LMFDB - Embedded newform 175.4.a.e.1.1

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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{41}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 10 $ x^2 - x - 10
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.70156$ 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)

$f(q)$	$=$	\(q-2.70156 q^{2} +0.701562 q^{3} -0.701562 q^{4} -1.89531 q^{6} +7.00000 q^{7} +23.5078 q^{8} -26.5078 q^{9} +4.01562 q^{11} -0.492189 q^{12} +51.6125 q^{13} -18.9109 q^{14} -57.8953 q^{16} -67.5078 q^{17} +71.6125 q^{18} -50.9109 q^{19} +4.91093 q^{21} -10.8485 q^{22} +0.507811 q^{23} +16.4922 q^{24} -139.434 q^{26} -37.5391 q^{27} -4.91093 q^{28} -120.058 q^{29} -292.303 q^{31} -31.6547 q^{32} +2.81721 q^{33} +182.377 q^{34} +18.5969 q^{36} -144.989 q^{37} +137.539 q^{38} +36.2094 q^{39} -57.2047 q^{41} -13.2672 q^{42} -283.020 q^{43} -2.81721 q^{44} -1.37188 q^{46} -233.769 q^{47} -40.6172 q^{48} +49.0000 q^{49} -47.3609 q^{51} -36.2094 q^{52} -406.334 q^{53} +101.414 q^{54} +164.555 q^{56} -35.7172 q^{57} +324.344 q^{58} -577.328 q^{59} +322.116 q^{61} +789.675 q^{62} -185.555 q^{63} +548.680 q^{64} -7.61086 q^{66} +985.459 q^{67} +47.3609 q^{68} +0.356261 q^{69} +1033.57 q^{71} -623.141 q^{72} +692.720 q^{73} +391.697 q^{74} +35.7172 q^{76} +28.1093 q^{77} -97.8219 q^{78} -428.236 q^{79} +689.375 q^{81} +154.542 q^{82} -537.592 q^{83} -3.44533 q^{84} +764.597 q^{86} -84.2280 q^{87} +94.3985 q^{88} -802.073 q^{89} +361.287 q^{91} -0.356261 q^{92} -205.069 q^{93} +631.541 q^{94} -22.2077 q^{96} +1752.82 q^{97} -132.377 q^{98} -106.445 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - 5 q^{3} + 5 q^{4} - 23 q^{6} + 14 q^{7} + 15 q^{8} - 21 q^{9} - 56 q^{11} - 33 q^{12} + 52 q^{13} + 7 q^{14} - 135 q^{16} - 103 q^{17} + 92 q^{18} - 57 q^{19} - 35 q^{21} - 233 q^{22} - 31 q^{23}+ \cdots - 437 q^{99}+O(q^{100}) $ 2 * q + q^2 - 5 * q^3 + 5 * q^4 - 23 * q^6 + 14 * q^7 + 15 * q^8 - 21 * q^9 - 56 * q^11 - 33 * q^12 + 52 * q^13 + 7 * q^14 - 135 * q^16 - 103 * q^17 + 92 * q^18 - 57 * q^19 - 35 * q^21 - 233 * q^22 - 31 * q^23 + 65 * q^24 - 138 * q^26 + 85 * q^27 + 35 * q^28 - 413 * q^29 - 162 * q^31 - 249 * q^32 + 345 * q^33 + 51 * q^34 + 50 * q^36 + 75 * q^37 + 115 * q^38 + 34 * q^39 - 505 * q^41 - 161 * q^42 - 73 * q^43 - 345 * q^44 - 118 * q^46 + 224 * q^47 + 399 * q^48 + 98 * q^49 + 155 * q^51 - 34 * q^52 - 262 * q^53 + 555 * q^54 + 105 * q^56 - q^57 - 760 * q^58 + 190 * q^59 + 990 * q^61 + 1272 * q^62 - 147 * q^63 + 361 * q^64 + 1259 * q^66 + 908 * q^67 - 155 * q^68 + 180 * q^69 + 127 * q^71 - 670 * q^72 - 337 * q^73 + 1206 * q^74 + q^76 - 392 * q^77 - 106 * q^78 - 1119 * q^79 - 158 * q^81 - 1503 * q^82 - 1517 * q^83 - 231 * q^84 + 1542 * q^86 + 1586 * q^87 + 605 * q^88 - 1713 * q^89 + 364 * q^91 - 180 * q^92 - 948 * q^93 + 2326 * q^94 + 1217 * q^96 + 1764 * q^97 + 49 * q^98 - 437 * 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 $

$2$	−2.70156	−0.955146	−0.477573	−	0.878592i	$-0.658483\pi$
$2$	−2.70156	−0.955146	−0.477573	+	0.878592i	$0.658483\pi$
$3$	0.701562	0.135016	0.0675078	−	0.997719i	$-0.478495\pi$
$3$	0.701562	0.135016	0.0675078	+	0.997719i	$0.478495\pi$
$4$	−0.701562	−0.0876953
$5$	0	0
$5$	0	0
$6$	−1.89531	−0.128960
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	23.5078	1.03891
$9$	−26.5078	−0.981771
$10$	0	0
$11$	4.01562	0.110069	0.0550343	−	0.998484i	$-0.482473\pi$
$11$	4.01562	0.110069	0.0550343	+	0.998484i	$0.482473\pi$
$12$	−0.492189	−0.0118402
$13$	51.6125	1.10113	0.550567	−	0.834791i	$-0.314412\pi$
$13$	51.6125	1.10113	0.550567	+	0.834791i	$0.314412\pi$
$14$	−18.9109	−0.361011
$15$	0	0
$16$	−57.8953	−0.904614
$17$	−67.5078	−0.963121	−0.481560	−	0.876413i	$-0.659930\pi$
$17$	−67.5078	−0.963121	−0.481560	+	0.876413i	$0.659930\pi$
$18$	71.6125	0.937735
$19$	−50.9109	−0.614725	−0.307362	−	0.951593i	$-0.599446\pi$
$19$	−50.9109	−0.614725	−0.307362	+	0.951593i	$0.599446\pi$
$20$	0	0
$21$	4.91093	0.0510311
$22$	−10.8485	−0.105132
$23$	0.507811	0.00460373	0.00230187	−	0.999997i	$-0.499267\pi$
$23$	0.507811	0.00460373	0.00230187	+	0.999997i	$0.499267\pi$
$24$	16.4922	0.140269
$25$	0	0
$26$	−139.434	−1.05174
$27$	−37.5391	−0.267570
$28$	−4.91093	−0.0331457
$29$	−120.058	−0.768765	−0.384382	−	0.923174i	$-0.625586\pi$
$29$	−120.058	−0.768765	−0.384382	+	0.923174i	$0.625586\pi$
$30$	0	0
$31$	−292.303	−1.69352	−0.846761	−	0.531973i	$-0.821451\pi$
$31$	−292.303	−1.69352	−0.846761	+	0.531973i	$0.821451\pi$
$32$	−31.6547	−0.174869
$33$	2.81721	0.0148610
$34$	182.377	0.919921
$35$	0	0
$36$	18.5969	0.0860966
$37$	−144.989	−0.644218	−0.322109	−	0.946703i	$-0.604392\pi$
$37$	−144.989	−0.644218	−0.322109	+	0.946703i	$0.604392\pi$
$38$	137.539	0.587152
$39$	36.2094	0.148670
$40$	0	0
$41$	−57.2047	−0.217899	−0.108950	−	0.994047i	$-0.534749\pi$
$41$	−57.2047	−0.217899	−0.108950	+	0.994047i	$0.534749\pi$
$42$	−13.2672	−0.0487422
$43$	−283.020	−1.00373	−0.501863	−	0.864947i	$-0.667352\pi$
$43$	−283.020	−1.00373	−0.501863	+	0.864947i	$0.667352\pi$
$44$	−2.81721	−0.00965250
$45$	0	0
$46$	−1.37188	−0.00439724
$47$	−233.769	−0.725504	−0.362752	−	0.931886i	$-0.618163\pi$
$47$	−233.769	−0.725504	−0.362752	+	0.931886i	$0.618163\pi$
$48$	−40.6172	−0.122137
$49$	49.0000	0.142857
$50$	0	0
$51$	−47.3609	−0.130036
$52$	−36.2094	−0.0965642
$53$	−406.334	−1.05310	−0.526550	−	0.850144i	$-0.676515\pi$
$53$	−406.334	−1.05310	−0.526550	+	0.850144i	$0.676515\pi$
$54$	101.414	0.255569
$55$	0	0
$56$	164.555	0.392670
$57$	−35.7172	−0.0829975
$58$	324.344	0.734283
$59$	−577.328	−1.27393	−0.636964	−	0.770894i	$-0.719810\pi$
$59$	−577.328	−1.27393	−0.636964	+	0.770894i	$0.719810\pi$
$60$	0	0
$61$	322.116	0.676110	0.338055	−	0.941126i	$-0.390231\pi$
$61$	322.116	0.676110	0.338055	+	0.941126i	$0.390231\pi$
$62$	789.675	1.61756
$63$	−185.555	−0.371074
$64$	548.680	1.07164
$65$	0	0
$66$	−7.61086	−0.0141944
$67$	985.459	1.79691	0.898455	−	0.439065i	$-0.144690\pi$
$67$	985.459	1.79691	0.898455	+	0.439065i	$0.144690\pi$
$68$	47.3609	0.0844611
$69$	0.356261	0.000621576 0
$70$	0	0
$71$	1033.57	1.72764	0.863821	−	0.503799i	$-0.168065\pi$
$71$	1033.57	1.72764	0.863821	+	0.503799i	$0.168065\pi$
$72$	−623.141	−1.01997
$73$	692.720	1.11064	0.555320	−	0.831637i	$-0.312596\pi$
$73$	692.720	1.11064	0.555320	+	0.831637i	$0.312596\pi$
$74$	391.697	0.615322
$75$	0	0
$76$	35.7172	0.0539084
$77$	28.1093	0.0416020
$78$	−97.8219	−0.142002
$79$	−428.236	−0.609877	−0.304939	−	0.952372i	$-0.598636\pi$
$79$	−428.236	−0.609877	−0.304939	+	0.952372i	$0.598636\pi$
$80$	0	0
$81$	689.375	0.945645
$82$	154.542	0.208126
$83$	−537.592	−0.710945	−0.355472	−	0.934687i	$-0.615680\pi$
$83$	−537.592	−0.710945	−0.355472	+	0.934687i	$0.615680\pi$
$84$	−3.44533	−0.00447519
$85$	0	0
$86$	764.597	0.958705
$87$	−84.2280	−0.103795
$88$	94.3985	0.114351
$89$	−802.073	−0.955277	−0.477638	−	0.878557i	$-0.658507\pi$
$89$	−802.073	−0.955277	−0.477638	+	0.878557i	$0.658507\pi$
$90$	0	0
$91$	361.287	0.416189
$92$	−0.356261	−0.000403725 0
$93$	−205.069	−0.228652
$94$	631.541	0.692962
$95$	0	0
$96$	−22.2077	−0.0236101
$97$	1752.82	1.83477	0.917384	−	0.398004i	$-0.130297\pi$
$97$	1752.82	1.83477	0.917384	+	0.398004i	$0.130297\pi$
$98$	−132.377	−0.136449
$99$	−106.445	−0.108062
$100$	0	0
$101$	−1987.85	−1.95840	−0.979200	−	0.202896i	$-0.934965\pi$
$101$	−1987.85	−1.95840	−0.979200	+	0.202896i	$0.934965\pi$
$102$	127.948	0.124204
$103$	−389.122	−0.372246	−0.186123	−	0.982526i	$-0.559592\pi$
$103$	−389.122	−0.372246	−0.186123	+	0.982526i	$0.559592\pi$
$104$	1213.30	1.14398
$105$	0	0
$106$	1097.74	1.00586
$107$	−508.595	−0.459512	−0.229756	−	0.973248i	$-0.573793\pi$
$107$	−508.595	−0.459512	−0.229756	+	0.973248i	$0.573793\pi$
$108$	26.3360	0.0234646
$109$	344.130	0.302400	0.151200	−	0.988503i	$-0.451686\pi$
$109$	344.130	0.302400	0.151200	+	0.988503i	$0.451686\pi$
$110$	0	0
$111$	−101.719	−0.0869795
$112$	−405.267	−0.341912
$113$	−1218.12	−1.01408	−0.507040	−	0.861923i	$-0.669260\pi$
$113$	−1218.12	−1.01408	−0.507040	+	0.861923i	$0.669260\pi$
$114$	96.4922	0.0792748
$115$	0	0
$116$	84.2280	0.0674170
$117$	−1368.13	−1.08106
$118$	1559.69	1.21679
$119$	−472.555	−0.364025
$120$	0	0
$121$	−1314.87	−0.987885
$122$	−870.215	−0.645784
$123$	−40.1327	−0.0294198
$124$	205.069	0.148514
$125$	0	0
$126$	501.287	0.354430
$127$	−281.652	−0.196792	−0.0983958	−	0.995147i	$-0.531371\pi$
$127$	−281.652	−0.196792	−0.0983958	+	0.995147i	$0.531371\pi$
$128$	−1229.05	−0.848704
$129$	−198.556	−0.135519
$130$	0	0
$131$	−699.706	−0.466669	−0.233334	−	0.972397i	$-0.574964\pi$
$131$	−699.706	−0.466669	−0.233334	+	0.972397i	$0.574964\pi$
$132$	−1.97645	−0.00130324
$133$	−356.377	−0.232344
$134$	−2662.28	−1.71631
$135$	0	0
$136$	−1586.96	−1.00059
$137$	1588.77	0.990789	0.495394	−	0.868668i	$-0.335024\pi$
$137$	1588.77	0.990789	0.495394	+	0.868668i	$0.335024\pi$
$138$	−0.962460	−0.000593696 0
$139$	2564.09	1.56463	0.782315	−	0.622884i	$-0.214039\pi$
$139$	2564.09	1.56463	0.782315	+	0.622884i	$0.214039\pi$
$140$	0	0
$141$	−164.003	−0.0979544
$142$	−2792.26	−1.65015
$143$	207.256	0.121200
$144$	1534.68	0.888124
$145$	0	0
$146$	−1871.43	−1.06082
$147$	34.3765	0.0192880
$148$	101.719	0.0564948
$149$	−3386.03	−1.86171	−0.930854	−	0.365390i	$-0.880935\pi$
$149$	−3386.03	−1.86171	−0.930854	+	0.365390i	$0.880935\pi$
$150$	0	0
$151$	−2301.98	−1.24061	−0.620306	−	0.784360i	$-0.712992\pi$
$151$	−2301.98	−1.24061	−0.620306	+	0.784360i	$0.712992\pi$
$152$	−1196.80	−0.638643
$153$	1789.48	0.945564
$154$	−75.9392	−0.0397360
$155$	0	0
$156$	−25.4031	−0.0130377
$157$	587.325	0.298558	0.149279	−	0.988795i	$-0.452305\pi$
$157$	587.325	0.298558	0.149279	+	0.988795i	$0.452305\pi$
$158$	1156.91	0.582522
$159$	−285.069	−0.142185
$160$	0	0
$161$	3.55467	0.00174005
$162$	−1862.39	−0.903229
$163$	914.189	0.439293	0.219647	−	0.975579i	$-0.429510\pi$
$163$	914.189	0.439293	0.219647	+	0.975579i	$0.429510\pi$
$164$	40.1327	0.0191087
$165$	0	0
$166$	1452.34	0.679056
$167$	3316.66	1.53683	0.768416	−	0.639951i	$-0.221045\pi$
$167$	3316.66	1.53683	0.768416	+	0.639951i	$0.221045\pi$
$168$	115.445	0.0530167
$169$	466.850	0.212494
$170$	0	0
$171$	1349.54	0.603519
$172$	198.556	0.0880220
$173$	1542.16	0.677737	0.338868	−	0.940834i	$-0.389956\pi$
$173$	1542.16	0.677737	0.338868	+	0.940834i	$0.389956\pi$
$174$	227.547	0.0991397
$175$	0	0
$176$	−232.486	−0.0995697
$177$	−405.031	−0.172000
$178$	2166.85	0.912429
$179$	−550.755	−0.229974	−0.114987	−	0.993367i	$-0.536683\pi$
$179$	−550.755	−0.229974	−0.114987	+	0.993367i	$0.536683\pi$
$180$	0	0
$181$	−3562.15	−1.46283	−0.731416	−	0.681931i	$-0.761140\pi$
$181$	−3562.15	−1.46283	−0.731416	+	0.681931i	$0.761140\pi$
$182$	−976.041	−0.397522
$183$	225.984	0.0912854
$184$	11.9375	0.00478285
$185$	0	0
$186$	554.006	0.218396
$187$	−271.086	−0.106009
$188$	164.003	0.0636232
$189$	−262.773	−0.101132
$190$	0	0
$191$	3436.00	1.30168	0.650838	−	0.759217i	$-0.274418\pi$
$191$	3436.00	1.30168	0.650838	+	0.759217i	$0.274418\pi$
$192$	384.933	0.144688
$193$	1047.12	0.390534	0.195267	−	0.980750i	$-0.437443\pi$
$193$	1047.12	0.390534	0.195267	+	0.980750i	$0.437443\pi$
$194$	−4735.37	−1.75247
$195$	0	0
$196$	−34.3765	−0.0125279
$197$	−3205.61	−1.15934	−0.579670	−	0.814851i	$-0.696819\pi$
$197$	−3205.61	−1.15934	−0.579670	+	0.814851i	$0.696819\pi$
$198$	287.569	0.103215
$199$	22.0532	0.00785581	0.00392791	−	0.999992i	$-0.498750\pi$
$199$	22.0532	0.00785581	0.00392791	+	0.999992i	$0.498750\pi$
$200$	0	0
$201$	691.361	0.242611
$202$	5370.30	1.87056
$203$	−840.405	−0.290566
$204$	33.2266	0.0114036
$205$	0	0
$206$	1051.24	0.355549
$207$	−13.4609	−0.00451981
$208$	−2988.12	−0.996101
$209$	−204.439	−0.0676619
$210$	0	0
$211$	2362.52	0.770819	0.385410	−	0.922746i	$-0.374060\pi$
$211$	2362.52	0.770819	0.385410	+	0.922746i	$0.374060\pi$
$212$	285.069	0.0923519
$213$	725.116	0.233259
$214$	1374.00	0.438901
$215$	0	0
$216$	−882.461	−0.277981
$217$	−2046.12	−0.640091
$218$	−929.688	−0.288837
$219$	485.986	0.149954
$220$	0	0
$221$	−3484.25	−1.06052
$222$	274.800	0.0830781
$223$	−1428.67	−0.429016	−0.214508	−	0.976722i	$-0.568815\pi$
$223$	−1428.67	−0.429016	−0.214508	+	0.976722i	$0.568815\pi$
$224$	−221.583	−0.0660943
$225$	0	0
$226$	3290.82	0.968594
$227$	5979.38	1.74831	0.874153	−	0.485651i	$-0.161418\pi$
$227$	5979.38	1.74831	0.874153	+	0.485651i	$0.161418\pi$
$228$	25.0578	0.00727849
$229$	6377.77	1.84042	0.920208	−	0.391430i	$-0.128020\pi$
$229$	6377.77	1.84042	0.920208	+	0.391430i	$0.128020\pi$
$230$	0	0
$231$	19.7205	0.00561693
$232$	−2822.30	−0.798676
$233$	−4947.18	−1.39099	−0.695495	−	0.718531i	$-0.744815\pi$
$233$	−4947.18	−1.39099	−0.695495	+	0.718531i	$0.744815\pi$
$234$	3696.10	1.03257
$235$	0	0
$236$	405.031	0.111717
$237$	−300.434	−0.0823430
$238$	1276.64	0.347698
$239$	−348.213	−0.0942427	−0.0471213	−	0.998889i	$-0.515005\pi$
$239$	−348.213	−0.0942427	−0.0471213	+	0.998889i	$0.515005\pi$
$240$	0	0
$241$	4702.23	1.25683	0.628417	−	0.777876i	$-0.283703\pi$
$241$	4702.23	1.25683	0.628417	+	0.777876i	$0.283703\pi$
$242$	3552.22	0.943575
$243$	1497.19	0.395247
$244$	−225.984	−0.0592916
$245$	0	0
$246$	108.421	0.0281003
$247$	−2627.64	−0.676894
$248$	−6871.41	−1.75941
$249$	−377.154	−0.0959887
$250$	0	0
$251$	−2431.64	−0.611489	−0.305745	−	0.952114i	$-0.598905\pi$
$251$	−2431.64	−0.611489	−0.305745	+	0.952114i	$0.598905\pi$
$252$	130.178	0.0325415
$253$	2.03917	0.000506727 0
$254$	760.899	0.187965
$255$	0	0
$256$	−1069.07	−0.261003
$257$	−1631.76	−0.396055	−0.198028	−	0.980196i	$-0.563454\pi$
$257$	−1631.76	−0.396055	−0.198028	+	0.980196i	$0.563454\pi$
$258$	536.412	0.129440
$259$	−1014.92	−0.243491
$260$	0	0
$261$	3182.47	0.754751
$262$	1890.30	0.445737
$263$	−3563.67	−0.835534	−0.417767	−	0.908554i	$-0.637187\pi$
$263$	−3563.67	−0.835534	−0.417767	+	0.908554i	$0.637187\pi$
$264$	66.2264	0.0154392
$265$	0	0
$266$	962.773	0.221923
$267$	−562.704	−0.128977
$268$	−691.361	−0.157581
$269$	791.631	0.179430	0.0897149	−	0.995967i	$-0.471404\pi$
$269$	791.631	0.179430	0.0897149	+	0.995967i	$0.471404\pi$
$270$	0	0
$271$	7823.17	1.75359	0.876796	−	0.480862i	$-0.159676\pi$
$271$	7823.17	1.75359	0.876796	+	0.480862i	$0.159676\pi$
$272$	3908.39	0.871253
$273$	253.466	0.0561921
$274$	−4292.17	−0.946348
$275$	0	0
$276$	−0.249939	−5.45093e−5 0
$277$	−1554.16	−0.337114	−0.168557	−	0.985692i	$-0.553911\pi$
$277$	−1554.16	−0.337114	−0.168557	+	0.985692i	$0.553911\pi$
$278$	−6927.05	−1.49445
$279$	7748.32	1.66265
$280$	0	0
$281$	−5043.72	−1.07076	−0.535379	−	0.844612i	$-0.679831\pi$
$281$	−5043.72	−1.07076	−0.535379	+	0.844612i	$0.679831\pi$
$282$	443.065	0.0935608
$283$	−5897.15	−1.23869	−0.619345	−	0.785119i	$-0.712602\pi$
$283$	−5897.15	−1.23869	−0.619345	+	0.785119i	$0.712602\pi$
$284$	−725.116	−0.151506
$285$	0	0
$286$	−559.916	−0.115764
$287$	−400.433	−0.0823582
$288$	839.097	0.171681
$289$	−355.696	−0.0723988
$290$	0	0
$291$	1229.72	0.247722
$292$	−485.986	−0.0973979
$293$	8592.25	1.71319	0.856595	−	0.515990i	$-0.172576\pi$
$293$	8592.25	1.71319	0.856595	+	0.515990i	$0.172576\pi$
$294$	−92.8704	−0.0184228
$295$	0	0
$296$	−3408.37	−0.669283
$297$	−150.743	−0.0294511
$298$	9147.58	1.77820
$299$	26.2094	0.00506932
$300$	0	0
$301$	−1981.14	−0.379373
$302$	6218.94	1.18497
$303$	−1394.60	−0.264415
$304$	2947.50	0.556089
$305$	0	0
$306$	−4834.40	−0.903152
$307$	−3682.20	−0.684542	−0.342271	−	0.939601i	$-0.611196\pi$
$307$	−3682.20	−0.684542	−0.342271	+	0.939601i	$0.611196\pi$
$308$	−19.7205	−0.00364830
$309$	−272.993	−0.0502590
$310$	0	0
$311$	6866.18	1.25191	0.625957	−	0.779858i	$-0.284709\pi$
$311$	6866.18	1.25191	0.625957	+	0.779858i	$0.284709\pi$
$312$	851.203	0.154455
$313$	2958.60	0.534281	0.267140	−	0.963658i	$-0.413921\pi$
$313$	2958.60	0.534281	0.267140	+	0.963658i	$0.413921\pi$
$314$	−1586.69	−0.285167
$315$	0	0
$316$	300.434	0.0534834
$317$	1585.21	0.280866	0.140433	−	0.990090i	$-0.455151\pi$
$317$	1585.21	0.280866	0.140433	+	0.990090i	$0.455151\pi$
$318$	770.131	0.135808
$319$	−482.107	−0.0846169
$320$	0	0
$321$	−356.811	−0.0620413
$322$	−9.60317	−0.00166200
$323$	3436.89	0.592054
$324$	−483.639	−0.0829286
$325$	0	0
$326$	−2469.74	−0.419589
$327$	241.428	0.0408288
$328$	−1344.76	−0.226377
$329$	−1636.38	−0.274215
$330$	0	0
$331$	6045.84	1.00396	0.501978	−	0.864880i	$-0.332606\pi$
$331$	6045.84	1.00396	0.501978	+	0.864880i	$0.332606\pi$
$332$	377.154	0.0623465
$333$	3843.34	0.632474
$334$	−8960.16	−1.46790
$335$	0	0
$336$	−284.320	−0.0461635
$337$	−9205.64	−1.48802	−0.744010	−	0.668168i	$-0.767079\pi$
$337$	−9205.64	−1.48802	−0.744010	+	0.668168i	$0.767079\pi$
$338$	−1261.22	−0.202963
$339$	−854.586	−0.136917
$340$	0	0
$341$	−1173.78	−0.186404
$342$	−3645.86	−0.576449
$343$	343.000	0.0539949
$344$	−6653.19	−1.04278
$345$	0	0
$346$	−4166.25	−0.647338
$347$	−3092.28	−0.478393	−0.239196	−	0.970971i	$-0.576884\pi$
$347$	−3092.28	−0.478393	−0.239196	+	0.970971i	$0.576884\pi$
$348$	59.0912	0.00910236
$349$	−5231.61	−0.802412	−0.401206	−	0.915988i	$-0.631409\pi$
$349$	−5231.61	−0.802412	−0.401206	+	0.915988i	$0.631409\pi$
$350$	0	0
$351$	−1937.48	−0.294630
$352$	−127.113	−0.0192476
$353$	9013.71	1.35907	0.679534	−	0.733644i	$-0.262182\pi$
$353$	9013.71	1.35907	0.679534	+	0.733644i	$0.262182\pi$
$354$	1094.22	0.164285
$355$	0	0
$356$	562.704	0.0837732
$357$	−331.526	−0.0491491
$358$	1487.90	0.219659
$359$	−4779.38	−0.702635	−0.351317	−	0.936256i	$-0.614266\pi$
$359$	−4779.38	−0.702635	−0.351317	+	0.936256i	$0.614266\pi$
$360$	0	0
$361$	−4267.08	−0.622114
$362$	9623.38	1.39722
$363$	−922.466	−0.133380
$364$	−253.466	−0.0364978
$365$	0	0
$366$	−610.510	−0.0871909
$367$	175.769	0.0250001	0.0125001	−	0.999922i	$-0.496021\pi$
$367$	175.769	0.0250001	0.0125001	+	0.999922i	$0.496021\pi$
$368$	−29.3999	−0.00416460
$369$	1516.37	0.213927
$370$	0	0
$371$	−2844.34	−0.398034
$372$	143.868	0.0200517
$373$	7982.98	1.10816	0.554079	−	0.832464i	$-0.313070\pi$
$373$	7982.98	1.10816	0.554079	+	0.832464i	$0.313070\pi$
$374$	732.355	0.101254
$375$	0	0
$376$	−5495.39	−0.753732
$377$	−6196.48	−0.846512
$378$	709.899	0.0965959
$379$	12663.1	1.71626	0.858129	−	0.513434i	$-0.171627\pi$
$379$	12663.1	1.71626	0.858129	+	0.513434i	$0.171627\pi$
$380$	0	0
$381$	−197.596	−0.0265700
$382$	−9282.56	−1.24329
$383$	4678.68	0.624202	0.312101	−	0.950049i	$-0.398967\pi$
$383$	4678.68	0.624202	0.312101	+	0.950049i	$0.398967\pi$
$384$	−862.258	−0.114588
$385$	0	0
$386$	−2828.85	−0.373017
$387$	7502.25	0.985428
$388$	−1229.72	−0.160900
$389$	−50.2546	−0.00655015	−0.00327508	−	0.999995i	$-0.501042\pi$
$389$	−50.2546	−0.00655015	−0.00327508	+	0.999995i	$0.501042\pi$
$390$	0	0
$391$	−34.2812	−0.00443395
$392$	1151.88	0.148415
$393$	−490.887	−0.0630076
$394$	8660.15	1.10734
$395$	0	0
$396$	74.6780	0.00947654
$397$	−11059.9	−1.39819	−0.699095	−	0.715029i	$-0.746413\pi$
$397$	−11059.9	−1.39819	−0.699095	+	0.715029i	$0.746413\pi$
$398$	−59.5780	−0.00750345
$399$	−250.020	−0.0313701
$400$	0	0
$401$	−13667.8	−1.70209	−0.851046	−	0.525092i	$-0.824031\pi$
$401$	−13667.8	−1.70209	−0.851046	+	0.525092i	$0.824031\pi$
$402$	−1867.75	−0.231729
$403$	−15086.5	−1.86479
$404$	1394.60	0.171742
$405$	0	0
$406$	2270.41	0.277533
$407$	−582.221	−0.0709082
$408$	−1113.35	−0.135096
$409$	−10990.5	−1.32872	−0.664359	−	0.747414i	$-0.731295\pi$
$409$	−10990.5	−1.32872	−0.664359	+	0.747414i	$0.731295\pi$
$410$	0	0
$411$	1114.62	0.133772
$412$	272.993	0.0326442
$413$	−4041.30	−0.481499
$414$	36.3656	0.00431708
$415$	0	0
$416$	−1633.78	−0.192554
$417$	1798.87	0.211249
$418$	552.305	0.0646271
$419$	1115.39	0.130049	0.0650246	−	0.997884i	$-0.479287\pi$
$419$	1115.39	0.130049	0.0650246	+	0.997884i	$0.479287\pi$
$420$	0	0
$421$	−2395.26	−0.277287	−0.138643	−	0.990342i	$-0.544274\pi$
$421$	−2395.26	−0.277287	−0.138643	+	0.990342i	$0.544274\pi$
$422$	−6382.50	−0.736245
$423$	6196.70	0.712278
$424$	−9552.03	−1.09407
$425$	0	0
$426$	−1958.95	−0.222796
$427$	2254.81	0.255545
$428$	356.811	0.0402970
$429$	145.403	0.0163639
$430$	0	0
$431$	1684.62	0.188272	0.0941360	−	0.995559i	$-0.469991\pi$
$431$	1684.62	0.188272	0.0941360	+	0.995559i	$0.469991\pi$
$432$	2173.34	0.242048
$433$	−13355.7	−1.48230	−0.741150	−	0.671340i	$-0.765719\pi$
$433$	−13355.7	−1.48230	−0.741150	+	0.671340i	$0.765719\pi$
$434$	5527.72	0.611381
$435$	0	0
$436$	−241.428	−0.0265191
$437$	−25.8531	−0.00283003
$438$	−1312.92	−0.143228
$439$	13817.4	1.50220	0.751101	−	0.660188i	$-0.229523\pi$
$439$	13817.4	1.50220	0.751101	+	0.660188i	$0.229523\pi$
$440$	0	0
$441$	−1298.88	−0.140253
$442$	9412.91	1.01296
$443$	6305.19	0.676227	0.338114	−	0.941105i	$-0.390211\pi$
$443$	6305.19	0.676227	0.338114	+	0.941105i	$0.390211\pi$
$444$	71.3621	0.00762769
$445$	0	0
$446$	3859.63	0.409773
$447$	−2375.51	−0.251360
$448$	3840.76	0.405042
$449$	5446.85	0.572501	0.286250	−	0.958155i	$-0.407591\pi$
$449$	5446.85	0.572501	0.286250	+	0.958155i	$0.407591\pi$
$450$	0	0
$451$	−229.712	−0.0239839
$452$	854.586	0.0889300
$453$	−1614.98	−0.167502
$454$	−16153.7	−1.66989
$455$	0	0
$456$	−839.633	−0.0862268
$457$	1221.02	0.124982	0.0624910	−	0.998046i	$-0.480096\pi$
$457$	1221.02	0.124982	0.0624910	+	0.998046i	$0.480096\pi$
$458$	−17230.0	−1.75787
$459$	2534.18	0.257702
$460$	0	0
$461$	9967.46	1.00701	0.503504	−	0.863993i	$-0.332044\pi$
$461$	9967.46	1.00701	0.503504	+	0.863993i	$0.332044\pi$
$462$	−53.2760	−0.00536499
$463$	−6309.54	−0.633324	−0.316662	−	0.948538i	$-0.602562\pi$
$463$	−6309.54	−0.633324	−0.316662	+	0.948538i	$0.602562\pi$
$464$	6950.79	0.695436
$465$	0	0
$466$	13365.1	1.32860
$467$	7784.79	0.771386	0.385693	−	0.922627i	$-0.373962\pi$
$467$	7784.79	0.771386	0.385693	+	0.922627i	$0.373962\pi$
$468$	959.831	0.0948039
$469$	6898.22	0.679168
$470$	0	0
$471$	412.045	0.0403100
$472$	−13571.7	−1.32349
$473$	−1136.50	−0.110479
$474$	811.641	0.0786496
$475$	0	0
$476$	331.526	0.0319233
$477$	10771.0	1.03390
$478$	940.718	0.0900156
$479$	−4425.28	−0.422122	−0.211061	−	0.977473i	$-0.567692\pi$
$479$	−4425.28	−0.422122	−0.211061	+	0.977473i	$0.567692\pi$
$480$	0	0
$481$	−7483.25	−0.709369
$482$	−12703.4	−1.20046
$483$	2.49382	0.000234934 0
$484$	922.466	0.0866328
$485$	0	0
$486$	−4044.76	−0.377519
$487$	13075.3	1.21663	0.608315	−	0.793695i	$-0.291846\pi$
$487$	13075.3	1.21663	0.608315	+	0.793695i	$0.291846\pi$
$488$	7572.23	0.702416
$489$	641.360	0.0593115
$490$	0	0
$491$	11455.7	1.05293	0.526463	−	0.850198i	$-0.323518\pi$
$491$	11455.7	1.05293	0.526463	+	0.850198i	$0.323518\pi$
$492$	28.1556	0.00257998
$493$	8104.84	0.740413
$494$	7098.73	0.646533
$495$	0	0
$496$	16923.0	1.53198
$497$	7235.01	0.652987
$498$	1018.91	0.0916833
$499$	5521.10	0.495307	0.247654	−	0.968849i	$-0.420340\pi$
$499$	5521.10	0.495307	0.247654	+	0.968849i	$0.420340\pi$
$500$	0	0
$501$	2326.84	0.207496
$502$	6569.23	0.584062
$503$	−11491.6	−1.01866	−0.509328	−	0.860573i	$-0.670106\pi$
$503$	−11491.6	−1.01866	−0.509328	+	0.860573i	$0.670106\pi$
$504$	−4361.98	−0.385512
$505$	0	0
$506$	−5.50896	−0.000483998 0
$507$	327.524	0.0286901
$508$	197.596	0.0172577
$509$	−9207.84	−0.801828	−0.400914	−	0.916116i	$-0.631307\pi$
$509$	−9207.84	−0.801828	−0.400914	+	0.916116i	$0.631307\pi$
$510$	0	0
$511$	4849.04	0.419783
$512$	12720.6	1.09800
$513$	1911.15	0.164482
$514$	4408.29	0.378291
$515$	0	0
$516$	139.300	0.0118843
$517$	−938.727	−0.0798552
$518$	2741.88	0.232570
$519$	1081.92	0.0915051
$520$	0	0
$521$	−11598.0	−0.975271	−0.487636	−	0.873047i	$-0.662141\pi$
$521$	−11598.0	−0.975271	−0.487636	+	0.873047i	$0.662141\pi$
$522$	−8597.64	−0.720898
$523$	4596.93	0.384340	0.192170	−	0.981362i	$-0.438448\pi$
$523$	4596.93	0.384340	0.192170	+	0.981362i	$0.438448\pi$
$524$	490.887	0.0409246
$525$	0	0
$526$	9627.49	0.798058
$527$	19732.7	1.63107
$528$	−163.103	−0.0134435
$529$	−12166.7	−0.999979
$530$	0	0
$531$	15303.7	1.25070
$532$	250.020	0.0203755
$533$	−2952.48	−0.239936
$534$	1520.18	0.123192
$535$	0	0
$536$	23166.0	1.86683
$537$	−386.389	−0.0310501
$538$	−2138.64	−0.171382
$539$	196.765	0.0157241
$540$	0	0
$541$	−8427.65	−0.669747	−0.334874	−	0.942263i	$-0.608694\pi$
$541$	−8427.65	−0.669747	−0.334874	+	0.942263i	$0.608694\pi$
$542$	−21134.8	−1.67494
$543$	−2499.07	−0.197505
$544$	2136.94	0.168420
$545$	0	0
$546$	−684.753	−0.0536717
$547$	12864.6	1.00557	0.502787	−	0.864410i	$-0.332308\pi$
$547$	12864.6	1.00557	0.502787	+	0.864410i	$0.332308\pi$
$548$	−1114.62	−0.0868875
$549$	−8538.58	−0.663785
$550$	0	0
$551$	6112.26	0.472579
$552$	8.37491	0.000645760 0
$553$	−2997.65	−0.230512
$554$	4198.67	0.321993
$555$	0	0
$556$	−1798.87	−0.137211
$557$	−19219.4	−1.46203	−0.731016	−	0.682360i	$-0.760954\pi$
$557$	−19219.4	−1.46203	−0.731016	+	0.682360i	$0.760954\pi$
$558$	−20932.6	−1.58807
$559$	−14607.4	−1.10524
$560$	0	0
$561$	−190.184	−0.0143129
$562$	13625.9	1.02273
$563$	17253.3	1.29155	0.645774	−	0.763529i	$-0.276535\pi$
$563$	17253.3	1.29155	0.645774	+	0.763529i	$0.276535\pi$
$564$	115.058	0.00859013
$565$	0	0
$566$	15931.5	1.18313
$567$	4825.62	0.357420
$568$	24297.0	1.79486
$569$	−6242.80	−0.459950	−0.229975	−	0.973197i	$-0.573864\pi$
$569$	−6242.80	−0.459950	−0.229975	+	0.973197i	$0.573864\pi$
$570$	0	0
$571$	5904.17	0.432718	0.216359	−	0.976314i	$-0.430582\pi$
$571$	5904.17	0.432718	0.216359	+	0.976314i	$0.430582\pi$
$572$	−145.403	−0.0106287
$573$	2410.57	0.175747
$574$	1081.79	0.0786642
$575$	0	0
$576$	−14544.3	−1.05210
$577$	11390.2	0.821806	0.410903	−	0.911679i	$-0.365214\pi$
$577$	11390.2	0.821806	0.410903	+	0.911679i	$0.365214\pi$
$578$	960.934	0.0691515
$579$	734.617	0.0527282
$580$	0	0
$581$	−3763.15	−0.268712
$582$	−3322.15	−0.236611
$583$	−1631.68	−0.115913
$584$	16284.3	1.15385
$585$	0	0
$586$	−23212.5	−1.63635
$587$	−20459.5	−1.43859	−0.719295	−	0.694704i	$-0.755535\pi$
$587$	−20459.5	−1.43859	−0.719295	+	0.694704i	$0.755535\pi$
$588$	−24.1173	−0.00169146
$589$	14881.4	1.04105
$590$	0	0
$591$	−2248.93	−0.156529
$592$	8394.19	0.582768
$593$	20051.1	1.38853	0.694266	−	0.719719i	$-0.255729\pi$
$593$	20051.1	1.38853	0.694266	+	0.719719i	$0.255729\pi$
$594$	407.241	0.0281301
$595$	0	0
$596$	2375.51	0.163263
$597$	15.4717	0.00106066
$598$	−70.8062	−0.00484194
$599$	10578.4	0.721572	0.360786	−	0.932649i	$-0.382508\pi$
$599$	10578.4	0.721572	0.360786	+	0.932649i	$0.382508\pi$
$600$	0	0
$601$	−4619.46	−0.313530	−0.156765	−	0.987636i	$-0.550107\pi$
$601$	−4619.46	−0.313530	−0.156765	+	0.987636i	$0.550107\pi$
$602$	5352.18	0.362356
$603$	−26122.4	−1.76415
$604$	1614.98	0.108796
$605$	0	0
$606$	3767.60	0.252555
$607$	5724.50	0.382785	0.191392	−	0.981514i	$-0.438700\pi$
$607$	5724.50	0.382785	0.191392	+	0.981514i	$0.438700\pi$
$608$	1611.57	0.107496
$609$	−589.596	−0.0392309
$610$	0	0
$611$	−12065.4	−0.798876
$612$	−1255.43	−0.0829214
$613$	−19286.4	−1.27075	−0.635375	−	0.772204i	$-0.719154\pi$
$613$	−19286.4	−1.27075	−0.635375	+	0.772204i	$0.719154\pi$
$614$	9947.70	0.653838
$615$	0	0
$616$	660.789	0.0432207
$617$	6864.84	0.447922	0.223961	−	0.974598i	$-0.428101\pi$
$617$	6864.84	0.447922	0.223961	+	0.974598i	$0.428101\pi$
$618$	737.508	0.0480047
$619$	−17559.4	−1.14018	−0.570090	−	0.821582i	$-0.693092\pi$
$619$	−17559.4	−1.14018	−0.570090	+	0.821582i	$0.693092\pi$
$620$	0	0
$621$	−19.0627	−0.00123182
$622$	−18549.4	−1.19576
$623$	−5614.51	−0.361061
$624$	−2096.35	−0.134489
$625$	0	0
$626$	−7992.84	−0.510317
$627$	−143.427	−0.00913542
$628$	−412.045	−0.0261821
$629$	9787.89	0.620459
$630$	0	0
$631$	−24780.6	−1.56339	−0.781694	−	0.623662i	$-0.785644\pi$
$631$	−24780.6	−1.56339	−0.781694	+	0.623662i	$0.785644\pi$
$632$	−10066.9	−0.633607
$633$	1657.46	0.104073
$634$	−4282.55	−0.268268
$635$	0	0
$636$	199.993	0.0124690
$637$	2529.01	0.157305
$638$	1302.44	0.0808215
$639$	−27397.8	−1.69615
$640$	0	0
$641$	−21724.3	−1.33863	−0.669313	−	0.742980i	$-0.733412\pi$
$641$	−21724.3	−1.33863	−0.669313	+	0.742980i	$0.733412\pi$
$642$	963.947	0.0592585
$643$	−18736.1	−1.14911	−0.574555	−	0.818466i	$-0.694825\pi$
$643$	−18736.1	−1.14911	−0.574555	+	0.818466i	$0.694825\pi$
$644$	−2.49382	−0.000152594 0
$645$	0	0
$646$	−9284.96	−0.565498
$647$	−25687.8	−1.56088	−0.780442	−	0.625228i	$-0.785006\pi$
$647$	−25687.8	−1.56088	−0.780442	+	0.625228i	$0.785006\pi$
$648$	16205.7	0.982438
$649$	−2318.33	−0.140219
$650$	0	0
$651$	−1435.48	−0.0864224
$652$	−641.360	−0.0385239
$653$	15450.7	0.925929	0.462964	−	0.886377i	$-0.346786\pi$
$653$	15450.7	0.925929	0.462964	+	0.886377i	$0.346786\pi$
$654$	−652.234	−0.0389975
$655$	0	0
$656$	3311.88	0.197115
$657$	−18362.5	−1.09039
$658$	4420.78	0.261915
$659$	−1402.85	−0.0829246	−0.0414623	−	0.999140i	$-0.513202\pi$
$659$	−1402.85	−0.0829246	−0.0414623	+	0.999140i	$0.513202\pi$
$660$	0	0
$661$	6896.75	0.405828	0.202914	−	0.979197i	$-0.434959\pi$
$661$	6896.75	0.405828	0.202914	+	0.979197i	$0.434959\pi$
$662$	−16333.2	−0.958926
$663$	−2444.42	−0.143187
$664$	−12637.6	−0.738606
$665$	0	0
$666$	−10383.0	−0.604105
$667$	−60.9666	−0.00353919
$668$	−2326.84	−0.134773
$669$	−1002.30	−0.0579239
$670$	0	0
$671$	1293.49	0.0744185
$672$	−155.454	−0.00892377
$673$	−3869.29	−0.221620	−0.110810	−	0.993842i	$-0.535344\pi$
$673$	−3869.29	−0.221620	−0.110810	+	0.993842i	$0.535344\pi$
$674$	24869.6	1.42128
$675$	0	0
$676$	−327.524	−0.0186347
$677$	711.604	0.0403976	0.0201988	−	0.999796i	$-0.493570\pi$
$677$	711.604	0.0403976	0.0201988	+	0.999796i	$0.493570\pi$
$678$	2308.72	0.130775
$679$	12269.8	0.693477
$680$	0	0
$681$	4194.91	0.236049
$682$	3171.04	0.178043
$683$	−13112.0	−0.734577	−0.367288	−	0.930107i	$-0.619714\pi$
$683$	−13112.0	−0.734577	−0.367288	+	0.930107i	$0.619714\pi$
$684$	−946.784	−0.0529257
$685$	0	0
$686$	−926.636	−0.0515731
$687$	4474.40	0.248485
$688$	16385.5	0.907984
$689$	−20971.9	−1.15960
$690$	0	0
$691$	−12573.5	−0.692211	−0.346105	−	0.938196i	$-0.612496\pi$
$691$	−12573.5	−0.692211	−0.346105	+	0.938196i	$0.612496\pi$
$692$	−1081.92	−0.0594343
$693$	−745.117	−0.0408437
$694$	8353.98	0.456935
$695$	0	0
$696$	−1980.02	−0.107834
$697$	3861.76	0.209863
$698$	14133.5	0.766421
$699$	−3470.75	−0.187805
$700$	0	0
$701$	−7489.24	−0.403516	−0.201758	−	0.979435i	$-0.564665\pi$
$701$	−7489.24	−0.403516	−0.201758	+	0.979435i	$0.564665\pi$
$702$	5234.23	0.281415
$703$	7381.53	0.396016
$704$	2203.29	0.117954
$705$	0	0
$706$	−24351.1	−1.29811
$707$	−13914.9	−0.740206
$708$	284.155	0.0150836
$709$	20869.6	1.10547	0.552733	−	0.833358i	$-0.313585\pi$
$709$	20869.6	1.10547	0.552733	+	0.833358i	$0.313585\pi$
$710$	0	0
$711$	11351.6	0.598760
$712$	−18855.0	−0.992445
$713$	−148.435	−0.00779652
$714$	895.639	0.0469446
$715$	0	0
$716$	386.389	0.0201676
$717$	−244.293	−0.0127242
$718$	12911.8	0.671119
$719$	−5889.90	−0.305503	−0.152751	−	0.988265i	$-0.548813\pi$
$719$	−5889.90	−0.305503	−0.152751	+	0.988265i	$0.548813\pi$
$720$	0	0
$721$	−2723.85	−0.140696
$722$	11527.8	0.594210
$723$	3298.91	0.169692
$724$	2499.07	0.128284
$725$	0	0
$726$	2492.10	0.127397
$727$	24760.2	1.26314	0.631571	−	0.775318i	$-0.282411\pi$
$727$	24760.2	1.26314	0.631571	+	0.775318i	$0.282411\pi$
$728$	8493.08	0.432382
$729$	−17562.7	−0.892280
$730$	0	0
$731$	19106.1	0.966708
$732$	−158.542	−0.00800530
$733$	−4537.53	−0.228646	−0.114323	−	0.993444i	$-0.536470\pi$
$733$	−4537.53	−0.228646	−0.114323	+	0.993444i	$0.536470\pi$
$734$	−474.850	−0.0238788
$735$	0	0
$736$	−16.0746	−0.000805051 0
$737$	3957.23	0.197784
$738$	−4096.57	−0.204332
$739$	−24010.0	−1.19516	−0.597579	−	0.801810i	$-0.703870\pi$
$739$	−24010.0	−1.19516	−0.597579	+	0.801810i	$0.703870\pi$
$740$	0	0
$741$	−1843.45	−0.0913913
$742$	7684.16	0.380181
$743$	25175.4	1.24306	0.621532	−	0.783389i	$-0.286511\pi$
$743$	25175.4	1.24306	0.621532	+	0.783389i	$0.286511\pi$
$744$	−4820.72	−0.237549
$745$	0	0
$746$	−21566.5	−1.05845
$747$	14250.4	0.697985
$748$	190.184	0.00929652
$749$	−3560.17	−0.173679
$750$	0	0
$751$	24920.0	1.21085	0.605423	−	0.795904i	$-0.293004\pi$
$751$	24920.0	1.21085	0.605423	+	0.795904i	$0.293004\pi$
$752$	13534.1	0.656301
$753$	−1705.95	−0.0825607
$754$	16740.2	0.808543
$755$	0	0
$756$	184.352	0.00886880
$757$	−28274.4	−1.35753	−0.678765	−	0.734356i	$-0.737484\pi$
$757$	−28274.4	−1.35753	−0.678765	+	0.734356i	$0.737484\pi$
$758$	−34210.3	−1.63928
$759$	1.43061	6.84160e−5 0
$760$	0	0
$761$	12377.9	0.589616	0.294808	−	0.955557i	$-0.404744\pi$
$761$	12377.9	0.589616	0.294808	+	0.955557i	$0.404744\pi$
$762$	533.818	0.0253782
$763$	2408.91	0.114297
$764$	−2410.57	−0.114151
$765$	0	0
$766$	−12639.7	−0.596204
$767$	−29797.3	−1.40276
$768$	−750.019	−0.0352396
$769$	−31845.7	−1.49335	−0.746674	−	0.665190i	$-0.768350\pi$
$769$	−31845.7	−1.49335	−0.746674	+	0.665190i	$0.768350\pi$
$770$	0	0
$771$	−1144.78	−0.0534736
$772$	−734.617	−0.0342480
$773$	7342.21	0.341631	0.170816	−	0.985303i	$-0.445360\pi$
$773$	7342.21	0.341631	0.170816	+	0.985303i	$0.445360\pi$
$774$	−20267.8	−0.941228
$775$	0	0
$776$	41205.1	1.90615
$777$	−712.032	−0.0328752
$778$	135.766	0.00625636
$779$	2912.35	0.133948
$780$	0	0
$781$	4150.44	0.190159
$782$	92.6127	0.00423507
$783$	4506.86	0.205699
$784$	−2836.87	−0.129231
$785$	0	0
$786$	1326.16	0.0601815
$787$	36457.4	1.65129	0.825646	−	0.564188i	$-0.190811\pi$
$787$	36457.4	1.65129	0.825646	+	0.564188i	$0.190811\pi$
$788$	2248.93	0.101669
$789$	−2500.14	−0.112810
$790$	0	0
$791$	−8526.83	−0.383286
$792$	−2502.30	−0.112267
$793$	16625.2	0.744487
$794$	29879.0	1.33548
$795$	0	0
$796$	−15.4717	−0.000688918 0
$797$	−9358.08	−0.415910	−0.207955	−	0.978138i	$-0.566681\pi$
$797$	−9358.08	−0.415910	−0.207955	+	0.978138i	$0.566681\pi$
$798$	675.445	0.0299630
$799$	15781.2	0.698747
$800$	0	0
$801$	21261.2	0.937863
$802$	36924.5	1.62575
$803$	2781.70	0.122247
$804$	−485.033	−0.0212758
$805$	0	0
$806$	40757.1	1.78115
$807$	555.378	0.0242258
$808$	−46730.0	−2.03460
$809$	28189.6	1.22509	0.612543	−	0.790437i	$-0.290147\pi$
$809$	28189.6	1.22509	0.612543	+	0.790437i	$0.290147\pi$
$810$	0	0
$811$	−22909.7	−0.991946	−0.495973	−	0.868338i	$-0.665189\pi$
$811$	−22909.7	−0.991946	−0.495973	+	0.868338i	$0.665189\pi$
$812$	589.596	0.0254812
$813$	5488.44	0.236763
$814$	1572.91	0.0677277
$815$	0	0
$816$	2741.98	0.117633
$817$	14408.8	0.617015
$818$	29691.5	1.26912
$819$	−9576.94	−0.408602
$820$	0	0
$821$	20512.6	0.871980	0.435990	−	0.899952i	$-0.356398\pi$
$821$	20512.6	0.871980	0.435990	+	0.899952i	$0.356398\pi$
$822$	−3011.22	−0.127772
$823$	7882.78	0.333872	0.166936	−	0.985968i	$-0.446613\pi$
$823$	7882.78	0.333872	0.166936	+	0.985968i	$0.446613\pi$
$824$	−9147.40	−0.386729
$825$	0	0
$826$	10917.8	0.459902
$827$	−19276.5	−0.810531	−0.405265	−	0.914199i	$-0.632821\pi$
$827$	−19276.5	−0.810531	−0.405265	+	0.914199i	$0.632821\pi$
$828$	9.44369	0.000396366 0
$829$	13771.7	0.576972	0.288486	−	0.957484i	$-0.406848\pi$
$829$	13771.7	0.576972	0.288486	+	0.957484i	$0.406848\pi$
$830$	0	0
$831$	−1090.34	−0.0455157
$832$	28318.7	1.18002
$833$	−3307.88	−0.137589
$834$	−4859.76	−0.201774
$835$	0	0
$836$	143.427	0.00593363
$837$	10972.8	0.453136
$838$	−3013.31	−0.124216
$839$	23175.4	0.953640	0.476820	−	0.879001i	$-0.341789\pi$
$839$	23175.4	0.953640	0.476820	+	0.879001i	$0.341789\pi$
$840$	0	0
$841$	−9975.12	−0.409001
$842$	6470.94	0.264849
$843$	−3538.48	−0.144569
$844$	−1657.46	−0.0675972
$845$	0	0
$846$	−16740.8	−0.680330
$847$	−9204.12	−0.373385
$848$	23524.9	0.952650
$849$	−4137.22	−0.167243
$850$	0	0
$851$	−73.6270	−0.00296581
$852$	−508.714	−0.0204557
$853$	−31431.7	−1.26166	−0.630832	−	0.775920i	$-0.717286\pi$
$853$	−31431.7	−1.26166	−0.630832	+	0.775920i	$0.717286\pi$
$854$	−6091.51	−0.244083
$855$	0	0
$856$	−11956.0	−0.477391
$857$	3992.52	0.159139	0.0795693	−	0.996829i	$-0.474646\pi$
$857$	3992.52	0.159139	0.0795693	+	0.996829i	$0.474646\pi$
$858$	−392.816	−0.0156300
$859$	12909.8	0.512778	0.256389	−	0.966574i	$-0.417467\pi$
$859$	12909.8	0.512778	0.256389	+	0.966574i	$0.417467\pi$
$860$	0	0
$861$	−280.929	−0.0111197
$862$	−4551.10	−0.179827
$863$	−40128.5	−1.58284	−0.791419	−	0.611274i	$-0.790657\pi$
$863$	−40128.5	−1.58284	−0.791419	+	0.611274i	$0.790657\pi$
$864$	1188.29	0.0467898
$865$	0	0
$866$	36081.3	1.41581
$867$	−249.542	−0.00977498
$868$	1435.48	0.0561330
$869$	−1719.63	−0.0671284
$870$	0	0
$871$	50862.0	1.97864
$872$	8089.73	0.314166
$873$	−46463.6	−1.80132
$874$	69.8438	0.00270309
$875$	0	0
$876$	−340.950	−0.0131502
$877$	3222.42	0.124074	0.0620372	−	0.998074i	$-0.480240\pi$
$877$	3222.42	0.124074	0.0620372	+	0.998074i	$0.480240\pi$
$878$	−37328.4	−1.43482
$879$	6028.00	0.231307
$880$	0	0
$881$	−18712.4	−0.715591	−0.357795	−	0.933800i	$-0.616471\pi$
$881$	−18712.4	−0.715591	−0.357795	+	0.933800i	$0.616471\pi$
$882$	3509.01	0.133962
$883$	−17924.0	−0.683114	−0.341557	−	0.939861i	$-0.610954\pi$
$883$	−17924.0	−0.683114	−0.341557	+	0.939861i	$0.610954\pi$
$884$	2444.42	0.0930029
$885$	0	0
$886$	−17033.9	−0.645896
$887$	9873.21	0.373743	0.186871	−	0.982384i	$-0.440165\pi$
$887$	9873.21	0.373743	0.186871	+	0.982384i	$0.440165\pi$
$888$	−2391.19	−0.0903637
$889$	−1971.56	−0.0743803
$890$	0	0
$891$	2768.27	0.104086
$892$	1002.30	0.0376226
$893$	11901.4	0.445985
$894$	6417.59	0.240086
$895$	0	0
$896$	−8603.38	−0.320780
$897$	18.3875	0.000684438 0
$898$	−14715.0	−0.546822
$899$	35093.3	1.30192
$900$	0	0
$901$	27430.7	1.01426
$902$	620.582	0.0229081
$903$	−1389.89	−0.0512212
$904$	−28635.3	−1.05354
$905$	0	0
$906$	4362.97	0.159989
$907$	−30129.2	−1.10300	−0.551502	−	0.834174i	$-0.685945\pi$
$907$	−30129.2	−1.10300	−0.551502	+	0.834174i	$0.685945\pi$
$908$	−4194.91	−0.153318
$909$	52693.5	1.92270
$910$	0	0
$911$	−37831.8	−1.37588	−0.687938	−	0.725769i	$-0.741484\pi$
$911$	−37831.8	−1.37588	−0.687938	+	0.725769i	$0.741484\pi$
$912$	2067.86	0.0750807
$913$	−2158.77	−0.0782527
$914$	−3298.65	−0.119376
$915$	0	0
$916$	−4474.40	−0.161396
$917$	−4897.94	−0.176384
$918$	−6846.24	−0.246143
$919$	4771.81	0.171281	0.0856407	−	0.996326i	$-0.472706\pi$
$919$	4771.81	0.171281	0.0856407	+	0.996326i	$0.472706\pi$
$920$	0	0
$921$	−2583.30	−0.0924240
$922$	−26927.7	−0.961841
$923$	53345.3	1.90236
$924$	−13.8351	−0.000492578 0
$925$	0	0
$926$	17045.6	0.604917
$927$	10314.8	0.365460
$928$	3800.39	0.134433
$929$	−45138.3	−1.59412	−0.797061	−	0.603899i	$-0.793613\pi$
$929$	−45138.3	−1.59412	−0.797061	+	0.603899i	$0.793613\pi$
$930$	0	0
$931$	−2494.64	−0.0878178
$932$	3470.75	0.121983
$933$	4817.05	0.169028
$934$	−21031.1	−0.736786
$935$	0	0
$936$	−32161.8	−1.12312
$937$	22634.7	0.789161	0.394580	−	0.918861i	$-0.370890\pi$
$937$	22634.7	0.789161	0.394580	+	0.918861i	$0.370890\pi$
$938$	−18636.0	−0.648705
$939$	2075.64	0.0721363
$940$	0	0
$941$	11695.3	0.405160	0.202580	−	0.979266i	$-0.435067\pi$
$941$	11695.3	0.405160	0.202580	+	0.979266i	$0.435067\pi$
$942$	−1113.16	−0.0385020
$943$	−29.0492	−0.00100315
$944$	33424.6	1.15241
$945$	0	0
$946$	3070.33	0.105523
$947$	−45429.1	−1.55887	−0.779434	−	0.626485i	$-0.784493\pi$
$947$	−45429.1	−1.55887	−0.779434	+	0.626485i	$0.784493\pi$
$948$	210.773	0.00722109
$949$	35753.0	1.22296
$950$	0	0
$951$	1112.13	0.0379213
$952$	−11108.7	−0.378189
$953$	−39025.9	−1.32652	−0.663259	−	0.748390i	$-0.730827\pi$
$953$	−39025.9	−1.32652	−0.663259	+	0.748390i	$0.730827\pi$
$954$	−29098.6	−0.987529
$955$	0	0
$956$	244.293	0.00826464
$957$	−338.228	−0.0114246
$958$	11955.2	0.403188
$959$	11121.4	0.374483
$960$	0	0
$961$	55650.1	1.86802
$962$	20216.5	0.677552
$963$	13481.7	0.451135
$964$	−3298.91	−0.110218
$965$	0	0
$966$	−6.73722	−0.000224396 0
$967$	−12986.1	−0.431857	−0.215929	−	0.976409i	$-0.569278\pi$
$967$	−12986.1	−0.431857	−0.215929	+	0.976409i	$0.569278\pi$
$968$	−30909.8	−1.02632
$969$	2411.19	0.0799366
$970$	0	0
$971$	−15044.2	−0.497210	−0.248605	−	0.968605i	$-0.579972\pi$
$971$	−15044.2	−0.497210	−0.248605	+	0.968605i	$0.579972\pi$
$972$	−1050.37	−0.0346613
$973$	17948.6	0.591374
$974$	−35323.8	−1.16206
$975$	0	0
$976$	−18649.0	−0.611618
$977$	26593.4	0.870828	0.435414	−	0.900230i	$-0.356602\pi$
$977$	26593.4	0.870828	0.435414	+	0.900230i	$0.356602\pi$
$978$	−1732.68	−0.0566512
$979$	−3220.82	−0.105146
$980$	0	0
$981$	−9122.12	−0.296888
$982$	−30948.2	−1.00570
$983$	36050.3	1.16971	0.584856	−	0.811137i	$-0.301151\pi$
$983$	36050.3	1.16971	0.584856	+	0.811137i	$0.301151\pi$
$984$	−943.431	−0.0305645
$985$	0	0
$986$	−21895.7	−0.707203
$987$	−1148.02	−0.0370233
$988$	1843.45	0.0593604
$989$	−143.721	−0.00462088
$990$	0	0
$991$	−54789.8	−1.75626	−0.878131	−	0.478420i	$-0.841210\pi$
$991$	−54789.8	−1.75626	−0.878131	+	0.478420i	$0.841210\pi$
$992$	9252.77	0.296145
$993$	4241.53	0.135550
$994$	−19545.8	−0.623699
$995$	0	0
$996$	264.597	0.00841775
$997$	23309.0	0.740426	0.370213	−	0.928947i	$-0.379285\pi$
$997$	23309.0	0.740426	0.370213	+	0.928947i	$0.379285\pi$
$998$	−14915.6	−0.473091
$999$	5442.75	0.172373

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	175.4.a.e.1.1	yes	2
3.2	odd	2		1575.4.a.s.1.2		2
5.2	odd	4		175.4.b.d.99.2		4
5.3	odd	4		175.4.b.d.99.3		4
5.4	even	2		175.4.a.d.1.2	✓	2
7.6	odd	2		1225.4.a.t.1.1		2
15.14	odd	2		1575.4.a.v.1.1		2
35.34	odd	2		1225.4.a.r.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
175.4.a.d.1.2	✓	2	5.4	even	2
175.4.a.e.1.1	yes	2	1.1	even	1	trivial
175.4.b.d.99.2		4	5.2	odd	4
175.4.b.d.99.3		4	5.3	odd	4
1225.4.a.r.1.2		2	35.34	odd	2
1225.4.a.t.1.1		2	7.6	odd	2
1575.4.a.s.1.2		2	3.2	odd	2
1575.4.a.v.1.1		2	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)

\(f(q)\)	\(=\)	\(q-2.70156 q^{2} +0.701562 q^{3} -0.701562 q^{4} -1.89531 q^{6} +7.00000 q^{7} +23.5078 q^{8} -26.5078 q^{9} +4.01562 q^{11} -0.492189 q^{12} +51.6125 q^{13} -18.9109 q^{14} -57.8953 q^{16} -67.5078 q^{17} +71.6125 q^{18} -50.9109 q^{19} +4.91093 q^{21} -10.8485 q^{22} +0.507811 q^{23} +16.4922 q^{24} -139.434 q^{26} -37.5391 q^{27} -4.91093 q^{28} -120.058 q^{29} -292.303 q^{31} -31.6547 q^{32} +2.81721 q^{33} +182.377 q^{34} +18.5969 q^{36} -144.989 q^{37} +137.539 q^{38} +36.2094 q^{39} -57.2047 q^{41} -13.2672 q^{42} -283.020 q^{43} -2.81721 q^{44} -1.37188 q^{46} -233.769 q^{47} -40.6172 q^{48} +49.0000 q^{49} -47.3609 q^{51} -36.2094 q^{52} -406.334 q^{53} +101.414 q^{54} +164.555 q^{56} -35.7172 q^{57} +324.344 q^{58} -577.328 q^{59} +322.116 q^{61} +789.675 q^{62} -185.555 q^{63} +548.680 q^{64} -7.61086 q^{66} +985.459 q^{67} +47.3609 q^{68} +0.356261 q^{69} +1033.57 q^{71} -623.141 q^{72} +692.720 q^{73} +391.697 q^{74} +35.7172 q^{76} +28.1093 q^{77} -97.8219 q^{78} -428.236 q^{79} +689.375 q^{81} +154.542 q^{82} -537.592 q^{83} -3.44533 q^{84} +764.597 q^{86} -84.2280 q^{87} +94.3985 q^{88} -802.073 q^{89} +361.287 q^{91} -0.356261 q^{92} -205.069 q^{93} +631.541 q^{94} -22.2077 q^{96} +1752.82 q^{97} -132.377 q^{98} -106.445 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - 5 q^{3} + 5 q^{4} - 23 q^{6} + 14 q^{7} + 15 q^{8} - 21 q^{9} - 56 q^{11} - 33 q^{12} + 52 q^{13} + 7 q^{14} - 135 q^{16} - 103 q^{17} + 92 q^{18} - 57 q^{19} - 35 q^{21} - 233 q^{22} - 31 q^{23}+ \cdots - 437 q^{99}+O(q^{100}) \) 2 * q + q^2 - 5 * q^3 + 5 * q^4 - 23 * q^6 + 14 * q^7 + 15 * q^8 - 21 * q^9 - 56 * q^11 - 33 * q^12 + 52 * q^13 + 7 * q^14 - 135 * q^16 - 103 * q^17 + 92 * q^18 - 57 * q^19 - 35 * q^21 - 233 * q^22 - 31 * q^23 + 65 * q^24 - 138 * q^26 + 85 * q^27 + 35 * q^28 - 413 * q^29 - 162 * q^31 - 249 * q^32 + 345 * q^33 + 51 * q^34 + 50 * q^36 + 75 * q^37 + 115 * q^38 + 34 * q^39 - 505 * q^41 - 161 * q^42 - 73 * q^43 - 345 * q^44 - 118 * q^46 + 224 * q^47 + 399 * q^48 + 98 * q^49 + 155 * q^51 - 34 * q^52 - 262 * q^53 + 555 * q^54 + 105 * q^56 - q^57 - 760 * q^58 + 190 * q^59 + 990 * q^61 + 1272 * q^62 - 147 * q^63 + 361 * q^64 + 1259 * q^66 + 908 * q^67 - 155 * q^68 + 180 * q^69 + 127 * q^71 - 670 * q^72 - 337 * q^73 + 1206 * q^74 + q^76 - 392 * q^77 - 106 * q^78 - 1119 * q^79 - 158 * q^81 - 1503 * q^82 - 1517 * q^83 - 231 * q^84 + 1542 * q^86 + 1586 * q^87 + 605 * q^88 - 1713 * q^89 + 364 * q^91 - 180 * q^92 - 948 * q^93 + 2326 * q^94 + 1217 * q^96 + 1764 * q^97 + 49 * q^98 - 437 * q^99

Introduction

L-functions

Modular forms

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