$78$	1875.86	2.72306
$79$	−465.390	−0.662791	−0.331396	−	0.943492i	$-0.607519\pi$
$79$	−465.390	−0.662791	−0.331396	+	0.943492i	$0.607519\pi$
$80$	0	0
$81$	2745.10	3.76557
$82$	93.6037	0.126058
$83$	−691.289	−0.914202	−0.457101	−	0.889415i	$-0.651112\pi$
$83$	−691.289	−0.914202	−0.457101	+	0.889415i	$0.651112\pi$
$84$	0	0
$85$	0	0
$86$	688.498	0.863287
$87$	2278.26	2.80753
$88$	−340.156	−0.412054
$89$	1013.18	1.20671	0.603353	−	0.797474i	$-0.293831\pi$
$89$	1013.18	1.20671	0.603353	+	0.797474i	$0.293831\pi$
$90$	0	0
$91$	0	0
$92$	12.8317	0.0145413
$93$	−1438.21	−1.60361
$94$	−421.482	−0.462474
$95$	0	0
$96$	−241.495	−0.256745
$97$	1151.67	1.20551	0.602757	−	0.797925i	$-0.294069\pi$
$97$	1151.67	1.20551	0.602757	+	0.797925i	$0.294069\pi$
$98$	0	0
$99$	1153.42	1.17094
$100$	0	0
$101$	−1201.03	−1.18324	−0.591618	−	0.806219i	$-0.701510\pi$
$101$	−1201.03	−1.18324	−0.591618	+	0.806219i	$0.701510\pi$
$102$	−310.460	−0.301373
$103$	1029.05	0.984421	0.492211	−	0.870476i	$-0.336189\pi$
$103$	1029.05	0.984421	0.492211	+	0.870476i	$0.336189\pi$
$104$	−1394.17	−1.31452
$105$	0	0
$106$	−930.559	−0.852678
$107$	−576.737	−0.521078	−0.260539	−	0.965463i	$-0.583900\pi$
$107$	−576.737	−0.521078	−0.260539	+	0.965463i	$0.583900\pi$
$108$	251.043	0.223673
$109$	−191.325	−0.168125	−0.0840623	−	0.996461i	$-0.526789\pi$
$109$	−191.325	−0.168125	−0.0840623	+	0.996461i	$0.526789\pi$
$110$	0	0
$111$	2866.30	2.45096
$112$	0	0
$113$	−2091.69	−1.74132	−0.870662	−	0.491882i	$-0.836309\pi$
$113$	−2091.69	−1.74132	−0.870662	+	0.491882i	$0.836309\pi$
$114$	−1138.00	−0.934941
$115$	0	0
$116$	120.606	0.0965343
$117$	4727.45	3.73549
$118$	1309.52	1.02162
$119$	0	0
$120$	0	0
$121$	−1087.84	−0.817310
$122$	486.906	0.361331
$123$	322.003	0.236049
$124$	−76.1357	−0.0551386
$125$	0	0
$126$	0	0
$127$	498.841	0.348543	0.174271	−	0.984698i	$-0.444243\pi$
$127$	498.841	0.348543	0.174271	+	0.984698i	$0.444243\pi$
$128$	1575.57	1.08798
$129$	2368.48	1.61654
$130$	0	0
$131$	311.650	0.207855	0.103928	−	0.994585i	$-0.466859\pi$
$131$	311.650	0.207855	0.103928	+	0.994585i	$0.466859\pi$
$132$	83.3481	0.0549585
$133$	0	0
$134$	−2164.55	−1.39543
$135$	0	0
$136$	230.739	0.145483
$137$	1022.69	0.637768	0.318884	−	0.947794i	$-0.396692\pi$
$137$	1022.69	0.637768	0.318884	+	0.947794i	$0.396692\pi$
$138$	708.013	0.436740
$139$	1141.29	0.696424	0.348212	−	0.937416i	$-0.386789\pi$
$139$	1141.29	0.696424	0.348212	+	0.937416i	$0.386789\pi$
$140$	0	0
$141$	−1449.93	−0.866000
$142$	−1091.42	−0.644997
$143$	996.623	0.582810
$144$	−5027.78	−2.90959
$145$	0	0
$146$	−693.182	−0.392932
$147$	0	0
$148$	151.736	0.0842743
$149$	1290.97	0.709801	0.354900	−	0.934904i	$-0.384515\pi$
$149$	1290.97	0.709801	0.354900	+	0.934904i	$0.384515\pi$
$150$	0	0
$151$	−1149.95	−0.619745	−0.309873	−	0.950778i	$-0.600286\pi$
$151$	−1149.95	−0.619745	−0.309873	+	0.950778i	$0.600286\pi$
$152$	845.781	0.451329
$153$	−782.407	−0.413424
$154$	0	0
$155$	0	0
$156$	341.612	0.175326
$157$	−3622.34	−1.84136	−0.920681	−	0.390316i	$-0.872366\pi$
$157$	−3622.34	−1.84136	−0.920681	+	0.390316i	$0.872366\pi$
$158$	−1359.38	−0.684472
$159$	−3201.19	−1.59667
$160$	0	0
$161$	0	0
$162$	8018.30	3.88875
$163$	−2705.71	−1.30017	−0.650086	−	0.759861i	$-0.725267\pi$
$163$	−2705.71	−1.30017	−0.650086	+	0.759861i	$0.725267\pi$
$164$	17.0462	0.00811635
$165$	0	0
$166$	−2019.22	−0.944107
$167$	3397.98	1.57451	0.787256	−	0.616627i	$-0.211501\pi$
$167$	3397.98	1.57451	0.787256	+	0.616627i	$0.211501\pi$
$168$	0	0
$169$	1887.78	0.859253
$170$	0	0
$171$	−2867.93	−1.28255
$172$	125.382	0.0555832
$173$	−4186.81	−1.83998	−0.919991	−	0.391939i	$-0.871804\pi$
$173$	−4186.81	−1.83998	−0.919991	+	0.391939i	$0.871804\pi$
$174$	6654.67	2.89936
$175$	0	0
$176$	−1059.94	−0.453953
$177$	4504.86	1.91303
$178$	2959.45	1.24618
$179$	917.853	0.383260	0.191630	−	0.981467i	$-0.438623\pi$
$179$	917.853	0.383260	0.191630	+	0.981467i	$0.438623\pi$
$180$	0	0
$181$	262.339	0.107732	0.0538661	−	0.998548i	$-0.482846\pi$
$181$	262.339	0.107732	0.0538661	+	0.998548i	$0.482846\pi$
$182$	0	0
$183$	1674.99	0.676607
$184$	−526.209	−0.210830
$185$	0	0
$186$	−4200.94	−1.65606
$187$	−164.944	−0.0645021
$188$	−76.7560	−0.0297766
$189$	0	0
$190$	0	0
$191$	1446.18	0.547864	0.273932	−	0.961749i	$-0.411676\pi$
$191$	1446.18	0.547864	0.273932	+	0.961749i	$0.411676\pi$
$192$	4758.66	1.78868
$193$	−4065.85	−1.51641	−0.758204	−	0.652018i	$-0.773923\pi$
$193$	−4065.85	−1.51641	−0.758204	+	0.652018i	$0.773923\pi$
$194$	3363.98	1.24495
$195$	0	0
$196$	0	0
$197$	1073.07	0.388088	0.194044	−	0.980993i	$-0.437840\pi$
$197$	1073.07	0.388088	0.194044	+	0.980993i	$0.437840\pi$
$198$	3369.09	1.20925
$199$	−2977.02	−1.06048	−0.530240	−	0.847848i	$-0.677898\pi$
$199$	−2977.02	−1.06048	−0.530240	+	0.847848i	$0.677898\pi$
$200$	0	0
$201$	−7446.19	−2.61300
$202$	−3508.14	−1.22194
$203$	0	0
$204$	−56.5378	−0.0194041
$205$	0	0
$206$	3005.80	1.01662
$207$	1784.30	0.599119
$208$	−4344.28	−1.44818
$209$	−604.607	−0.200103
$210$	0	0
$211$	−521.629	−0.170192	−0.0850958	−	0.996373i	$-0.527120\pi$
$211$	−521.629	−0.170192	−0.0850958	+	0.996373i	$0.527120\pi$
$212$	−169.464	−0.0549002
$213$	−3754.55	−1.20778
$214$	−1684.62	−0.538123
$215$	0	0
$216$	−10294.9	−3.24297
$217$	0	0
$218$	−558.849	−0.173624
$219$	−2384.60	−0.735781
$220$	0	0
$221$	−676.043	−0.205772
$222$	8372.31	2.53114
$223$	2381.79	0.715229	0.357615	−	0.933869i	$-0.383590\pi$
$223$	2381.79	0.715229	0.357615	+	0.933869i	$0.383590\pi$
$224$	0	0
$225$	0	0
$226$	−6109.71	−1.79828
$227$	−341.033	−0.0997144	−0.0498572	−	0.998756i	$-0.515877\pi$
$227$	−341.033	−0.0997144	−0.0498572	+	0.998756i	$0.515877\pi$
$228$	−207.241	−0.0601967
$229$	775.044	0.223652	0.111826	−	0.993728i	$-0.464330\pi$
$229$	775.044	0.223652	0.111826	+	0.993728i	$0.464330\pi$
$230$	0	0
$231$	0	0
$232$	−4945.87	−1.39962
$233$	−2499.84	−0.702876	−0.351438	−	0.936211i	$-0.614307\pi$
$233$	−2499.84	−0.702876	−0.351438	+	0.936211i	$0.614307\pi$
$234$	13808.6	3.85769
$235$	0	0
$236$	238.478	0.0657778
$237$	−4676.37	−1.28170
$238$	0	0
$239$	−2181.70	−0.590470	−0.295235	−	0.955425i	$-0.595398\pi$
$239$	−2181.70	−0.590470	−0.295235	+	0.955425i	$0.595398\pi$
$240$	0	0
$241$	−2270.14	−0.606774	−0.303387	−	0.952867i	$-0.598117\pi$
$241$	−2270.14	−0.606774	−0.303387	+	0.952867i	$0.598117\pi$
$242$	−3177.52	−0.844045
$243$	14841.0	3.91791
$244$	88.6705	0.0232645
$245$	0	0
$246$	940.555	0.243771
$247$	−2478.05	−0.638359
$248$	3122.22	0.799439
$249$	−6946.26	−1.76788
$250$	0	0
$251$	2943.81	0.740286	0.370143	−	0.928975i	$-0.379309\pi$
$251$	2943.81	0.740286	0.370143	+	0.928975i	$0.379309\pi$
$252$	0	0
$253$	376.160	0.0934743
$254$	1457.09	0.359944
$255$	0	0
$256$	813.518	0.198613
$257$	−3623.62	−0.879515	−0.439757	−	0.898117i	$-0.644936\pi$
$257$	−3623.62	−0.879515	−0.439757	+	0.898117i	$0.644936\pi$
$258$	6918.22	1.66942
$259$	0	0
$260$	0	0
$261$	16770.8	3.97734
$262$	910.314	0.214654
$263$	−461.230	−0.108139	−0.0540696	−	0.998537i	$-0.517219\pi$
$263$	−461.230	−0.108139	−0.0540696	+	0.998537i	$0.517219\pi$
$264$	−3417.98	−0.796827
$265$	0	0
$266$	0	0
$267$	10180.7	2.33352
$268$	−394.185	−0.0898459
$269$	−5730.33	−1.29883	−0.649413	−	0.760436i	$-0.724985\pi$
$269$	−5730.33	−1.29883	−0.649413	+	0.760436i	$0.724985\pi$
$270$	0	0
$271$	−435.216	−0.0975554	−0.0487777	−	0.998810i	$-0.515533\pi$
$271$	−435.216	−0.0975554	−0.0487777	+	0.998810i	$0.515533\pi$
$272$	718.991	0.160277
$273$	0	0
$274$	2987.22	0.658630
$275$	0	0
$276$	128.936	0.0281198
$277$	107.500	0.0233179	0.0116590	−	0.999932i	$-0.496289\pi$
$277$	107.500	0.0233179	0.0116590	+	0.999932i	$0.496289\pi$
$278$	3333.65	0.719204
$279$	−10587.0	−2.27179
$280$	0	0
$281$	−8058.94	−1.71088	−0.855438	−	0.517905i	$-0.826712\pi$
$281$	−8058.94	−1.71088	−0.855438	+	0.517905i	$0.826712\pi$
$282$	−4235.16	−0.894327
$283$	5796.37	1.21752	0.608760	−	0.793354i	$-0.291667\pi$
$283$	5796.37	1.21752	0.608760	+	0.793354i	$0.291667\pi$
$284$	−198.758	−0.0415285
$285$	0	0
$286$	2911.08	0.601874
$287$	0	0
$288$	−1777.71	−0.363723
$289$	−4801.11	−0.977226
$290$	0	0
$291$	11572.3	2.33121
$292$	−126.235	−0.0252992
$293$	1250.96	0.249427	0.124713	−	0.992193i	$-0.460199\pi$
$293$	1250.96	0.249427	0.124713	+	0.992193i	$0.460199\pi$
$294$	0	0
$295$	0	0
$296$	−6222.46	−1.22187
$297$	7359.33	1.43782
$298$	3770.85	0.733019
$299$	1541.74	0.298197
$300$	0	0
$301$	0	0
$302$	−3358.94	−0.640018
$303$	−12068.3	−2.28813
$304$	2635.48	0.497221
$305$	0	0
$306$	−2285.37	−0.426947
$307$	−493.727	−0.0917865	−0.0458933	−	0.998946i	$-0.514613\pi$
$307$	−493.727	−0.0917865	−0.0458933	+	0.998946i	$0.514613\pi$
$308$	0	0
$309$	10340.2	1.90367
$310$	0	0
$311$	−105.464	−0.0192293	−0.00961466	−	0.999954i	$-0.503060\pi$
$311$	−105.464	−0.0192293	−0.00961466	+	0.999954i	$0.503060\pi$
$312$	−14009.0	−2.54200
$313$	−3710.63	−0.670087	−0.335044	−	0.942203i	$-0.608751\pi$
$313$	−3710.63	−0.670087	−0.335044	+	0.942203i	$0.608751\pi$
$314$	−10580.7	−1.90159
$315$	0	0
$316$	−247.557	−0.0440701
$317$	−6501.40	−1.15191	−0.575955	−	0.817482i	$-0.695370\pi$
$317$	−6501.40	−1.15191	−0.575955	+	0.817482i	$0.695370\pi$
$318$	−9350.51	−1.64890
$319$	3535.56	0.620543
$320$	0	0
$321$	−5795.21	−1.00766
$322$	0	0
$323$	410.125	0.0706500
$324$	1460.21	0.250379
$325$	0	0
$326$	−7903.25	−1.34270
$327$	−1922.48	−0.325118
$328$	−699.038	−0.117677
$329$	0	0
$330$	0	0
$331$	−108.688	−0.0180485	−0.00902424	−	0.999959i	$-0.502873\pi$
$331$	−108.688	−0.0180485	−0.00902424	+	0.999959i	$0.502873\pi$
$332$	−367.720	−0.0607869
$333$	21099.5	3.47221
$334$	9925.31	1.62601
$335$	0	0
$336$	0	0
$337$	−8207.88	−1.32674	−0.663370	−	0.748291i	$-0.730875\pi$
$337$	−8207.88	−1.32674	−0.663370	+	0.748291i	$0.730875\pi$
$338$	5514.10	0.887360
$339$	−21017.9	−3.36735
$340$	0	0
$341$	−2231.92	−0.354443
$342$	−8377.08	−1.32450
$343$	0	0
$344$	−5141.75	−0.805885
$345$	0	0
$346$	−12229.4	−1.90017
$347$	−9402.55	−1.45463	−0.727313	−	0.686306i	$-0.759231\pi$
$347$	−9402.55	−1.45463	−0.727313	+	0.686306i	$0.759231\pi$
$348$	1211.88	0.186677
$349$	−690.530	−0.105912	−0.0529559	−	0.998597i	$-0.516864\pi$
$349$	−690.530	−0.105912	−0.0529559	+	0.998597i	$0.516864\pi$
$350$	0	0
$351$	30163.1	4.58685
$352$	−374.769	−0.0567479
$353$	3601.39	0.543010	0.271505	−	0.962437i	$-0.412479\pi$
$353$	3601.39	0.543010	0.271505	+	0.962437i	$0.412479\pi$
$354$	13158.5	1.97561
$355$	0	0
$356$	538.945	0.0802360
$357$	0	0
$358$	2681.00	0.395797
$359$	8063.90	1.18551	0.592753	−	0.805384i	$-0.298041\pi$
$359$	8063.90	1.18551	0.592753	+	0.805384i	$0.298041\pi$
$360$	0	0
$361$	−5355.68	−0.780825
$362$	766.280	0.111256
$363$	−10930.9	−1.58051
$364$	0	0
$365$	0	0
$366$	4892.57	0.698739
$367$	3389.62	0.482117	0.241059	−	0.970511i	$-0.422505\pi$
$367$	3389.62	0.482117	0.241059	+	0.970511i	$0.422505\pi$
$368$	−1639.68	−0.232267
$369$	2370.35	0.334405
$370$	0	0
$371$	0	0
$372$	−765.033	−0.106627
$373$	8841.24	1.22730	0.613649	−	0.789579i	$-0.289701\pi$
$373$	8841.24	1.22730	0.613649	+	0.789579i	$0.289701\pi$
$374$	−481.793	−0.0666121
$375$	0	0
$376$	3147.65	0.431723
$377$	14490.9	1.97963
$378$	0	0
$379$	3038.06	0.411754	0.205877	−	0.978578i	$-0.433995\pi$
$379$	3038.06	0.411754	0.205877	+	0.978578i	$0.433995\pi$
$380$	0	0
$381$	5012.49	0.674009
$382$	4224.22	0.565785
$383$	2818.68	0.376052	0.188026	−	0.982164i	$-0.439791\pi$
$383$	2818.68	0.376052	0.188026	+	0.982164i	$0.439791\pi$
$384$	15831.7	2.10393
$385$	0	0
$386$	−11876.1	−1.56601
$387$	17435.0	2.29010
$388$	612.614	0.0801567
$389$	12188.7	1.58867	0.794334	−	0.607481i	$-0.207820\pi$
$389$	12188.7	1.58867	0.794334	+	0.607481i	$0.207820\pi$
$390$	0	0
$391$	−255.162	−0.0330028
$392$	0	0
$393$	3131.55	0.401948
$394$	3134.39	0.400782
$395$	0	0
$396$	613.545	0.0778581
$397$	−10390.3	−1.31354	−0.656771	−	0.754090i	$-0.728078\pi$
$397$	−10390.3	−1.31354	−0.656771	+	0.754090i	$0.728078\pi$
$398$	−8695.73	−1.09517
$399$	0	0
$400$	0	0
$401$	1502.28	0.187083	0.0935413	−	0.995615i	$-0.470181\pi$
$401$	1502.28	0.187083	0.0935413	+	0.995615i	$0.470181\pi$
$402$	−21749.9	−2.69848
$403$	−9147.77	−1.13073
$404$	−638.867	−0.0786753
$405$	0	0
$406$	0	0
$407$	4448.12	0.541733
$408$	2318.53	0.281334
$409$	14097.2	1.70431	0.852155	−	0.523290i	$-0.175295\pi$
$409$	14097.2	1.70431	0.852155	+	0.523290i	$0.175295\pi$
$410$	0	0
$411$	10276.3	1.23331
$412$	547.387	0.0654559
$413$	0	0
$414$	5211.86	0.618717
$415$	0	0
$416$	−1536.04	−0.181035
$417$	11468.0	1.34674
$418$	−1766.02	−0.206648
$419$	11529.9	1.34433	0.672163	−	0.740403i	$-0.265365\pi$
$419$	11529.9	1.34433	0.672163	+	0.740403i	$0.265365\pi$
$420$	0	0
$421$	−3370.05	−0.390134	−0.195067	−	0.980790i	$-0.562492\pi$
$421$	−3370.05	−0.390134	−0.195067	+	0.980790i	$0.562492\pi$
$422$	−1523.65	−0.175759
$423$	−10673.3	−1.22684
$424$	6949.48	0.795982
$425$	0	0
$426$	−10966.9	−1.24729
$427$	0	0
$428$	−306.786	−0.0346473
$429$	10014.3	1.12703
$430$	0	0
$431$	−8327.07	−0.930628	−0.465314	−	0.885146i	$-0.654059\pi$
$431$	−8327.07	−0.930628	−0.465314	+	0.885146i	$0.654059\pi$
$432$	−32079.3	−3.57272
$433$	818.149	0.0908031	0.0454015	−	0.998969i	$-0.485543\pi$
$433$	818.149	0.0908031	0.0454015	+	0.998969i	$0.485543\pi$
$434$	0	0
$435$	0	0
$436$	−101.772	−0.0111789
$437$	−935.303	−0.102384
$438$	−6965.28	−0.759849
$439$	8465.78	0.920386	0.460193	−	0.887819i	$-0.347780\pi$
$439$	8465.78	0.920386	0.460193	+	0.887819i	$0.347780\pi$
$440$	0	0
$441$	0	0
$442$	−1974.69	−0.212503
$443$	14568.9	1.56251	0.781254	−	0.624213i	$-0.214580\pi$
$443$	14568.9	1.56251	0.781254	+	0.624213i	$0.214580\pi$
$444$	1524.68	0.162969
$445$	0	0
$446$	6957.07	0.738625
$447$	12972.0	1.37261
$448$	0	0
$449$	−841.211	−0.0884169	−0.0442084	−	0.999022i	$-0.514077\pi$
$449$	−841.211	−0.0884169	−0.0442084	+	0.999022i	$0.514077\pi$
$450$	0	0
$451$	499.707	0.0521736
$452$	−1112.64	−0.115784
$453$	−11555.0	−1.19846
$454$	−996.140	−0.102976
$455$	0	0
$456$	8498.64	0.872775
$457$	13876.7	1.42040	0.710200	−	0.704000i	$-0.248604\pi$
$457$	13876.7	1.42040	0.710200	+	0.704000i	$0.248604\pi$
$458$	2263.86	0.230968
$459$	−4992.08	−0.507648
$460$	0	0
$461$	3053.09	0.308452	0.154226	−	0.988036i	$-0.450712\pi$
$461$	3053.09	0.308452	0.154226	+	0.988036i	$0.450712\pi$
$462$	0	0
$463$	9935.80	0.997312	0.498656	−	0.866800i	$-0.333827\pi$
$463$	9935.80	0.997312	0.498656	+	0.866800i	$0.333827\pi$
$464$	−15411.5	−1.54194
$465$	0	0
$466$	−7301.91	−0.725868
$467$	9927.26	0.983681	0.491840	−	0.870685i	$-0.336324\pi$
$467$	9927.26	0.983681	0.491840	+	0.870685i	$0.336324\pi$
$468$	2514.69	0.248379
$469$	0	0
$470$	0	0
$471$	−36398.2	−3.56081
$472$	−9779.62	−0.953694
$473$	3675.58	0.357301
$474$	−13659.4	−1.32363
$475$	0	0
$476$	0	0
$477$	−23564.8	−2.26196
$478$	−6372.63	−0.609785
$479$	−13174.2	−1.25667	−0.628336	−	0.777942i	$-0.716264\pi$
$479$	−13174.2	−1.25667	−0.628336	+	0.777942i	$0.716264\pi$
$480$	0	0
$481$	18231.2	1.72821
$482$	−6630.96	−0.626622
$483$	0	0
$484$	−578.658	−0.0543443
$485$	0	0
$486$	43349.8	4.04607
$487$	8473.76	0.788466	0.394233	−	0.919011i	$-0.371010\pi$
$487$	8473.76	0.788466	0.394233	+	0.919011i	$0.371010\pi$
$488$	−3636.25	−0.337306
$489$	−27187.8	−2.51426
$490$	0	0
$491$	−10831.1	−0.995523	−0.497762	−	0.867314i	$-0.665845\pi$
$491$	−10831.1	−0.995523	−0.497762	+	0.867314i	$0.665845\pi$
$492$	171.284	0.0156953
$493$	−2398.29	−0.219094
$494$	−7238.26	−0.659240
$495$	0	0
$496$	9728.92	0.880728
$497$	0	0
$498$	−20289.7	−1.82571
$499$	−12092.1	−1.08480	−0.542402	−	0.840119i	$-0.682485\pi$
$499$	−12092.1	−1.08480	−0.542402	+	0.840119i	$0.682485\pi$
$500$	0	0
$501$	34143.8	3.04478
$502$	8598.72	0.764501
$503$	18398.5	1.63091	0.815455	−	0.578821i	$-0.196487\pi$
$503$	18398.5	1.63091	0.815455	+	0.578821i	$0.196487\pi$
$504$	0	0
$505$	0	0
$506$	1098.74	0.0965319
$507$	18968.9	1.66162
$508$	265.350	0.0231752
$509$	−20702.7	−1.80281	−0.901406	−	0.432975i	$-0.857464\pi$
$509$	−20702.7	−1.80281	−0.901406	+	0.432975i	$0.857464\pi$
$510$	0	0
$511$	0	0
$512$	−10228.3	−0.882874
$513$	−18298.6	−1.57486
$514$	−10584.4	−0.908285
$515$	0	0
$516$	1259.88	0.107486
$517$	−2250.10	−0.191411
$518$	0	0
$519$	−42070.2	−3.55814
$520$	0	0
$521$	−14473.1	−1.21704	−0.608518	−	0.793540i	$-0.708236\pi$
$521$	−14473.1	−1.21704	−0.608518	+	0.793540i	$0.708236\pi$
$522$	48986.6	4.10744
$523$	15917.9	1.33087	0.665433	−	0.746457i	$-0.268247\pi$
$523$	15917.9	1.33087	0.665433	+	0.746457i	$0.268247\pi$
$524$	165.777	0.0138206
$525$	0	0
$526$	−1347.23	−0.111677
$527$	1513.98	0.125143
$528$	−10650.5	−0.877850
$529$	−11585.1	−0.952173
$530$	0	0
$531$	33161.4	2.71013
$532$	0	0
$533$	2048.11	0.166442
$534$	29737.3	2.40985
$535$	0	0
$536$	16165.0	1.30265
$537$	9222.83	0.741145
$538$	−16738.0	−1.34131
$539$	0	0
$540$	0	0
$541$	−8839.33	−0.702463	−0.351232	−	0.936289i	$-0.614237\pi$
$541$	−8839.33	−0.702463	−0.351232	+	0.936289i	$0.614237\pi$
$542$	−1271.24	−0.100746
$543$	2636.06	0.208332
$544$	254.219	0.0200359
$545$	0	0
$546$	0	0
$547$	−11349.6	−0.887155	−0.443577	−	0.896236i	$-0.646291\pi$
$547$	−11349.6	−0.887155	−0.443577	+	0.896236i	$0.646291\pi$
$548$	544.003	0.0424063
$549$	12330.0	0.958530
$550$	0	0
$551$	−8790.98	−0.679689
$552$	−5287.49	−0.407700
$553$	0	0
$554$	314.003	0.0240807
$555$	0	0
$556$	607.090	0.0463064
$557$	6610.86	0.502893	0.251446	−	0.967871i	$-0.419094\pi$
$557$	6610.86	0.502893	0.251446	+	0.967871i	$0.419094\pi$
$558$	−30924.1	−2.34610
$559$	15064.8	1.13984
$560$	0	0
$561$	−1657.40	−0.124734
$562$	−23539.8	−1.76684
$563$	12849.9	0.961918	0.480959	−	0.876743i	$-0.340289\pi$
$563$	12849.9	0.961918	0.480959	+	0.876743i	$0.340289\pi$
$564$	−771.266	−0.0575818
$565$	0	0
$566$	16930.9	1.25735
$567$	0	0
$568$	8150.77	0.602110
$569$	−8973.04	−0.661106	−0.330553	−	0.943787i	$-0.607235\pi$
$569$	−8973.04	−0.661106	−0.330553	+	0.943787i	$0.607235\pi$
$570$	0	0
$571$	1322.04	0.0968923	0.0484461	−	0.998826i	$-0.484573\pi$
$571$	1322.04	0.0968923	0.0484461	+	0.998826i	$0.484573\pi$
$572$	530.137	0.0387520
$573$	14531.6	1.05945
$574$	0	0
$575$	0	0
$576$	35029.6	2.53397
$577$	−23232.6	−1.67623	−0.838116	−	0.545491i	$-0.816343\pi$
$577$	−23232.6	−1.67623	−0.838116	+	0.545491i	$0.816343\pi$
$578$	−14023.8	−1.00919
$579$	−40854.8	−2.93241
$580$	0	0
$581$	0	0
$582$	33802.2	2.40747
$583$	−4967.83	−0.352910
$584$	5176.73	0.366806
$585$	0	0
$586$	3654.00	0.257586
$587$	−13229.8	−0.930244	−0.465122	−	0.885247i	$-0.653990\pi$
$587$	−13229.8	−0.930244	−0.465122	+	0.885247i	$0.653990\pi$
$588$	0	0
$589$	5549.55	0.388226
$590$	0	0
$591$	10782.5	0.750480
$592$	−19389.4	−1.34611
$593$	−26345.3	−1.82440	−0.912201	−	0.409744i	$-0.865618\pi$
$593$	−26345.3	−1.82440	−0.912201	+	0.409744i	$0.865618\pi$
$594$	21496.2	1.48485
$595$	0	0
$596$	686.710	0.0471959
$597$	−29913.9	−2.05075
$598$	4503.34	0.307952
$599$	−12588.6	−0.858690	−0.429345	−	0.903141i	$-0.641255\pi$
$599$	−12588.6	−0.858690	−0.429345	+	0.903141i	$0.641255\pi$
$600$	0	0
$601$	−5027.54	−0.341227	−0.170614	−	0.985338i	$-0.554575\pi$
$601$	−5027.54	−0.341227	−0.170614	+	0.985338i	$0.554575\pi$
$602$	0	0
$603$	−54813.2	−3.70177
$604$	−611.697	−0.0412079
$605$	0	0
$606$	−35250.7	−2.36298
$607$	−1961.59	−0.131167	−0.0655835	−	0.997847i	$-0.520891\pi$
$607$	−1961.59	−0.131167	−0.0655835	+	0.997847i	$0.520891\pi$
$608$	931.845	0.0621567
$609$	0	0
$610$	0	0
$611$	−9222.30	−0.610629
$612$	−416.188	−0.0274892
$613$	−9314.66	−0.613728	−0.306864	−	0.951753i	$-0.599280\pi$
$613$	−9314.66	−0.613728	−0.306864	+	0.951753i	$0.599280\pi$
$614$	−1442.15	−0.0947890
$615$	0	0
$616$	0	0
$617$	−20829.3	−1.35909	−0.679544	−	0.733635i	$-0.737822\pi$
$617$	−20829.3	−1.35909	−0.679544	+	0.733635i	$0.737822\pi$
$618$	30203.1	1.96594
$619$	13364.8	0.867813	0.433906	−	0.900958i	$-0.357135\pi$
$619$	13364.8	0.867813	0.433906	+	0.900958i	$0.357135\pi$
$620$	0	0
$621$	11384.6	0.735665
$622$	−308.055	−0.0198583
$623$	0	0
$624$	−43652.5	−2.80048
$625$	0	0
$626$	−10838.6	−0.692006
$627$	−6075.25	−0.386957
$628$	−1926.84	−0.122435
$629$	−3017.31	−0.191269
$630$	0	0
$631$	−29261.9	−1.84612	−0.923058	−	0.384661i	$-0.874318\pi$
$631$	−29261.9	−1.84612	−0.923058	+	0.384661i	$0.874318\pi$
$632$	10151.9	0.638960
$633$	−5241.47	−0.329115
$634$	−18990.3	−1.18959
$635$	0	0
$636$	−1702.82	−0.106166
$637$	0	0
$638$	10327.2	0.640841
$639$	−27638.2	−1.71103
$640$	0	0
$641$	−12448.5	−0.767063	−0.383531	−	0.923528i	$-0.625292\pi$
$641$	−12448.5	−0.767063	−0.383531	+	0.923528i	$0.625292\pi$
$642$	−16927.5	−1.04062
$643$	−13458.0	−0.825397	−0.412698	−	0.910868i	$-0.635414\pi$
$643$	−13458.0	−0.825397	−0.412698	+	0.910868i	$0.635414\pi$
$644$	0	0
$645$	0	0
$646$	1197.95	0.0729611
$647$	−1837.80	−0.111671	−0.0558356	−	0.998440i	$-0.517782\pi$
$647$	−1837.80	−0.111671	−0.0558356	+	0.998440i	$0.517782\pi$
$648$	−59881.2	−3.63018
$649$	6990.96	0.422834
$650$	0	0
$651$	0	0
$652$	−1439.26	−0.0864506
$653$	12128.4	0.726830	0.363415	−	0.931627i	$-0.381611\pi$
$653$	12128.4	0.726830	0.363415	+	0.931627i	$0.381611\pi$
$654$	−5615.47	−0.335753
$655$	0	0
$656$	−2178.22	−0.129642
$657$	−17553.6	−1.04236
$658$	0	0
$659$	4131.14	0.244198	0.122099	−	0.992518i	$-0.461038\pi$
$659$	4131.14	0.244198	0.122099	+	0.992518i	$0.461038\pi$
$660$	0	0
$661$	21362.6	1.25705	0.628525	−	0.777790i	$-0.283659\pi$
$661$	21362.6	1.25705	0.628525	+	0.777790i	$0.283659\pi$
$662$	−317.473	−0.0186389
$663$	−6793.06	−0.397920
$664$	15079.7	0.881331
$665$	0	0
$666$	61630.6	3.58579
$667$	5469.37	0.317504
$668$	1807.50	0.104692
$669$	23932.8	1.38310
$670$	0	0
$671$	2599.37	0.149549
$672$	0	0
$673$	26260.3	1.50410	0.752052	−	0.659104i	$-0.229064\pi$
$673$	26260.3	1.50410	0.752052	+	0.659104i	$0.229064\pi$
$674$	−23974.8	−1.37014
$675$	0	0
$676$	1004.17	0.0571332
$677$	30516.5	1.73242	0.866208	−	0.499684i	$-0.166551\pi$
$677$	30516.5	1.73242	0.866208	+	0.499684i	$0.166551\pi$
$678$	−61392.1	−3.47750
$679$	0	0
$680$	0	0
$681$	−3426.80	−0.192827
$682$	−6519.31	−0.366037
$683$	29207.4	1.63629	0.818147	−	0.575009i	$-0.195002\pi$
$683$	29207.4	1.63629	0.818147	+	0.575009i	$0.195002\pi$
$684$	−1525.55	−0.0852790
$685$	0	0
$686$	0	0
$687$	7787.85	0.432497
$688$	−16021.8	−0.887830
$689$	−20361.3	−1.12584
$690$	0	0
$691$	10624.8	0.584928	0.292464	−	0.956276i	$-0.405525\pi$
$691$	10624.8	0.584928	0.292464	+	0.956276i	$0.405525\pi$
$692$	−2227.10	−0.122344
$693$	0	0
$694$	−27464.4	−1.50221
$695$	0	0
$696$	−49697.5	−2.70658
$697$	−338.968	−0.0184209
$698$	−2017.00	−0.109376
$699$	−25119.1	−1.35922
$700$	0	0
$701$	22033.0	1.18712	0.593562	−	0.804788i	$-0.297721\pi$
$701$	22033.0	1.18712	0.593562	+	0.804788i	$0.297721\pi$
$702$	88104.8	4.73689
$703$	−11060.0	−0.593367
$704$	7384.81	0.395349
$705$	0	0
$706$	10519.5	0.560772
$707$	0	0
$708$	2396.29	0.127201
$709$	24867.4	1.31723	0.658614	−	0.752481i	$-0.271143\pi$
$709$	24867.4	1.31723	0.658614	+	0.752481i	$0.271143\pi$
$710$	0	0
$711$	−34423.9	−1.81575
$712$	−22101.3	−1.16332
$713$	−3452.69	−0.181352
$714$	0	0
$715$	0	0
$716$	488.237	0.0254836
$717$	−21922.3	−1.14185
$718$	23554.2	1.22428
$719$	−1404.96	−0.0728739	−0.0364370	−	0.999336i	$-0.511601\pi$
$719$	−1404.96	−0.0728739	−0.0364370	+	0.999336i	$0.511601\pi$
$720$	0	0
$721$	0	0
$722$	−15643.6	−0.806366
$723$	−22811.0	−1.17337
$724$	139.547	0.00716330
$725$	0	0
$726$	−31928.6	−1.63221
$727$	28384.6	1.44804	0.724020	−	0.689779i	$-0.242292\pi$
$727$	28384.6	1.44804	0.724020	+	0.689779i	$0.242292\pi$
$728$	0	0
$729$	75008.8	3.81084
$730$	0	0
$731$	−2493.27	−0.126152
$732$	890.985	0.0449887
$733$	20471.8	1.03157	0.515787	−	0.856717i	$-0.327500\pi$
$733$	20471.8	1.03157	0.515787	+	0.856717i	$0.327500\pi$
$734$	9900.91	0.497887
$735$	0	0
$736$	−579.754	−0.0290353
$737$	−11555.5	−0.577548
$738$	6923.65	0.345343
$739$	−33181.0	−1.65167	−0.825834	−	0.563913i	$-0.809295\pi$
$739$	−33181.0	−1.65167	−0.825834	+	0.563913i	$0.809295\pi$
$740$	0	0
$741$	−24900.1	−1.23445
$742$	0	0
$743$	−30844.1	−1.52296	−0.761481	−	0.648188i	$-0.775527\pi$
$743$	−30844.1	−1.52296	−0.761481	+	0.648188i	$0.775527\pi$
$744$	31372.9	1.54595
$745$	0	0
$746$	25824.8	1.26744
$747$	−51133.1	−2.50450
$748$	−87.7393	−0.00428886
$749$	0	0
$750$	0	0
$751$	−24693.0	−1.19981	−0.599906	−	0.800070i	$-0.704796\pi$
$751$	−24693.0	−1.19981	−0.599906	+	0.800070i	$0.704796\pi$
$752$	9808.18	0.475622
$753$	29580.2	1.43156
$754$	42327.2	2.04438
$755$	0	0
$756$	0	0
$757$	6870.57	0.329875	0.164937	−	0.986304i	$-0.447258\pi$
$757$	6870.57	0.329875	0.164937	+	0.986304i	$0.447258\pi$
$758$	8874.01	0.425222
$759$	3779.76	0.180760
$760$	0	0
$761$	30882.6	1.47108	0.735540	−	0.677481i	$-0.236928\pi$
$761$	30882.6	1.47108	0.735540	+	0.677481i	$0.236928\pi$
$762$	14641.2	0.696057
$763$	0	0
$764$	769.273	0.0364284
$765$	0	0
$766$	8233.23	0.388353
$767$	28653.3	1.34890
$768$	8174.45	0.384076
$769$	−13536.3	−0.634761	−0.317380	−	0.948298i	$-0.602803\pi$
$769$	−13536.3	−0.634761	−0.317380	+	0.948298i	$0.602803\pi$
$770$	0	0
$771$	−36411.1	−1.70080
$772$	−2162.77	−0.100829
$773$	14285.7	0.664712	0.332356	−	0.943154i	$-0.392157\pi$
$773$	14285.7	0.664712	0.332356	+	0.943154i	$0.392157\pi$
$774$	50926.7	2.36502
$775$	0	0
$776$	−25122.4	−1.16217
$777$	0	0
$778$	35602.6	1.64064
$779$	−1242.50	−0.0571464
$780$	0	0
$781$	−5826.58	−0.266954
$782$	−745.315	−0.0340824
$783$	107005.	4.88382
$784$	0	0
$785$	0	0
$786$	9147.08	0.415096
$787$	−8453.63	−0.382896	−0.191448	−	0.981503i	$-0.561318\pi$
$787$	−8453.63	−0.382896	−0.191448	+	0.981503i	$0.561318\pi$
$788$	570.803	0.0258046
$789$	−4634.56	−0.209119
$790$	0	0
$791$	0	0
$792$	−25160.6	−1.12884
$793$	10653.8	0.477085
$794$	−30349.6	−1.35651
$795$	0	0
$796$	−1583.58	−0.0705131
$797$	5465.35	0.242902	0.121451	−	0.992597i	$-0.461245\pi$
$797$	5465.35	0.242902	0.121451	+	0.992597i	$0.461245\pi$
$798$	0	0
$799$	1526.32	0.0675810
$800$	0	0
$801$	74942.7	3.30583
$802$	4388.07	0.193202
$803$	−3700.58	−0.162628
$804$	−3960.88	−0.173743
$805$	0	0
$806$	−26720.2	−1.16771
$807$	−57579.9	−2.51166
$808$	26199.0	1.14069
$809$	16601.1	0.721462	0.360731	−	0.932670i	$-0.382527\pi$
$809$	16601.1	0.721462	0.360731	+	0.932670i	$0.382527\pi$
$810$	0	0
$811$	−24613.6	−1.06572	−0.532861	−	0.846203i	$-0.678883\pi$
$811$	−24613.6	−1.06572	−0.532861	+	0.846203i	$0.678883\pi$
$812$	0	0
$813$	−4373.17	−0.188652
$814$	12992.7	0.559453
$815$	0	0
$816$	7224.61	0.309941
$817$	−9139.14	−0.391356
$818$	41177.2	1.76006
$819$	0	0
$820$	0	0
$821$	−17940.6	−0.762644	−0.381322	−	0.924442i	$-0.624531\pi$
$821$	−17940.6	−0.762644	−0.381322	+	0.924442i	$0.624531\pi$
$822$	30016.4	1.27365
$823$	45909.1	1.94446	0.972229	−	0.234030i	$-0.0751914\pi$
$823$	45909.1	1.94446	0.972229	+	0.234030i	$0.0751914\pi$
$824$	−22447.5	−0.949026
$825$	0	0
$826$	0	0
$827$	36696.8	1.54301	0.771507	−	0.636221i	$-0.219503\pi$
$827$	36696.8	1.54301	0.771507	+	0.636221i	$0.219503\pi$
$828$	949.131	0.0398365
$829$	28920.5	1.21164	0.605820	−	0.795601i	$-0.292845\pi$
$829$	28920.5	1.21164	0.605820	+	0.795601i	$0.292845\pi$
$830$	0	0
$831$	1080.19	0.0450920
$832$	30267.5	1.26122
$833$	0	0
$834$	33497.4	1.39079
$835$	0	0
$836$	−321.611	−0.0133052
$837$	−67549.6	−2.78955
$838$	33678.2	1.38830
$839$	5852.41	0.240820	0.120410	−	0.992724i	$-0.461579\pi$
$839$	5852.41	0.240820	0.120410	+	0.992724i	$0.461579\pi$
$840$	0	0
$841$	27018.0	1.10779
$842$	−9843.74	−0.402895
$843$	−80978.5	−3.30848
$844$	−277.472	−0.0113163
$845$	0	0
$846$	−31176.1	−1.26697
$847$	0	0
$848$	21654.8	0.876920
$849$	58243.5	2.35443
$850$	0	0
$851$	6881.08	0.277180
$852$	−1997.17	−0.0803075
$853$	18715.7	0.751245	0.375623	−	0.926773i	$-0.377429\pi$
$853$	18715.7	0.751245	0.375623	+	0.926773i	$0.377429\pi$
$854$	0	0
$855$	0	0
$856$	12580.8	0.502342
$857$	−17297.1	−0.689450	−0.344725	−	0.938704i	$-0.612028\pi$
$857$	−17297.1	−0.689450	−0.344725	+	0.938704i	$0.612028\pi$
$858$	29251.4	1.16390
$859$	30732.8	1.22071	0.610354	−	0.792129i	$-0.291027\pi$
$859$	30732.8	1.22071	0.610354	+	0.792129i	$0.291027\pi$
$860$	0	0
$861$	0	0
$862$	−24322.9	−0.961070
$863$	−10793.0	−0.425722	−0.212861	−	0.977082i	$-0.568278\pi$
$863$	−10793.0	−0.425722	−0.212861	+	0.977082i	$0.568278\pi$
$864$	−11342.5	−0.446620
$865$	0	0
$866$	2389.77	0.0937733
$867$	−48242.9	−1.88975
$868$	0	0
$869$	−7257.12	−0.283292
$870$	0	0
$871$	−47361.7	−1.84247
$872$	4173.52	0.162080
$873$	85186.8	3.30256
$874$	−2731.97	−0.105733
$875$	0	0
$876$	−1268.45	−0.0489233
$877$	19566.7	0.753388	0.376694	−	0.926338i	$-0.377061\pi$
$877$	19566.7	0.753388	0.376694	+	0.926338i	$0.377061\pi$
$878$	24728.1	0.950492
$879$	12570.0	0.482339
$880$	0	0
$881$	27700.5	1.05931	0.529656	−	0.848213i	$-0.322321\pi$
$881$	27700.5	1.05931	0.529656	+	0.848213i	$0.322321\pi$
$882$	0	0
$883$	−5071.61	−0.193288	−0.0966440	−	0.995319i	$-0.530811\pi$
$883$	−5071.61	−0.193288	−0.0966440	+	0.995319i	$0.530811\pi$
$884$	−359.610	−0.0136821
$885$	0	0
$886$	42555.1	1.61362
$887$	22548.8	0.853569	0.426784	−	0.904353i	$-0.359646\pi$
$887$	22548.8	0.853569	0.426784	+	0.904353i	$0.359646\pi$
$888$	−62525.0	−2.36284
$889$	0	0
$890$	0	0
$891$	42806.0	1.60949
$892$	1266.95	0.0475568
$893$	5594.76	0.209654
$894$	37890.6	1.41751
$895$	0	0
$896$	0	0
$897$	15491.8	0.576651
$898$	−2457.13	−0.0913091
$899$	−32452.1	−1.20393
$900$	0	0
$901$	3369.85	0.124601
$902$	1459.62	0.0538803
$903$	0	0
$904$	45627.7	1.67871
$905$	0	0
$906$	−33751.5	−1.23766
$907$	24242.2	0.887484	0.443742	−	0.896155i	$-0.353651\pi$
$907$	24242.2	0.887484	0.443742	+	0.896155i	$0.353651\pi$
$908$	−181.407	−0.00663018
$909$	−88837.4	−3.24153
$910$	0	0
$911$	6574.69	0.239110	0.119555	−	0.992828i	$-0.461853\pi$
$911$	6574.69	0.239110	0.119555	+	0.992828i	$0.461853\pi$
$912$	26482.0	0.961521
$913$	−10779.7	−0.390751
$914$	40533.0	1.46686
$915$	0	0
$916$	412.272	0.0148710
$917$	0	0
$918$	−14581.6	−0.524253
$919$	−41912.3	−1.50442	−0.752208	−	0.658925i	$-0.771011\pi$
$919$	−41912.3	−1.50442	−0.752208	+	0.658925i	$0.771011\pi$
$920$	0	0
$921$	−4961.10	−0.177496
$922$	8917.91	0.318542
$923$	−23880.9	−0.851625
$924$	0	0
$925$	0	0
$926$	29021.9	1.02994
$927$	76116.6	2.69687
$928$	−5449.15	−0.192755
$929$	−48171.4	−1.70124	−0.850621	−	0.525780i	$-0.823774\pi$
$929$	−48171.4	−1.70124	−0.850621	+	0.525780i	$0.823774\pi$
$930$	0	0
$931$	0	0
$932$	−1329.75	−0.0467355
$933$	−1059.73	−0.0371855
$934$	28997.0	1.01586
$935$	0	0
$936$	−103124.	−3.60118
$937$	−48696.8	−1.69782	−0.848909	−	0.528539i	$-0.822740\pi$
$937$	−48696.8	−1.69782	−0.848909	+	0.528539i	$0.822740\pi$
$938$	0	0
$939$	−37285.4	−1.29581
$940$	0	0
$941$	−2000.30	−0.0692964	−0.0346482	−	0.999400i	$-0.511031\pi$
$941$	−2000.30	−0.0692964	−0.0346482	+	0.999400i	$0.511031\pi$
$942$	−106317.	−3.67729
$943$	773.028	0.0266949
$944$	−30473.6	−1.05067
$945$	0	0
$946$	10736.2	0.368988
$947$	42495.4	1.45820	0.729101	−	0.684407i	$-0.239939\pi$
$947$	42495.4	1.45820	0.729101	+	0.684407i	$0.239939\pi$
$948$	−2487.52	−0.0852224
$949$	−15167.3	−0.518810
$950$	0	0
$951$	−65327.9	−2.22755
$952$	0	0
$953$	51166.5	1.73919	0.869594	−	0.493768i	$-0.164381\pi$
$953$	51166.5	1.73919	0.869594	+	0.493768i	$0.164381\pi$
$954$	−68831.4	−2.33595
$955$	0	0
$956$	−1160.52	−0.0392614
$957$	35526.2	1.20000
$958$	−38481.3	−1.29778
$959$	0	0
$960$	0	0
$961$	−9304.76	−0.312335
$962$	53252.3	1.78474
$963$	−42660.0	−1.42752
$964$	−1207.56	−0.0403454
$965$	0	0
$966$	0	0
$967$	−26324.2	−0.875418	−0.437709	−	0.899117i	$-0.644210\pi$
$967$	−26324.2	−0.875418	−0.437709	+	0.899117i	$0.644210\pi$
$968$	23729.9	0.787923
$969$	4121.05	0.136622
$970$	0	0
$971$	−5234.57	−0.173002	−0.0865012	−	0.996252i	$-0.527569\pi$
$971$	−5234.57	−0.173002	−0.0865012	+	0.996252i	$0.527569\pi$
$972$	7894.44	0.260508
$973$	0	0
$974$	24751.4	0.814257
$975$	0	0
$976$	−11330.7	−0.371604
$977$	−44980.6	−1.47293	−0.736467	−	0.676474i	$-0.763507\pi$
$977$	−44980.6	−1.47293	−0.736467	+	0.676474i	$0.763507\pi$
$978$	−79414.0	−2.59650
$979$	15799.1	0.515774
$980$	0	0
$981$	−14151.9	−0.460585
$982$	−31637.1	−1.02809
$983$	16393.3	0.531908	0.265954	−	0.963986i	$-0.414313\pi$
$983$	16393.3	0.531908	0.265954	+	0.963986i	$0.414313\pi$
$984$	−7024.13	−0.227562
$985$	0	0
$986$	−7005.27	−0.226261
$987$	0	0
$988$	−1318.16	−0.0424456
$989$	5685.98	0.182815
$990$	0	0
$991$	−3883.34	−0.124479	−0.0622393	−	0.998061i	$-0.519824\pi$
$991$	−3883.34	−0.124479	−0.0622393	+	0.998061i	$0.519824\pi$
$992$	3439.92	0.110098
$993$	−1092.13	−0.0349020
$994$	0	0
$995$	0	0
$996$	−3694.95	−0.117549
$997$	−13963.6	−0.443563	−0.221781	−	0.975096i	$-0.571187\pi$
$997$	−13963.6	−0.443563	−0.221781	+	0.975096i	$0.571187\pi$
$998$	−35320.4	−1.12029
$999$	134624.	4.26357

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1225.4.a.bl.1.7		8
5.4	even	2		1225.4.a.bn.1.2		8
7.2	even	3		175.4.e.f.151.2	yes	16
7.4	even	3		175.4.e.f.51.2	yes	16
7.6	odd	2		1225.4.a.bk.1.7		8
35.2	odd	12		175.4.k.e.74.13		32
35.4	even	6		175.4.e.e.51.7	✓	16
35.9	even	6		175.4.e.e.151.7	yes	16
35.18	odd	12		175.4.k.e.149.13		32
35.23	odd	12		175.4.k.e.74.4		32
35.32	odd	12		175.4.k.e.149.4		32
35.34	odd	2		1225.4.a.bo.1.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
175.4.e.e.51.7	✓	16	35.4	even	6
175.4.e.e.151.7	yes	16	35.9	even	6
175.4.e.f.51.2	yes	16	7.4	even	3
175.4.e.f.151.2	yes	16	7.2	even	3
175.4.k.e.74.4		32	35.23	odd	12
175.4.k.e.74.13		32	35.2	odd	12
175.4.k.e.149.4		32	35.32	odd	12
175.4.k.e.149.13		32	35.18	odd	12
1225.4.a.bk.1.7		8	7.6	odd	2
1225.4.a.bl.1.7		8	1.1	even	1	trivial
1225.4.a.bn.1.2		8	5.4	even	2
1225.4.a.bo.1.2		8	35.34	odd	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:	\( N \)	\(=\)	\( 1225 = 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1225.a (trivial)

\(f(q)\)	\(=\)	\(q+2.92095 q^{2} +10.0483 q^{3} +0.531934 q^{4} +29.3505 q^{6} -21.8138 q^{8} +73.9678 q^{9} +15.5936 q^{11} +5.34501 q^{12} +63.9123 q^{13} -67.9725 q^{16} -10.5777 q^{17} +216.056 q^{18} -38.7727 q^{19} +45.5481 q^{22} +24.1227 q^{23} -219.191 q^{24} +186.684 q^{26} +471.945 q^{27} +226.731 q^{29} -143.130 q^{31} -24.0335 q^{32} +156.689 q^{33} -30.8968 q^{34} +39.3460 q^{36} +285.253 q^{37} -113.253 q^{38} +642.208 q^{39} +32.0457 q^{41} +235.711 q^{43} +8.29477 q^{44} +70.4612 q^{46} -144.296 q^{47} -683.006 q^{48} -106.287 q^{51} +33.9971 q^{52} -318.581 q^{53} +1378.53 q^{54} -389.599 q^{57} +662.270 q^{58} +448.322 q^{59} +166.695 q^{61} -418.076 q^{62} +473.580 q^{64} +457.680 q^{66} -741.042 q^{67} -5.62662 q^{68} +242.392 q^{69} -373.652 q^{71} -1613.52 q^{72} -237.314 q^{73} +833.209 q^{74} -20.6245 q^{76} +1875.86 q^{78} -465.390 q^{79} +2745.10 q^{81} +93.6037 q^{82} -691.289 q^{83} +688.498 q^{86} +2278.26 q^{87} -340.156 q^{88} +1013.18 q^{89} +12.8317 q^{92} -1438.21 q^{93} -421.482 q^{94} -241.495 q^{96} +1151.67 q^{97} +1153.42 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{2} + 6 q^{3} + 27 q^{4} + 4 q^{6} - 42 q^{8} + 42 q^{9} + 103 q^{12} + 138 q^{13} + 191 q^{16} + 138 q^{17} + 81 q^{18} - 42 q^{19} + 127 q^{22} + 124 q^{23} - 292 q^{24} + 23 q^{26} + 204 q^{27}+ \cdots + 4086 q^{99}+O(q^{100}) \) 8 * q - q^2 + 6 * q^3 + 27 * q^4 + 4 * q^6 - 42 * q^8 + 42 * q^9 + 103 * q^12 + 138 * q^13 + 191 * q^16 + 138 * q^17 + 81 * q^18 - 42 * q^19 + 127 * q^22 + 124 * q^23 - 292 * q^24 + 23 * q^26 + 204 * q^27 + 288 * q^29 + 18 * q^31 - 562 * q^32 + 236 * q^33 - 488 * q^34 - 711 * q^36 + 220 * q^37 + 1104 * q^38 + 22 * q^39 + 168 * q^41 + 60 * q^43 - 880 * q^44 - 98 * q^46 + 1098 * q^47 + 338 * q^48 + 628 * q^51 + 1370 * q^52 - 734 * q^53 + 2163 * q^54 + 914 * q^57 + 924 * q^58 + 558 * q^59 - 1496 * q^61 + 801 * q^62 + 2436 * q^64 - 682 * q^66 - 1166 * q^67 + 2455 * q^68 + 822 * q^69 - 724 * q^71 - 2191 * q^72 + 892 * q^73 + 235 * q^74 - 1509 * q^76 + 2122 * q^78 - 840 * q^79 + 2392 * q^81 + 4154 * q^82 + 1146 * q^83 + 1816 * q^86 + 4286 * q^87 - 788 * q^88 + 2678 * q^89 + 6932 * q^92 - 1986 * q^93 + 3625 * q^94 - 6601 * q^96 + 664 * q^97 + 4086 * q^99

Introduction

L-functions

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