LMFDB - Embedded newform 175.4.a.j.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [175,4,Mod(1,175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(175, 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("175.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 175 = 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	175.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:	$10.3253342510$
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	$\chi$	$=$	175.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+5.31366 q^{2} +1.93939 q^{3} +20.2350 q^{4} +10.3052 q^{6} +7.00000 q^{7} +65.0123 q^{8} -23.2388 q^{9} +25.5420 q^{11} +39.2434 q^{12} -64.1014 q^{13} +37.1956 q^{14} +183.574 q^{16} -27.6952 q^{17} -123.483 q^{18} +0.792436 q^{19} +13.5757 q^{21} +135.721 q^{22} +108.606 q^{23} +126.084 q^{24} -340.613 q^{26} -97.4324 q^{27} +141.645 q^{28} -234.000 q^{29} +129.204 q^{31} +455.349 q^{32} +49.5357 q^{33} -147.163 q^{34} -470.236 q^{36} +38.3108 q^{37} +4.21073 q^{38} -124.317 q^{39} -403.216 q^{41} +72.1366 q^{42} +172.895 q^{43} +516.840 q^{44} +577.097 q^{46} +206.943 q^{47} +356.020 q^{48} +49.0000 q^{49} -53.7117 q^{51} -1297.09 q^{52} +144.031 q^{53} -517.722 q^{54} +455.086 q^{56} +1.53684 q^{57} -1243.40 q^{58} -679.086 q^{59} -574.717 q^{61} +686.544 q^{62} -162.671 q^{63} +950.977 q^{64} +263.216 q^{66} +515.640 q^{67} -560.411 q^{68} +210.630 q^{69} +556.612 q^{71} -1510.81 q^{72} -173.243 q^{73} +203.571 q^{74} +16.0349 q^{76} +178.794 q^{77} -660.580 q^{78} +79.3290 q^{79} +438.488 q^{81} -2142.55 q^{82} +1043.56 q^{83} +274.704 q^{84} +918.703 q^{86} -453.817 q^{87} +1660.54 q^{88} +652.060 q^{89} -448.710 q^{91} +2197.65 q^{92} +250.576 q^{93} +1099.62 q^{94} +883.097 q^{96} +515.714 q^{97} +260.369 q^{98} -593.564 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 4 q^{2} + 10 q^{3} + 18 q^{4} + 6 q^{6} + 35 q^{7} + 42 q^{8} + 23 q^{9} + 42 q^{11} + 136 q^{12} + 34 q^{13} + 28 q^{14} + 74 q^{16} + 238 q^{17} - 2 q^{18} - 36 q^{19} + 70 q^{21} + 358 q^{22} + 152 q^{23}+ \cdots + 2652 q^{99}+O(q^{100}) $ 5 * q + 4 * q^2 + 10 * q^3 + 18 * q^4 + 6 * q^6 + 35 * q^7 + 42 * q^8 + 23 * q^9 + 42 * q^11 + 136 * q^12 + 34 * q^13 + 28 * q^14 + 74 * q^16 + 238 * q^17 - 2 * q^18 - 36 * q^19 + 70 * q^21 + 358 * q^22 + 152 * q^23 - 36 * q^24 - 310 * q^26 + 334 * q^27 + 126 * q^28 - 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 + 42 * q^42 + 304 * q^43 + 712 * q^44 + 88 * q^46 + 370 * q^47 - 696 * q^48 + 245 * q^49 + 638 * q^51 - 1156 * q^52 + 976 * q^53 + 498 * q^54 + 294 * q^56 - 588 * q^57 - 2722 * q^58 - 432 * q^59 - 442 * q^61 - 956 * q^62 + 161 * q^63 + 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 + 294 * q^77 - 2742 * q^78 + 1790 * q^79 - 151 * q^81 - 3480 * q^82 + 1656 * q^83 + 952 * q^84 + 1216 * q^86 - 1674 * q^87 - 1092 * q^88 + 746 * q^89 + 238 * q^91 + 572 * q^92 + 676 * q^93 - 826 * q^94 + 2040 * q^96 + 518 * q^97 + 196 * q^98 + 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	$-0.440247\pi$
$3$	1.93939	0.373235	0.186617	+	0.982433i	$0.440247\pi$
$4$	20.2350	2.52937
$5$	0	0
$5$	0	0
$6$	10.3052	0.701182
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	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	$-0.739670\pi$
$13$	−64.1014	−1.36758	−0.683790	+	0.729679i	$0.739670\pi$
$14$	37.1956	0.710067
$15$	0	0
$16$	183.574	2.86834
$17$	−27.6952	−0.395122	−0.197561	−	0.980291i	$-0.563302\pi$
$17$	−27.6952	−0.395122	−0.197561	+	0.980291i	$0.563302\pi$
$18$	−123.483	−1.61696
$19$	0.792436	0.00956828	0.00478414	−	0.999989i	$-0.498477\pi$
$19$	0.792436	0.00956828	0.00478414	+	0.999989i	$0.498477\pi$
$20$	0	0
$21$	13.5757	0.141070
$22$	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
$28$	141.645	0.956012
$29$	−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	$-0.377888\pi$
$31$	129.204	0.748570	0.374285	+	0.927314i	$0.377888\pi$
$32$	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	$-0.778722\pi$
$41$	−403.216	−1.53590	−0.767949	+	0.640511i	$0.778722\pi$
$42$	72.1366	0.265022
$43$	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	$-0.395939\pi$
$47$	206.943	0.642250	0.321125	+	0.947037i	$0.395939\pi$
$48$	356.020	1.07056
$49$	49.0000	0.142857
$50$	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
$56$	455.086	1.08595
$57$	1.53684	0.00357122
$58$	−1243.40	−2.81493
$59$	−679.086	−1.49846	−0.749232	−	0.662307i	$-0.769577\pi$
$59$	−679.086	−1.49846	−0.749232	+	0.662307i	$0.769577\pi$
$60$	0	0
$61$	−574.717	−1.20631	−0.603155	−	0.797624i	$-0.706090\pi$
$61$	−574.717	−1.20631	−0.603155	+	0.797624i	$0.706090\pi$
$62$	686.544	1.40631
$63$	−162.671	−0.325312
$64$	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	$-0.544350\pi$
$73$	−173.243	−0.277762	−0.138881	+	0.990309i	$0.544350\pi$
$74$	203.571	0.319792
$75$	0	0
$76$	16.0349	0.0242017
$77$	178.794	0.264616
$78$	−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	$-0.257596\pi$
$83$	1043.56	1.38007	0.690034	+	0.723777i	$0.257596\pi$
$84$	274.704	0.356817
$85$	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	$-0.373061\pi$
$89$	652.060	0.776609	0.388304	+	0.921531i	$0.373061\pi$
$90$	0	0
$91$	−448.710	−0.516897
$92$	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	$-0.413005\pi$
$97$	515.714	0.539823	0.269912	+	0.962885i	$0.413005\pi$
$98$	260.369	0.268380
$99$	−593.564	−0.602580
$100$	0	0
$101$	−536.339	−0.528394	−0.264197	−	0.964469i	$-0.585107\pi$
$101$	−536.339	−0.528394	−0.264197	+	0.964469i	$0.585107\pi$
$102$	−285.405	−0.277052
$103$	381.693	0.365139	0.182570	−	0.983193i	$-0.441559\pi$
$103$	381.693	0.365139	0.182570	+	0.983193i	$0.441559\pi$
$104$	−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
$112$	1285.02	1.08413
$113$	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
$119$	−193.866	−0.149342
$120$	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
$126$	−864.380	−0.611152
$127$	−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	$-0.255584\pi$
$131$	2082.90	1.38919	0.694594	+	0.719402i	$0.255584\pi$
$132$	1002.35	0.660936
$133$	5.54705	0.00361647
$134$	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	$-0.360199\pi$
$139$	1393.67	0.850426	0.425213	+	0.905093i	$0.360199\pi$
$140$	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
$147$	95.0299	0.0533193
$148$	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
$154$	950.048	0.497124
$155$	0	0
$156$	−2515.56	−1.29106
$157$	1673.50	0.850701	0.425351	−	0.905029i	$-0.360151\pi$
$157$	1673.50	0.850701	0.425351	+	0.905029i	$0.360151\pi$
$158$	421.527	0.212246
$159$	279.332	0.139324
$160$	0	0
$161$	760.245	0.372147
$162$	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	$-0.496569\pi$
$167$	46.5250	0.0215581	0.0107791	+	0.999942i	$0.496569\pi$
$168$	882.588	0.405316
$169$	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	$-0.315191\pi$
$173$	2496.28	1.09704	0.548521	+	0.836137i	$0.315191\pi$
$174$	−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	$-0.436424\pi$
$181$	966.273	0.396809	0.198405	+	0.980120i	$0.436424\pi$
$182$	−2384.29	−0.971074
$183$	−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
$189$	−682.027	−0.262488
$190$	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
$196$	991.513	0.361338
$197$	−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	$-0.279871\pi$
$199$	3580.56	1.27547	0.637736	+	0.770255i	$0.279871\pi$
$200$	0	0
$201$	1000.02	0.350927
$202$	−2849.92	−0.992673
$203$	−1638.00	−0.566331
$204$	−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
$217$	904.426	0.282933
$218$	−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	$-0.384626\pi$
$223$	2361.52	0.709145	0.354573	+	0.935028i	$0.384626\pi$
$224$	3187.44	0.950758
$225$	0	0
$226$	8731.14	2.56985
$227$	586.877	0.171596	0.0857982	−	0.996313i	$-0.472656\pi$
$227$	586.877	0.171596	0.0857982	+	0.996313i	$0.472656\pi$
$228$	31.0979	0.00903292
$229$	−4619.55	−1.33305	−0.666526	−	0.745482i	$-0.732219\pi$
$229$	−4619.55	−1.33305	−0.666526	+	0.745482i	$0.732219\pi$
$230$	0	0
$231$	346.750	0.0987639
$232$	−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
$238$	−1030.14	−0.280563
$239$	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	$-0.657323\pi$
$241$	−3549.53	−0.948736	−0.474368	+	0.880327i	$0.657323\pi$
$242$	−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	$-0.297896\pi$
$251$	4717.19	1.18624	0.593120	+	0.805114i	$0.297896\pi$
$252$	−3291.65	−0.822835
$253$	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	$-0.225325\pi$
$257$	6260.31	1.51948	0.759742	+	0.650224i	$0.225325\pi$
$258$	1781.72	0.429942
$259$	268.176	0.0643383
$260$	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
$266$	29.4751	0.00679412
$267$	1264.60	0.289858
$268$	10433.9	2.37819
$269$	−7059.21	−1.60003	−0.800014	−	0.599982i	$-0.795174\pi$
$269$	−7059.21	−1.60003	−0.800014	+	0.599982i	$0.795174\pi$
$270$	0	0
$271$	−8534.52	−1.91305	−0.956523	−	0.291658i	$-0.905793\pi$
$271$	−8534.52	−1.91305	−0.956523	+	0.291658i	$0.905793\pi$
$272$	−5084.11	−1.13334
$273$	−870.222	−0.192924
$274$	−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	$-0.902113\pi$
$283$	−9074.90	−1.90617	−0.953087	+	0.302697i	$0.902113\pi$
$284$	11263.0	2.35330
$285$	0	0
$286$	−8699.92	−1.79873
$287$	−2822.51	−0.580515
$288$	−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	$-0.588093\pi$
$293$	−2740.72	−0.546466	−0.273233	+	0.961948i	$0.588093\pi$
$294$	504.956	0.100169
$295$	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
$301$	1210.26	0.231755
$302$	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	$-0.724946\pi$
$307$	−6985.46	−1.29864	−0.649318	+	0.760517i	$0.724946\pi$
$308$	3617.88	0.669311
$309$	740.250	0.136283
$310$	0	0
$311$	−356.841	−0.0650630	−0.0325315	−	0.999471i	$-0.510357\pi$
$311$	−356.841	−0.0650630	−0.0325315	+	0.999471i	$0.510357\pi$
$312$	−8082.16	−1.46655
$313$	6630.12	1.19731	0.598653	−	0.801009i	$-0.295703\pi$
$313$	6630.12	1.19731	0.598653	+	0.801009i	$0.295703\pi$
$314$	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
$322$	4039.68	0.699139
$323$	−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
$329$	1448.60	0.242748
$330$	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
$336$	2492.14	0.404635
$337$	−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
$343$	343.000	0.0539949
$344$	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	$-0.525354\pi$
$349$	−1037.55	−0.159137	−0.0795683	+	0.996829i	$0.525354\pi$
$350$	0	0
$351$	6245.56	0.949752
$352$	11630.5	1.76110
$353$	4087.08	0.616242	0.308121	−	0.951347i	$-0.400300\pi$
$353$	4087.08	0.616242	0.308121	+	0.951347i	$0.400300\pi$
$354$	−6998.13	−1.05070
$355$	0	0
$356$	13194.4	1.96433
$357$	−375.982	−0.0557397
$358$	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
$364$	−9079.63	−1.30742
$365$	0	0
$366$	−5922.59	−0.845844
$367$	−8769.14	−1.24726	−0.623631	−	0.781719i	$-0.714343\pi$
$367$	−8769.14	−1.24726	−0.623631	+	0.781719i	$0.714343\pi$
$368$	19937.3	2.82419
$369$	9370.25	1.32194
$370$	0	0
$371$	1008.22	0.141089
$372$	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
$378$	−3624.06	−0.493125
$379$	−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	$-0.271245\pi$
$383$	9869.61	1.31675	0.658373	+	0.752692i	$0.271245\pi$
$384$	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
$392$	3185.60	0.410452
$393$	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	$-0.344240\pi$
$397$	7436.17	0.940078	0.470039	+	0.882646i	$0.344240\pi$
$398$	19025.8	2.39618
$399$	10.7579	0.00134979
$400$	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
$406$	−8703.79	−1.06394
$407$	978.533	0.119175
$408$	−3491.92	−0.423715
$409$	−1708.72	−0.206578	−0.103289	−	0.994651i	$-0.532937\pi$
$409$	−1708.72	−0.206578	−0.103289	+	0.994651i	$0.532937\pi$
$410$	0	0
$411$	−151.604	−0.0181948
$412$	7723.54	0.923571
$413$	−4753.60	−0.566366
$414$	−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	$-0.287518\pi$
$419$	10618.8	1.23810	0.619050	+	0.785352i	$0.287518\pi$
$420$	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
$427$	−4023.02	−0.455943
$428$	−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	$-0.420877\pi$
$433$	4433.34	0.492038	0.246019	+	0.969265i	$0.420877\pi$
$434$	4805.81	0.531535
$435$	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	$-0.251213\pi$
$439$	12958.4	1.40882	0.704408	+	0.709796i	$0.251213\pi$
$440$	0	0
$441$	−1138.70	−0.122957
$442$	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
$448$	6656.84	0.702023
$449$	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	$-0.521618\pi$
$461$	−1343.41	−0.135724	−0.0678621	+	0.997695i	$0.521618\pi$
$462$	1842.51	0.185544
$463$	−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	$-0.759436\pi$
$467$	−14688.9	−1.45551	−0.727755	+	0.685838i	$0.759436\pi$
$468$	30142.8	2.97724
$469$	3609.48	0.355374
$470$	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
$476$	−3922.88	−0.377741
$477$	−3347.11	−0.321286
$478$	5991.22	0.573288
$479$	−15298.6	−1.45931	−0.729657	−	0.683813i	$-0.760320\pi$
$479$	−15298.6	−1.45931	−0.729657	+	0.683813i	$0.760320\pi$
$480$	0	0
$481$	−2455.78	−0.232794
$482$	−18861.0	−1.78235
$483$	1474.41	0.138898
$484$	−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
$497$	3896.29	0.351655
$498$	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	$-0.469109\pi$
$503$	2186.17	0.193791	0.0968953	+	0.995295i	$0.469109\pi$
$504$	−10575.7	−0.934677
$505$	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	$-0.450024\pi$
$509$	3591.11	0.312718	0.156359	+	0.987700i	$0.450024\pi$
$510$	0	0
$511$	−1212.70	−0.104984
$512$	−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
$518$	1424.99	0.120870
$519$	4841.24	0.409455
$520$	0	0
$521$	−8605.34	−0.723621	−0.361811	−	0.932252i	$-0.617841\pi$
$521$	−8605.34	−0.723621	−0.361811	+	0.932252i	$0.617841\pi$
$522$	28895.1	2.42280
$523$	22536.4	1.88423	0.942113	−	0.335297i	$-0.108837\pi$
$523$	22536.4	1.88423	0.942113	+	0.335297i	$0.108837\pi$
$524$	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
$532$	112.244	0.00914738
$533$	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
$539$	1251.56	0.100015
$540$	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
$546$	−4624.06	−0.362439
$547$	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
$553$	555.303	0.0427014
$554$	−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	$-0.453954\pi$
$563$	3851.42	0.288309	0.144154	+	0.989555i	$0.453954\pi$
$564$	8121.14	0.606315
$565$	0	0
$566$	−48220.9	−3.58105
$567$	3069.42	0.227343
$568$	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
$574$	−14997.9	−1.09059
$575$	0	0
$576$	−22099.6	−1.59864
$577$	−11059.1	−0.797912	−0.398956	−	0.916970i	$-0.630627\pi$
$577$	−11059.1	−0.797912	−0.398956	+	0.916970i	$0.630627\pi$
$578$	−22030.3	−1.58536
$579$	−4468.44	−0.320729
$580$	0	0
$581$	7304.92	0.521617
$582$	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	$-0.589251\pi$
$587$	−7871.25	−0.553461	−0.276730	+	0.960948i	$0.589251\pi$
$588$	1922.93	0.134864
$589$	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	$-0.522260\pi$
$593$	−2018.06	−0.139750	−0.0698750	+	0.997556i	$0.522260\pi$
$594$	−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	$-0.376145\pi$
$601$	11178.7	0.758715	0.379358	+	0.925250i	$0.376145\pi$
$602$	6430.92	0.435390
$603$	−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	$-0.398185\pi$
$607$	9404.67	0.628870	0.314435	+	0.949279i	$0.398185\pi$
$608$	360.835	0.0240687
$609$	−3176.72	−0.211375
$610$	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
$616$	11623.8	0.760286
$617$	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	$-0.564939\pi$
$619$	−6240.33	−0.405202	−0.202601	+	0.979261i	$0.564939\pi$
$620$	0	0
$621$	−10581.8	−0.683788
$622$	−1896.13	−0.122231
$623$	4564.42	0.293531
$624$	−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
$637$	−3140.97	−0.195369
$638$	−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	$-0.522349\pi$
$643$	−2287.70	−0.140308	−0.0701541	+	0.997536i	$0.522349\pi$
$644$	15383.5	0.941298
$645$	0	0
$646$	−116.617	−0.00710253
$647$	1769.31	0.107510	0.0537548	−	0.998554i	$-0.482881\pi$
$647$	1769.31	0.107510	0.0537548	+	0.998554i	$0.482881\pi$
$648$	28507.1	1.72819
$649$	−17345.2	−1.04909
$650$	0	0
$651$	1754.03	0.105600
$652$	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
$658$	7697.37	0.456040
$659$	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	$-0.794810\pi$
$661$	−27167.9	−1.59865	−0.799326	+	0.600898i	$0.794810\pi$
$662$	−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
$672$	6181.68	0.354856
$673$	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	$-0.460512\pi$
$677$	4359.22	0.247472	0.123736	+	0.992315i	$0.460512\pi$
$678$	16933.0	0.959159
$679$	3610.00	0.204034
$680$	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
$686$	1822.58	0.101438
$687$	−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	$-0.543323\pi$
$691$	−4929.13	−0.271365	−0.135682	+	0.990752i	$0.543323\pi$
$692$	50512.0	2.77482
$693$	−4154.95	−0.227754
$694$	−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
$707$	−3754.38	−0.199714
$708$	−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
$714$	−1997.84	−0.104716
$715$	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	$-0.628147\pi$
$719$	−15107.3	−0.783598	−0.391799	+	0.920051i	$0.628147\pi$
$720$	0	0
$721$	2671.85	0.138010
$722$	−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	$-0.632402\pi$
$727$	−15840.9	−0.808124	−0.404062	+	0.914732i	$0.632402\pi$
$728$	−29171.7	−1.48513
$729$	−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	$-0.745208\pi$
$733$	−27639.7	−1.39276	−0.696382	+	0.717671i	$0.745208\pi$
$734$	−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
$742$	5357.32	0.265059
$743$	−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
$749$	−9667.83	−0.471635
$750$	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
$756$	−13800.8	−0.663928
$757$	−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	$-0.288282\pi$
$761$	25912.3	1.23432	0.617162	+	0.786836i	$0.288282\pi$
$762$	−1978.93	−0.0940803
$763$	−2734.07	−0.129725
$764$	−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	$-0.683355\pi$
$769$	−23231.3	−1.08939	−0.544697	+	0.838633i	$0.683355\pi$
$770$	0	0
$771$	12141.2	0.567125
$772$	−46622.4	−2.17354
$773$	836.306	0.0389131	0.0194566	−	0.999811i	$-0.493806\pi$
$773$	836.306	0.0389131	0.0194566	+	0.999811i	$0.493806\pi$
$774$	−21349.5	−0.991465
$775$	0	0
$776$	33527.8	1.55100
$777$	520.096	0.0240133
$778$	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
$784$	8995.11	0.409763
$785$	0	0
$786$	21464.7	0.974074
$787$	31945.6	1.44693	0.723467	−	0.690359i	$-0.242547\pi$
$787$	31945.6	1.44693	0.723467	+	0.690359i	$0.242547\pi$
$788$	−4492.58	−0.203098
$789$	−11157.4	−0.503438
$790$	0	0
$791$	11502.1	0.517024
$792$	−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	$-0.460080\pi$
$797$	5628.83	0.250168	0.125084	+	0.992146i	$0.460080\pi$
$798$	57.1637	0.00253580
$799$	−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	$-0.584868\pi$
$811$	−12170.3	−0.526949	−0.263474	+	0.964666i	$0.584868\pi$
$812$	−33144.9	−1.43246
$813$	−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
$819$	10427.5	0.444891
$820$	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
$826$	−25259.0	−1.06401
$827$	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	$-0.571617\pi$
$829$	−10650.1	−0.446194	−0.223097	+	0.974796i	$0.571617\pi$
$830$	0	0
$831$	−2548.24	−0.106375
$832$	−60959.0	−2.54011
$833$	−1357.06	−0.0564460
$834$	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	$-0.290232\pi$
$839$	29761.9	1.22466	0.612332	+	0.790601i	$0.290232\pi$
$840$	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
$847$	−4750.26	−0.192705
$848$	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	$-0.662971\pi$
$853$	−24410.2	−0.979824	−0.489912	+	0.871772i	$0.662971\pi$
$854$	−21376.9	−0.856562
$855$	0	0
$856$	−89789.8	−3.58522
$857$	15189.3	0.605433	0.302717	−	0.953081i	$-0.402106\pi$
$857$	15189.3	0.605433	0.302717	+	0.953081i	$0.402106\pi$
$858$	−16872.5	−0.671349
$859$	−10339.1	−0.410668	−0.205334	−	0.978692i	$-0.565828\pi$
$859$	−10339.1	−0.410668	−0.205334	+	0.978692i	$0.565828\pi$
$860$	0	0
$861$	−5473.94	−0.216668
$862$	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
$868$	18301.0	0.715642
$869$	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	$-0.178911\pi$
$881$	44253.2	1.69231	0.846156	+	0.532935i	$0.178911\pi$
$882$	−6050.66	−0.230994
$883$	−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	$-0.385278\pi$
$887$	18632.5	0.705318	0.352659	+	0.935752i	$0.385278\pi$
$888$	4830.38	0.182542
$889$	−1344.22	−0.0507130
$890$	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
$896$	9872.65	0.368105
$897$	−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
$903$	2347.17	0.0864993
$904$	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
$917$	14580.3	0.525064
$918$	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
$924$	7016.47	0.249810
$925$	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	$-0.235309\pi$
$929$	41849.0	1.47795	0.738977	+	0.673730i	$0.235309\pi$
$930$	0	0
$931$	38.8294	0.00136690
$932$	−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	$-0.697021\pi$
$937$	−33281.9	−1.16038	−0.580188	+	0.814482i	$0.697021\pi$
$938$	19179.5	0.667627
$939$	12858.4	0.446876
$940$	0	0
$941$	12740.6	0.441373	0.220687	−	0.975345i	$-0.429170\pi$
$941$	12740.6	0.441373	0.220687	+	0.975345i	$0.429170\pi$
$942$	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
$952$	−12603.7	−0.429084
$953$	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
$959$	−547.196	−0.0184253
$960$	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
$966$	7834.50	0.260943
$967$	−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	$-0.584027\pi$
$971$	−15789.7	−0.521850	−0.260925	+	0.965359i	$0.584027\pi$
$972$	70439.3	2.32443
$973$	9755.67	0.321431
$974$	−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	$-0.610922\pi$
$983$	−21047.5	−0.682922	−0.341461	+	0.939896i	$0.610922\pi$
$984$	−50839.1	−1.64704
$985$	0	0
$986$	34436.1	1.11224
$987$	2809.40	0.0906019
$988$	−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
$994$	20703.5	0.660640
$995$	0	0
$996$	40952.8	1.30285
$997$	7853.62	0.249475	0.124738	−	0.992190i	$-0.460191\pi$
$997$	7853.62	0.249475	0.124738	+	0.992190i	$0.460191\pi$
$998$	−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	175.4.a.j.1.5		5
3.2	odd	2		1575.4.a.bn.1.1		5
5.2	odd	4		35.4.b.a.29.10	yes	10
5.3	odd	4		35.4.b.a.29.1	✓	10
5.4	even	2		175.4.a.i.1.1		5
7.6	odd	2		1225.4.a.bh.1.5		5
15.2	even	4		315.4.d.c.64.1		10
15.8	even	4		315.4.d.c.64.10		10
15.14	odd	2		1575.4.a.bq.1.5		5
20.3	even	4		560.4.g.f.449.5		10
20.7	even	4		560.4.g.f.449.6		10
35.2	odd	12		245.4.j.e.214.10		20
35.3	even	12		245.4.j.f.79.10		20
35.12	even	12		245.4.j.f.214.10		20
35.13	even	4		245.4.b.d.99.1		10
35.17	even	12		245.4.j.f.79.1		20
35.18	odd	12		245.4.j.e.79.10		20
35.23	odd	12		245.4.j.e.214.1		20
35.27	even	4		245.4.b.d.99.10		10
35.32	odd	12		245.4.j.e.79.1		20
35.33	even	12		245.4.j.f.214.1		20
35.34	odd	2		1225.4.a.be.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.4.b.a.29.1	✓	10	5.3	odd	4
35.4.b.a.29.10	yes	10	5.2	odd	4
175.4.a.i.1.1		5	5.4	even	2
175.4.a.j.1.5		5	1.1	even	1	trivial
245.4.b.d.99.1		10	35.13	even	4
245.4.b.d.99.10		10	35.27	even	4
245.4.j.e.79.1		20	35.32	odd	12
245.4.j.e.79.10		20	35.18	odd	12
245.4.j.e.214.1		20	35.23	odd	12
245.4.j.e.214.10		20	35.2	odd	12
245.4.j.f.79.1		20	35.17	even	12
245.4.j.f.79.10		20	35.3	even	12
245.4.j.f.214.1		20	35.33	even	12
245.4.j.f.214.10		20	35.12	even	12
315.4.d.c.64.1		10	15.2	even	4
315.4.d.c.64.10		10	15.8	even	4
560.4.g.f.449.5		10	20.3	even	4
560.4.g.f.449.6		10	20.7	even	4
1225.4.a.be.1.1		5	35.34	odd	2
1225.4.a.bh.1.5		5	7.6	odd	2
1575.4.a.bn.1.1		5	3.2	odd	2
1575.4.a.bq.1.5		5	15.14	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:	\( N \)	\(=\)	\( 175 = 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	175.a (trivial)

\(f(q)\)	\(=\)	\(q+5.31366 q^{2} +1.93939 q^{3} +20.2350 q^{4} +10.3052 q^{6} +7.00000 q^{7} +65.0123 q^{8} -23.2388 q^{9} +25.5420 q^{11} +39.2434 q^{12} -64.1014 q^{13} +37.1956 q^{14} +183.574 q^{16} -27.6952 q^{17} -123.483 q^{18} +0.792436 q^{19} +13.5757 q^{21} +135.721 q^{22} +108.606 q^{23} +126.084 q^{24} -340.613 q^{26} -97.4324 q^{27} +141.645 q^{28} -234.000 q^{29} +129.204 q^{31} +455.349 q^{32} +49.5357 q^{33} -147.163 q^{34} -470.236 q^{36} +38.3108 q^{37} +4.21073 q^{38} -124.317 q^{39} -403.216 q^{41} +72.1366 q^{42} +172.895 q^{43} +516.840 q^{44} +577.097 q^{46} +206.943 q^{47} +356.020 q^{48} +49.0000 q^{49} -53.7117 q^{51} -1297.09 q^{52} +144.031 q^{53} -517.722 q^{54} +455.086 q^{56} +1.53684 q^{57} -1243.40 q^{58} -679.086 q^{59} -574.717 q^{61} +686.544 q^{62} -162.671 q^{63} +950.977 q^{64} +263.216 q^{66} +515.640 q^{67} -560.411 q^{68} +210.630 q^{69} +556.612 q^{71} -1510.81 q^{72} -173.243 q^{73} +203.571 q^{74} +16.0349 q^{76} +178.794 q^{77} -660.580 q^{78} +79.3290 q^{79} +438.488 q^{81} -2142.55 q^{82} +1043.56 q^{83} +274.704 q^{84} +918.703 q^{86} -453.817 q^{87} +1660.54 q^{88} +652.060 q^{89} -448.710 q^{91} +2197.65 q^{92} +250.576 q^{93} +1099.62 q^{94} +883.097 q^{96} +515.714 q^{97} +260.369 q^{98} -593.564 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 4 q^{2} + 10 q^{3} + 18 q^{4} + 6 q^{6} + 35 q^{7} + 42 q^{8} + 23 q^{9} + 42 q^{11} + 136 q^{12} + 34 q^{13} + 28 q^{14} + 74 q^{16} + 238 q^{17} - 2 q^{18} - 36 q^{19} + 70 q^{21} + 358 q^{22} + 152 q^{23}+ \cdots + 2652 q^{99}+O(q^{100}) \) 5 * q + 4 * q^2 + 10 * q^3 + 18 * q^4 + 6 * q^6 + 35 * q^7 + 42 * q^8 + 23 * q^9 + 42 * q^11 + 136 * q^12 + 34 * q^13 + 28 * q^14 + 74 * q^16 + 238 * q^17 - 2 * q^18 - 36 * q^19 + 70 * q^21 + 358 * q^22 + 152 * q^23 - 36 * q^24 - 310 * q^26 + 334 * q^27 + 126 * q^28 - 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 + 42 * q^42 + 304 * q^43 + 712 * q^44 + 88 * q^46 + 370 * q^47 - 696 * q^48 + 245 * q^49 + 638 * q^51 - 1156 * q^52 + 976 * q^53 + 498 * q^54 + 294 * q^56 - 588 * q^57 - 2722 * q^58 - 432 * q^59 - 442 * q^61 - 956 * q^62 + 161 * q^63 + 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 + 294 * q^77 - 2742 * q^78 + 1790 * q^79 - 151 * q^81 - 3480 * q^82 + 1656 * q^83 + 952 * q^84 + 1216 * q^86 - 1674 * q^87 - 1092 * q^88 + 746 * q^89 + 238 * q^91 + 572 * q^92 + 676 * q^93 - 826 * q^94 + 2040 * q^96 + 518 * q^97 + 196 * q^98 + 2652 * q^99

Introduction

L-functions

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