LMFDB - Embedded newform 1225.4.a.r.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("1225.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1225 = 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1225.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$72.2773397570$
Analytic rank:	$0$
Dimension:	$2$
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			1.2
Root			$-2.70156$ of defining polynomial
Character	$\chi$	$=$	1225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.70156 q^{2} +0.701562 q^{3} -0.701562 q^{4} +1.89531 q^{6} -23.5078 q^{8} -26.5078 q^{9} +4.01562 q^{11} -0.492189 q^{12} +51.6125 q^{13} -57.8953 q^{16} -67.5078 q^{17} -71.6125 q^{18} +50.9109 q^{19} +10.8485 q^{22} -0.507811 q^{23} -16.4922 q^{24} +139.434 q^{26} -37.5391 q^{27} -120.058 q^{29} +292.303 q^{31} +31.6547 q^{32} +2.81721 q^{33} -182.377 q^{34} +18.5969 q^{36} +144.989 q^{37} +137.539 q^{38} +36.2094 q^{39} +57.2047 q^{41} +283.020 q^{43} -2.81721 q^{44} -1.37188 q^{46} -233.769 q^{47} -40.6172 q^{48} -47.3609 q^{51} -36.2094 q^{52} +406.334 q^{53} -101.414 q^{54} +35.7172 q^{57} -324.344 q^{58} +577.328 q^{59} -322.116 q^{61} +789.675 q^{62} +548.680 q^{64} +7.61086 q^{66} -985.459 q^{67} +47.3609 q^{68} -0.356261 q^{69} +1033.57 q^{71} +623.141 q^{72} +692.720 q^{73} +391.697 q^{74} -35.7172 q^{76} +97.8219 q^{78} -428.236 q^{79} +689.375 q^{81} +154.542 q^{82} -537.592 q^{83} +764.597 q^{86} -84.2280 q^{87} -94.3985 q^{88} +802.073 q^{89} +0.356261 q^{92} +205.069 q^{93} -631.541 q^{94} +22.2077 q^{96} +1752.82 q^{97} -106.445 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - 5 q^{3} + 5 q^{4} + 23 q^{6} - 15 q^{8} - 21 q^{9} - 56 q^{11} - 33 q^{12} + 52 q^{13} - 135 q^{16} - 103 q^{17} - 92 q^{18} + 57 q^{19} + 233 q^{22} + 31 q^{23} - 65 q^{24} + 138 q^{26}+ \cdots - 437 q^{99}+O(q^{100}) $ 2 * q - q^2 - 5 * q^3 + 5 * q^4 + 23 * q^6 - 15 * q^8 - 21 * q^9 - 56 * q^11 - 33 * q^12 + 52 * q^13 - 135 * q^16 - 103 * q^17 - 92 * q^18 + 57 * q^19 + 233 * q^22 + 31 * q^23 - 65 * q^24 + 138 * q^26 + 85 * q^27 - 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 + 73 * q^43 - 345 * q^44 - 118 * q^46 + 224 * q^47 + 399 * q^48 + 155 * q^51 - 34 * q^52 + 262 * q^53 - 555 * q^54 + q^57 + 760 * q^58 - 190 * q^59 - 990 * q^61 + 1272 * q^62 + 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 + 106 * q^78 - 1119 * q^79 - 158 * q^81 - 1503 * q^82 - 1517 * q^83 + 1542 * q^86 + 1586 * q^87 - 605 * q^88 + 1713 * q^89 + 180 * q^92 + 948 * q^93 - 2326 * q^94 - 1217 * q^96 + 1764 * q^97 - 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	$-0.341517\pi$
$2$	2.70156	0.955146	0.477573	+	0.878592i	$0.341517\pi$
$3$	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
$7$	0	0
$7$	0	0
$8$	−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	$0.314412\pi$
$14$	0	0
$15$	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	$-0.400554\pi$
$19$	50.9109	0.614725	0.307362	+	0.951593i	$0.400554\pi$
$20$	0	0
$21$	0	0
$22$	10.8485	0.105132
$23$	−0.507811	−0.00460373	−0.00230187	−	0.999997i	$-0.500733\pi$
$23$	−0.507811	−0.00460373	−0.00230187	+	0.999997i	$0.500733\pi$
$24$	−16.4922	−0.140269
$25$	0	0
$26$	139.434	1.05174
$27$	−37.5391	−0.267570
$28$	0	0
$29$	−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	$-0.178549\pi$
$31$	292.303	1.69352	0.846761	+	0.531973i	$0.178549\pi$
$32$	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	$-0.395608\pi$
$37$	144.989	0.644218	0.322109	+	0.946703i	$0.395608\pi$
$38$	137.539	0.587152
$39$	36.2094	0.148670
$40$	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	$0.618163\pi$
$48$	−40.6172	−0.122137
$49$	0	0
$50$	0	0
$51$	−47.3609	−0.130036
$52$	−36.2094	−0.0965642
$53$	406.334	1.05310	0.526550	−	0.850144i	$-0.323485\pi$
$53$	406.334	1.05310	0.526550	+	0.850144i	$0.323485\pi$
$54$	−101.414	−0.255569
$55$	0	0
$56$	0	0
$57$	35.7172	0.0829975
$58$	−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	$-0.609769\pi$
$61$	−322.116	−0.676110	−0.338055	+	0.941126i	$0.609769\pi$
$62$	789.675	1.61756
$63$	0	0
$64$	548.680	1.07164
$65$	0	0
$66$	7.61086	0.0141944
$67$	−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
$69$	−0.356261	−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
$77$	0	0
$78$	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	$0.615680\pi$
$84$	0	0
$85$	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	$-0.341493\pi$
$89$	802.073	0.955277	0.477638	+	0.878557i	$0.341493\pi$
$90$	0	0
$91$	0	0
$92$	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	$0.130297\pi$
$98$	0	0
$99$	−106.445	−0.108062
$100$	0	0
$101$	1987.85	1.95840	0.979200	−	0.202896i	$-0.0650354\pi$
$101$	1987.85	1.95840	0.979200	+	0.202896i	$0.0650354\pi$
$102$	−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	$-0.426207\pi$
$107$	508.595	0.459512	0.229756	+	0.973248i	$0.426207\pi$
$108$	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
$112$	0	0
$113$	1218.12	1.01408	0.507040	−	0.861923i	$-0.330740\pi$
$113$	1218.12	1.01408	0.507040	+	0.861923i	$0.330740\pi$
$114$	96.4922	0.0792748
$115$	0	0
$116$	84.2280	0.0674170
$117$	−1368.13	−1.08106
$118$	1559.69	1.21679
$119$	0	0
$120$	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
$126$	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
$129$	198.556	0.135519
$130$	0	0
$131$	699.706	0.466669	0.233334	−	0.972397i	$-0.425036\pi$
$131$	699.706	0.466669	0.233334	+	0.972397i	$0.425036\pi$
$132$	−1.97645	−0.00130324
$133$	0	0
$134$	−2662.28	−1.71631
$135$	0	0
$136$	1586.96	1.00059
$137$	−1588.77	−0.990789	−0.495394	−	0.868668i	$-0.664976\pi$
$137$	−1588.77	−0.990789	−0.495394	+	0.868668i	$0.664976\pi$
$138$	−0.962460	−0.000593696 0
$139$	−2564.09	−1.56463	−0.782315	−	0.622884i	$-0.785961\pi$
$139$	−2564.09	−1.56463	−0.782315	+	0.622884i	$0.785961\pi$
$140$	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
$147$	0	0
$148$	−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
$154$	0	0
$155$	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
$161$	0	0
$162$	1862.39	0.903229
$163$	−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
$171$	−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	$-0.238860\pi$
$181$	3562.15	1.46283	0.731416	+	0.681931i	$0.238860\pi$
$182$	0	0
$183$	−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
$189$	0	0
$190$	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	$-0.562557\pi$
$193$	−1047.12	−0.390534	−0.195267	+	0.980750i	$0.562557\pi$
$194$	4735.37	1.75247
$195$	0	0
$196$	0	0
$197$	3205.61	1.15934	0.579670	−	0.814851i	$-0.303181\pi$
$197$	3205.61	1.15934	0.579670	+	0.814851i	$0.303181\pi$
$198$	−287.569	−0.103215
$199$	−22.0532	−0.00785581	−0.00392791	−	0.999992i	$-0.501250\pi$
$199$	−22.0532	−0.00785581	−0.00392791	+	0.999992i	$0.501250\pi$
$200$	0	0
$201$	−691.361	−0.242611
$202$	5370.30	1.87056
$203$	0	0
$204$	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
$217$	0	0
$218$	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	$0.568815\pi$
$224$	0	0
$225$	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	$-0.871980\pi$
$229$	−6377.77	−1.84042	−0.920208	+	0.391430i	$0.871980\pi$
$230$	0	0
$231$	0	0
$232$	2822.30	0.798676
$233$	4947.18	1.39099	0.695495	−	0.718531i	$-0.255185\pi$
$233$	4947.18	1.39099	0.695495	+	0.718531i	$0.255185\pi$
$234$	−3696.10	−1.03257
$235$	0	0
$236$	−405.031	−0.111717
$237$	−300.434	−0.0823430
$238$	0	0
$239$	−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	$-0.716297\pi$
$241$	−4702.23	−1.25683	−0.628417	+	0.777876i	$0.716297\pi$
$242$	−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	$-0.401095\pi$
$251$	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	$0.563454\pi$
$258$	536.412	0.129440
$259$	0	0
$260$	0	0
$261$	3182.47	0.754751
$262$	1890.30	0.445737
$263$	3563.67	0.835534	0.417767	−	0.908554i	$-0.362813\pi$
$263$	3563.67	0.835534	0.417767	+	0.908554i	$0.362813\pi$
$264$	−66.2264	−0.0154392
$265$	0	0
$266$	0	0
$267$	562.704	0.128977
$268$	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	$-0.840324\pi$
$271$	−7823.17	−1.75359	−0.876796	+	0.480862i	$0.840324\pi$
$272$	3908.39	0.871253
$273$	0	0
$274$	−4292.17	−0.946348
$275$	0	0
$276$	0.249939	5.45093e−5 0
$277$	1554.16	0.337114	0.168557	−	0.985692i	$-0.446089\pi$
$277$	1554.16	0.337114	0.168557	+	0.985692i	$0.446089\pi$
$278$	−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
$287$	0	0
$288$	−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	$0.172576\pi$
$294$	0	0
$295$	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
$301$	0	0
$302$	−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	$0.611196\pi$
$308$	0	0
$309$	−272.993	−0.0502590
$310$	0	0
$311$	−6866.18	−1.25191	−0.625957	−	0.779858i	$-0.715291\pi$
$311$	−6866.18	−1.25191	−0.625957	+	0.779858i	$0.715291\pi$
$312$	−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	$-0.544849\pi$
$317$	−1585.21	−0.280866	−0.140433	+	0.990090i	$0.544849\pi$
$318$	770.131	0.135808
$319$	−482.107	−0.0846169
$320$	0	0
$321$	356.811	0.0620413
$322$	0	0
$323$	−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
$329$	0	0
$330$	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
$336$	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
$339$	854.586	0.136917
$340$	0	0
$341$	1173.78	0.186404
$342$	−3645.86	−0.576449
$343$	0	0
$344$	−6653.19	−1.04278
$345$	0	0
$346$	4166.25	0.647338
$347$	3092.28	0.478393	0.239196	−	0.970971i	$-0.423116\pi$
$347$	3092.28	0.478393	0.239196	+	0.970971i	$0.423116\pi$
$348$	59.0912	0.00910236
$349$	5231.61	0.802412	0.401206	−	0.915988i	$-0.368591\pi$
$349$	5231.61	0.802412	0.401206	+	0.915988i	$0.368591\pi$
$350$	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
$357$	0	0
$358$	−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
$364$	0	0
$365$	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
$371$	0	0
$372$	−143.868	−0.0200517
$373$	−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
$381$	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
$392$	0	0
$393$	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
$399$	0	0
$400$	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
$406$	0	0
$407$	582.221	0.0709082
$408$	1113.35	0.135096
$409$	10990.5	1.32872	0.664359	−	0.747414i	$-0.268705\pi$
$409$	10990.5	1.32872	0.664359	+	0.747414i	$0.268705\pi$
$410$	0	0
$411$	−1114.62	−0.133772
$412$	272.993	0.0326442
$413$	0	0
$414$	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	$-0.520713\pi$
$419$	−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
$423$	6196.70	0.712278
$424$	−9552.03	−1.09407
$425$	0	0
$426$	1958.95	0.222796
$427$	0	0
$428$	−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	$0.765719\pi$
$434$	0	0
$435$	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	$-0.770477\pi$
$439$	−13817.4	−1.50220	−0.751101	+	0.660188i	$0.770477\pi$
$440$	0	0
$441$	0	0
$442$	−9412.91	−1.01296
$443$	−6305.19	−0.676227	−0.338114	−	0.941105i	$-0.609789\pi$
$443$	−6305.19	−0.676227	−0.338114	+	0.941105i	$0.609789\pi$
$444$	−71.3621	−0.00762769
$445$	0	0
$446$	−3859.63	−0.409773
$447$	−2375.51	−0.251360
$448$	0	0
$449$	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	$-0.519904\pi$
$457$	−1221.02	−0.124982	−0.0624910	+	0.998046i	$0.519904\pi$
$458$	−17230.0	−1.75787
$459$	2534.18	0.257702
$460$	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	$0.373962\pi$
$468$	959.831	0.0948039
$469$	0	0
$470$	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
$476$	0	0
$477$	−10771.0	−1.03390
$478$	−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
$486$	4044.76	0.377519
$487$	−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
$489$	−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
$497$	0	0
$498$	−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	$0.670106\pi$
$504$	0	0
$505$	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	$-0.368693\pi$
$509$	9207.84	0.801828	0.400914	+	0.916116i	$0.368693\pi$
$510$	0	0
$511$	0	0
$512$	−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
$518$	0	0
$519$	1081.92	0.0915051
$520$	0	0
$521$	11598.0	0.975271	0.487636	−	0.873047i	$-0.337859\pi$
$521$	11598.0	0.975271	0.487636	+	0.873047i	$0.337859\pi$
$522$	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
$532$	0	0
$533$	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
$539$	0	0
$540$	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
$546$	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
$549$	8538.58	0.663785
$550$	0	0
$551$	−6112.26	−0.472579
$552$	8.37491	0.000645760 0
$553$	0	0
$554$	4198.67	0.321993
$555$	0	0
$556$	1798.87	0.137211
$557$	19219.4	1.46203	0.731016	−	0.682360i	$-0.239046\pi$
$557$	19219.4	1.46203	0.731016	+	0.682360i	$0.239046\pi$
$558$	−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
$567$	0	0
$568$	−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
$574$	0	0
$575$	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
$581$	0	0
$582$	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	$0.755535\pi$
$588$	0	0
$589$	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	$-0.449893\pi$
$601$	4619.46	0.313530	0.156765	+	0.987636i	$0.449893\pi$
$602$	0	0
$603$	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
$609$	0	0
$610$	0	0
$611$	−12065.4	−0.798876
$612$	−1255.43	−0.0829214
$613$	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
$616$	0	0
$617$	−6864.84	−0.447922	−0.223961	−	0.974598i	$-0.571899\pi$
$617$	−6864.84	−0.447922	−0.223961	+	0.974598i	$0.571899\pi$
$618$	−737.508	−0.0480047
$619$	17559.4	1.14018	0.570090	−	0.821582i	$-0.306908\pi$
$619$	17559.4	1.14018	0.570090	+	0.821582i	$0.306908\pi$
$620$	0	0
$621$	19.0627	0.00123182
$622$	−18549.4	−1.19576
$623$	0	0
$624$	−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
$637$	0	0
$638$	−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	$0.694825\pi$
$644$	0	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
$651$	0	0
$652$	641.360	0.0385239
$653$	−15450.7	−0.925929	−0.462964	−	0.886377i	$-0.653214\pi$
$653$	−15450.7	−0.925929	−0.462964	+	0.886377i	$0.653214\pi$
$654$	652.234	0.0389975
$655$	0	0
$656$	−3311.88	−0.197115
$657$	−18362.5	−1.09039
$658$	0	0
$659$	−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	$-0.565041\pi$
$661$	−6896.75	−0.405828	−0.202914	+	0.979197i	$0.565041\pi$
$662$	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
$672$	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	$0.493570\pi$
$678$	2308.72	0.130775
$679$	0	0
$680$	0	0
$681$	4194.91	0.236049
$682$	3171.04	0.178043
$683$	13112.0	0.734577	0.367288	−	0.930107i	$-0.380286\pi$
$683$	13112.0	0.734577	0.367288	+	0.930107i	$0.380286\pi$
$684$	946.784	0.0529257
$685$	0	0
$686$	0	0
$687$	−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	$-0.387504\pi$
$691$	12573.5	0.692211	0.346105	+	0.938196i	$0.387504\pi$
$692$	−1081.92	−0.0594343
$693$	0	0
$694$	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
$707$	0	0
$708$	−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
$714$	0	0
$715$	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	$-0.451187\pi$
$719$	5889.90	0.305503	0.152751	+	0.988265i	$0.451187\pi$
$720$	0	0
$721$	0	0
$722$	−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	$0.282411\pi$
$728$	0	0
$729$	−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
$742$	0	0
$743$	−25175.4	−1.24306	−0.621532	−	0.783389i	$-0.713489\pi$
$743$	−25175.4	−1.24306	−0.621532	+	0.783389i	$0.713489\pi$
$744$	−4820.72	−0.237549
$745$	0	0
$746$	−21566.5	−1.05845
$747$	14250.4	0.697985
$748$	190.184	0.00929652
$749$	0	0
$750$	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
$756$	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
$759$	−1.43061	−6.84160e−5 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	$0.595256\pi$
$762$	533.818	0.0253782
$763$	0	0
$764$	−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	$-0.231650\pi$
$769$	31845.7	1.49335	0.746674	+	0.665190i	$0.231650\pi$
$770$	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
$777$	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
$783$	4506.86	0.205699
$784$	0	0
$785$	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
$791$	0	0
$792$	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	$0.566681\pi$
$798$	0	0
$799$	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	$-0.334811\pi$
$811$	22909.7	0.991946	0.495973	+	0.868338i	$0.334811\pi$
$812$	0	0
$813$	−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
$819$	0	0
$820$	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	$-0.553387\pi$
$823$	−7882.78	−0.333872	−0.166936	+	0.985968i	$0.553387\pi$
$824$	9147.40	0.386729
$825$	0	0
$826$	0	0
$827$	19276.5	0.810531	0.405265	−	0.914199i	$-0.367179\pi$
$827$	19276.5	0.810531	0.405265	+	0.914199i	$0.367179\pi$
$828$	−9.44369	−0.000396366 0
$829$	−13771.7	−0.576972	−0.288486	−	0.957484i	$-0.593152\pi$
$829$	−13771.7	−0.576972	−0.288486	+	0.957484i	$0.593152\pi$
$830$	0	0
$831$	1090.34	0.0455157
$832$	28318.7	1.18002
$833$	0	0
$834$	−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	$-0.658211\pi$
$839$	−23175.4	−0.953640	−0.476820	+	0.879001i	$0.658211\pi$
$840$	0	0
$841$	−9975.12	−0.409001
$842$	−6470.94	−0.264849
$843$	−3538.48	−0.144569
$844$	−1657.46	−0.0675972
$845$	0	0
$846$	16740.8	0.680330
$847$	0	0
$848$	−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	$0.717286\pi$
$854$	0	0
$855$	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	$-0.582533\pi$
$859$	−12909.8	−0.512778	−0.256389	+	0.966574i	$0.582533\pi$
$860$	0	0
$861$	0	0
$862$	4551.10	0.179827
$863$	40128.5	1.58284	0.791419	−	0.611274i	$-0.209343\pi$
$863$	40128.5	1.58284	0.791419	+	0.611274i	$0.209343\pi$
$864$	−1188.29	−0.0467898
$865$	0	0
$866$	−36081.3	−1.41581
$867$	−249.542	−0.00977498
$868$	0	0
$869$	−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	$-0.519760\pi$
$877$	−3222.42	−0.124074	−0.0620372	+	0.998074i	$0.519760\pi$
$878$	−37328.4	−1.43482
$879$	6028.00	0.231307
$880$	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	$0.440165\pi$
$888$	−2391.19	−0.0903637
$889$	0	0
$890$	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
$896$	0	0
$897$	−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
$903$	0	0
$904$	−28635.3	−1.05354
$905$	0	0
$906$	−4362.97	−0.159989
$907$	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
$909$	−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
$917$	0	0
$918$	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
$924$	0	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	$-0.206387\pi$
$929$	45138.3	1.59412	0.797061	+	0.603899i	$0.206387\pi$
$930$	0	0
$931$	0	0
$932$	−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	$0.370890\pi$
$938$	0	0
$939$	2075.64	0.0721363
$940$	0	0
$941$	−11695.3	−0.405160	−0.202580	−	0.979266i	$-0.564933\pi$
$941$	−11695.3	−0.405160	−0.202580	+	0.979266i	$0.564933\pi$
$942$	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	$-0.215507\pi$
$947$	45429.1	1.55887	0.779434	+	0.626485i	$0.215507\pi$
$948$	210.773	0.00722109
$949$	35753.0	1.22296
$950$	0	0
$951$	−1112.13	−0.0379213
$952$	0	0
$953$	39025.9	1.32652	0.663259	−	0.748390i	$-0.269173\pi$
$953$	39025.9	1.32652	0.663259	+	0.748390i	$0.269173\pi$
$954$	−29098.6	−0.987529
$955$	0	0
$956$	244.293	0.00826464
$957$	−338.228	−0.0114246
$958$	11955.2	0.403188
$959$	0	0
$960$	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
$966$	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
$969$	−2411.19	−0.0799366
$970$	0	0
$971$	15044.2	0.497210	0.248605	−	0.968605i	$-0.420028\pi$
$971$	15044.2	0.497210	0.248605	+	0.968605i	$0.420028\pi$
$972$	−1050.37	−0.0346613
$973$	0	0
$974$	−35323.8	−1.16206
$975$	0	0
$976$	18649.0	0.611618
$977$	−26593.4	−0.870828	−0.435414	−	0.900230i	$-0.643398\pi$
$977$	−26593.4	−0.870828	−0.435414	+	0.900230i	$0.643398\pi$
$978$	−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
$987$	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
$993$	4241.53	0.135550
$994$	0	0
$995$	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	1225.4.a.r.1.2		2
5.4	even	2		1225.4.a.t.1.1		2
7.6	odd	2		175.4.a.d.1.2	✓	2
21.20	even	2		1575.4.a.v.1.1		2
35.13	even	4		175.4.b.d.99.2		4
35.27	even	4		175.4.b.d.99.3		4
35.34	odd	2		175.4.a.e.1.1	yes	2
105.104	even	2		1575.4.a.s.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
175.4.a.d.1.2	✓	2	7.6	odd	2
175.4.a.e.1.1	yes	2	35.34	odd	2
175.4.b.d.99.2		4	35.13	even	4
175.4.b.d.99.3		4	35.27	even	4
1225.4.a.r.1.2		2	1.1	even	1	trivial
1225.4.a.t.1.1		2	5.4	even	2
1575.4.a.s.1.2		2	105.104	even	2
1575.4.a.v.1.1		2	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1225 = 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1225.a (trivial)

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

Introduction

L-functions

Modular forms

Varieties

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