LMFDB - Embedded newform 1575.4.a.v.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1575,4,Mod(1,1575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1575, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1575.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1575 = 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1575.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:	$92.9280082590$
Analytic rank:	$0$
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:	no (minimal twist has level 175)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.70156$ of defining polynomial
Character	$\chi$	$=$	1575.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.70156 q^{2} -0.701562 q^{4} -7.00000 q^{7} +23.5078 q^{8} -4.01562 q^{11} -51.6125 q^{13} +18.9109 q^{14} -57.8953 q^{16} -67.5078 q^{17} -50.9109 q^{19} +10.8485 q^{22} +0.507811 q^{23} +139.434 q^{26} +4.91093 q^{28} +120.058 q^{29} -292.303 q^{31} -31.6547 q^{32} +182.377 q^{34} +144.989 q^{37} +137.539 q^{38} +57.2047 q^{41} +283.020 q^{43} +2.81721 q^{44} -1.37188 q^{46} -233.769 q^{47} +49.0000 q^{49} +36.2094 q^{52} -406.334 q^{53} -164.555 q^{56} -324.344 q^{58} +577.328 q^{59} +322.116 q^{61} +789.675 q^{62} +548.680 q^{64} -985.459 q^{67} +47.3609 q^{68} -1033.57 q^{71} -692.720 q^{73} -391.697 q^{74} +35.7172 q^{76} +28.1093 q^{77} -428.236 q^{79} -154.542 q^{82} -537.592 q^{83} -764.597 q^{86} -94.3985 q^{88} +802.073 q^{89} +361.287 q^{91} -0.356261 q^{92} +631.541 q^{94} -1752.82 q^{97} -132.377 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} + 5 q^{4} - 14 q^{7} + 15 q^{8} + 56 q^{11} - 52 q^{13} - 7 q^{14} - 135 q^{16} - 103 q^{17} - 57 q^{19} + 233 q^{22} - 31 q^{23} + 138 q^{26} - 35 q^{28} + 413 q^{29} - 162 q^{31} - 249 q^{32}+ \cdots + 49 q^{98}+O(q^{100}) $ 2 * q + q^2 + 5 * q^4 - 14 * q^7 + 15 * q^8 + 56 * q^11 - 52 * q^13 - 7 * q^14 - 135 * q^16 - 103 * q^17 - 57 * q^19 + 233 * q^22 - 31 * q^23 + 138 * q^26 - 35 * q^28 + 413 * q^29 - 162 * q^31 - 249 * q^32 + 51 * q^34 - 75 * q^37 + 115 * q^38 + 505 * q^41 + 73 * q^43 + 345 * q^44 - 118 * q^46 + 224 * q^47 + 98 * q^49 + 34 * q^52 - 262 * q^53 - 105 * q^56 + 760 * q^58 - 190 * q^59 + 990 * q^61 + 1272 * q^62 + 361 * q^64 - 908 * q^67 - 155 * q^68 - 127 * q^71 + 337 * q^73 - 1206 * q^74 + q^76 - 392 * q^77 - 1119 * q^79 + 1503 * q^82 - 1517 * q^83 - 1542 * q^86 - 605 * q^88 + 1713 * q^89 + 364 * q^91 - 180 * q^92 + 2326 * q^94 - 1764 * q^97 + 49 * q^98

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	0
$3$	0	0
$4$	−0.701562	−0.0876953
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	23.5078	1.03891
$9$	0	0
$10$	0	0
$11$	−4.01562	−0.110069	−0.0550343	−	0.998484i	$-0.517527\pi$
$11$	−4.01562	−0.110069	−0.0550343	+	0.998484i	$0.517527\pi$
$12$	0	0
$13$	−51.6125	−1.10113	−0.550567	−	0.834791i	$-0.685588\pi$
$13$	−51.6125	−1.10113	−0.550567	+	0.834791i	$0.685588\pi$
$14$	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	$0.659930\pi$
$18$	0	0
$19$	−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
$21$	0	0
$22$	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	$0.499267\pi$
$24$	0	0
$25$	0	0
$26$	139.434	1.05174
$27$	0	0
$28$	4.91093	0.0331457
$29$	120.058	0.768765	0.384382	−	0.923174i	$-0.374414\pi$
$29$	120.058	0.768765	0.384382	+	0.923174i	$0.374414\pi$
$30$	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
$33$	0	0
$34$	182.377	0.919921
$35$	0	0
$36$	0	0
$37$	144.989	0.644218	0.322109	−	0.946703i	$-0.395608\pi$
$37$	144.989	0.644218	0.322109	+	0.946703i	$0.395608\pi$
$38$	137.539	0.587152
$39$	0	0
$40$	0	0
$41$	57.2047	0.217899	0.108950	−	0.994047i	$-0.465251\pi$
$41$	57.2047	0.217899	0.108950	+	0.994047i	$0.465251\pi$
$42$	0	0
$43$	283.020	1.00373	0.501863	−	0.864947i	$-0.332648\pi$
$43$	283.020	1.00373	0.501863	+	0.864947i	$0.332648\pi$
$44$	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	$0.618163\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	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	$0.676515\pi$
$54$	0	0
$55$	0	0
$56$	−164.555	−0.392670
$57$	0	0
$58$	−324.344	−0.734283
$59$	577.328	1.27393	0.636964	−	0.770894i	$-0.280190\pi$
$59$	577.328	1.27393	0.636964	+	0.770894i	$0.280190\pi$
$60$	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
$63$	0	0
$64$	548.680	1.07164
$65$	0	0
$66$	0	0
$67$	−985.459	−1.79691	−0.898455	−	0.439065i	$-0.855310\pi$
$67$	−985.459	−1.79691	−0.898455	+	0.439065i	$0.855310\pi$
$68$	47.3609	0.0844611
$69$	0	0
$70$	0	0
$71$	−1033.57	−1.72764	−0.863821	−	0.503799i	$-0.831935\pi$
$71$	−1033.57	−1.72764	−0.863821	+	0.503799i	$0.831935\pi$
$72$	0	0
$73$	−692.720	−1.11064	−0.555320	−	0.831637i	$-0.687404\pi$
$73$	−692.720	−1.11064	−0.555320	+	0.831637i	$0.687404\pi$
$74$	−391.697	−0.615322
$75$	0	0
$76$	35.7172	0.0539084
$77$	28.1093	0.0416020
$78$	0	0
$79$	−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
$81$	0	0
$82$	−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	$0.615680\pi$
$84$	0	0
$85$	0	0
$86$	−764.597	−0.958705
$87$	0	0
$88$	−94.3985	−0.114351
$89$	802.073	0.955277	0.477638	−	0.878557i	$-0.341493\pi$
$89$	802.073	0.955277	0.477638	+	0.878557i	$0.341493\pi$
$90$	0	0
$91$	361.287	0.416189
$92$	−0.356261	−0.000403725 0
$93$	0	0
$94$	631.541	0.692962
$95$	0	0
$96$	0	0
$97$	−1752.82	−1.83477	−0.917384	−	0.398004i	$-0.869703\pi$
$97$	−1752.82	−1.83477	−0.917384	+	0.398004i	$0.869703\pi$
$98$	−132.377	−0.136449
$99$	0	0
$100$	0	0
$101$	1987.85	1.95840	0.979200	−	0.202896i	$-0.0650354\pi$
$101$	1987.85	1.95840	0.979200	+	0.202896i	$0.0650354\pi$
$102$	0	0
$103$	389.122	0.372246	0.186123	−	0.982526i	$-0.440408\pi$
$103$	389.122	0.372246	0.186123	+	0.982526i	$0.440408\pi$
$104$	−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	$0.573793\pi$
$108$	0	0
$109$	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
$111$	0	0
$112$	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	$0.669260\pi$
$114$	0	0
$115$	0	0
$116$	−84.2280	−0.0674170
$117$	0	0
$118$	−1559.69	−1.21679
$119$	472.555	0.364025
$120$	0	0
$121$	−1314.87	−0.987885
$122$	−870.215	−0.645784
$123$	0	0
$124$	205.069	0.148514
$125$	0	0
$126$	0	0
$127$	281.652	0.196792	0.0983958	−	0.995147i	$-0.468629\pi$
$127$	281.652	0.196792	0.0983958	+	0.995147i	$0.468629\pi$
$128$	−1229.05	−0.848704
$129$	0	0
$130$	0	0
$131$	699.706	0.466669	0.233334	−	0.972397i	$-0.425036\pi$
$131$	699.706	0.466669	0.233334	+	0.972397i	$0.425036\pi$
$132$	0	0
$133$	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	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
$141$	0	0
$142$	2792.26	1.65015
$143$	207.256	0.121200
$144$	0	0
$145$	0	0
$146$	1871.43	1.06082
$147$	0	0
$148$	−101.719	−0.0564948
$149$	3386.03	1.86171	0.930854	−	0.365390i	$-0.119065\pi$
$149$	3386.03	1.86171	0.930854	+	0.365390i	$0.119065\pi$
$150$	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
$153$	0	0
$154$	−75.9392	−0.0397360
$155$	0	0
$156$	0	0
$157$	−587.325	−0.298558	−0.149279	−	0.988795i	$-0.547695\pi$
$157$	−587.325	−0.298558	−0.149279	+	0.988795i	$0.547695\pi$
$158$	1156.91	0.582522
$159$	0	0
$160$	0	0
$161$	−3.55467	−0.00174005
$162$	0	0
$163$	−914.189	−0.439293	−0.219647	−	0.975579i	$-0.570490\pi$
$163$	−914.189	−0.439293	−0.219647	+	0.975579i	$0.570490\pi$
$164$	−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	$0.221045\pi$
$168$	0	0
$169$	466.850	0.212494
$170$	0	0
$171$	0	0
$172$	−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	$0.389956\pi$
$174$	0	0
$175$	0	0
$176$	232.486	0.0995697
$177$	0	0
$178$	−2166.85	−0.912429
$179$	550.755	0.229974	0.114987	−	0.993367i	$-0.463317\pi$
$179$	550.755	0.229974	0.114987	+	0.993367i	$0.463317\pi$
$180$	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
$183$	0	0
$184$	11.9375	0.00478285
$185$	0	0
$186$	0	0
$187$	271.086	0.106009
$188$	164.003	0.0636232
$189$	0	0
$190$	0	0
$191$	−3436.00	−1.30168	−0.650838	−	0.759217i	$-0.725582\pi$
$191$	−3436.00	−1.30168	−0.650838	+	0.759217i	$0.725582\pi$
$192$	0	0
$193$	−1047.12	−0.390534	−0.195267	−	0.980750i	$-0.562557\pi$
$193$	−1047.12	−0.390534	−0.195267	+	0.980750i	$0.562557\pi$
$194$	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	$0.696819\pi$
$198$	0	0
$199$	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
$201$	0	0
$202$	−5370.30	−1.87056
$203$	−840.405	−0.290566
$204$	0	0
$205$	0	0
$206$	−1051.24	−0.355549
$207$	0	0
$208$	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
$213$	0	0
$214$	1374.00	0.438901
$215$	0	0
$216$	0	0
$217$	2046.12	0.640091
$218$	−929.688	−0.288837
$219$	0	0
$220$	0	0
$221$	3484.25	1.06052
$222$	0	0
$223$	1428.67	0.429016	0.214508	−	0.976722i	$-0.431185\pi$
$223$	1428.67	0.429016	0.214508	+	0.976722i	$0.431185\pi$
$224$	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	$0.161418\pi$
$228$	0	0
$229$	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
$231$	0	0
$232$	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	$0.744815\pi$
$234$	0	0
$235$	0	0
$236$	−405.031	−0.111717
$237$	0	0
$238$	−1276.64	−0.347698
$239$	348.213	0.0942427	0.0471213	−	0.998889i	$-0.484995\pi$
$239$	348.213	0.0942427	0.0471213	+	0.998889i	$0.484995\pi$
$240$	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
$243$	0	0
$244$	−225.984	−0.0592916
$245$	0	0
$246$	0	0
$247$	2627.64	0.676894
$248$	−6871.41	−1.75941
$249$	0	0
$250$	0	0
$251$	2431.64	0.611489	0.305745	−	0.952114i	$-0.401095\pi$
$251$	2431.64	0.611489	0.305745	+	0.952114i	$0.401095\pi$
$252$	0	0
$253$	−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	$0.563454\pi$
$258$	0	0
$259$	−1014.92	−0.243491
$260$	0	0
$261$	0	0
$262$	−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	$0.637187\pi$
$264$	0	0
$265$	0	0
$266$	−962.773	−0.221923
$267$	0	0
$268$	691.361	0.157581
$269$	−791.631	−0.179430	−0.0897149	−	0.995967i	$-0.528596\pi$
$269$	−791.631	−0.179430	−0.0897149	+	0.995967i	$0.528596\pi$
$270$	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
$273$	0	0
$274$	−4292.17	−0.946348
$275$	0	0
$276$	0	0
$277$	1554.16	0.337114	0.168557	−	0.985692i	$-0.446089\pi$
$277$	1554.16	0.337114	0.168557	+	0.985692i	$0.446089\pi$
$278$	−6927.05	−1.49445
$279$	0	0
$280$	0	0
$281$	5043.72	1.07076	0.535379	−	0.844612i	$-0.320169\pi$
$281$	5043.72	1.07076	0.535379	+	0.844612i	$0.320169\pi$
$282$	0	0
$283$	5897.15	1.23869	0.619345	−	0.785119i	$-0.287398\pi$
$283$	5897.15	1.23869	0.619345	+	0.785119i	$0.287398\pi$
$284$	725.116	0.151506
$285$	0	0
$286$	−559.916	−0.115764
$287$	−400.433	−0.0823582
$288$	0	0
$289$	−355.696	−0.0723988
$290$	0	0
$291$	0	0
$292$	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	$0.172576\pi$
$294$	0	0
$295$	0	0
$296$	3408.37	0.669283
$297$	0	0
$298$	−9147.58	−1.77820
$299$	−26.2094	−0.00506932
$300$	0	0
$301$	−1981.14	−0.379373
$302$	6218.94	1.18497
$303$	0	0
$304$	2947.50	0.556089
$305$	0	0
$306$	0	0
$307$	3682.20	0.684542	0.342271	−	0.939601i	$-0.388804\pi$
$307$	3682.20	0.684542	0.342271	+	0.939601i	$0.388804\pi$
$308$	−19.7205	−0.00364830
$309$	0	0
$310$	0	0
$311$	−6866.18	−1.25191	−0.625957	−	0.779858i	$-0.715291\pi$
$311$	−6866.18	−1.25191	−0.625957	+	0.779858i	$0.715291\pi$
$312$	0	0
$313$	−2958.60	−0.534281	−0.267140	−	0.963658i	$-0.586079\pi$
$313$	−2958.60	−0.534281	−0.267140	+	0.963658i	$0.586079\pi$
$314$	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	$0.455151\pi$
$318$	0	0
$319$	−482.107	−0.0846169
$320$	0	0
$321$	0	0
$322$	9.60317	0.00166200
$323$	3436.89	0.592054
$324$	0	0
$325$	0	0
$326$	2469.74	0.419589
$327$	0	0
$328$	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
$333$	0	0
$334$	−8960.16	−1.46790
$335$	0	0
$336$	0	0
$337$	9205.64	1.48802	0.744010	−	0.668168i	$-0.232921\pi$
$337$	9205.64	1.48802	0.744010	+	0.668168i	$0.232921\pi$
$338$	−1261.22	−0.202963
$339$	0	0
$340$	0	0
$341$	1173.78	0.186404
$342$	0	0
$343$	−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	$0.576884\pi$
$348$	0	0
$349$	−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
$351$	0	0
$352$	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	$0.262182\pi$
$354$	0	0
$355$	0	0
$356$	−562.704	−0.0837732
$357$	0	0
$358$	−1487.90	−0.219659
$359$	4779.38	0.702635	0.351317	−	0.936256i	$-0.385734\pi$
$359$	4779.38	0.702635	0.351317	+	0.936256i	$0.385734\pi$
$360$	0	0
$361$	−4267.08	−0.622114
$362$	9623.38	1.39722
$363$	0	0
$364$	−253.466	−0.0364978
$365$	0	0
$366$	0	0
$367$	−175.769	−0.0250001	−0.0125001	−	0.999922i	$-0.503979\pi$
$367$	−175.769	−0.0250001	−0.0125001	+	0.999922i	$0.503979\pi$
$368$	−29.3999	−0.00416460
$369$	0	0
$370$	0	0
$371$	2844.34	0.398034
$372$	0	0
$373$	−7982.98	−1.10816	−0.554079	−	0.832464i	$-0.686930\pi$
$373$	−7982.98	−1.10816	−0.554079	+	0.832464i	$0.686930\pi$
$374$	−732.355	−0.101254
$375$	0	0
$376$	−5495.39	−0.753732
$377$	−6196.48	−0.846512
$378$	0	0
$379$	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
$381$	0	0
$382$	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	$0.398967\pi$
$384$	0	0
$385$	0	0
$386$	2828.85	0.373017
$387$	0	0
$388$	1229.72	0.160900
$389$	50.2546	0.00655015	0.00327508	−	0.999995i	$-0.498958\pi$
$389$	50.2546	0.00655015	0.00327508	+	0.999995i	$0.498958\pi$
$390$	0	0
$391$	−34.2812	−0.00443395
$392$	1151.88	0.148415
$393$	0	0
$394$	8660.15	1.10734
$395$	0	0
$396$	0	0
$397$	11059.9	1.39819	0.699095	−	0.715029i	$-0.253587\pi$
$397$	11059.9	1.39819	0.699095	+	0.715029i	$0.253587\pi$
$398$	−59.5780	−0.00750345
$399$	0	0
$400$	0	0
$401$	13667.8	1.70209	0.851046	−	0.525092i	$-0.175969\pi$
$401$	13667.8	1.70209	0.851046	+	0.525092i	$0.175969\pi$
$402$	0	0
$403$	15086.5	1.86479
$404$	−1394.60	−0.171742
$405$	0	0
$406$	2270.41	0.277533
$407$	−582.221	−0.0709082
$408$	0	0
$409$	−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
$411$	0	0
$412$	−272.993	−0.0326442
$413$	−4041.30	−0.481499
$414$	0	0
$415$	0	0
$416$	1633.78	0.192554
$417$	0	0
$418$	−552.305	−0.0646271
$419$	−1115.39	−0.130049	−0.0650246	−	0.997884i	$-0.520713\pi$
$419$	−1115.39	−0.130049	−0.0650246	+	0.997884i	$0.520713\pi$
$420$	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
$423$	0	0
$424$	−9552.03	−1.09407
$425$	0	0
$426$	0	0
$427$	−2254.81	−0.255545
$428$	356.811	0.0402970
$429$	0	0
$430$	0	0
$431$	−1684.62	−0.188272	−0.0941360	−	0.995559i	$-0.530009\pi$
$431$	−1684.62	−0.188272	−0.0941360	+	0.995559i	$0.530009\pi$
$432$	0	0
$433$	13355.7	1.48230	0.741150	−	0.671340i	$-0.234281\pi$
$433$	13355.7	1.48230	0.741150	+	0.671340i	$0.234281\pi$
$434$	−5527.72	−0.611381
$435$	0	0
$436$	−241.428	−0.0265191
$437$	−25.8531	−0.00283003
$438$	0	0
$439$	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
$441$	0	0
$442$	−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	$0.390211\pi$
$444$	0	0
$445$	0	0
$446$	−3859.63	−0.409773
$447$	0	0
$448$	−3840.76	−0.405042
$449$	−5446.85	−0.572501	−0.286250	−	0.958155i	$-0.592409\pi$
$449$	−5446.85	−0.572501	−0.286250	+	0.958155i	$0.592409\pi$
$450$	0	0
$451$	−229.712	−0.0239839
$452$	854.586	0.0889300
$453$	0	0
$454$	−16153.7	−1.66989
$455$	0	0
$456$	0	0
$457$	−1221.02	−0.124982	−0.0624910	−	0.998046i	$-0.519904\pi$
$457$	−1221.02	−0.124982	−0.0624910	+	0.998046i	$0.519904\pi$
$458$	−17230.0	−1.75787
$459$	0	0
$460$	0	0
$461$	−9967.46	−1.00701	−0.503504	−	0.863993i	$-0.667956\pi$
$461$	−9967.46	−1.00701	−0.503504	+	0.863993i	$0.667956\pi$
$462$	0	0
$463$	6309.54	0.633324	0.316662	−	0.948538i	$-0.397438\pi$
$463$	6309.54	0.633324	0.316662	+	0.948538i	$0.397438\pi$
$464$	−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	$0.373962\pi$
$468$	0	0
$469$	6898.22	0.679168
$470$	0	0
$471$	0	0
$472$	13571.7	1.32349
$473$	−1136.50	−0.110479
$474$	0	0
$475$	0	0
$476$	−331.526	−0.0319233
$477$	0	0
$478$	−940.718	−0.0900156
$479$	4425.28	0.422122	0.211061	−	0.977473i	$-0.432308\pi$
$479$	4425.28	0.422122	0.211061	+	0.977473i	$0.432308\pi$
$480$	0	0
$481$	−7483.25	−0.709369
$482$	−12703.4	−1.20046
$483$	0	0
$484$	922.466	0.0866328
$485$	0	0
$486$	0	0
$487$	−13075.3	−1.21663	−0.608315	−	0.793695i	$-0.708154\pi$
$487$	−13075.3	−1.21663	−0.608315	+	0.793695i	$0.708154\pi$
$488$	7572.23	0.702416
$489$	0	0
$490$	0	0
$491$	−11455.7	−1.05293	−0.526463	−	0.850198i	$-0.676482\pi$
$491$	−11455.7	−1.05293	−0.526463	+	0.850198i	$0.676482\pi$
$492$	0	0
$493$	−8104.84	−0.740413
$494$	−7098.73	−0.646533
$495$	0	0
$496$	16923.0	1.53198
$497$	7235.01	0.652987
$498$	0	0
$499$	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
$501$	0	0
$502$	−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	$0.670106\pi$
$504$	0	0
$505$	0	0
$506$	5.50896	0.000483998 0
$507$	0	0
$508$	−197.596	−0.0172577
$509$	9207.84	0.801828	0.400914	−	0.916116i	$-0.368693\pi$
$509$	9207.84	0.801828	0.400914	+	0.916116i	$0.368693\pi$
$510$	0	0
$511$	4849.04	0.419783
$512$	12720.6	1.09800
$513$	0	0
$514$	4408.29	0.378291
$515$	0	0
$516$	0	0
$517$	938.727	0.0798552
$518$	2741.88	0.232570
$519$	0	0
$520$	0	0
$521$	11598.0	0.975271	0.487636	−	0.873047i	$-0.337859\pi$
$521$	11598.0	0.975271	0.487636	+	0.873047i	$0.337859\pi$
$522$	0	0
$523$	−4596.93	−0.384340	−0.192170	−	0.981362i	$-0.561552\pi$
$523$	−4596.93	−0.384340	−0.192170	+	0.981362i	$0.561552\pi$
$524$	−490.887	−0.0409246
$525$	0	0
$526$	9627.49	0.798058
$527$	19732.7	1.63107
$528$	0	0
$529$	−12166.7	−0.999979
$530$	0	0
$531$	0	0
$532$	−250.020	−0.0203755
$533$	−2952.48	−0.239936
$534$	0	0
$535$	0	0
$536$	−23166.0	−1.86683
$537$	0	0
$538$	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
$543$	0	0
$544$	2136.94	0.168420
$545$	0	0
$546$	0	0
$547$	−12864.6	−1.00557	−0.502787	−	0.864410i	$-0.667692\pi$
$547$	−12864.6	−1.00557	−0.502787	+	0.864410i	$0.667692\pi$
$548$	−1114.62	−0.0868875
$549$	0	0
$550$	0	0
$551$	−6112.26	−0.472579
$552$	0	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	$0.760954\pi$
$558$	0	0
$559$	−14607.4	−1.10524
$560$	0	0
$561$	0	0
$562$	−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	$0.276535\pi$
$564$	0	0
$565$	0	0
$566$	−15931.5	−1.18313
$567$	0	0
$568$	−24297.0	−1.79486
$569$	6242.80	0.459950	0.229975	−	0.973197i	$-0.426136\pi$
$569$	6242.80	0.459950	0.229975	+	0.973197i	$0.426136\pi$
$570$	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
$573$	0	0
$574$	1081.79	0.0786642
$575$	0	0
$576$	0	0
$577$	−11390.2	−0.821806	−0.410903	−	0.911679i	$-0.634786\pi$
$577$	−11390.2	−0.821806	−0.410903	+	0.911679i	$0.634786\pi$
$578$	960.934	0.0691515
$579$	0	0
$580$	0	0
$581$	3763.15	0.268712
$582$	0	0
$583$	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	$0.755535\pi$
$588$	0	0
$589$	14881.4	1.04105
$590$	0	0
$591$	0	0
$592$	−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	$0.255729\pi$
$594$	0	0
$595$	0	0
$596$	−2375.51	−0.163263
$597$	0	0
$598$	70.8062	0.00484194
$599$	−10578.4	−0.721572	−0.360786	−	0.932649i	$-0.617492\pi$
$599$	−10578.4	−0.721572	−0.360786	+	0.932649i	$0.617492\pi$
$600$	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
$603$	0	0
$604$	1614.98	0.108796
$605$	0	0
$606$	0	0
$607$	−5724.50	−0.382785	−0.191392	−	0.981514i	$-0.561300\pi$
$607$	−5724.50	−0.382785	−0.191392	+	0.981514i	$0.561300\pi$
$608$	1611.57	0.107496
$609$	0	0
$610$	0	0
$611$	12065.4	0.798876
$612$	0	0
$613$	19286.4	1.27075	0.635375	−	0.772204i	$-0.280846\pi$
$613$	19286.4	1.27075	0.635375	+	0.772204i	$0.280846\pi$
$614$	−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	$0.428101\pi$
$618$	0	0
$619$	−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
$621$	0	0
$622$	18549.4	1.19576
$623$	−5614.51	−0.361061
$624$	0	0
$625$	0	0
$626$	7992.84	0.510317
$627$	0	0
$628$	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
$633$	0	0
$634$	−4282.55	−0.268268
$635$	0	0
$636$	0	0
$637$	−2529.01	−0.157305
$638$	1302.44	0.0808215
$639$	0	0
$640$	0	0
$641$	21724.3	1.33863	0.669313	−	0.742980i	$-0.266588\pi$
$641$	21724.3	1.33863	0.669313	+	0.742980i	$0.266588\pi$
$642$	0	0
$643$	18736.1	1.14911	0.574555	−	0.818466i	$-0.305175\pi$
$643$	18736.1	1.14911	0.574555	+	0.818466i	$0.305175\pi$
$644$	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	$0.785006\pi$
$648$	0	0
$649$	−2318.33	−0.140219
$650$	0	0
$651$	0	0
$652$	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	$0.346786\pi$
$654$	0	0
$655$	0	0
$656$	−3311.88	−0.197115
$657$	0	0
$658$	−4420.78	−0.261915
$659$	1402.85	0.0829246	0.0414623	−	0.999140i	$-0.486798\pi$
$659$	1402.85	0.0829246	0.0414623	+	0.999140i	$0.486798\pi$
$660$	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
$663$	0	0
$664$	−12637.6	−0.738606
$665$	0	0
$666$	0	0
$667$	60.9666	0.00353919
$668$	−2326.84	−0.134773
$669$	0	0
$670$	0	0
$671$	−1293.49	−0.0744185
$672$	0	0
$673$	3869.29	0.221620	0.110810	−	0.993842i	$-0.464656\pi$
$673$	3869.29	0.221620	0.110810	+	0.993842i	$0.464656\pi$
$674$	−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	$0.493570\pi$
$678$	0	0
$679$	12269.8	0.693477
$680$	0	0
$681$	0	0
$682$	−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	$0.619714\pi$
$684$	0	0
$685$	0	0
$686$	926.636	0.0515731
$687$	0	0
$688$	−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
$693$	0	0
$694$	8353.98	0.456935
$695$	0	0
$696$	0	0
$697$	−3861.76	−0.209863
$698$	14133.5	0.766421
$699$	0	0
$700$	0	0
$701$	7489.24	0.403516	0.201758	−	0.979435i	$-0.435335\pi$
$701$	7489.24	0.403516	0.201758	+	0.979435i	$0.435335\pi$
$702$	0	0
$703$	−7381.53	−0.396016
$704$	−2203.29	−0.117954
$705$	0	0
$706$	−24351.1	−1.29811
$707$	−13914.9	−0.740206
$708$	0	0
$709$	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
$711$	0	0
$712$	18855.0	0.992445
$713$	−148.435	−0.00779652
$714$	0	0
$715$	0	0
$716$	−386.389	−0.0201676
$717$	0	0
$718$	−12911.8	−0.671119
$719$	5889.90	0.305503	0.152751	−	0.988265i	$-0.451187\pi$
$719$	5889.90	0.305503	0.152751	+	0.988265i	$0.451187\pi$
$720$	0	0
$721$	−2723.85	−0.140696
$722$	11527.8	0.594210
$723$	0	0
$724$	2499.07	0.128284
$725$	0	0
$726$	0	0
$727$	−24760.2	−1.26314	−0.631571	−	0.775318i	$-0.717589\pi$
$727$	−24760.2	−1.26314	−0.631571	+	0.775318i	$0.717589\pi$
$728$	8493.08	0.432382
$729$	0	0
$730$	0	0
$731$	−19106.1	−0.966708
$732$	0	0
$733$	4537.53	0.228646	0.114323	−	0.993444i	$-0.463530\pi$
$733$	4537.53	0.228646	0.114323	+	0.993444i	$0.463530\pi$
$734$	474.850	0.0238788
$735$	0	0
$736$	−16.0746	−0.000805051 0
$737$	3957.23	0.197784
$738$	0	0
$739$	−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
$741$	0	0
$742$	−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	$0.286511\pi$
$744$	0	0
$745$	0	0
$746$	21566.5	1.05845
$747$	0	0
$748$	−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
$753$	0	0
$754$	16740.2	0.808543
$755$	0	0
$756$	0	0
$757$	28274.4	1.35753	0.678765	−	0.734356i	$-0.262516\pi$
$757$	28274.4	1.35753	0.678765	+	0.734356i	$0.262516\pi$
$758$	−34210.3	−1.63928
$759$	0	0
$760$	0	0
$761$	−12377.9	−0.589616	−0.294808	−	0.955557i	$-0.595256\pi$
$761$	−12377.9	−0.589616	−0.294808	+	0.955557i	$0.595256\pi$
$762$	0	0
$763$	−2408.91	−0.114297
$764$	2410.57	0.114151
$765$	0	0
$766$	−12639.7	−0.596204
$767$	−29797.3	−1.40276
$768$	0	0
$769$	−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
$771$	0	0
$772$	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	$0.445360\pi$
$774$	0	0
$775$	0	0
$776$	−41205.1	−1.90615
$777$	0	0
$778$	−135.766	−0.00625636
$779$	−2912.35	−0.133948
$780$	0	0
$781$	4150.44	0.190159
$782$	92.6127	0.00423507
$783$	0	0
$784$	−2836.87	−0.129231
$785$	0	0
$786$	0	0
$787$	−36457.4	−1.65129	−0.825646	−	0.564188i	$-0.809189\pi$
$787$	−36457.4	−1.65129	−0.825646	+	0.564188i	$0.809189\pi$
$788$	2248.93	0.101669
$789$	0	0
$790$	0	0
$791$	8526.83	0.383286
$792$	0	0
$793$	−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	$0.566681\pi$
$798$	0	0
$799$	15781.2	0.698747
$800$	0	0
$801$	0	0
$802$	−36924.5	−1.62575
$803$	2781.70	0.122247
$804$	0	0
$805$	0	0
$806$	−40757.1	−1.78115
$807$	0	0
$808$	46730.0	2.03460
$809$	−28189.6	−1.22509	−0.612543	−	0.790437i	$-0.709853\pi$
$809$	−28189.6	−1.22509	−0.612543	+	0.790437i	$0.709853\pi$
$810$	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
$813$	0	0
$814$	1572.91	0.0677277
$815$	0	0
$816$	0	0
$817$	−14408.8	−0.617015
$818$	29691.5	1.26912
$819$	0	0
$820$	0	0
$821$	−20512.6	−0.871980	−0.435990	−	0.899952i	$-0.643602\pi$
$821$	−20512.6	−0.871980	−0.435990	+	0.899952i	$0.643602\pi$
$822$	0	0
$823$	−7882.78	−0.333872	−0.166936	−	0.985968i	$-0.553387\pi$
$823$	−7882.78	−0.333872	−0.166936	+	0.985968i	$0.553387\pi$
$824$	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	$0.632821\pi$
$828$	0	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
$831$	0	0
$832$	−28318.7	−1.18002
$833$	−3307.88	−0.137589
$834$	0	0
$835$	0	0
$836$	−143.427	−0.00593363
$837$	0	0
$838$	3013.31	0.124216
$839$	−23175.4	−0.953640	−0.476820	−	0.879001i	$-0.658211\pi$
$839$	−23175.4	−0.953640	−0.476820	+	0.879001i	$0.658211\pi$
$840$	0	0
$841$	−9975.12	−0.409001
$842$	6470.94	0.264849
$843$	0	0
$844$	−1657.46	−0.0675972
$845$	0	0
$846$	0	0
$847$	9204.12	0.373385
$848$	23524.9	0.952650
$849$	0	0
$850$	0	0
$851$	73.6270	0.00296581
$852$	0	0
$853$	31431.7	1.26166	0.630832	−	0.775920i	$-0.282714\pi$
$853$	31431.7	1.26166	0.630832	+	0.775920i	$0.282714\pi$
$854$	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	$0.474646\pi$
$858$	0	0
$859$	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
$861$	0	0
$862$	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	$0.790657\pi$
$864$	0	0
$865$	0	0
$866$	−36081.3	−1.41581
$867$	0	0
$868$	−1435.48	−0.0561330
$869$	1719.63	0.0671284
$870$	0	0
$871$	50862.0	1.97864
$872$	8089.73	0.314166
$873$	0	0
$874$	69.8438	0.00270309
$875$	0	0
$876$	0	0
$877$	−3222.42	−0.124074	−0.0620372	−	0.998074i	$-0.519760\pi$
$877$	−3222.42	−0.124074	−0.0620372	+	0.998074i	$0.519760\pi$
$878$	−37328.4	−1.43482
$879$	0	0
$880$	0	0
$881$	18712.4	0.715591	0.357795	−	0.933800i	$-0.383529\pi$
$881$	18712.4	0.715591	0.357795	+	0.933800i	$0.383529\pi$
$882$	0	0
$883$	17924.0	0.683114	0.341557	−	0.939861i	$-0.389046\pi$
$883$	17924.0	0.683114	0.341557	+	0.939861i	$0.389046\pi$
$884$	−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	$0.440165\pi$
$888$	0	0
$889$	−1971.56	−0.0743803
$890$	0	0
$891$	0	0
$892$	−1002.30	−0.0376226
$893$	11901.4	0.445985
$894$	0	0
$895$	0	0
$896$	8603.38	0.320780
$897$	0	0
$898$	14715.0	0.546822
$899$	−35093.3	−1.30192
$900$	0	0
$901$	27430.7	1.01426
$902$	620.582	0.0229081
$903$	0	0
$904$	−28635.3	−1.05354
$905$	0	0
$906$	0	0
$907$	30129.2	1.10300	0.551502	−	0.834174i	$-0.314055\pi$
$907$	30129.2	1.10300	0.551502	+	0.834174i	$0.314055\pi$
$908$	−4194.91	−0.153318
$909$	0	0
$910$	0	0
$911$	37831.8	1.37588	0.687938	−	0.725769i	$-0.258516\pi$
$911$	37831.8	1.37588	0.687938	+	0.725769i	$0.258516\pi$
$912$	0	0
$913$	2158.77	0.0782527
$914$	3298.65	0.119376
$915$	0	0
$916$	−4474.40	−0.161396
$917$	−4897.94	−0.176384
$918$	0	0
$919$	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
$921$	0	0
$922$	26927.7	0.961841
$923$	53345.3	1.90236
$924$	0	0
$925$	0	0
$926$	−17045.6	−0.604917
$927$	0	0
$928$	−3800.39	−0.134433
$929$	45138.3	1.59412	0.797061	−	0.603899i	$-0.206387\pi$
$929$	45138.3	1.59412	0.797061	+	0.603899i	$0.206387\pi$
$930$	0	0
$931$	−2494.64	−0.0878178
$932$	3470.75	0.121983
$933$	0	0
$934$	−21031.1	−0.736786
$935$	0	0
$936$	0	0
$937$	−22634.7	−0.789161	−0.394580	−	0.918861i	$-0.629110\pi$
$937$	−22634.7	−0.789161	−0.394580	+	0.918861i	$0.629110\pi$
$938$	−18636.0	−0.648705
$939$	0	0
$940$	0	0
$941$	−11695.3	−0.405160	−0.202580	−	0.979266i	$-0.564933\pi$
$941$	−11695.3	−0.405160	−0.202580	+	0.979266i	$0.564933\pi$
$942$	0	0
$943$	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	$0.784493\pi$
$948$	0	0
$949$	35753.0	1.22296
$950$	0	0
$951$	0	0
$952$	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	$0.730827\pi$
$954$	0	0
$955$	0	0
$956$	−244.293	−0.00826464
$957$	0	0
$958$	−11955.2	−0.403188
$959$	−11121.4	−0.374483
$960$	0	0
$961$	55650.1	1.86802
$962$	20216.5	0.677552
$963$	0	0
$964$	−3298.91	−0.110218
$965$	0	0
$966$	0	0
$967$	12986.1	0.431857	0.215929	−	0.976409i	$-0.430722\pi$
$967$	12986.1	0.431857	0.215929	+	0.976409i	$0.430722\pi$
$968$	−30909.8	−1.02632
$969$	0	0
$970$	0	0
$971$	15044.2	0.497210	0.248605	−	0.968605i	$-0.420028\pi$
$971$	15044.2	0.497210	0.248605	+	0.968605i	$0.420028\pi$
$972$	0	0
$973$	−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	$0.356602\pi$
$978$	0	0
$979$	−3220.82	−0.105146
$980$	0	0
$981$	0	0
$982$	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	$0.301151\pi$
$984$	0	0
$985$	0	0
$986$	21895.7	0.707203
$987$	0	0
$988$	−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
$993$	0	0
$994$	−19545.8	−0.623699
$995$	0	0
$996$	0	0
$997$	−23309.0	−0.740426	−0.370213	−	0.928947i	$-0.620715\pi$
$997$	−23309.0	−0.740426	−0.370213	+	0.928947i	$0.620715\pi$
$998$	−14915.6	−0.473091
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
175.4.a.d.1.2	✓	2	3.2	odd	2
175.4.a.e.1.1	yes	2	15.14	odd	2
175.4.b.d.99.2		4	15.8	even	4
175.4.b.d.99.3		4	15.2	even	4
1225.4.a.r.1.2		2	21.20	even	2
1225.4.a.t.1.1		2	105.104	even	2
1575.4.a.s.1.2		2	5.4	even	2
1575.4.a.v.1.1		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1575.a (trivial)

\(f(q)\)	\(=\)	\(q-2.70156 q^{2} -0.701562 q^{4} -7.00000 q^{7} +23.5078 q^{8} -4.01562 q^{11} -51.6125 q^{13} +18.9109 q^{14} -57.8953 q^{16} -67.5078 q^{17} -50.9109 q^{19} +10.8485 q^{22} +0.507811 q^{23} +139.434 q^{26} +4.91093 q^{28} +120.058 q^{29} -292.303 q^{31} -31.6547 q^{32} +182.377 q^{34} +144.989 q^{37} +137.539 q^{38} +57.2047 q^{41} +283.020 q^{43} +2.81721 q^{44} -1.37188 q^{46} -233.769 q^{47} +49.0000 q^{49} +36.2094 q^{52} -406.334 q^{53} -164.555 q^{56} -324.344 q^{58} +577.328 q^{59} +322.116 q^{61} +789.675 q^{62} +548.680 q^{64} -985.459 q^{67} +47.3609 q^{68} -1033.57 q^{71} -692.720 q^{73} -391.697 q^{74} +35.7172 q^{76} +28.1093 q^{77} -428.236 q^{79} -154.542 q^{82} -537.592 q^{83} -764.597 q^{86} -94.3985 q^{88} +802.073 q^{89} +361.287 q^{91} -0.356261 q^{92} +631.541 q^{94} -1752.82 q^{97} -132.377 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + 5 q^{4} - 14 q^{7} + 15 q^{8} + 56 q^{11} - 52 q^{13} - 7 q^{14} - 135 q^{16} - 103 q^{17} - 57 q^{19} + 233 q^{22} - 31 q^{23} + 138 q^{26} - 35 q^{28} + 413 q^{29} - 162 q^{31} - 249 q^{32}+ \cdots + 49 q^{98}+O(q^{100}) \) 2 * q + q^2 + 5 * q^4 - 14 * q^7 + 15 * q^8 + 56 * q^11 - 52 * q^13 - 7 * q^14 - 135 * q^16 - 103 * q^17 - 57 * q^19 + 233 * q^22 - 31 * q^23 + 138 * q^26 - 35 * q^28 + 413 * q^29 - 162 * q^31 - 249 * q^32 + 51 * q^34 - 75 * q^37 + 115 * q^38 + 505 * q^41 + 73 * q^43 + 345 * q^44 - 118 * q^46 + 224 * q^47 + 98 * q^49 + 34 * q^52 - 262 * q^53 - 105 * q^56 + 760 * q^58 - 190 * q^59 + 990 * q^61 + 1272 * q^62 + 361 * q^64 - 908 * q^67 - 155 * q^68 - 127 * q^71 + 337 * q^73 - 1206 * q^74 + q^76 - 392 * q^77 - 1119 * q^79 + 1503 * q^82 - 1517 * q^83 - 1542 * q^86 - 605 * q^88 + 1713 * q^89 + 364 * q^91 - 180 * q^92 + 2326 * q^94 - 1764 * q^97 + 49 * q^98

Introduction

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