LMFDB - Embedded newform 1225.4.a.bh.1.5

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:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 27x^{3} + 7x^{2} + 120x + 60 $ x^5 - x^4 - 27x^3 + 7x^2 + 120*x + 60
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 35)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-4.31366$ 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)

$f(q)$	$=$	\(q+5.31366 q^{2} -1.93939 q^{3} +20.2350 q^{4} -10.3052 q^{6} +65.0123 q^{8} -23.2388 q^{9} +25.5420 q^{11} -39.2434 q^{12} +64.1014 q^{13} +183.574 q^{16} +27.6952 q^{17} -123.483 q^{18} -0.792436 q^{19} +135.721 q^{22} +108.606 q^{23} -126.084 q^{24} +340.613 q^{26} +97.4324 q^{27} -234.000 q^{29} -129.204 q^{31} +455.349 q^{32} -49.5357 q^{33} +147.163 q^{34} -470.236 q^{36} +38.3108 q^{37} -4.21073 q^{38} -124.317 q^{39} +403.216 q^{41} +172.895 q^{43} +516.840 q^{44} +577.097 q^{46} -206.943 q^{47} -356.020 q^{48} -53.7117 q^{51} +1297.09 q^{52} +144.031 q^{53} +517.722 q^{54} +1.53684 q^{57} -1243.40 q^{58} +679.086 q^{59} +574.717 q^{61} -686.544 q^{62} +950.977 q^{64} -263.216 q^{66} +515.640 q^{67} +560.411 q^{68} -210.630 q^{69} +556.612 q^{71} -1510.81 q^{72} +173.243 q^{73} +203.571 q^{74} -16.0349 q^{76} -660.580 q^{78} +79.3290 q^{79} +438.488 q^{81} +2142.55 q^{82} -1043.56 q^{83} +918.703 q^{86} +453.817 q^{87} +1660.54 q^{88} -652.060 q^{89} +2197.65 q^{92} +250.576 q^{93} -1099.62 q^{94} -883.097 q^{96} -515.714 q^{97} -593.564 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 4 q^{2} - 10 q^{3} + 18 q^{4} - 6 q^{6} + 42 q^{8} + 23 q^{9} + 42 q^{11} - 136 q^{12} - 34 q^{13} + 74 q^{16} - 238 q^{17} - 2 q^{18} + 36 q^{19} + 358 q^{22} + 152 q^{23} + 36 q^{24} + 310 q^{26}+ \cdots + 2652 q^{99}+O(q^{100}) $ 5 * q + 4 * q^2 - 10 * q^3 + 18 * q^4 - 6 * q^6 + 42 * q^8 + 23 * q^9 + 42 * q^11 - 136 * q^12 - 34 * q^13 + 74 * q^16 - 238 * q^17 - 2 * q^18 + 36 * q^19 + 358 * q^22 + 152 * q^23 + 36 * q^24 + 310 * q^26 - 334 * q^27 - 44 * q^29 - 60 * q^31 + 710 * q^32 - 426 * q^33 + 482 * q^34 - 210 * q^36 + 312 * q^37 - 280 * q^38 - 106 * q^39 + 426 * q^41 + 304 * q^43 + 712 * q^44 + 88 * q^46 - 370 * q^47 + 696 * q^48 + 638 * q^51 + 1156 * q^52 + 976 * q^53 - 498 * q^54 - 588 * q^57 - 2722 * q^58 + 432 * q^59 + 442 * q^61 + 956 * q^62 + 1362 * q^64 - 574 * q^66 - 804 * q^67 + 420 * q^68 + 2404 * q^69 + 440 * q^71 - 3150 * q^72 - 564 * q^73 - 1512 * q^74 + 1336 * q^76 - 2742 * q^78 + 1790 * q^79 - 151 * q^81 + 3480 * q^82 - 1656 * q^83 + 1216 * q^86 + 1674 * q^87 - 1092 * q^88 - 746 * q^89 + 572 * q^92 + 676 * q^93 + 826 * q^94 - 2040 * q^96 - 518 * q^97 + 2652 * 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 $

$2$	5.31366	1.87866	0.939331	−	0.343013i	$-0.111447\pi$
$2$	5.31366	1.87866	0.939331	+	0.343013i	$0.111447\pi$
$3$	−1.93939	−0.373235	−0.186617	−	0.982433i	$-0.559753\pi$
$3$	−1.93939	−0.373235	−0.186617	+	0.982433i	$0.559753\pi$
$4$	20.2350	2.52937
$5$	0	0
$5$	0	0
$6$	−10.3052	−0.701182
$7$	0	0
$7$	0	0
$8$	65.0123	2.87317
$9$	−23.2388	−0.860696
$10$	0	0
$11$	25.5420	0.700108	0.350054	−	0.936730i	$-0.386163\pi$
$11$	25.5420	0.700108	0.350054	+	0.936730i	$0.386163\pi$
$12$	−39.2434	−0.944049
$13$	64.1014	1.36758	0.683790	−	0.729679i	$-0.260330\pi$
$13$	64.1014	1.36758	0.683790	+	0.729679i	$0.260330\pi$
$14$	0	0
$15$	0	0
$16$	183.574	2.86834
$17$	27.6952	0.395122	0.197561	−	0.980291i	$-0.436698\pi$
$17$	27.6952	0.395122	0.197561	+	0.980291i	$0.436698\pi$
$18$	−123.483	−1.61696
$19$	−0.792436	−0.00956828	−0.00478414	−	0.999989i	$-0.501523\pi$
$19$	−0.792436	−0.00956828	−0.00478414	+	0.999989i	$0.501523\pi$
$20$	0	0
$21$	0	0
$22$	135.721	1.31527
$23$	108.606	0.984609	0.492305	−	0.870423i	$-0.336155\pi$
$23$	108.606	0.984609	0.492305	+	0.870423i	$0.336155\pi$
$24$	−126.084	−1.07237
$25$	0	0
$26$	340.613	2.56922
$27$	97.4324	0.694477
$28$	0	0
$29$	−234.000	−1.49837	−0.749186	−	0.662360i	$-0.769555\pi$
$29$	−234.000	−1.49837	−0.749186	+	0.662360i	$0.769555\pi$
$30$	0	0
$31$	−129.204	−0.748570	−0.374285	−	0.927314i	$-0.622112\pi$
$31$	−129.204	−0.748570	−0.374285	+	0.927314i	$0.622112\pi$
$32$	455.349	2.51547
$33$	−49.5357	−0.261305
$34$	147.163	0.742300
$35$	0	0
$36$	−470.236	−2.17702
$37$	38.3108	0.170223	0.0851116	−	0.996371i	$-0.472875\pi$
$37$	38.3108	0.170223	0.0851116	+	0.996371i	$0.472875\pi$
$38$	−4.21073	−0.0179756
$39$	−124.317	−0.510429
$40$	0	0
$41$	403.216	1.53590	0.767949	−	0.640511i	$-0.221278\pi$
$41$	403.216	1.53590	0.767949	+	0.640511i	$0.221278\pi$
$42$	0	0
$43$	172.895	0.613167	0.306584	−	0.951844i	$-0.400814\pi$
$43$	172.895	0.613167	0.306584	+	0.951844i	$0.400814\pi$
$44$	516.840	1.77083
$45$	0	0
$46$	577.097	1.84975
$47$	−206.943	−0.642250	−0.321125	−	0.947037i	$-0.604061\pi$
$47$	−206.943	−0.642250	−0.321125	+	0.947037i	$0.604061\pi$
$48$	−356.020	−1.07056
$49$	0	0
$50$	0	0
$51$	−53.7117	−0.147473
$52$	1297.09	3.45911
$53$	144.031	0.373287	0.186643	−	0.982428i	$-0.440239\pi$
$53$	144.031	0.373287	0.186643	+	0.982428i	$0.440239\pi$
$54$	517.722	1.30469
$55$	0	0
$56$	0	0
$57$	1.53684	0.00357122
$58$	−1243.40	−2.81493
$59$	679.086	1.49846	0.749232	−	0.662307i	$-0.230423\pi$
$59$	679.086	1.49846	0.749232	+	0.662307i	$0.230423\pi$
$60$	0	0
$61$	574.717	1.20631	0.603155	−	0.797624i	$-0.293910\pi$
$61$	574.717	1.20631	0.603155	+	0.797624i	$0.293910\pi$
$62$	−686.544	−1.40631
$63$	0	0
$64$	950.977	1.85738
$65$	0	0
$66$	−263.216	−0.490903
$67$	515.640	0.940230	0.470115	−	0.882605i	$-0.344212\pi$
$67$	515.640	0.940230	0.470115	+	0.882605i	$0.344212\pi$
$68$	560.411	0.999409
$69$	−210.630	−0.367491
$70$	0	0
$71$	556.612	0.930391	0.465195	−	0.885208i	$-0.345984\pi$
$71$	556.612	0.930391	0.465195	+	0.885208i	$0.345984\pi$
$72$	−1510.81	−2.47292
$73$	173.243	0.277762	0.138881	−	0.990309i	$-0.455650\pi$
$73$	173.243	0.277762	0.138881	+	0.990309i	$0.455650\pi$
$74$	203.571	0.319792
$75$	0	0
$76$	−16.0349	−0.0242017
$77$	0	0
$78$	−660.580	−0.958923
$79$	79.3290	0.112977	0.0564887	−	0.998403i	$-0.482010\pi$
$79$	79.3290	0.112977	0.0564887	+	0.998403i	$0.482010\pi$
$80$	0	0
$81$	438.488	0.601493
$82$	2142.55	2.88543
$83$	−1043.56	−1.38007	−0.690034	−	0.723777i	$-0.742404\pi$
$83$	−1043.56	−1.38007	−0.690034	+	0.723777i	$0.742404\pi$
$84$	0	0
$85$	0	0
$86$	918.703	1.15193
$87$	453.817	0.559245
$88$	1660.54	2.01153
$89$	−652.060	−0.776609	−0.388304	−	0.921531i	$-0.626939\pi$
$89$	−652.060	−0.776609	−0.388304	+	0.921531i	$0.626939\pi$
$90$	0	0
$91$	0	0
$92$	2197.65	2.49044
$93$	250.576	0.279393
$94$	−1099.62	−1.20657
$95$	0	0
$96$	−883.097	−0.938861
$97$	−515.714	−0.539823	−0.269912	−	0.962885i	$-0.586995\pi$
$97$	−515.714	−0.539823	−0.269912	+	0.962885i	$0.586995\pi$
$98$	0	0
$99$	−593.564	−0.602580
$100$	0	0
$101$	536.339	0.528394	0.264197	−	0.964469i	$-0.414893\pi$
$101$	536.339	0.528394	0.264197	+	0.964469i	$0.414893\pi$
$102$	−285.405	−0.277052
$103$	−381.693	−0.365139	−0.182570	−	0.983193i	$-0.558441\pi$
$103$	−381.693	−0.365139	−0.182570	+	0.983193i	$0.558441\pi$
$104$	4167.38	3.92928
$105$	0	0
$106$	765.332	0.701279
$107$	−1381.12	−1.24783	−0.623915	−	0.781492i	$-0.714459\pi$
$107$	−1381.12	−1.24783	−0.623915	+	0.781492i	$0.714459\pi$
$108$	1971.54	1.75659
$109$	−390.582	−0.343220	−0.171610	−	0.985165i	$-0.554897\pi$
$109$	−390.582	−0.343220	−0.171610	+	0.985165i	$0.554897\pi$
$110$	0	0
$111$	−74.2995	−0.0635333
$112$	0	0
$113$	1643.15	1.36792	0.683958	−	0.729521i	$-0.260257\pi$
$113$	1643.15	1.36792	0.683958	+	0.729521i	$0.260257\pi$
$114$	8.16624	0.00670911
$115$	0	0
$116$	−4734.99	−3.78994
$117$	−1489.64	−1.17707
$118$	3608.43	2.81511
$119$	0	0
$120$	0	0
$121$	−678.609	−0.509849
$122$	3053.85	2.26625
$123$	−781.992	−0.573251
$124$	−2614.43	−1.89341
$125$	0	0
$126$	0	0
$127$	−192.032	−0.134174	−0.0670869	−	0.997747i	$-0.521370\pi$
$127$	−192.032	−0.134174	−0.0670869	+	0.997747i	$0.521370\pi$
$128$	1410.38	0.973914
$129$	−335.310	−0.228856
$130$	0	0
$131$	−2082.90	−1.38919	−0.694594	−	0.719402i	$-0.744416\pi$
$131$	−2082.90	−1.38919	−0.694594	+	0.719402i	$0.744416\pi$
$132$	−1002.35	−0.660936
$133$	0	0
$134$	2739.93	1.76637
$135$	0	0
$136$	1800.53	1.13525
$137$	−78.1709	−0.0487488	−0.0243744	−	0.999703i	$-0.507759\pi$
$137$	−78.1709	−0.0487488	−0.0243744	+	0.999703i	$0.507759\pi$
$138$	−1119.21	−0.690390
$139$	−1393.67	−0.850426	−0.425213	−	0.905093i	$-0.639801\pi$
$139$	−1393.67	−0.850426	−0.425213	+	0.905093i	$0.639801\pi$
$140$	0	0
$141$	401.342	0.239710
$142$	2957.65	1.74789
$143$	1637.28	0.957454
$144$	−4266.03	−2.46877
$145$	0	0
$146$	920.556	0.521820
$147$	0	0
$148$	775.217	0.430557
$149$	32.5002	0.0178693	0.00893463	−	0.999960i	$-0.497156\pi$
$149$	32.5002	0.0178693	0.00893463	+	0.999960i	$0.497156\pi$
$150$	0	0
$151$	466.762	0.251553	0.125777	−	0.992059i	$-0.459858\pi$
$151$	466.762	0.251553	0.125777	+	0.992059i	$0.459858\pi$
$152$	−51.5181	−0.0274913
$153$	−643.602	−0.340080
$154$	0	0
$155$	0	0
$156$	−2515.56	−1.29106
$157$	−1673.50	−0.850701	−0.425351	−	0.905029i	$-0.639849\pi$
$157$	−1673.50	−0.850701	−0.425351	+	0.905029i	$0.639849\pi$
$158$	421.527	0.212246
$159$	−279.332	−0.139324
$160$	0	0
$161$	0	0
$162$	2329.98	1.13000
$163$	1869.36	0.898279	0.449139	−	0.893462i	$-0.351731\pi$
$163$	1869.36	0.898279	0.449139	+	0.893462i	$0.351731\pi$
$164$	8159.06	3.88485
$165$	0	0
$166$	−5545.12	−2.59268
$167$	−46.5250	−0.0215581	−0.0107791	−	0.999942i	$-0.503431\pi$
$167$	−46.5250	−0.0215581	−0.0107791	+	0.999942i	$0.503431\pi$
$168$	0	0
$169$	1911.99	0.870275
$170$	0	0
$171$	18.4152	0.00823538
$172$	3498.52	1.55093
$173$	−2496.28	−1.09704	−0.548521	−	0.836137i	$-0.684809\pi$
$173$	−2496.28	−1.09704	−0.548521	+	0.836137i	$0.684809\pi$
$174$	2411.43	1.05063
$175$	0	0
$176$	4688.83	2.00815
$177$	−1317.01	−0.559279
$178$	−3464.82	−1.45898
$179$	2975.70	1.24254	0.621269	−	0.783598i	$-0.286618\pi$
$179$	2975.70	1.24254	0.621269	+	0.783598i	$0.286618\pi$
$180$	0	0
$181$	−966.273	−0.396809	−0.198405	−	0.980120i	$-0.563576\pi$
$181$	−966.273	−0.396809	−0.198405	+	0.980120i	$0.563576\pi$
$182$	0	0
$183$	−1114.60	−0.450237
$184$	7060.76	2.82895
$185$	0	0
$186$	1331.47	0.524884
$187$	707.389	0.276628
$188$	−4187.48	−1.62449
$189$	0	0
$190$	0	0
$191$	−1545.50	−0.585488	−0.292744	−	0.956191i	$-0.594568\pi$
$191$	−1545.50	−0.585488	−0.292744	+	0.956191i	$0.594568\pi$
$192$	−1844.31	−0.693238
$193$	−2304.05	−0.859322	−0.429661	−	0.902990i	$-0.641367\pi$
$193$	−2304.05	−0.859322	−0.429661	+	0.902990i	$0.641367\pi$
$194$	−2740.33	−1.01415
$195$	0	0
$196$	0	0
$197$	−222.021	−0.0802960	−0.0401480	−	0.999194i	$-0.512783\pi$
$197$	−222.021	−0.0802960	−0.0401480	+	0.999194i	$0.512783\pi$
$198$	−3153.99	−1.13204
$199$	−3580.56	−1.27547	−0.637736	−	0.770255i	$-0.720129\pi$
$199$	−3580.56	−1.27547	−0.637736	+	0.770255i	$0.720129\pi$
$200$	0	0
$201$	−1000.02	−0.350927
$202$	2849.92	0.992673
$203$	0	0
$204$	−1086.85	−0.373014
$205$	0	0
$206$	−2028.19	−0.685973
$207$	−2523.88	−0.847449
$208$	11767.3	3.92268
$209$	−20.2404	−0.00669883
$210$	0	0
$211$	−4181.04	−1.36415	−0.682073	−	0.731284i	$-0.738921\pi$
$211$	−4181.04	−1.36415	−0.682073	+	0.731284i	$0.738921\pi$
$212$	2914.46	0.944180
$213$	−1079.49	−0.347254
$214$	−7338.79	−2.34425
$215$	0	0
$216$	6334.31	1.99535
$217$	0	0
$218$	−2075.42	−0.644794
$219$	−335.986	−0.103670
$220$	0	0
$221$	1775.30	0.540361
$222$	−394.802	−0.119357
$223$	−2361.52	−0.709145	−0.354573	−	0.935028i	$-0.615374\pi$
$223$	−2361.52	−0.709145	−0.354573	+	0.935028i	$0.615374\pi$
$224$	0	0
$225$	0	0
$226$	8731.14	2.56985
$227$	−586.877	−0.171596	−0.0857982	−	0.996313i	$-0.527344\pi$
$227$	−586.877	−0.171596	−0.0857982	+	0.996313i	$0.527344\pi$
$228$	31.0979	0.00903292
$229$	4619.55	1.33305	0.666526	−	0.745482i	$-0.267781\pi$
$229$	4619.55	1.33305	0.666526	+	0.745482i	$0.267781\pi$
$230$	0	0
$231$	0	0
$232$	−15212.9	−4.30507
$233$	−5120.44	−1.43971	−0.719853	−	0.694127i	$-0.755791\pi$
$233$	−5120.44	−1.43971	−0.719853	+	0.694127i	$0.755791\pi$
$234$	−7915.43	−2.21132
$235$	0	0
$236$	13741.3	3.79017
$237$	−153.850	−0.0421671
$238$	0	0
$239$	1127.51	0.305158	0.152579	−	0.988291i	$-0.451242\pi$
$239$	1127.51	0.305158	0.152579	+	0.988291i	$0.451242\pi$
$240$	0	0
$241$	3549.53	0.948736	0.474368	−	0.880327i	$-0.342677\pi$
$241$	3549.53	0.948736	0.474368	+	0.880327i	$0.342677\pi$
$242$	−3605.89	−0.957833
$243$	−3481.07	−0.918975
$244$	11629.4	3.05120
$245$	0	0
$246$	−4155.24	−1.07694
$247$	−50.7963	−0.0130854
$248$	−8399.84	−2.15077
$249$	2023.87	0.515090
$250$	0	0
$251$	−4717.19	−1.18624	−0.593120	−	0.805114i	$-0.702104\pi$
$251$	−4717.19	−1.18624	−0.593120	+	0.805114i	$0.702104\pi$
$252$	0	0
$253$	2774.02	0.689333
$254$	−1020.39	−0.252067
$255$	0	0
$256$	−113.552	−0.0277227
$257$	−6260.31	−1.51948	−0.759742	−	0.650224i	$-0.774675\pi$
$257$	−6260.31	−1.51948	−0.759742	+	0.650224i	$0.774675\pi$
$258$	−1781.72	−0.429942
$259$	0	0
$260$	0	0
$261$	5437.88	1.28964
$262$	−11067.8	−2.60981
$263$	−5753.04	−1.34885	−0.674425	−	0.738344i	$-0.735608\pi$
$263$	−5753.04	−1.34885	−0.674425	+	0.738344i	$0.735608\pi$
$264$	−3220.43	−0.750772
$265$	0	0
$266$	0	0
$267$	1264.60	0.289858
$268$	10433.9	2.37819
$269$	7059.21	1.60003	0.800014	−	0.599982i	$-0.204826\pi$
$269$	7059.21	1.60003	0.800014	+	0.599982i	$0.204826\pi$
$270$	0	0
$271$	8534.52	1.91305	0.956523	−	0.291658i	$-0.0942069\pi$
$271$	8534.52	1.91305	0.956523	+	0.291658i	$0.0942069\pi$
$272$	5084.11	1.13334
$273$	0	0
$274$	−415.373	−0.0915826
$275$	0	0
$276$	−4262.08	−0.929519
$277$	−1313.94	−0.285008	−0.142504	−	0.989794i	$-0.545515\pi$
$277$	−1313.94	−0.285008	−0.142504	+	0.989794i	$0.545515\pi$
$278$	−7405.47	−1.59766
$279$	3002.54	0.644291
$280$	0	0
$281$	−247.229	−0.0524856	−0.0262428	−	0.999656i	$-0.508354\pi$
$281$	−247.229	−0.0524856	−0.0262428	+	0.999656i	$0.508354\pi$
$282$	2132.60	0.450334
$283$	9074.90	1.90617	0.953087	−	0.302697i	$-0.0978872\pi$
$283$	9074.90	1.90617	0.953087	+	0.302697i	$0.0978872\pi$
$284$	11263.0	2.35330
$285$	0	0
$286$	8699.92	1.79873
$287$	0	0
$288$	−10581.7	−2.16505
$289$	−4145.98	−0.843879
$290$	0	0
$291$	1000.17	0.201481
$292$	3505.57	0.702562
$293$	2740.72	0.546466	0.273233	−	0.961948i	$-0.411907\pi$
$293$	2740.72	0.546466	0.273233	+	0.961948i	$0.411907\pi$
$294$	0	0
$295$	0	0
$296$	2490.68	0.489080
$297$	2488.61	0.486209
$298$	172.695	0.0335703
$299$	6961.83	1.34653
$300$	0	0
$301$	0	0
$302$	2480.21	0.472584
$303$	−1040.17	−0.197215
$304$	−145.470	−0.0274451
$305$	0	0
$306$	−3419.88	−0.638894
$307$	6985.46	1.29864	0.649318	−	0.760517i	$-0.275054\pi$
$307$	6985.46	1.29864	0.649318	+	0.760517i	$0.275054\pi$
$308$	0	0
$309$	740.250	0.136283
$310$	0	0
$311$	356.841	0.0650630	0.0325315	−	0.999471i	$-0.489643\pi$
$311$	356.841	0.0650630	0.0325315	+	0.999471i	$0.489643\pi$
$312$	−8082.16	−1.46655
$313$	−6630.12	−1.19731	−0.598653	−	0.801009i	$-0.704297\pi$
$313$	−6630.12	−1.19731	−0.598653	+	0.801009i	$0.704297\pi$
$314$	−8892.42	−1.59818
$315$	0	0
$316$	1605.22	0.285762
$317$	2494.05	0.441891	0.220946	−	0.975286i	$-0.429086\pi$
$317$	2494.05	0.441891	0.220946	+	0.975286i	$0.429086\pi$
$318$	−1484.27	−0.261742
$319$	−5976.83	−1.04902
$320$	0	0
$321$	2678.52	0.465734
$322$	0	0
$323$	−21.9467	−0.00378063
$324$	8872.79	1.52140
$325$	0	0
$326$	9933.13	1.68756
$327$	757.489	0.128102
$328$	26214.0	4.41289
$329$	0	0
$330$	0	0
$331$	−4682.47	−0.777558	−0.388779	−	0.921331i	$-0.627103\pi$
$331$	−4682.47	−0.777558	−0.388779	+	0.921331i	$0.627103\pi$
$332$	−21116.4	−3.49070
$333$	−890.297	−0.146510
$334$	−247.218	−0.0405005
$335$	0	0
$336$	0	0
$337$	−3596.60	−0.581363	−0.290681	−	0.956820i	$-0.593882\pi$
$337$	−3596.60	−0.581363	−0.290681	+	0.956820i	$0.593882\pi$
$338$	10159.7	1.63495
$339$	−3186.70	−0.510554
$340$	0	0
$341$	−3300.12	−0.524080
$342$	97.8523	0.0154715
$343$	0	0
$344$	11240.3	1.76173
$345$	0	0
$346$	−13264.4	−2.06097
$347$	−1899.51	−0.293865	−0.146932	−	0.989147i	$-0.546940\pi$
$347$	−1899.51	−0.293865	−0.146932	+	0.989147i	$0.546940\pi$
$348$	9182.97	1.41454
$349$	1037.55	0.159137	0.0795683	−	0.996829i	$-0.474646\pi$
$349$	1037.55	0.159137	0.0795683	+	0.996829i	$0.474646\pi$
$350$	0	0
$351$	6245.56	0.949752
$352$	11630.5	1.76110
$353$	−4087.08	−0.616242	−0.308121	−	0.951347i	$-0.599700\pi$
$353$	−4087.08	−0.616242	−0.308121	+	0.951347i	$0.599700\pi$
$354$	−6998.13	−1.05070
$355$	0	0
$356$	−13194.4	−1.96433
$357$	0	0
$358$	15811.8	2.33431
$359$	3472.67	0.510530	0.255265	−	0.966871i	$-0.417837\pi$
$359$	3472.67	0.510530	0.255265	+	0.966871i	$0.417837\pi$
$360$	0	0
$361$	−6858.37	−0.999908
$362$	−5134.44	−0.745470
$363$	1316.08	0.190293
$364$	0	0
$365$	0	0
$366$	−5922.59	−0.845844
$367$	8769.14	1.24726	0.623631	−	0.781719i	$-0.285657\pi$
$367$	8769.14	1.24726	0.623631	+	0.781719i	$0.285657\pi$
$368$	19937.3	2.82419
$369$	−9370.25	−1.32194
$370$	0	0
$371$	0	0
$372$	5070.39	0.706687
$373$	−11368.9	−1.57817	−0.789085	−	0.614284i	$-0.789445\pi$
$373$	−11368.9	−1.57817	−0.789085	+	0.614284i	$0.789445\pi$
$374$	3758.82	0.519690
$375$	0	0
$376$	−13453.8	−1.84529
$377$	−14999.8	−2.04914
$378$	0	0
$379$	−12137.4	−1.64500	−0.822501	−	0.568764i	$-0.807422\pi$
$379$	−12137.4	−1.64500	−0.822501	+	0.568764i	$0.807422\pi$
$380$	0	0
$381$	372.424	0.0500784
$382$	−8212.24	−1.09993
$383$	−9869.61	−1.31675	−0.658373	−	0.752692i	$-0.728755\pi$
$383$	−9869.61	−1.31675	−0.658373	+	0.752692i	$0.728755\pi$
$384$	−2735.27	−0.363499
$385$	0	0
$386$	−12242.9	−1.61438
$387$	−4017.86	−0.527751
$388$	−10435.5	−1.36541
$389$	57.1166	0.00744454	0.00372227	−	0.999993i	$-0.498815\pi$
$389$	57.1166	0.00744454	0.00372227	+	0.999993i	$0.498815\pi$
$390$	0	0
$391$	3007.88	0.389041
$392$	0	0
$393$	4039.54	0.518494
$394$	−1179.74	−0.150849
$395$	0	0
$396$	−12010.7	−1.52415
$397$	−7436.17	−0.940078	−0.470039	−	0.882646i	$-0.655760\pi$
$397$	−7436.17	−0.940078	−0.470039	+	0.882646i	$0.655760\pi$
$398$	−19025.8	−2.39618
$399$	0	0
$400$	0	0
$401$	−12465.0	−1.55230	−0.776149	−	0.630550i	$-0.782829\pi$
$401$	−12465.0	−1.55230	−0.776149	+	0.630550i	$0.782829\pi$
$402$	−5313.79	−0.659273
$403$	−8282.15	−1.02373
$404$	10852.8	1.33650
$405$	0	0
$406$	0	0
$407$	978.533	0.119175
$408$	−3491.92	−0.423715
$409$	1708.72	0.206578	0.103289	−	0.994651i	$-0.467063\pi$
$409$	1708.72	0.206578	0.103289	+	0.994651i	$0.467063\pi$
$410$	0	0
$411$	151.604	0.0181948
$412$	−7723.54	−0.923571
$413$	0	0
$414$	−13411.0	−1.59207
$415$	0	0
$416$	29188.5	3.44011
$417$	2702.86	0.317409
$418$	−107.550	−0.0125848
$419$	−10618.8	−1.23810	−0.619050	−	0.785352i	$-0.712482\pi$
$419$	−10618.8	−1.23810	−0.619050	+	0.785352i	$0.712482\pi$
$420$	0	0
$421$	13273.5	1.53661	0.768304	−	0.640085i	$-0.221101\pi$
$421$	13273.5	1.53661	0.768304	+	0.640085i	$0.221101\pi$
$422$	−22216.6	−2.56277
$423$	4809.10	0.552781
$424$	9363.80	1.07251
$425$	0	0
$426$	−5736.02	−0.652373
$427$	0	0
$428$	−27946.9	−3.15622
$429$	−3175.31	−0.357355
$430$	0	0
$431$	7918.20	0.884934	0.442467	−	0.896785i	$-0.354103\pi$
$431$	7918.20	0.884934	0.442467	+	0.896785i	$0.354103\pi$
$432$	17886.0	1.99199
$433$	−4433.34	−0.492038	−0.246019	−	0.969265i	$-0.579123\pi$
$433$	−4433.34	−0.492038	−0.246019	+	0.969265i	$0.579123\pi$
$434$	0	0
$435$	0	0
$436$	−7903.40	−0.868129
$437$	−86.0637	−0.00942102
$438$	−1785.31	−0.194761
$439$	−12958.4	−1.40882	−0.704408	−	0.709796i	$-0.748787\pi$
$439$	−12958.4	−1.40882	−0.704408	+	0.709796i	$0.748787\pi$
$440$	0	0
$441$	0	0
$442$	9433.34	1.01515
$443$	−12040.4	−1.29133	−0.645664	−	0.763621i	$-0.723420\pi$
$443$	−12040.4	−1.29133	−0.645664	+	0.763621i	$0.723420\pi$
$444$	−1503.45	−0.160699
$445$	0	0
$446$	−12548.3	−1.33224
$447$	−63.0304	−0.00666943
$448$	0	0
$449$	11586.3	1.21780	0.608899	−	0.793247i	$-0.291611\pi$
$449$	11586.3	1.21780	0.608899	+	0.793247i	$0.291611\pi$
$450$	0	0
$451$	10298.9	1.07529
$452$	33249.1	3.45997
$453$	−905.232	−0.0938886
$454$	−3118.46	−0.322371
$455$	0	0
$456$	99.9135	0.0102607
$457$	9734.34	0.996396	0.498198	−	0.867063i	$-0.333995\pi$
$457$	9734.34	0.996396	0.498198	+	0.867063i	$0.333995\pi$
$458$	24546.7	2.50435
$459$	2698.41	0.274403
$460$	0	0
$461$	1343.41	0.135724	0.0678621	−	0.997695i	$-0.478382\pi$
$461$	1343.41	0.135724	0.0678621	+	0.997695i	$0.478382\pi$
$462$	0	0
$463$	−6613.72	−0.663857	−0.331929	−	0.943305i	$-0.607699\pi$
$463$	−6613.72	−0.663857	−0.331929	+	0.943305i	$0.607699\pi$
$464$	−42956.3	−4.29784
$465$	0	0
$466$	−27208.3	−2.70472
$467$	14688.9	1.45551	0.727755	−	0.685838i	$-0.240564\pi$
$467$	14688.9	1.45551	0.727755	+	0.685838i	$0.240564\pi$
$468$	−30142.8	−2.97724
$469$	0	0
$470$	0	0
$471$	3245.57	0.317511
$472$	44148.9	4.30534
$473$	4416.07	0.429283
$474$	−817.504	−0.0792177
$475$	0	0
$476$	0	0
$477$	−3347.11	−0.321286
$478$	5991.22	0.573288
$479$	15298.6	1.45931	0.729657	−	0.683813i	$-0.239680\pi$
$479$	15298.6	1.45931	0.729657	+	0.683813i	$0.239680\pi$
$480$	0	0
$481$	2455.78	0.232794
$482$	18861.0	1.78235
$483$	0	0
$484$	−13731.6	−1.28960
$485$	0	0
$486$	−18497.2	−1.72644
$487$	−9653.80	−0.898266	−0.449133	−	0.893465i	$-0.648267\pi$
$487$	−9653.80	−0.898266	−0.449133	+	0.893465i	$0.648267\pi$
$488$	37363.7	3.46593
$489$	−3625.41	−0.335269
$490$	0	0
$491$	20142.6	1.85137	0.925684	−	0.378297i	$-0.123490\pi$
$491$	20142.6	1.85137	0.925684	+	0.378297i	$0.123490\pi$
$492$	−15823.6	−1.44996
$493$	−6480.69	−0.592039
$494$	−269.914	−0.0245830
$495$	0	0
$496$	−23718.4	−2.14715
$497$	0	0
$498$	10754.1	0.967679
$499$	−1309.29	−0.117459	−0.0587293	−	0.998274i	$-0.518705\pi$
$499$	−1309.29	−0.117459	−0.0587293	+	0.998274i	$0.518705\pi$
$500$	0	0
$501$	90.2299	0.00804625
$502$	−25065.5	−2.22855
$503$	−2186.17	−0.193791	−0.0968953	−	0.995295i	$-0.530891\pi$
$503$	−2186.17	−0.193791	−0.0968953	+	0.995295i	$0.530891\pi$
$504$	0	0
$505$	0	0
$506$	14740.2	1.29502
$507$	−3708.09	−0.324817
$508$	−3885.76	−0.339375
$509$	−3591.11	−0.312718	−0.156359	−	0.987700i	$-0.549976\pi$
$509$	−3591.11	−0.312718	−0.156359	+	0.987700i	$0.549976\pi$
$510$	0	0
$511$	0	0
$512$	−11886.4	−1.02600
$513$	−77.2089	−0.00664495
$514$	−33265.2	−2.85460
$515$	0	0
$516$	−6784.97	−0.578860
$517$	−5285.73	−0.449644
$518$	0	0
$519$	4841.24	0.409455
$520$	0	0
$521$	8605.34	0.723621	0.361811	−	0.932252i	$-0.382159\pi$
$521$	8605.34	0.723621	0.361811	+	0.932252i	$0.382159\pi$
$522$	28895.1	2.42280
$523$	−22536.4	−1.88423	−0.942113	−	0.335297i	$-0.891163\pi$
$523$	−22536.4	−1.88423	−0.942113	+	0.335297i	$0.891163\pi$
$524$	−42147.3	−3.51377
$525$	0	0
$526$	−30569.7	−2.53403
$527$	−3578.32	−0.295776
$528$	−9093.45	−0.749510
$529$	−371.637	−0.0305447
$530$	0	0
$531$	−15781.1	−1.28972
$532$	0	0
$533$	25846.7	2.10046
$534$	6719.63	0.544544
$535$	0	0
$536$	33522.9	2.70144
$537$	−5771.03	−0.463758
$538$	37510.2	3.00591
$539$	0	0
$540$	0	0
$541$	8782.44	0.697942	0.348971	−	0.937134i	$-0.386531\pi$
$541$	8782.44	0.697942	0.348971	+	0.937134i	$0.386531\pi$
$542$	45349.5	3.59396
$543$	1873.98	0.148103
$544$	12611.0	0.993917
$545$	0	0
$546$	0	0
$547$	22593.0	1.76601	0.883004	−	0.469366i	$-0.155518\pi$
$547$	22593.0	1.76601	0.883004	+	0.469366i	$0.155518\pi$
$548$	−1581.78	−0.123304
$549$	−13355.7	−1.03827
$550$	0	0
$551$	185.430	0.0143368
$552$	−13693.5	−1.05586
$553$	0	0
$554$	−6981.84	−0.535433
$555$	0	0
$556$	−28200.8	−2.15104
$557$	−7264.13	−0.552587	−0.276294	−	0.961073i	$-0.589106\pi$
$557$	−7264.13	−0.552587	−0.276294	+	0.961073i	$0.589106\pi$
$558$	15954.5	1.21040
$559$	11082.8	0.838555
$560$	0	0
$561$	−1371.90	−0.103247
$562$	−1313.69	−0.0986027
$563$	−3851.42	−0.288309	−0.144154	−	0.989555i	$-0.546046\pi$
$563$	−3851.42	−0.288309	−0.144154	+	0.989555i	$0.546046\pi$
$564$	8121.14	0.606315
$565$	0	0
$566$	48220.9	3.58105
$567$	0	0
$568$	36186.7	2.67317
$569$	16580.3	1.22158	0.610792	−	0.791791i	$-0.290851\pi$
$569$	16580.3	1.22158	0.610792	+	0.791791i	$0.290851\pi$
$570$	0	0
$571$	6385.86	0.468021	0.234010	−	0.972234i	$-0.424815\pi$
$571$	6385.86	0.468021	0.234010	+	0.972234i	$0.424815\pi$
$572$	33130.2	2.42175
$573$	2997.32	0.218525
$574$	0	0
$575$	0	0
$576$	−22099.6	−1.59864
$577$	11059.1	0.797912	0.398956	−	0.916970i	$-0.369373\pi$
$577$	11059.1	0.797912	0.398956	+	0.916970i	$0.369373\pi$
$578$	−22030.3	−1.58536
$579$	4468.44	0.320729
$580$	0	0
$581$	0	0
$582$	5314.56	0.378515
$583$	3678.84	0.261341
$584$	11263.0	0.798055
$585$	0	0
$586$	14563.2	1.02662
$587$	7871.25	0.553461	0.276730	−	0.960948i	$-0.410749\pi$
$587$	7871.25	0.553461	0.276730	+	0.960948i	$0.410749\pi$
$588$	0	0
$589$	102.386	0.00716253
$590$	0	0
$591$	430.584	0.0299693
$592$	7032.85	0.488258
$593$	2018.06	0.139750	0.0698750	−	0.997556i	$-0.477740\pi$
$593$	2018.06	0.139750	0.0698750	+	0.997556i	$0.477740\pi$
$594$	13223.6	0.913422
$595$	0	0
$596$	657.640	0.0451980
$597$	6944.08	0.476051
$598$	36992.8	2.52968
$599$	−1356.67	−0.0925409	−0.0462705	−	0.998929i	$-0.514734\pi$
$599$	−1356.67	−0.0925409	−0.0462705	+	0.998929i	$0.514734\pi$
$600$	0	0
$601$	−11178.7	−0.758715	−0.379358	−	0.925250i	$-0.623855\pi$
$601$	−11178.7	−0.758715	−0.379358	+	0.925250i	$0.623855\pi$
$602$	0	0
$603$	−11982.8	−0.809252
$604$	9444.91	0.636272
$605$	0	0
$606$	−5527.10	−0.370500
$607$	−9404.67	−0.628870	−0.314435	−	0.949279i	$-0.601815\pi$
$607$	−9404.67	−0.628870	−0.314435	+	0.949279i	$0.601815\pi$
$608$	−360.835	−0.0240687
$609$	0	0
$610$	0	0
$611$	−13265.3	−0.878328
$612$	−13023.3	−0.860187
$613$	−18938.0	−1.24779	−0.623897	−	0.781507i	$-0.714452\pi$
$613$	−18938.0	−1.24779	−0.623897	+	0.781507i	$0.714452\pi$
$614$	37118.4	2.43970
$615$	0	0
$616$	0	0
$617$	17716.9	1.15600	0.578001	−	0.816036i	$-0.303833\pi$
$617$	17716.9	1.15600	0.578001	+	0.816036i	$0.303833\pi$
$618$	3933.43	0.256029
$619$	6240.33	0.405202	0.202601	−	0.979261i	$-0.435061\pi$
$619$	6240.33	0.405202	0.202601	+	0.979261i	$0.435061\pi$
$620$	0	0
$621$	10581.8	0.683788
$622$	1896.13	0.122231
$623$	0	0
$624$	−22821.4	−1.46408
$625$	0	0
$626$	−35230.2	−2.24933
$627$	39.2539	0.00250024
$628$	−33863.2	−2.15174
$629$	1061.03	0.0672589
$630$	0	0
$631$	−25887.7	−1.63323	−0.816617	−	0.577179i	$-0.804153\pi$
$631$	−25887.7	−1.63323	−0.816617	+	0.577179i	$0.804153\pi$
$632$	5157.37	0.324603
$633$	8108.65	0.509147
$634$	13252.5	0.830165
$635$	0	0
$636$	−5652.27	−0.352401
$637$	0	0
$638$	−31758.8	−1.97076
$639$	−12935.0	−0.800783
$640$	0	0
$641$	18798.4	1.15834	0.579168	−	0.815208i	$-0.303378\pi$
$641$	18798.4	1.15834	0.579168	+	0.815208i	$0.303378\pi$
$642$	14232.8	0.874956
$643$	2287.70	0.140308	0.0701541	−	0.997536i	$-0.477651\pi$
$643$	2287.70	0.140308	0.0701541	+	0.997536i	$0.477651\pi$
$644$	0	0
$645$	0	0
$646$	−116.617	−0.00710253
$647$	−1769.31	−0.107510	−0.0537548	−	0.998554i	$-0.517119\pi$
$647$	−1769.31	−0.107510	−0.0537548	+	0.998554i	$0.517119\pi$
$648$	28507.1	1.72819
$649$	17345.2	1.04909
$650$	0	0
$651$	0	0
$652$	37826.4	2.27208
$653$	3891.48	0.233209	0.116604	−	0.993178i	$-0.462799\pi$
$653$	3891.48	0.233209	0.116604	+	0.993178i	$0.462799\pi$
$654$	4025.04	0.240660
$655$	0	0
$656$	74019.8	4.40547
$657$	−4025.96	−0.239068
$658$	0	0
$659$	20097.6	1.18800	0.594001	−	0.804465i	$-0.297548\pi$
$659$	20097.6	1.18800	0.594001	+	0.804465i	$0.297548\pi$
$660$	0	0
$661$	27167.9	1.59865	0.799326	−	0.600898i	$-0.205190\pi$
$661$	27167.9	1.59865	0.799326	+	0.600898i	$0.205190\pi$
$662$	−24881.0	−1.46077
$663$	−3442.99	−0.201681
$664$	−67844.3	−3.96516
$665$	0	0
$666$	−4730.73	−0.275243
$667$	−25414.0	−1.47531
$668$	−941.430	−0.0545285
$669$	4579.91	0.264678
$670$	0	0
$671$	14679.4	0.844548
$672$	0	0
$673$	25909.7	1.48402	0.742010	−	0.670389i	$-0.233873\pi$
$673$	25909.7	1.48402	0.742010	+	0.670389i	$0.233873\pi$
$674$	−19111.1	−1.09218
$675$	0	0
$676$	38689.1	2.20125
$677$	−4359.22	−0.247472	−0.123736	−	0.992315i	$-0.539488\pi$
$677$	−4359.22	−0.247472	−0.123736	+	0.992315i	$0.539488\pi$
$678$	−16933.0	−0.959159
$679$	0	0
$680$	0	0
$681$	1138.18	0.0640458
$682$	−17535.7	−0.984569
$683$	29721.9	1.66512	0.832559	−	0.553937i	$-0.186875\pi$
$683$	29721.9	1.66512	0.832559	+	0.553937i	$0.186875\pi$
$684$	372.632	0.0208303
$685$	0	0
$686$	0	0
$687$	−8959.10	−0.497541
$688$	31738.9	1.75877
$689$	9232.60	0.510499
$690$	0	0
$691$	4929.13	0.271365	0.135682	−	0.990752i	$-0.456677\pi$
$691$	4929.13	0.271365	0.135682	+	0.990752i	$0.456677\pi$
$692$	−50512.0	−2.77482
$693$	0	0
$694$	−10093.4	−0.552073
$695$	0	0
$696$	29503.7	1.60680
$697$	11167.1	0.606866
$698$	5513.18	0.298964
$699$	9930.51	0.537348
$700$	0	0
$701$	−19358.8	−1.04304	−0.521520	−	0.853239i	$-0.674635\pi$
$701$	−19358.8	−1.04304	−0.521520	+	0.853239i	$0.674635\pi$
$702$	33186.7	1.78426
$703$	−30.3589	−0.00162874
$704$	24289.8	1.30036
$705$	0	0
$706$	−21717.3	−1.15771
$707$	0	0
$708$	−26649.6	−1.41462
$709$	−17186.3	−0.910359	−0.455180	−	0.890400i	$-0.650425\pi$
$709$	−17186.3	−0.910359	−0.455180	+	0.890400i	$0.650425\pi$
$710$	0	0
$711$	−1843.51	−0.0972392
$712$	−42391.9	−2.23133
$713$	−14032.4	−0.737049
$714$	0	0
$715$	0	0
$716$	60213.1	3.14284
$717$	−2186.68	−0.113896
$718$	18452.6	0.959113
$719$	15107.3	0.783598	0.391799	−	0.920051i	$-0.371853\pi$
$719$	15107.3	0.783598	0.391799	+	0.920051i	$0.371853\pi$
$720$	0	0
$721$	0	0
$722$	−36443.0	−1.87849
$723$	−6883.91	−0.354102
$724$	−19552.5	−1.00368
$725$	0	0
$726$	6993.22	0.357497
$727$	15840.9	0.808124	0.404062	−	0.914732i	$-0.367598\pi$
$727$	15840.9	0.808124	0.404062	+	0.914732i	$0.367598\pi$
$728$	0	0
$729$	−5088.04	−0.258499
$730$	0	0
$731$	4788.35	0.242276
$732$	−22553.8	−1.13882
$733$	27639.7	1.39276	0.696382	−	0.717671i	$-0.254792\pi$
$733$	27639.7	1.39276	0.696382	+	0.717671i	$0.254792\pi$
$734$	46596.2	2.34318
$735$	0	0
$736$	49453.8	2.47675
$737$	13170.4	0.658263
$738$	−49790.3	−2.48348
$739$	34874.4	1.73596	0.867982	−	0.496597i	$-0.165417\pi$
$739$	34874.4	1.73596	0.867982	+	0.496597i	$0.165417\pi$
$740$	0	0
$741$	98.5136	0.00488392
$742$	0	0
$743$	−27686.7	−1.36706	−0.683530	−	0.729922i	$-0.739556\pi$
$743$	−27686.7	−1.36706	−0.683530	+	0.729922i	$0.739556\pi$
$744$	16290.5	0.802741
$745$	0	0
$746$	−60410.2	−2.96485
$747$	24251.1	1.18782
$748$	14314.0	0.699694
$749$	0	0
$750$	0	0
$751$	4806.85	0.233561	0.116781	−	0.993158i	$-0.462743\pi$
$751$	4806.85	0.233561	0.116781	+	0.993158i	$0.462743\pi$
$752$	−37989.3	−1.84219
$753$	9148.45	0.442747
$754$	−79703.6	−3.84965
$755$	0	0
$756$	0	0
$757$	−40166.6	−1.92851	−0.964253	−	0.264983i	$-0.914634\pi$
$757$	−40166.6	−1.92851	−0.964253	+	0.264983i	$0.914634\pi$
$758$	−64493.9	−3.09040
$759$	−5379.90	−0.257283
$760$	0	0
$761$	−25912.3	−1.23432	−0.617162	−	0.786836i	$-0.711718\pi$
$761$	−25912.3	−1.23432	−0.617162	+	0.786836i	$0.711718\pi$
$762$	1978.93	0.0940803
$763$	0	0
$764$	−31273.1	−1.48092
$765$	0	0
$766$	−52443.7	−2.47372
$767$	43530.4	2.04927
$768$	220.221	0.0103471
$769$	23231.3	1.08939	0.544697	−	0.838633i	$-0.316645\pi$
$769$	23231.3	1.08939	0.544697	+	0.838633i	$0.316645\pi$
$770$	0	0
$771$	12141.2	0.567125
$772$	−46622.4	−2.17354
$773$	−836.306	−0.0389131	−0.0194566	−	0.999811i	$-0.506194\pi$
$773$	−836.306	−0.0389131	−0.0194566	+	0.999811i	$0.506194\pi$
$774$	−21349.5	−0.991465
$775$	0	0
$776$	−33527.8	−1.55100
$777$	0	0
$778$	303.498	0.0139858
$779$	−319.523	−0.0146959
$780$	0	0
$781$	14217.0	0.651374
$782$	15982.8	0.730875
$783$	−22799.2	−1.04058
$784$	0	0
$785$	0	0
$786$	21464.7	0.974074
$787$	−31945.6	−1.44693	−0.723467	−	0.690359i	$-0.757453\pi$
$787$	−31945.6	−1.44693	−0.723467	+	0.690359i	$0.757453\pi$
$788$	−4492.58	−0.203098
$789$	11157.4	0.503438
$790$	0	0
$791$	0	0
$792$	−38589.0	−1.73131
$793$	36840.2	1.64973
$794$	−39513.3	−1.76609
$795$	0	0
$796$	−72452.4	−3.22614
$797$	−5628.83	−0.250168	−0.125084	−	0.992146i	$-0.539920\pi$
$797$	−5628.83	−0.250168	−0.125084	+	0.992146i	$0.539920\pi$
$798$	0	0
$799$	−5731.32	−0.253767
$800$	0	0
$801$	15153.1	0.668424
$802$	−66234.6	−2.91624
$803$	4424.97	0.194463
$804$	−20235.4	−0.887623
$805$	0	0
$806$	−44008.5	−1.92324
$807$	−13690.5	−0.597186
$808$	34868.7	1.51816
$809$	−27890.7	−1.21210	−0.606048	−	0.795428i	$-0.707246\pi$
$809$	−27890.7	−1.21210	−0.606048	+	0.795428i	$0.707246\pi$
$810$	0	0
$811$	12170.3	0.526949	0.263474	−	0.964666i	$-0.415132\pi$
$811$	12170.3	0.526949	0.263474	+	0.964666i	$0.415132\pi$
$812$	0	0
$813$	−16551.7	−0.714015
$814$	5199.59	0.223889
$815$	0	0
$816$	−9860.04	−0.423003
$817$	−137.008	−0.00586696
$818$	9079.53	0.388091
$819$	0	0
$820$	0	0
$821$	−30932.3	−1.31491	−0.657457	−	0.753492i	$-0.728368\pi$
$821$	−30932.3	−1.31491	−0.657457	+	0.753492i	$0.728368\pi$
$822$	805.569	0.0341818
$823$	−25949.0	−1.09906	−0.549528	−	0.835475i	$-0.685193\pi$
$823$	−25949.0	−1.09906	−0.549528	+	0.835475i	$0.685193\pi$
$824$	−24814.7	−1.04911
$825$	0	0
$826$	0	0
$827$	27728.5	1.16592	0.582959	−	0.812502i	$-0.301895\pi$
$827$	27728.5	1.16592	0.582959	+	0.812502i	$0.301895\pi$
$828$	−51070.6	−2.14351
$829$	10650.1	0.446194	0.223097	−	0.974796i	$-0.428383\pi$
$829$	10650.1	0.446194	0.223097	+	0.974796i	$0.428383\pi$
$830$	0	0
$831$	2548.24	0.106375
$832$	60959.0	2.54011
$833$	0	0
$834$	14362.1	0.596304
$835$	0	0
$836$	−409.563	−0.0169438
$837$	−12588.6	−0.519865
$838$	−56424.8	−2.32597
$839$	−29761.9	−1.22466	−0.612332	−	0.790601i	$-0.709768\pi$
$839$	−29761.9	−1.22466	−0.612332	+	0.790601i	$0.709768\pi$
$840$	0	0
$841$	30367.2	1.24512
$842$	70531.0	2.88677
$843$	479.473	0.0195895
$844$	−84603.2	−3.45043
$845$	0	0
$846$	25553.9	1.03849
$847$	0	0
$848$	26440.3	1.07071
$849$	−17599.7	−0.711451
$850$	0	0
$851$	4160.80	0.167603
$852$	−21843.4	−0.878334
$853$	24410.2	0.979824	0.489912	−	0.871772i	$-0.337029\pi$
$853$	24410.2	0.979824	0.489912	+	0.871772i	$0.337029\pi$
$854$	0	0
$855$	0	0
$856$	−89789.8	−3.58522
$857$	−15189.3	−0.605433	−0.302717	−	0.953081i	$-0.597894\pi$
$857$	−15189.3	−0.605433	−0.302717	+	0.953081i	$0.597894\pi$
$858$	−16872.5	−0.671349
$859$	10339.1	0.410668	0.205334	−	0.978692i	$-0.434172\pi$
$859$	10339.1	0.410668	0.205334	+	0.978692i	$0.434172\pi$
$860$	0	0
$861$	0	0
$862$	42074.6	1.66249
$863$	27566.6	1.08734	0.543671	−	0.839298i	$-0.317034\pi$
$863$	27566.6	1.08734	0.543671	+	0.839298i	$0.317034\pi$
$864$	44365.7	1.74693
$865$	0	0
$866$	−23557.2	−0.924373
$867$	8040.65	0.314965
$868$	0	0
$869$	2026.22	0.0790964
$870$	0	0
$871$	33053.2	1.28584
$872$	−25392.6	−0.986127
$873$	11984.6	0.464624
$874$	−457.313	−0.0176989
$875$	0	0
$876$	−6798.65	−0.262221
$877$	−6032.49	−0.232272	−0.116136	−	0.993233i	$-0.537051\pi$
$877$	−6032.49	−0.232272	−0.116136	+	0.993233i	$0.537051\pi$
$878$	−68856.4	−2.64669
$879$	−5315.31	−0.203960
$880$	0	0
$881$	−44253.2	−1.69231	−0.846156	−	0.532935i	$-0.821089\pi$
$881$	−44253.2	−1.69231	−0.846156	+	0.532935i	$0.821089\pi$
$882$	0	0
$883$	−7744.13	−0.295142	−0.147571	−	0.989051i	$-0.547146\pi$
$883$	−7744.13	−0.295142	−0.147571	+	0.989051i	$0.547146\pi$
$884$	35923.1	1.36677
$885$	0	0
$886$	−63978.8	−2.42597
$887$	−18632.5	−0.705318	−0.352659	−	0.935752i	$-0.614722\pi$
$887$	−18632.5	−0.705318	−0.352659	+	0.935752i	$0.614722\pi$
$888$	−4830.38	−0.182542
$889$	0	0
$890$	0	0
$891$	11199.8	0.421110
$892$	−47785.3	−1.79369
$893$	163.989	0.00614522
$894$	−334.922	−0.0125296
$895$	0	0
$896$	0	0
$897$	−13501.7	−0.502573
$898$	61565.7	2.28783
$899$	30233.7	1.12164
$900$	0	0
$901$	3988.97	0.147494
$902$	54725.0	2.02011
$903$	0	0
$904$	106825.	3.93025
$905$	0	0
$906$	−4810.09	−0.176385
$907$	1258.19	0.0460612	0.0230306	−	0.999735i	$-0.492668\pi$
$907$	1258.19	0.0460612	0.0230306	+	0.999735i	$0.492668\pi$
$908$	−11875.4	−0.434030
$909$	−12463.9	−0.454786
$910$	0	0
$911$	27133.6	0.986800	0.493400	−	0.869802i	$-0.335754\pi$
$911$	27133.6	0.986800	0.493400	+	0.869802i	$0.335754\pi$
$912$	282.123	0.0102435
$913$	−26654.6	−0.966197
$914$	51724.9	1.87189
$915$	0	0
$916$	93476.5	3.37178
$917$	0	0
$918$	14338.4	0.515510
$919$	42534.1	1.52673	0.763367	−	0.645965i	$-0.223545\pi$
$919$	42534.1	1.52673	0.763367	+	0.645965i	$0.223545\pi$
$920$	0	0
$921$	−13547.5	−0.484697
$922$	7138.42	0.254980
$923$	35679.7	1.27238
$924$	0	0
$925$	0	0
$926$	−35143.1	−1.24716
$927$	8870.08	0.314274
$928$	−106552.	−3.76911
$929$	−41849.0	−1.47795	−0.738977	−	0.673730i	$-0.764691\pi$
$929$	−41849.0	−1.47795	−0.738977	+	0.673730i	$0.764691\pi$
$930$	0	0
$931$	0	0
$932$	−103612.	−3.64155
$933$	−692.053	−0.0242838
$934$	78051.9	2.73441
$935$	0	0
$936$	−96844.9	−3.38192
$937$	33281.9	1.16038	0.580188	−	0.814482i	$-0.302979\pi$
$937$	33281.9	1.16038	0.580188	+	0.814482i	$0.302979\pi$
$938$	0	0
$939$	12858.4	0.446876
$940$	0	0
$941$	−12740.6	−0.441373	−0.220687	−	0.975345i	$-0.570830\pi$
$941$	−12740.6	−0.441373	−0.220687	+	0.975345i	$0.570830\pi$
$942$	17245.8	0.596496
$943$	43791.9	1.51226
$944$	124662.	4.29810
$945$	0	0
$946$	23465.5	0.806478
$947$	20681.5	0.709670	0.354835	−	0.934929i	$-0.384537\pi$
$947$	20681.5	0.709670	0.354835	+	0.934929i	$0.384537\pi$
$948$	−3113.14	−0.106656
$949$	11105.1	0.379861
$950$	0	0
$951$	−4836.92	−0.164929
$952$	0	0
$953$	6702.50	0.227823	0.113912	−	0.993491i	$-0.463662\pi$
$953$	6702.50	0.227823	0.113912	+	0.993491i	$0.463662\pi$
$954$	−17785.4	−0.603588
$955$	0	0
$956$	22815.2	0.771857
$957$	11591.4	0.391532
$958$	81291.6	2.74156
$959$	0	0
$960$	0	0
$961$	−13097.4	−0.439643
$962$	13049.2	0.437341
$963$	32095.5	1.07400
$964$	71824.6	2.39970
$965$	0	0
$966$	0	0
$967$	−36459.9	−1.21248	−0.606242	−	0.795280i	$-0.707324\pi$
$967$	−36459.9	−1.21248	−0.606242	+	0.795280i	$0.707324\pi$
$968$	−44117.9	−1.46488
$969$	42.5631	0.00141107
$970$	0	0
$971$	15789.7	0.521850	0.260925	−	0.965359i	$-0.415973\pi$
$971$	15789.7	0.521850	0.260925	+	0.965359i	$0.415973\pi$
$972$	−70439.3	−2.32443
$973$	0	0
$974$	−51297.0	−1.68754
$975$	0	0
$976$	105503.	3.46011
$977$	−34429.2	−1.12742	−0.563709	−	0.825974i	$-0.690626\pi$
$977$	−34429.2	−1.12742	−0.563709	+	0.825974i	$0.690626\pi$
$978$	−19264.2	−0.629857
$979$	−16654.9	−0.543710
$980$	0	0
$981$	9076.65	0.295408
$982$	107031.	3.47809
$983$	21047.5	0.682922	0.341461	−	0.939896i	$-0.389078\pi$
$983$	21047.5	0.682922	0.341461	+	0.939896i	$0.389078\pi$
$984$	−50839.1	−1.64704
$985$	0	0
$986$	−34436.1	−1.11224
$987$	0	0
$988$	−1027.86	−0.0330978
$989$	18777.5	0.603730
$990$	0	0
$991$	23839.1	0.764150	0.382075	−	0.924131i	$-0.375210\pi$
$991$	23839.1	0.764150	0.382075	+	0.924131i	$0.375210\pi$
$992$	−58832.7	−1.88301
$993$	9081.12	0.290212
$994$	0	0
$995$	0	0
$996$	40952.8	1.30285
$997$	−7853.62	−0.249475	−0.124738	−	0.992190i	$-0.539809\pi$
$997$	−7853.62	−0.249475	−0.124738	+	0.992190i	$0.539809\pi$
$998$	−6957.12	−0.220665
$999$	3732.71	0.118216

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1225.4.a.bh.1.5		5
5.2	odd	4		245.4.b.d.99.10		10
5.3	odd	4		245.4.b.d.99.1		10
5.4	even	2		1225.4.a.be.1.1		5
7.6	odd	2		175.4.a.j.1.5		5
21.20	even	2		1575.4.a.bn.1.1		5
35.2	odd	12		245.4.j.f.214.10		20
35.3	even	12		245.4.j.e.79.10		20
35.12	even	12		245.4.j.e.214.10		20
35.13	even	4		35.4.b.a.29.1	✓	10
35.17	even	12		245.4.j.e.79.1		20
35.18	odd	12		245.4.j.f.79.10		20
35.23	odd	12		245.4.j.f.214.1		20
35.27	even	4		35.4.b.a.29.10	yes	10
35.32	odd	12		245.4.j.f.79.1		20
35.33	even	12		245.4.j.e.214.1		20
35.34	odd	2		175.4.a.i.1.1		5
105.62	odd	4		315.4.d.c.64.1		10
105.83	odd	4		315.4.d.c.64.10		10
105.104	even	2		1575.4.a.bq.1.5		5
140.27	odd	4		560.4.g.f.449.6		10
140.83	odd	4		560.4.g.f.449.5		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.4.b.a.29.1	✓	10	35.13	even	4
35.4.b.a.29.10	yes	10	35.27	even	4
175.4.a.i.1.1		5	35.34	odd	2
175.4.a.j.1.5		5	7.6	odd	2
245.4.b.d.99.1		10	5.3	odd	4
245.4.b.d.99.10		10	5.2	odd	4
245.4.j.e.79.1		20	35.17	even	12
245.4.j.e.79.10		20	35.3	even	12
245.4.j.e.214.1		20	35.33	even	12
245.4.j.e.214.10		20	35.12	even	12
245.4.j.f.79.1		20	35.32	odd	12
245.4.j.f.79.10		20	35.18	odd	12
245.4.j.f.214.1		20	35.23	odd	12
245.4.j.f.214.10		20	35.2	odd	12
315.4.d.c.64.1		10	105.62	odd	4
315.4.d.c.64.10		10	105.83	odd	4
560.4.g.f.449.5		10	140.83	odd	4
560.4.g.f.449.6		10	140.27	odd	4
1225.4.a.be.1.1		5	5.4	even	2
1225.4.a.bh.1.5		5	1.1	even	1	trivial
1575.4.a.bn.1.1		5	21.20	even	2
1575.4.a.bq.1.5		5	105.104	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:	\( N \)	\(=\)	\( 1225 = 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1225.a (trivial)

\(f(q)\)	\(=\)	\(q+5.31366 q^{2} -1.93939 q^{3} +20.2350 q^{4} -10.3052 q^{6} +65.0123 q^{8} -23.2388 q^{9} +25.5420 q^{11} -39.2434 q^{12} +64.1014 q^{13} +183.574 q^{16} +27.6952 q^{17} -123.483 q^{18} -0.792436 q^{19} +135.721 q^{22} +108.606 q^{23} -126.084 q^{24} +340.613 q^{26} +97.4324 q^{27} -234.000 q^{29} -129.204 q^{31} +455.349 q^{32} -49.5357 q^{33} +147.163 q^{34} -470.236 q^{36} +38.3108 q^{37} -4.21073 q^{38} -124.317 q^{39} +403.216 q^{41} +172.895 q^{43} +516.840 q^{44} +577.097 q^{46} -206.943 q^{47} -356.020 q^{48} -53.7117 q^{51} +1297.09 q^{52} +144.031 q^{53} +517.722 q^{54} +1.53684 q^{57} -1243.40 q^{58} +679.086 q^{59} +574.717 q^{61} -686.544 q^{62} +950.977 q^{64} -263.216 q^{66} +515.640 q^{67} +560.411 q^{68} -210.630 q^{69} +556.612 q^{71} -1510.81 q^{72} +173.243 q^{73} +203.571 q^{74} -16.0349 q^{76} -660.580 q^{78} +79.3290 q^{79} +438.488 q^{81} +2142.55 q^{82} -1043.56 q^{83} +918.703 q^{86} +453.817 q^{87} +1660.54 q^{88} -652.060 q^{89} +2197.65 q^{92} +250.576 q^{93} -1099.62 q^{94} -883.097 q^{96} -515.714 q^{97} -593.564 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 4 q^{2} - 10 q^{3} + 18 q^{4} - 6 q^{6} + 42 q^{8} + 23 q^{9} + 42 q^{11} - 136 q^{12} - 34 q^{13} + 74 q^{16} - 238 q^{17} - 2 q^{18} + 36 q^{19} + 358 q^{22} + 152 q^{23} + 36 q^{24} + 310 q^{26}+ \cdots + 2652 q^{99}+O(q^{100}) \) 5 * q + 4 * q^2 - 10 * q^3 + 18 * q^4 - 6 * q^6 + 42 * q^8 + 23 * q^9 + 42 * q^11 - 136 * q^12 - 34 * q^13 + 74 * q^16 - 238 * q^17 - 2 * q^18 + 36 * q^19 + 358 * q^22 + 152 * q^23 + 36 * q^24 + 310 * q^26 - 334 * q^27 - 44 * q^29 - 60 * q^31 + 710 * q^32 - 426 * q^33 + 482 * q^34 - 210 * q^36 + 312 * q^37 - 280 * q^38 - 106 * q^39 + 426 * q^41 + 304 * q^43 + 712 * q^44 + 88 * q^46 - 370 * q^47 + 696 * q^48 + 638 * q^51 + 1156 * q^52 + 976 * q^53 - 498 * q^54 - 588 * q^57 - 2722 * q^58 + 432 * q^59 + 442 * q^61 + 956 * q^62 + 1362 * q^64 - 574 * q^66 - 804 * q^67 + 420 * q^68 + 2404 * q^69 + 440 * q^71 - 3150 * q^72 - 564 * q^73 - 1512 * q^74 + 1336 * q^76 - 2742 * q^78 + 1790 * q^79 - 151 * q^81 + 3480 * q^82 - 1656 * q^83 + 1216 * q^86 + 1674 * q^87 - 1092 * q^88 - 746 * q^89 + 572 * q^92 + 676 * q^93 + 826 * q^94 - 2040 * q^96 - 518 * q^97 + 2652 * q^99

Introduction

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