LMFDB - Embedded newform 1225.4.a.t.1.1

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:	$1$
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:	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	$\chi$	$=$	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	$-0.658483\pi$
$2$	−2.70156	−0.955146	−0.477573	+	0.878592i	$0.658483\pi$
$3$	−0.701562	−0.135016	−0.0675078	−	0.997719i	$-0.521505\pi$
$3$	−0.701562	−0.135016	−0.0675078	+	0.997719i	$0.521505\pi$
$4$	−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	$-0.685588\pi$
$13$	−51.6125	−1.10113	−0.550567	+	0.834791i	$0.685588\pi$
$14$	0	0
$15$	0	0
$16$	−57.8953	−0.904614
$17$	67.5078	0.963121	0.481560	−	0.876413i	$-0.340070\pi$
$17$	67.5078	0.963121	0.481560	+	0.876413i	$0.340070\pi$
$18$	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	$-0.499267\pi$
$23$	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
$28$	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	$-0.604392\pi$
$37$	−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	$-0.465251\pi$
$41$	57.2047	0.217899	0.108950	+	0.994047i	$0.465251\pi$
$42$	0	0
$43$	−283.020	−1.00373	−0.501863	−	0.864947i	$-0.667352\pi$
$43$	−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	$-0.381837\pi$
$47$	233.769	0.725504	0.362752	+	0.931886i	$0.381837\pi$
$48$	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	$-0.676515\pi$
$53$	−406.334	−1.05310	−0.526550	+	0.850144i	$0.676515\pi$
$54$	−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	$-0.144690\pi$
$67$	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	$-0.687404\pi$
$73$	−692.720	−1.11064	−0.555320	+	0.831637i	$0.687404\pi$
$74$	391.697	0.615322
$75$	0	0
$76$	−35.7172	−0.0539084
$77$	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	$-0.384320\pi$
$83$	537.592	0.710945	0.355472	+	0.934687i	$0.384320\pi$
$84$	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	$-0.869703\pi$
$97$	−1752.82	−1.83477	−0.917384	+	0.398004i	$0.869703\pi$
$98$	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	$-0.440408\pi$
$103$	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	$0.573793\pi$
$108$	−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	$-0.669260\pi$
$113$	−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
$119$	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	$-0.531371\pi$
$127$	−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	$-0.425036\pi$
$131$	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	$-0.335024\pi$
$137$	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	$-0.785961\pi$
$139$	−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	$-0.547695\pi$
$157$	−587.325	−0.298558	−0.149279	+	0.988795i	$0.547695\pi$
$158$	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	$-0.429510\pi$
$163$	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	$-0.778955\pi$
$167$	−3316.66	−1.53683	−0.768416	+	0.639951i	$0.778955\pi$
$168$	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	$-0.610044\pi$
$173$	−1542.16	−0.677737	−0.338868	+	0.940834i	$0.610044\pi$
$174$	−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	$-0.437443\pi$
$193$	1047.12	0.390534	0.195267	+	0.980750i	$0.437443\pi$
$194$	4735.37	1.75247
$195$	0	0
$196$	0	0
$197$	−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	$-0.501250\pi$
$199$	−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	$-0.431185\pi$
$223$	1428.67	0.429016	0.214508	+	0.976722i	$0.431185\pi$
$224$	0	0
$225$	0	0
$226$	3290.82	0.968594
$227$	−5979.38	−1.74831	−0.874153	−	0.485651i	$-0.838582\pi$
$227$	−5979.38	−1.74831	−0.874153	+	0.485651i	$0.838582\pi$
$228$	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	$-0.744815\pi$
$233$	−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
$238$	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	$-0.436546\pi$
$257$	1631.76	0.396055	0.198028	+	0.980196i	$0.436546\pi$
$258$	−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	$-0.637187\pi$
$263$	−3563.67	−0.835534	−0.417767	+	0.908554i	$0.637187\pi$
$264$	−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	$-0.553911\pi$
$277$	−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	$-0.287398\pi$
$283$	5897.15	1.23869	0.619345	+	0.785119i	$0.287398\pi$
$284$	−725.116	−0.151506
$285$	0	0
$286$	559.916	0.115764
$287$	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	$-0.827424\pi$
$293$	−8592.25	−1.71319	−0.856595	+	0.515990i	$0.827424\pi$
$294$	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	$-0.388804\pi$
$307$	3682.20	0.684542	0.342271	+	0.939601i	$0.388804\pi$
$308$	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	$-0.586079\pi$
$313$	−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	$0.455151\pi$
$318$	−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	$-0.767079\pi$
$337$	−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
$343$	0	0
$344$	−6653.19	−1.04278
$345$	0	0
$346$	4166.25	0.647338
$347$	−3092.28	−0.478393	−0.239196	−	0.970971i	$-0.576884\pi$
$347$	−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	$-0.368591\pi$
$349$	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	$-0.737818\pi$
$353$	−9013.71	−1.35907	−0.679534	+	0.733644i	$0.737818\pi$
$354$	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	$-0.503979\pi$
$367$	−175.769	−0.0250001	−0.0125001	+	0.999922i	$0.503979\pi$
$368$	−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	$-0.313070\pi$
$373$	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
$378$	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	$-0.601033\pi$
$383$	−4678.68	−0.624202	−0.312101	+	0.950049i	$0.601033\pi$
$384$	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	$-0.253587\pi$
$397$	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	$-0.824031\pi$
$401$	−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	$-0.234281\pi$
$433$	13355.7	1.48230	0.741150	+	0.671340i	$0.234281\pi$
$434$	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	$-0.390211\pi$
$443$	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
$448$	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	$-0.480096\pi$
$457$	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	$-0.667956\pi$
$461$	−9967.46	−1.00701	−0.503504	+	0.863993i	$0.667956\pi$
$462$	0	0
$463$	−6309.54	−0.633324	−0.316662	−	0.948538i	$-0.602562\pi$
$463$	−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	$-0.626038\pi$
$467$	−7784.79	−0.771386	−0.385693	+	0.922627i	$0.626038\pi$
$468$	−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	$-0.291846\pi$
$487$	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
$497$	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	$-0.329894\pi$
$503$	11491.6	1.01866	0.509328	+	0.860573i	$0.329894\pi$
$504$	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	$-0.561552\pi$
$523$	−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
$528$	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	$-0.332308\pi$
$547$	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
$553$	0	0
$554$	4198.67	0.321993
$555$	0	0
$556$	1798.87	0.137211
$557$	−19219.4	−1.46203	−0.731016	−	0.682360i	$-0.760954\pi$
$557$	−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	$-0.723465\pi$
$563$	−17253.3	−1.29155	−0.645774	+	0.763529i	$0.723465\pi$
$564$	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	$-0.634786\pi$
$577$	−11390.2	−0.821806	−0.410903	+	0.911679i	$0.634786\pi$
$578$	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	$-0.244465\pi$
$587$	20459.5	1.43859	0.719295	+	0.694704i	$0.244465\pi$
$588$	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	$-0.744271\pi$
$593$	−20051.1	−1.38853	−0.694266	+	0.719719i	$0.744271\pi$
$594$	−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	$-0.561300\pi$
$607$	−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
$612$	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
$616$	0	0
$617$	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	$-0.306908\pi$
$619$	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	$-0.305175\pi$
$643$	18736.1	1.14911	0.574555	+	0.818466i	$0.305175\pi$
$644$	0	0
$645$	0	0
$646$	−9284.96	−0.565498
$647$	25687.8	1.56088	0.780442	−	0.625228i	$-0.214994\pi$
$647$	25687.8	1.56088	0.780442	+	0.625228i	$0.214994\pi$
$648$	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	$-0.346786\pi$
$653$	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
$658$	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	$-0.535344\pi$
$673$	−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	$-0.506430\pi$
$677$	−711.604	−0.0403976	−0.0201988	+	0.999796i	$0.506430\pi$
$678$	−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	$-0.619714\pi$
$683$	−13112.0	−0.734577	−0.367288	+	0.930107i	$0.619714\pi$
$684$	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	$-0.717589\pi$
$727$	−24760.2	−1.26314	−0.631571	+	0.775318i	$0.717589\pi$
$728$	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	$-0.463530\pi$
$733$	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
$738$	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	$-0.286511\pi$
$743$	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
$749$	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	$-0.737484\pi$
$757$	−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	$-0.595256\pi$
$761$	−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	$-0.554640\pi$
$773$	−7342.21	−0.341631	−0.170816	+	0.985303i	$0.554640\pi$
$774$	−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	$-0.809189\pi$
$787$	−36457.4	−1.65129	−0.825646	+	0.564188i	$0.809189\pi$
$788$	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	$-0.433319\pi$
$797$	9358.08	0.415910	0.207955	+	0.978138i	$0.433319\pi$
$798$	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	$-0.446613\pi$
$823$	7882.78	0.333872	0.166936	+	0.985968i	$0.446613\pi$
$824$	9147.40	0.386729
$825$	0	0
$826$	0	0
$827$	−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	$-0.593152\pi$
$829$	−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	$-0.282714\pi$
$853$	31431.7	1.26166	0.630832	+	0.775920i	$0.282714\pi$
$854$	0	0
$855$	0	0
$856$	−11956.0	−0.477391
$857$	−3992.52	−0.159139	−0.0795693	−	0.996829i	$-0.525354\pi$
$857$	−3992.52	−0.159139	−0.0795693	+	0.996829i	$0.525354\pi$
$858$	−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	$-0.790657\pi$
$863$	−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
$868$	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	$-0.480240\pi$
$877$	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	$-0.383529\pi$
$881$	18712.4	0.715591	0.357795	+	0.933800i	$0.383529\pi$
$882$	0	0
$883$	−17924.0	−0.683114	−0.341557	−	0.939861i	$-0.610954\pi$
$883$	−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	$-0.559835\pi$
$887$	−9873.21	−0.373743	−0.186871	+	0.982384i	$0.559835\pi$
$888$	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	$-0.685945\pi$
$907$	−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
$917$	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	$-0.629110\pi$
$937$	−22634.7	−0.789161	−0.394580	+	0.918861i	$0.629110\pi$
$938$	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	$-0.784493\pi$
$947$	−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
$952$	0	0
$953$	−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
$959$	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	$-0.569278\pi$
$967$	−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	$-0.420028\pi$
$971$	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	$-0.356602\pi$
$977$	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	$-0.698849\pi$
$983$	−36050.3	−1.16971	−0.584856	+	0.811137i	$0.698849\pi$
$984$	−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	$-0.620715\pi$
$997$	−23309.0	−0.740426	−0.370213	+	0.928947i	$0.620715\pi$
$998$	−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.t.1.1		2
5.4	even	2		1225.4.a.r.1.2		2
7.6	odd	2		175.4.a.e.1.1	yes	2
21.20	even	2		1575.4.a.s.1.2		2
35.13	even	4		175.4.b.d.99.3		4
35.27	even	4		175.4.b.d.99.2		4
35.34	odd	2		175.4.a.d.1.2	✓	2
105.104	even	2		1575.4.a.v.1.1		2

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

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

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