LMFDB - Embedded newform 2205.4.a.y.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2205,4,Mod(1,2205)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2205, 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("2205.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2205 = 3^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2205.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:	$130.099211563$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 315)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	2205.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.56155 q^{2} -1.43845 q^{4} +5.00000 q^{5} +24.1771 q^{8} +O(q^{10})$	$q-2.56155 q^{2} -1.43845 q^{4} +5.00000 q^{5} +24.1771 q^{8} -12.8078 q^{10} -6.24621 q^{11} +56.3542 q^{13} -50.4233 q^{16} +24.6004 q^{17} +90.7083 q^{19} -7.19224 q^{20} +16.0000 q^{22} +69.8617 q^{23} +25.0000 q^{25} -144.354 q^{26} +228.847 q^{29} +67.8920 q^{31} -64.2547 q^{32} -63.0152 q^{34} -58.8466 q^{37} -232.354 q^{38} +120.885 q^{40} +19.2007 q^{41} +365.218 q^{43} +8.98485 q^{44} -178.955 q^{46} -195.153 q^{47} -64.0388 q^{50} -81.0625 q^{52} +511.201 q^{53} -31.2311 q^{55} -586.203 q^{58} -284.000 q^{59} +123.460 q^{61} -173.909 q^{62} +567.978 q^{64} +281.771 q^{65} +144.968 q^{67} -35.3863 q^{68} +73.0284 q^{71} -638.850 q^{73} +150.739 q^{74} -130.479 q^{76} +976.189 q^{79} -252.116 q^{80} -49.1837 q^{82} -484.466 q^{83} +123.002 q^{85} -935.525 q^{86} -151.015 q^{88} +1017.30 q^{89} -100.492 q^{92} +499.896 q^{94} +453.542 q^{95} -1806.67 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - 7 q^{4} + 10 q^{5} + 3 q^{8}+O(q^{10}) $ 2 * q - q^2 - 7 * q^4 + 10 * q^5 + 3 * q^8	$ 2 q - q^{2} - 7 q^{4} + 10 q^{5} + 3 q^{8} - 5 q^{10} + 4 q^{11} + 22 q^{13} - 39 q^{16} - 58 q^{17} - 35 q^{20} + 32 q^{22} + 82 q^{23} + 50 q^{25} - 198 q^{26} + 334 q^{29} + 210 q^{31} + 123 q^{32} - 192 q^{34} + 6 q^{37} - 374 q^{38} + 15 q^{40} - 176 q^{41} + 46 q^{43} - 48 q^{44} - 160 q^{46} - 514 q^{47} - 25 q^{50} + 110 q^{52} + 808 q^{53} + 20 q^{55} - 422 q^{58} - 568 q^{59} + 618 q^{61} + 48 q^{62} + 769 q^{64} + 110 q^{65} + 694 q^{67} + 424 q^{68} + 814 q^{71} - 82 q^{73} + 252 q^{74} + 374 q^{76} + 600 q^{79} - 195 q^{80} - 354 q^{82} + 268 q^{83} - 290 q^{85} - 1434 q^{86} - 368 q^{88} + 72 q^{89} - 168 q^{92} + 2 q^{94} - 1626 q^{97}+O(q^{100}) $ 2 * q - q^2 - 7 * q^4 + 10 * q^5 + 3 * q^8 - 5 * q^10 + 4 * q^11 + 22 * q^13 - 39 * q^16 - 58 * q^17 - 35 * q^20 + 32 * q^22 + 82 * q^23 + 50 * q^25 - 198 * q^26 + 334 * q^29 + 210 * q^31 + 123 * q^32 - 192 * q^34 + 6 * q^37 - 374 * q^38 + 15 * q^40 - 176 * q^41 + 46 * q^43 - 48 * q^44 - 160 * q^46 - 514 * q^47 - 25 * q^50 + 110 * q^52 + 808 * q^53 + 20 * q^55 - 422 * q^58 - 568 * q^59 + 618 * q^61 + 48 * q^62 + 769 * q^64 + 110 * q^65 + 694 * q^67 + 424 * q^68 + 814 * q^71 - 82 * q^73 + 252 * q^74 + 374 * q^76 + 600 * q^79 - 195 * q^80 - 354 * q^82 + 268 * q^83 - 290 * q^85 - 1434 * q^86 - 368 * q^88 + 72 * q^89 - 168 * q^92 + 2 * q^94 - 1626 * q^97

Display 100 coefficients 10 coefficients

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 $

$2$	−2.56155	−0.905646	−0.452823	−	0.891601i	$-0.649583\pi$
$2$	−2.56155	−0.905646	−0.452823	+	0.891601i	$0.649583\pi$
$3$	0	0
$3$	0	0
$4$	−1.43845	−0.179806
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	24.1771	1.06849
$9$	0	0
$10$	−12.8078	−0.405017
$11$	−6.24621	−0.171209	−0.0856047	−	0.996329i	$-0.527282\pi$
$11$	−6.24621	−0.171209	−0.0856047	+	0.996329i	$0.527282\pi$
$12$	0	0
$13$	56.3542	1.20229	0.601147	−	0.799138i	$-0.294710\pi$
$13$	56.3542	1.20229	0.601147	+	0.799138i	$0.294710\pi$
$14$	0	0
$15$	0	0
$16$	−50.4233	−0.787864
$17$	24.6004	0.350969	0.175484	−	0.984482i	$-0.443851\pi$
$17$	24.6004	0.350969	0.175484	+	0.984482i	$0.443851\pi$
$18$	0	0
$19$	90.7083	1.09526	0.547629	−	0.836721i	$-0.315530\pi$
$19$	90.7083	1.09526	0.547629	+	0.836721i	$0.315530\pi$
$20$	−7.19224	−0.0804116
$21$	0	0
$22$	16.0000	0.155055
$23$	69.8617	0.633356	0.316678	−	0.948533i	$-0.397433\pi$
$23$	69.8617	0.633356	0.316678	+	0.948533i	$0.397433\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−144.354	−1.08885
$27$	0	0
$28$	0	0
$29$	228.847	1.46537	0.732685	−	0.680568i	$-0.238267\pi$
$29$	228.847	1.46537	0.732685	+	0.680568i	$0.238267\pi$
$30$	0	0
$31$	67.8920	0.393347	0.196674	−	0.980469i	$-0.436986\pi$
$31$	67.8920	0.393347	0.196674	+	0.980469i	$0.436986\pi$
$32$	−64.2547	−0.354961
$33$	0	0
$34$	−63.0152	−0.317853
$35$	0	0
$36$	0	0
$37$	−58.8466	−0.261468	−0.130734	−	0.991417i	$-0.541733\pi$
$37$	−58.8466	−0.261468	−0.130734	+	0.991417i	$0.541733\pi$
$38$	−232.354	−0.991916
$39$	0	0
$40$	120.885	0.477842
$41$	19.2007	0.0731379	0.0365689	−	0.999331i	$-0.488357\pi$
$41$	19.2007	0.0731379	0.0365689	+	0.999331i	$0.488357\pi$
$42$	0	0
$43$	365.218	1.29524	0.647618	−	0.761965i	$-0.275765\pi$
$43$	365.218	1.29524	0.647618	+	0.761965i	$0.275765\pi$
$44$	8.98485	0.0307845
$45$	0	0
$46$	−178.955	−0.573596
$47$	−195.153	−0.605661	−0.302830	−	0.953044i	$-0.597932\pi$
$47$	−195.153	−0.605661	−0.302830	+	0.953044i	$0.597932\pi$
$48$	0	0
$49$	0	0
$50$	−64.0388	−0.181129
$51$	0	0
$52$	−81.0625	−0.216180
$53$	511.201	1.32488	0.662442	−	0.749113i	$-0.269520\pi$
$53$	511.201	1.32488	0.662442	+	0.749113i	$0.269520\pi$
$54$	0	0
$55$	−31.2311	−0.0765672
$56$	0	0
$57$	0	0
$58$	−586.203	−1.32711
$59$	−284.000	−0.626672	−0.313336	−	0.949642i	$-0.601447\pi$
$59$	−284.000	−0.626672	−0.313336	+	0.949642i	$0.601447\pi$
$60$	0	0
$61$	123.460	0.259139	0.129569	−	0.991570i	$-0.458641\pi$
$61$	123.460	0.259139	0.129569	+	0.991570i	$0.458641\pi$
$62$	−173.909	−0.356233
$63$	0	0
$64$	567.978	1.10933
$65$	281.771	0.537683
$66$	0	0
$67$	144.968	0.264338	0.132169	−	0.991227i	$-0.457806\pi$
$67$	144.968	0.264338	0.132169	+	0.991227i	$0.457806\pi$
$68$	−35.3863	−0.0631062
$69$	0	0
$70$	0	0
$71$	73.0284	0.122069	0.0610344	−	0.998136i	$-0.480560\pi$
$71$	73.0284	0.122069	0.0610344	+	0.998136i	$0.480560\pi$
$72$	0	0
$73$	−638.850	−1.02427	−0.512135	−	0.858905i	$-0.671145\pi$
$73$	−638.850	−1.02427	−0.512135	+	0.858905i	$0.671145\pi$
$74$	150.739	0.236797
$75$	0	0
$76$	−130.479	−0.196934
$77$	0	0
$78$	0	0
$79$	976.189	1.39025	0.695126	−	0.718888i	$-0.255349\pi$
$79$	976.189	1.39025	0.695126	+	0.718888i	$0.255349\pi$
$80$	−252.116	−0.352343
$81$	0	0
$82$	−49.1837	−0.0662370
$83$	−484.466	−0.640687	−0.320344	−	0.947301i	$-0.603798\pi$
$83$	−484.466	−0.640687	−0.320344	+	0.947301i	$0.603798\pi$
$84$	0	0
$85$	123.002	0.156958
$86$	−935.525	−1.17303
$87$	0	0
$88$	−151.015	−0.182935
$89$	1017.30	1.21161	0.605806	−	0.795612i	$-0.292851\pi$
$89$	1017.30	1.21161	0.605806	+	0.795612i	$0.292851\pi$
$90$	0	0
$91$	0	0
$92$	−100.492	−0.113881
$93$	0	0
$94$	499.896	0.548514
$95$	453.542	0.489815
$96$	0	0
$97$	−1806.67	−1.89113	−0.945564	−	0.325437i	$-0.894489\pi$
$97$	−1806.67	−1.89113	−0.945564	+	0.325437i	$0.894489\pi$
$98$	0	0
$99$	0	0
$100$	−35.9612	−0.0359612
$101$	483.053	0.475897	0.237948	−	0.971278i	$-0.423525\pi$
$101$	483.053	0.475897	0.237948	+	0.971278i	$0.423525\pi$
$102$	0	0
$103$	339.049	0.324345	0.162172	−	0.986762i	$-0.448150\pi$
$103$	339.049	0.324345	0.162172	+	0.986762i	$0.448150\pi$
$104$	1362.48	1.28464
$105$	0	0
$106$	−1309.47	−1.19987
$107$	−450.847	−0.407336	−0.203668	−	0.979040i	$-0.565286\pi$
$107$	−450.847	−0.407336	−0.203668	+	0.979040i	$0.565286\pi$
$108$	0	0
$109$	−1841.70	−1.61838	−0.809189	−	0.587548i	$-0.800093\pi$
$109$	−1841.70	−1.61838	−0.809189	+	0.587548i	$0.800093\pi$
$110$	80.0000	0.0693427
$111$	0	0
$112$	0	0
$113$	1874.72	1.56069	0.780347	−	0.625347i	$-0.215042\pi$
$113$	1874.72	1.56069	0.780347	+	0.625347i	$0.215042\pi$
$114$	0	0
$115$	349.309	0.283245
$116$	−329.184	−0.263482
$117$	0	0
$118$	727.481	0.567543
$119$	0	0
$120$	0	0
$121$	−1291.98	−0.970687
$122$	−316.250	−0.234688
$123$	0	0
$124$	−97.6591	−0.0707262
$125$	125.000	0.0894427
$126$	0	0
$127$	−38.0984	−0.0266196	−0.0133098	−	0.999911i	$-0.504237\pi$
$127$	−38.0984	−0.0266196	−0.0133098	+	0.999911i	$0.504237\pi$
$128$	−940.868	−0.649702
$129$	0	0
$130$	−721.771	−0.486950
$131$	1551.51	1.03478	0.517389	−	0.855750i	$-0.326904\pi$
$131$	1551.51	1.03478	0.517389	+	0.855750i	$0.326904\pi$
$132$	0	0
$133$	0	0
$134$	−371.343	−0.239396
$135$	0	0
$136$	594.765	0.375005
$137$	−1203.24	−0.750364	−0.375182	−	0.926951i	$-0.622420\pi$
$137$	−1203.24	−0.750364	−0.375182	+	0.926951i	$0.622420\pi$
$138$	0	0
$139$	−1897.00	−1.15756	−0.578781	−	0.815483i	$-0.696471\pi$
$139$	−1897.00	−1.15756	−0.578781	+	0.815483i	$0.696471\pi$
$140$	0	0
$141$	0	0
$142$	−187.066	−0.110551
$143$	−352.000	−0.205844
$144$	0	0
$145$	1144.23	0.655334
$146$	1636.45	0.927626
$147$	0	0
$148$	84.6477	0.0470135
$149$	704.888	0.387562	0.193781	−	0.981045i	$-0.437925\pi$
$149$	704.888	0.387562	0.193781	+	0.981045i	$0.437925\pi$
$150$	0	0
$151$	−3035.21	−1.63578	−0.817888	−	0.575378i	$-0.804855\pi$
$151$	−3035.21	−1.63578	−0.817888	+	0.575378i	$0.804855\pi$
$152$	2193.06	1.17027
$153$	0	0
$154$	0	0
$155$	339.460	0.175910
$156$	0	0
$157$	−2713.65	−1.37944	−0.689722	−	0.724074i	$-0.742267\pi$
$157$	−2713.65	−1.37944	−0.689722	+	0.724074i	$0.742267\pi$
$158$	−2500.56	−1.25908
$159$	0	0
$160$	−321.274	−0.158743
$161$	0	0
$162$	0	0
$163$	−465.259	−0.223570	−0.111785	−	0.993732i	$-0.535657\pi$
$163$	−465.259	−0.223570	−0.111785	+	0.993732i	$0.535657\pi$
$164$	−27.6193	−0.0131506
$165$	0	0
$166$	1240.98	0.580236
$167$	4156.06	1.92578	0.962891	−	0.269892i	$-0.0869880\pi$
$167$	4156.06	1.92578	0.962891	+	0.269892i	$0.0869880\pi$
$168$	0	0
$169$	978.792	0.445513
$170$	−315.076	−0.142148
$171$	0	0
$172$	−525.346	−0.232891
$173$	−4241.17	−1.86387	−0.931936	−	0.362622i	$-0.881881\pi$
$173$	−4241.17	−1.86387	−0.931936	+	0.362622i	$0.881881\pi$
$174$	0	0
$175$	0	0
$176$	314.955	0.134890
$177$	0	0
$178$	−2605.87	−1.09729
$179$	2940.35	1.22778	0.613889	−	0.789392i	$-0.289604\pi$
$179$	2940.35	1.22778	0.613889	+	0.789392i	$0.289604\pi$
$180$	0	0
$181$	1986.35	0.815716	0.407858	−	0.913045i	$-0.366276\pi$
$181$	1986.35	0.815716	0.407858	+	0.913045i	$0.366276\pi$
$182$	0	0
$183$	0	0
$184$	1689.05	0.676732
$185$	−294.233	−0.116932
$186$	0	0
$187$	−153.659	−0.0600891
$188$	280.718	0.108901
$189$	0	0
$190$	−1161.77	−0.443598
$191$	−1615.88	−0.612150	−0.306075	−	0.952007i	$-0.599016\pi$
$191$	−1615.88	−0.612150	−0.306075	+	0.952007i	$0.599016\pi$
$192$	0	0
$193$	−2052.44	−0.765482	−0.382741	−	0.923856i	$-0.625020\pi$
$193$	−2052.44	−0.765482	−0.382741	+	0.923856i	$0.625020\pi$
$194$	4627.88	1.71269
$195$	0	0
$196$	0	0
$197$	4468.58	1.61611	0.808054	−	0.589108i	$-0.200521\pi$
$197$	4468.58	1.61611	0.808054	+	0.589108i	$0.200521\pi$
$198$	0	0
$199$	−1543.07	−0.549675	−0.274838	−	0.961491i	$-0.588624\pi$
$199$	−1543.07	−0.549675	−0.274838	+	0.961491i	$0.588624\pi$
$200$	604.427	0.213697
$201$	0	0
$202$	−1237.37	−0.430994
$203$	0	0
$204$	0	0
$205$	96.0037	0.0327083
$206$	−868.492	−0.293741
$207$	0	0
$208$	−2841.56	−0.947245
$209$	−566.583	−0.187519
$210$	0	0
$211$	1284.02	0.418937	0.209469	−	0.977815i	$-0.432827\pi$
$211$	1284.02	0.418937	0.209469	+	0.977815i	$0.432827\pi$
$212$	−735.335	−0.238222
$213$	0	0
$214$	1154.87	0.368902
$215$	1826.09	0.579248
$216$	0	0
$217$	0	0
$218$	4717.62	1.46568
$219$	0	0
$220$	44.9242	0.0137672
$221$	1386.33	0.421968
$222$	0	0
$223$	−3815.31	−1.14571	−0.572853	−	0.819658i	$-0.694163\pi$
$223$	−3815.31	−1.14571	−0.572853	+	0.819658i	$0.694163\pi$
$224$	0	0
$225$	0	0
$226$	−4802.18	−1.41344
$227$	−2271.53	−0.664172	−0.332086	−	0.943249i	$-0.607752\pi$
$227$	−2271.53	−0.664172	−0.332086	+	0.943249i	$0.607752\pi$
$228$	0	0
$229$	2367.54	0.683195	0.341598	−	0.939846i	$-0.389032\pi$
$229$	2367.54	0.683195	0.341598	+	0.939846i	$0.389032\pi$
$230$	−894.773	−0.256520
$231$	0	0
$232$	5532.84	1.56573
$233$	−1617.71	−0.454849	−0.227425	−	0.973796i	$-0.573030\pi$
$233$	−1617.71	−0.454849	−0.227425	+	0.973796i	$0.573030\pi$
$234$	0	0
$235$	−975.767	−0.270860
$236$	408.519	0.112679
$237$	0	0
$238$	0	0
$239$	1935.29	0.523781	0.261891	−	0.965098i	$-0.415654\pi$
$239$	1935.29	0.523781	0.261891	+	0.965098i	$0.415654\pi$
$240$	0	0
$241$	−477.901	−0.127736	−0.0638679	−	0.997958i	$-0.520344\pi$
$241$	−477.901	−0.127736	−0.0638679	+	0.997958i	$0.520344\pi$
$242$	3309.49	0.879099
$243$	0	0
$244$	−177.591	−0.0465947
$245$	0	0
$246$	0	0
$247$	5111.79	1.31682
$248$	1641.43	0.420286
$249$	0	0
$250$	−320.194	−0.0810034
$251$	−4769.70	−1.19944	−0.599722	−	0.800208i	$-0.704722\pi$
$251$	−4769.70	−1.19944	−0.599722	+	0.800208i	$0.704722\pi$
$252$	0	0
$253$	−436.371	−0.108436
$254$	97.5910	0.0241079
$255$	0	0
$256$	−2133.74	−0.520933
$257$	−682.524	−0.165660	−0.0828302	−	0.996564i	$-0.526396\pi$
$257$	−682.524	−0.165660	−0.0828302	+	0.996564i	$0.526396\pi$
$258$	0	0
$259$	0	0
$260$	−405.312	−0.0966785
$261$	0	0
$262$	−3974.27	−0.937142
$263$	3029.11	0.710202	0.355101	−	0.934828i	$-0.384447\pi$
$263$	3029.11	0.710202	0.355101	+	0.934828i	$0.384447\pi$
$264$	0	0
$265$	2556.00	0.592506
$266$	0	0
$267$	0	0
$268$	−208.529	−0.0475295
$269$	−6187.33	−1.40241	−0.701205	−	0.712960i	$-0.747354\pi$
$269$	−6187.33	−1.40241	−0.701205	+	0.712960i	$0.747354\pi$
$270$	0	0
$271$	7558.90	1.69436	0.847178	−	0.531309i	$-0.178300\pi$
$271$	7558.90	1.69436	0.847178	+	0.531309i	$0.178300\pi$
$272$	−1240.43	−0.276516
$273$	0	0
$274$	3082.17	0.679564
$275$	−156.155	−0.0342419
$276$	0	0
$277$	3685.36	0.799393	0.399697	−	0.916647i	$-0.369115\pi$
$277$	3685.36	0.799393	0.399697	+	0.916647i	$0.369115\pi$
$278$	4859.26	1.04834
$279$	0	0
$280$	0	0
$281$	7969.11	1.69180	0.845902	−	0.533338i	$-0.179063\pi$
$281$	7969.11	1.69180	0.845902	+	0.533338i	$0.179063\pi$
$282$	0	0
$283$	2479.73	0.520864	0.260432	−	0.965492i	$-0.416135\pi$
$283$	2479.73	0.520864	0.260432	+	0.965492i	$0.416135\pi$
$284$	−105.048	−0.0219487
$285$	0	0
$286$	901.667	0.186422
$287$	0	0
$288$	0	0
$289$	−4307.82	−0.876821
$290$	−2931.01	−0.593500
$291$	0	0
$292$	918.952	0.184170
$293$	5950.02	1.18636	0.593181	−	0.805069i	$-0.297872\pi$
$293$	5950.02	1.18636	0.593181	+	0.805069i	$0.297872\pi$
$294$	0	0
$295$	−1420.00	−0.280256
$296$	−1422.74	−0.279375
$297$	0	0
$298$	−1805.61	−0.350994
$299$	3937.00	0.761480
$300$	0	0
$301$	0	0
$302$	7774.86	1.48143
$303$	0	0
$304$	−4573.81	−0.862915
$305$	617.301	0.115890
$306$	0	0
$307$	3129.90	0.581865	0.290933	−	0.956744i	$-0.406034\pi$
$307$	3129.90	0.581865	0.290933	+	0.956744i	$0.406034\pi$
$308$	0	0
$309$	0	0
$310$	−869.545	−0.159312
$311$	−7261.25	−1.32395	−0.661973	−	0.749527i	$-0.730281\pi$
$311$	−7261.25	−1.32395	−0.661973	+	0.749527i	$0.730281\pi$
$312$	0	0
$313$	2310.83	0.417303	0.208652	−	0.977990i	$-0.433093\pi$
$313$	2310.83	0.417303	0.208652	+	0.977990i	$0.433093\pi$
$314$	6951.16	1.24929
$315$	0	0
$316$	−1404.20	−0.249975
$317$	4701.40	0.832987	0.416494	−	0.909139i	$-0.363259\pi$
$317$	4701.40	0.832987	0.416494	+	0.909139i	$0.363259\pi$
$318$	0	0
$319$	−1429.42	−0.250885
$320$	2839.89	0.496109
$321$	0	0
$322$	0	0
$323$	2231.46	0.384401
$324$	0	0
$325$	1408.85	0.240459
$326$	1191.79	0.202475
$327$	0	0
$328$	464.218	0.0781468
$329$	0	0
$330$	0	0
$331$	1366.18	0.226864	0.113432	−	0.993546i	$-0.463816\pi$
$331$	1366.18	0.226864	0.113432	+	0.993546i	$0.463816\pi$
$332$	696.879	0.115199
$333$	0	0
$334$	−10646.0	−1.74408
$335$	724.839	0.118215
$336$	0	0
$337$	−740.632	−0.119718	−0.0598588	−	0.998207i	$-0.519065\pi$
$337$	−740.632	−0.119718	−0.0598588	+	0.998207i	$0.519065\pi$
$338$	−2507.23	−0.403477
$339$	0	0
$340$	−176.932	−0.0282220
$341$	−424.068	−0.0673448
$342$	0	0
$343$	0	0
$344$	8829.90	1.38394
$345$	0	0
$346$	10864.0	1.68801
$347$	−2605.56	−0.403094	−0.201547	−	0.979479i	$-0.564597\pi$
$347$	−2605.56	−0.403094	−0.201547	+	0.979479i	$0.564597\pi$
$348$	0	0
$349$	−4665.07	−0.715517	−0.357758	−	0.933814i	$-0.616459\pi$
$349$	−4665.07	−0.715517	−0.357758	+	0.933814i	$0.616459\pi$
$350$	0	0
$351$	0	0
$352$	401.349	0.0607726
$353$	−2964.75	−0.447019	−0.223509	−	0.974702i	$-0.571751\pi$
$353$	−2964.75	−0.447019	−0.223509	+	0.974702i	$0.571751\pi$
$354$	0	0
$355$	365.142	0.0545908
$356$	−1463.33	−0.217855
$357$	0	0
$358$	−7531.87	−1.11193
$359$	4267.55	0.627389	0.313695	−	0.949524i	$-0.398433\pi$
$359$	4267.55	0.627389	0.313695	+	0.949524i	$0.398433\pi$
$360$	0	0
$361$	1369.00	0.199592
$362$	−5088.15	−0.738749
$363$	0	0
$364$	0	0
$365$	−3194.25	−0.458068
$366$	0	0
$367$	9280.33	1.31997	0.659985	−	0.751279i	$-0.270562\pi$
$367$	9280.33	1.31997	0.659985	+	0.751279i	$0.270562\pi$
$368$	−3522.66	−0.498998
$369$	0	0
$370$	753.693	0.105899
$371$	0	0
$372$	0	0
$373$	10781.1	1.49657	0.748286	−	0.663376i	$-0.230877\pi$
$373$	10781.1	1.49657	0.748286	+	0.663376i	$0.230877\pi$
$374$	393.606	0.0544195
$375$	0	0
$376$	−4718.24	−0.647140
$377$	12896.5	1.76181
$378$	0	0
$379$	−5914.16	−0.801557	−0.400779	−	0.916175i	$-0.631260\pi$
$379$	−5914.16	−0.801557	−0.400779	+	0.916175i	$0.631260\pi$
$380$	−652.396	−0.0880716
$381$	0	0
$382$	4139.15	0.554391
$383$	11513.9	1.53612	0.768059	−	0.640379i	$-0.221223\pi$
$383$	11513.9	1.53612	0.768059	+	0.640379i	$0.221223\pi$
$384$	0	0
$385$	0	0
$386$	5257.44	0.693256
$387$	0	0
$388$	2598.80	0.340036
$389$	5399.73	0.703797	0.351898	−	0.936038i	$-0.385536\pi$
$389$	5399.73	0.703797	0.351898	+	0.936038i	$0.385536\pi$
$390$	0	0
$391$	1718.62	0.222288
$392$	0	0
$393$	0	0
$394$	−11446.5	−1.46362
$395$	4880.95	0.621739
$396$	0	0
$397$	−2622.13	−0.331488	−0.165744	−	0.986169i	$-0.553003\pi$
$397$	−2622.13	−0.331488	−0.165744	+	0.986169i	$0.553003\pi$
$398$	3952.66	0.497811
$399$	0	0
$400$	−1260.58	−0.157573
$401$	−11119.1	−1.38469	−0.692344	−	0.721568i	$-0.743422\pi$
$401$	−11119.1	−1.38469	−0.692344	+	0.721568i	$0.743422\pi$
$402$	0	0
$403$	3826.00	0.472920
$404$	−694.846	−0.0855690
$405$	0	0
$406$	0	0
$407$	367.568	0.0447658
$408$	0	0
$409$	6589.18	0.796611	0.398305	−	0.917253i	$-0.369598\pi$
$409$	6589.18	0.796611	0.398305	+	0.917253i	$0.369598\pi$
$410$	−245.919	−0.0296221
$411$	0	0
$412$	−487.704	−0.0583191
$413$	0	0
$414$	0	0
$415$	−2422.33	−0.286524
$416$	−3621.02	−0.426767
$417$	0	0
$418$	1451.33	0.169825
$419$	11871.6	1.38416	0.692081	−	0.721820i	$-0.256694\pi$
$419$	11871.6	1.38416	0.692081	+	0.721820i	$0.256694\pi$
$420$	0	0
$421$	−1731.57	−0.200455	−0.100227	−	0.994965i	$-0.531957\pi$
$421$	−1731.57	−0.200455	−0.100227	+	0.994965i	$0.531957\pi$
$422$	−3289.09	−0.379409
$423$	0	0
$424$	12359.3	1.41562
$425$	615.009	0.0701937
$426$	0	0
$427$	0	0
$428$	648.519	0.0732415
$429$	0	0
$430$	−4677.62	−0.524593
$431$	10653.8	1.19066	0.595330	−	0.803481i	$-0.297021\pi$
$431$	10653.8	1.19066	0.595330	+	0.803481i	$0.297021\pi$
$432$	0	0
$433$	−2642.01	−0.293226	−0.146613	−	0.989194i	$-0.546837\pi$
$433$	−2642.01	−0.293226	−0.146613	+	0.989194i	$0.546837\pi$
$434$	0	0
$435$	0	0
$436$	2649.19	0.290994
$437$	6337.04	0.693688
$438$	0	0
$439$	−8858.21	−0.963051	−0.481525	−	0.876432i	$-0.659917\pi$
$439$	−8858.21	−0.963051	−0.481525	+	0.876432i	$0.659917\pi$
$440$	−755.076	−0.0818110
$441$	0	0
$442$	−3551.17	−0.382153
$443$	−6621.73	−0.710176	−0.355088	−	0.934833i	$-0.615549\pi$
$443$	−6621.73	−0.710176	−0.355088	+	0.934833i	$0.615549\pi$
$444$	0	0
$445$	5086.50	0.541849
$446$	9773.13	1.03760
$447$	0	0
$448$	0	0
$449$	13081.7	1.37497	0.687487	−	0.726196i	$-0.258714\pi$
$449$	13081.7	1.37497	0.687487	+	0.726196i	$0.258714\pi$
$450$	0	0
$451$	−119.932	−0.0125219
$452$	−2696.68	−0.280622
$453$	0	0
$454$	5818.65	0.601504
$455$	0	0
$456$	0	0
$457$	12167.0	1.24540	0.622699	−	0.782461i	$-0.286036\pi$
$457$	12167.0	1.24540	0.622699	+	0.782461i	$0.286036\pi$
$458$	−6064.59	−0.618733
$459$	0	0
$460$	−502.462	−0.0509292
$461$	−11283.8	−1.14000	−0.570000	−	0.821645i	$-0.693057\pi$
$461$	−11283.8	−1.14000	−0.570000	+	0.821645i	$0.693057\pi$
$462$	0	0
$463$	16542.9	1.66051	0.830254	−	0.557385i	$-0.188195\pi$
$463$	16542.9	1.66051	0.830254	+	0.557385i	$0.188195\pi$
$464$	−11539.2	−1.15451
$465$	0	0
$466$	4143.85	0.411932
$467$	10266.9	1.01734	0.508668	−	0.860963i	$-0.330138\pi$
$467$	10266.9	1.01734	0.508668	+	0.860963i	$0.330138\pi$
$468$	0	0
$469$	0	0
$470$	2499.48	0.245303
$471$	0	0
$472$	−6866.29	−0.669590
$473$	−2281.23	−0.221757
$474$	0	0
$475$	2267.71	0.219052
$476$	0	0
$477$	0	0
$478$	−4957.36	−0.474360
$479$	7967.98	0.760055	0.380027	−	0.924975i	$-0.375915\pi$
$479$	7967.98	0.760055	0.380027	+	0.924975i	$0.375915\pi$
$480$	0	0
$481$	−3316.25	−0.314362
$482$	1224.17	0.115683
$483$	0	0
$484$	1858.45	0.174535
$485$	−9033.34	−0.845738
$486$	0	0
$487$	9956.62	0.926443	0.463221	−	0.886243i	$-0.346694\pi$
$487$	9956.62	0.926443	0.463221	+	0.886243i	$0.346694\pi$
$488$	2984.91	0.276886
$489$	0	0
$490$	0	0
$491$	−18660.8	−1.71518	−0.857589	−	0.514336i	$-0.828038\pi$
$491$	−18660.8	−1.71518	−0.857589	+	0.514336i	$0.828038\pi$
$492$	0	0
$493$	5629.71	0.514299
$494$	−13094.1	−1.19258
$495$	0	0
$496$	−3423.34	−0.309904
$497$	0	0
$498$	0	0
$499$	4074.21	0.365504	0.182752	−	0.983159i	$-0.441499\pi$
$499$	4074.21	0.365504	0.182752	+	0.983159i	$0.441499\pi$
$500$	−179.806	−0.0160823
$501$	0	0
$502$	12217.8	1.08627
$503$	4255.51	0.377224	0.188612	−	0.982052i	$-0.439601\pi$
$503$	4255.51	0.377224	0.188612	+	0.982052i	$0.439601\pi$
$504$	0	0
$505$	2415.26	0.212827
$506$	1117.79	0.0982050
$507$	0	0
$508$	54.8025	0.00478636
$509$	−10171.8	−0.885771	−0.442885	−	0.896578i	$-0.646045\pi$
$509$	−10171.8	−0.885771	−0.442885	+	0.896578i	$0.646045\pi$
$510$	0	0
$511$	0	0
$512$	12992.6	1.12148
$513$	0	0
$514$	1748.32	0.150030
$515$	1695.25	0.145051
$516$	0	0
$517$	1218.97	0.103695
$518$	0	0
$519$	0	0
$520$	6812.40	0.574506
$521$	−1680.39	−0.141303	−0.0706517	−	0.997501i	$-0.522508\pi$
$521$	−1680.39	−0.141303	−0.0706517	+	0.997501i	$0.522508\pi$
$522$	0	0
$523$	−13211.8	−1.10461	−0.552305	−	0.833642i	$-0.686251\pi$
$523$	−13211.8	−1.10461	−0.552305	+	0.833642i	$0.686251\pi$
$524$	−2231.76	−0.186059
$525$	0	0
$526$	−7759.23	−0.643191
$527$	1670.17	0.138053
$528$	0	0
$529$	−7286.34	−0.598861
$530$	−6547.34	−0.536600
$531$	0	0
$532$	0	0
$533$	1082.04	0.0879333
$534$	0	0
$535$	−2254.23	−0.182166
$536$	3504.90	0.282441
$537$	0	0
$538$	15849.2	1.27009
$539$	0	0
$540$	0	0
$541$	9650.84	0.766954	0.383477	−	0.923551i	$-0.374727\pi$
$541$	9650.84	0.766954	0.383477	+	0.923551i	$0.374727\pi$
$542$	−19362.5	−1.53449
$543$	0	0
$544$	−1580.69	−0.124580
$545$	−9208.52	−0.723761
$546$	0	0
$547$	−23864.3	−1.86538	−0.932689	−	0.360682i	$-0.882544\pi$
$547$	−23864.3	−1.86538	−0.932689	+	0.360682i	$0.882544\pi$
$548$	1730.80	0.134920
$549$	0	0
$550$	400.000	0.0310110
$551$	20758.3	1.60496
$552$	0	0
$553$	0	0
$554$	−9440.25	−0.723967
$555$	0	0
$556$	2728.73	0.208136
$557$	−2314.22	−0.176044	−0.0880221	−	0.996119i	$-0.528055\pi$
$557$	−2314.22	−0.176044	−0.0880221	+	0.996119i	$0.528055\pi$
$558$	0	0
$559$	20581.5	1.55726
$560$	0	0
$561$	0	0
$562$	−20413.3	−1.53218
$563$	7017.86	0.525342	0.262671	−	0.964885i	$-0.415397\pi$
$563$	7017.86	0.525342	0.262671	+	0.964885i	$0.415397\pi$
$564$	0	0
$565$	9373.58	0.697964
$566$	−6351.95	−0.471718
$567$	0	0
$568$	1765.61	0.130429
$569$	5302.37	0.390662	0.195331	−	0.980737i	$-0.437422\pi$
$569$	5302.37	0.390662	0.195331	+	0.980737i	$0.437422\pi$
$570$	0	0
$571$	17767.2	1.30216	0.651082	−	0.759008i	$-0.274315\pi$
$571$	17767.2	1.30216	0.651082	+	0.759008i	$0.274315\pi$
$572$	506.333	0.0370120
$573$	0	0
$574$	0	0
$575$	1746.54	0.126671
$576$	0	0
$577$	6089.57	0.439363	0.219681	−	0.975572i	$-0.429498\pi$
$577$	6089.57	0.439363	0.219681	+	0.975572i	$0.429498\pi$
$578$	11034.7	0.794089
$579$	0	0
$580$	−1645.92	−0.117833
$581$	0	0
$582$	0	0
$583$	−3193.07	−0.226833
$584$	−15445.5	−1.09442
$585$	0	0
$586$	−15241.3	−1.07442
$587$	26543.9	1.86641	0.933205	−	0.359344i	$-0.117000\pi$
$587$	26543.9	1.86641	0.933205	+	0.359344i	$0.117000\pi$
$588$	0	0
$589$	6158.37	0.430817
$590$	3637.40	0.253813
$591$	0	0
$592$	2967.24	0.206001
$593$	−16365.0	−1.13327	−0.566635	−	0.823969i	$-0.691755\pi$
$593$	−16365.0	−1.13327	−0.566635	+	0.823969i	$0.691755\pi$
$594$	0	0
$595$	0	0
$596$	−1013.94	−0.0696859
$597$	0	0
$598$	−10084.8	−0.689631
$599$	−17516.3	−1.19482	−0.597411	−	0.801935i	$-0.703804\pi$
$599$	−17516.3	−1.19482	−0.597411	+	0.801935i	$0.703804\pi$
$600$	0	0
$601$	−8693.80	−0.590062	−0.295031	−	0.955488i	$-0.595330\pi$
$601$	−8693.80	−0.590062	−0.295031	+	0.955488i	$0.595330\pi$
$602$	0	0
$603$	0	0
$604$	4365.99	0.294122
$605$	−6459.92	−0.434105
$606$	0	0
$607$	18096.9	1.21010	0.605051	−	0.796186i	$-0.293153\pi$
$607$	18096.9	1.21010	0.605051	+	0.796186i	$0.293153\pi$
$608$	−5828.44	−0.388774
$609$	0	0
$610$	−1581.25	−0.104956
$611$	−10997.7	−0.728183
$612$	0	0
$613$	4641.61	0.305828	0.152914	−	0.988239i	$-0.451134\pi$
$613$	4641.61	0.305828	0.152914	+	0.988239i	$0.451134\pi$
$614$	−8017.40	−0.526964
$615$	0	0
$616$	0	0
$617$	14676.1	0.957600	0.478800	−	0.877924i	$-0.341072\pi$
$617$	14676.1	0.957600	0.478800	+	0.877924i	$0.341072\pi$
$618$	0	0
$619$	19645.3	1.27563	0.637813	−	0.770192i	$-0.279839\pi$
$619$	19645.3	1.27563	0.637813	+	0.770192i	$0.279839\pi$
$620$	−488.296	−0.0316297
$621$	0	0
$622$	18600.1	1.19903
$623$	0	0
$624$	0	0
$625$	625.000	0.0400000
$626$	−5919.32	−0.377929
$627$	0	0
$628$	3903.44	0.248032
$629$	−1447.65	−0.0917671
$630$	0	0
$631$	26231.2	1.65491	0.827456	−	0.561531i	$-0.189788\pi$
$631$	26231.2	1.65491	0.827456	+	0.561531i	$0.189788\pi$
$632$	23601.4	1.48546
$633$	0	0
$634$	−12042.9	−0.754391
$635$	−190.492	−0.0119046
$636$	0	0
$637$	0	0
$638$	3661.55	0.227213
$639$	0	0
$640$	−4704.34	−0.290555
$641$	30882.4	1.90293	0.951466	−	0.307754i	$-0.0995773\pi$
$641$	30882.4	1.90293	0.951466	+	0.307754i	$0.0995773\pi$
$642$	0	0
$643$	6216.88	0.381290	0.190645	−	0.981659i	$-0.438942\pi$
$643$	6216.88	0.381290	0.190645	+	0.981659i	$0.438942\pi$
$644$	0	0
$645$	0	0
$646$	−5716.00	−0.348132
$647$	21210.4	1.28882	0.644410	−	0.764680i	$-0.277103\pi$
$647$	21210.4	1.28882	0.644410	+	0.764680i	$0.277103\pi$
$648$	0	0
$649$	1773.92	0.107292
$650$	−3608.85	−0.217771
$651$	0	0
$652$	669.251	0.0401992
$653$	32938.7	1.97395	0.986977	−	0.160864i	$-0.0514281\pi$
$653$	32938.7	1.97395	0.986977	+	0.160864i	$0.0514281\pi$
$654$	0	0
$655$	7757.54	0.462767
$656$	−968.165	−0.0576227
$657$	0	0
$658$	0	0
$659$	9543.51	0.564131	0.282066	−	0.959395i	$-0.408980\pi$
$659$	9543.51	0.564131	0.282066	+	0.959395i	$0.408980\pi$
$660$	0	0
$661$	−13274.5	−0.781116	−0.390558	−	0.920578i	$-0.627718\pi$
$661$	−13274.5	−0.781116	−0.390558	+	0.920578i	$0.627718\pi$
$662$	−3499.55	−0.205459
$663$	0	0
$664$	−11713.0	−0.684565
$665$	0	0
$666$	0	0
$667$	15987.6	0.928101
$668$	−5978.27	−0.346267
$669$	0	0
$670$	−1856.71	−0.107061
$671$	−771.159	−0.0443670
$672$	0	0
$673$	−13575.3	−0.777545	−0.388772	−	0.921334i	$-0.627101\pi$
$673$	−13575.3	−0.777545	−0.388772	+	0.921334i	$0.627101\pi$
$674$	1897.17	0.108422
$675$	0	0
$676$	−1407.94	−0.0801058
$677$	−12020.7	−0.682414	−0.341207	−	0.939988i	$-0.610836\pi$
$677$	−12020.7	−0.682414	−0.341207	+	0.939988i	$0.610836\pi$
$678$	0	0
$679$	0	0
$680$	2973.83	0.167707
$681$	0	0
$682$	1086.27	0.0609905
$683$	18391.9	1.03038	0.515188	−	0.857077i	$-0.327722\pi$
$683$	18391.9	1.03038	0.515188	+	0.857077i	$0.327722\pi$
$684$	0	0
$685$	−6016.21	−0.335573
$686$	0	0
$687$	0	0
$688$	−18415.5	−1.02047
$689$	28808.3	1.59290
$690$	0	0
$691$	−15594.1	−0.858508	−0.429254	−	0.903184i	$-0.641223\pi$
$691$	−15594.1	−0.858508	−0.429254	+	0.903184i	$0.641223\pi$
$692$	6100.69	0.335135
$693$	0	0
$694$	6674.28	0.365061
$695$	−9484.98	−0.517677
$696$	0	0
$697$	472.346	0.0256691
$698$	11949.8	0.648004
$699$	0	0
$700$	0	0
$701$	−31093.7	−1.67531	−0.837656	−	0.546198i	$-0.816075\pi$
$701$	−31093.7	−1.67531	−0.837656	+	0.546198i	$0.816075\pi$
$702$	0	0
$703$	−5337.88	−0.286375
$704$	−3547.71	−0.189928
$705$	0	0
$706$	7594.36	0.404841
$707$	0	0
$708$	0	0
$709$	−2494.67	−0.132143	−0.0660714	−	0.997815i	$-0.521047\pi$
$709$	−2494.67	−0.132143	−0.0660714	+	0.997815i	$0.521047\pi$
$710$	−935.331	−0.0494399
$711$	0	0
$712$	24595.3	1.29459
$713$	4743.06	0.249129
$714$	0	0
$715$	−1760.00	−0.0920563
$716$	−4229.54	−0.220762
$717$	0	0
$718$	−10931.6	−0.568192
$719$	−34467.1	−1.78777	−0.893885	−	0.448295i	$-0.852031\pi$
$719$	−34467.1	−1.78777	−0.893885	+	0.448295i	$0.852031\pi$
$720$	0	0
$721$	0	0
$722$	−3506.77	−0.180759
$723$	0	0
$724$	−2857.27	−0.146670
$725$	5721.16	0.293074
$726$	0	0
$727$	−9314.97	−0.475204	−0.237602	−	0.971363i	$-0.576361\pi$
$727$	−9314.97	−0.475204	−0.237602	+	0.971363i	$0.576361\pi$
$728$	0	0
$729$	0	0
$730$	8182.24	0.414847
$731$	8984.49	0.454588
$732$	0	0
$733$	−16146.3	−0.813611	−0.406805	−	0.913515i	$-0.633357\pi$
$733$	−16146.3	−0.813611	−0.406805	+	0.913515i	$0.633357\pi$
$734$	−23772.1	−1.19543
$735$	0	0
$736$	−4488.95	−0.224816
$737$	−905.500	−0.0452571
$738$	0	0
$739$	−36749.1	−1.82928	−0.914640	−	0.404268i	$-0.867526\pi$
$739$	−36749.1	−1.82928	−0.914640	+	0.404268i	$0.867526\pi$
$740$	423.239	0.0210251
$741$	0	0
$742$	0	0
$743$	−2527.09	−0.124778	−0.0623890	−	0.998052i	$-0.519872\pi$
$743$	−2527.09	−0.124778	−0.0623890	+	0.998052i	$0.519872\pi$
$744$	0	0
$745$	3524.44	0.173323
$746$	−27616.2	−1.35536
$747$	0	0
$748$	221.031	0.0108044
$749$	0	0
$750$	0	0
$751$	15828.0	0.769070	0.384535	−	0.923111i	$-0.374362\pi$
$751$	15828.0	0.769070	0.384535	+	0.923111i	$0.374362\pi$
$752$	9840.28	0.477178
$753$	0	0
$754$	−33035.0	−1.59557
$755$	−15176.1	−0.731541
$756$	0	0
$757$	−20845.9	−1.00087	−0.500435	−	0.865774i	$-0.666826\pi$
$757$	−20845.9	−1.00087	−0.500435	+	0.865774i	$0.666826\pi$
$758$	15149.4	0.725927
$759$	0	0
$760$	10965.3	0.523360
$761$	−2420.55	−0.115302	−0.0576511	−	0.998337i	$-0.518361\pi$
$761$	−2420.55	−0.115302	−0.0576511	+	0.998337i	$0.518361\pi$
$762$	0	0
$763$	0	0
$764$	2324.35	0.110068
$765$	0	0
$766$	−29493.5	−1.39118
$767$	−16004.6	−0.753445
$768$	0	0
$769$	−22646.3	−1.06196	−0.530980	−	0.847384i	$-0.678176\pi$
$769$	−22646.3	−1.06196	−0.530980	+	0.847384i	$0.678176\pi$
$770$	0	0
$771$	0	0
$772$	2952.33	0.137638
$773$	−27620.2	−1.28516	−0.642580	−	0.766219i	$-0.722136\pi$
$773$	−27620.2	−1.28516	−0.642580	+	0.766219i	$0.722136\pi$
$774$	0	0
$775$	1697.30	0.0786695
$776$	−43680.0	−2.02064
$777$	0	0
$778$	−13831.7	−0.637391
$779$	1741.67	0.0801049
$780$	0	0
$781$	−456.151	−0.0208993
$782$	−4402.35	−0.201314
$783$	0	0
$784$	0	0
$785$	−13568.2	−0.616906
$786$	0	0
$787$	−14767.1	−0.668859	−0.334429	−	0.942421i	$-0.608544\pi$
$787$	−14767.1	−0.668859	−0.334429	+	0.942421i	$0.608544\pi$
$788$	−6427.82	−0.290586
$789$	0	0
$790$	−12502.8	−0.563076
$791$	0	0
$792$	0	0
$793$	6957.50	0.311561
$794$	6716.71	0.300211
$795$	0	0
$796$	2219.62	0.0988348
$797$	7549.97	0.335551	0.167775	−	0.985825i	$-0.446342\pi$
$797$	7549.97	0.335551	0.167775	+	0.985825i	$0.446342\pi$
$798$	0	0
$799$	−4800.85	−0.212568
$800$	−1606.37	−0.0709921
$801$	0	0
$802$	28482.1	1.25404
$803$	3990.39	0.175365
$804$	0	0
$805$	0	0
$806$	−9800.50	−0.428298
$807$	0	0
$808$	11678.8	0.508489
$809$	17920.0	0.778783	0.389391	−	0.921072i	$-0.372685\pi$
$809$	17920.0	0.778783	0.389391	+	0.921072i	$0.372685\pi$
$810$	0	0
$811$	−25536.6	−1.10569	−0.552843	−	0.833285i	$-0.686457\pi$
$811$	−25536.6	−1.10569	−0.552843	+	0.833285i	$0.686457\pi$
$812$	0	0
$813$	0	0
$814$	−941.545	−0.0405420
$815$	−2326.30	−0.0999836
$816$	0	0
$817$	33128.3	1.41862
$818$	−16878.5	−0.721447
$819$	0	0
$820$	−138.096	−0.00588114
$821$	−13688.8	−0.581904	−0.290952	−	0.956738i	$-0.593972\pi$
$821$	−13688.8	−0.581904	−0.290952	+	0.956738i	$0.593972\pi$
$822$	0	0
$823$	−15102.9	−0.639678	−0.319839	−	0.947472i	$-0.603629\pi$
$823$	−15102.9	−0.639678	−0.319839	+	0.947472i	$0.603629\pi$
$824$	8197.22	0.346558
$825$	0	0
$826$	0	0
$827$	−36290.3	−1.52592	−0.762961	−	0.646445i	$-0.776255\pi$
$827$	−36290.3	−1.52592	−0.762961	+	0.646445i	$0.776255\pi$
$828$	0	0
$829$	39405.1	1.65090	0.825449	−	0.564477i	$-0.190922\pi$
$829$	39405.1	1.65090	0.825449	+	0.564477i	$0.190922\pi$
$830$	6204.92	0.259489
$831$	0	0
$832$	32007.9	1.33374
$833$	0	0
$834$	0	0
$835$	20780.3	0.861236
$836$	815.000	0.0337170
$837$	0	0
$838$	−30409.6	−1.25356
$839$	33093.9	1.36177	0.680886	−	0.732389i	$-0.261595\pi$
$839$	33093.9	1.36177	0.680886	+	0.732389i	$0.261595\pi$
$840$	0	0
$841$	27981.8	1.14731
$842$	4435.50	0.181541
$843$	0	0
$844$	−1847.00	−0.0753274
$845$	4893.96	0.199239
$846$	0	0
$847$	0	0
$848$	−25776.4	−1.04383
$849$	0	0
$850$	−1575.38	−0.0635706
$851$	−4111.12	−0.165602
$852$	0	0
$853$	−29441.6	−1.18178	−0.590892	−	0.806750i	$-0.701224\pi$
$853$	−29441.6	−1.18178	−0.590892	+	0.806750i	$0.701224\pi$
$854$	0	0
$855$	0	0
$856$	−10900.2	−0.435233
$857$	16012.9	0.638260	0.319130	−	0.947711i	$-0.396609\pi$
$857$	16012.9	0.638260	0.319130	+	0.947711i	$0.396609\pi$
$858$	0	0
$859$	13404.3	0.532421	0.266211	−	0.963915i	$-0.414228\pi$
$859$	13404.3	0.532421	0.266211	+	0.963915i	$0.414228\pi$
$860$	−2626.73	−0.104152
$861$	0	0
$862$	−27290.2	−1.07832
$863$	2058.73	0.0812050	0.0406025	−	0.999175i	$-0.487072\pi$
$863$	2058.73	0.0812050	0.0406025	+	0.999175i	$0.487072\pi$
$864$	0	0
$865$	−21205.8	−0.833549
$866$	6767.66	0.265559
$867$	0	0
$868$	0	0
$869$	−6097.48	−0.238024
$870$	0	0
$871$	8169.54	0.317812
$872$	−44527.0	−1.72922
$873$	0	0
$874$	−16232.7	−0.628236
$875$	0	0
$876$	0	0
$877$	−16477.0	−0.634422	−0.317211	−	0.948355i	$-0.602746\pi$
$877$	−16477.0	−0.634422	−0.317211	+	0.948355i	$0.602746\pi$
$878$	22690.8	0.872183
$879$	0	0
$880$	1574.77	0.0603245
$881$	−43307.7	−1.65616	−0.828079	−	0.560612i	$-0.810566\pi$
$881$	−43307.7	−1.65616	−0.828079	+	0.560612i	$0.810566\pi$
$882$	0	0
$883$	15197.9	0.579217	0.289608	−	0.957145i	$-0.406475\pi$
$883$	15197.9	0.579217	0.289608	+	0.957145i	$0.406475\pi$
$884$	−1994.17	−0.0758723
$885$	0	0
$886$	16961.9	0.643168
$887$	−44953.6	−1.70168	−0.850842	−	0.525422i	$-0.823907\pi$
$887$	−44953.6	−1.70168	−0.850842	+	0.525422i	$0.823907\pi$
$888$	0	0
$889$	0	0
$890$	−13029.3	−0.490724
$891$	0	0
$892$	5488.13	0.206005
$893$	−17702.0	−0.663355
$894$	0	0
$895$	14701.8	0.549079
$896$	0	0
$897$	0	0
$898$	−33509.5	−1.24524
$899$	15536.9	0.576400
$900$	0	0
$901$	12575.7	0.464993
$902$	307.212	0.0113404
$903$	0	0
$904$	45325.2	1.66758
$905$	9931.77	0.364799
$906$	0	0
$907$	−38388.7	−1.40537	−0.702687	−	0.711499i	$-0.748017\pi$
$907$	−38388.7	−1.40537	−0.702687	+	0.711499i	$0.748017\pi$
$908$	3267.48	0.119422
$909$	0	0
$910$	0	0
$911$	−46222.1	−1.68102	−0.840508	−	0.541799i	$-0.817743\pi$
$911$	−46222.1	−1.68102	−0.840508	+	0.541799i	$0.817743\pi$
$912$	0	0
$913$	3026.08	0.109692
$914$	−31166.3	−1.12789
$915$	0	0
$916$	−3405.59	−0.122842
$917$	0	0
$918$	0	0
$919$	−30946.4	−1.11080	−0.555402	−	0.831582i	$-0.687436\pi$
$919$	−30946.4	−1.11080	−0.555402	+	0.831582i	$0.687436\pi$
$920$	8445.26	0.302644
$921$	0	0
$922$	28904.1	1.03244
$923$	4115.46	0.146763
$924$	0	0
$925$	−1471.16	−0.0522936
$926$	−42375.6	−1.50383
$927$	0	0
$928$	−14704.5	−0.520149
$929$	1907.48	0.0673652	0.0336826	−	0.999433i	$-0.489276\pi$
$929$	1907.48	0.0673652	0.0336826	+	0.999433i	$0.489276\pi$
$930$	0	0
$931$	0	0
$932$	2326.99	0.0817845
$933$	0	0
$934$	−26299.2	−0.921347
$935$	−768.296	−0.0268727
$936$	0	0
$937$	3334.99	0.116275	0.0581374	−	0.998309i	$-0.481484\pi$
$937$	3334.99	0.116275	0.0581374	+	0.998309i	$0.481484\pi$
$938$	0	0
$939$	0	0
$940$	1403.59	0.0487022
$941$	−9632.00	−0.333681	−0.166841	−	0.985984i	$-0.553357\pi$
$941$	−9632.00	−0.333681	−0.166841	+	0.985984i	$0.553357\pi$
$942$	0	0
$943$	1341.40	0.0463223
$944$	14320.2	0.493732
$945$	0	0
$946$	5843.48	0.200833
$947$	53606.0	1.83945	0.919727	−	0.392559i	$-0.128410\pi$
$947$	53606.0	1.83945	0.919727	+	0.392559i	$0.128410\pi$
$948$	0	0
$949$	−36001.9	−1.23148
$950$	−5808.85	−0.198383
$951$	0	0
$952$	0	0
$953$	−28433.6	−0.966481	−0.483240	−	0.875488i	$-0.660540\pi$
$953$	−28433.6	−0.966481	−0.483240	+	0.875488i	$0.660540\pi$
$954$	0	0
$955$	−8079.38	−0.273762
$956$	−2783.82	−0.0941790
$957$	0	0
$958$	−20410.4	−0.688341
$959$	0	0
$960$	0	0
$961$	−25181.7	−0.845278
$962$	8494.75	0.284700
$963$	0	0
$964$	687.436	0.0229676
$965$	−10262.2	−0.342334
$966$	0	0
$967$	33877.6	1.12661	0.563304	−	0.826250i	$-0.309530\pi$
$967$	33877.6	1.12661	0.563304	+	0.826250i	$0.309530\pi$
$968$	−31236.4	−1.03717
$969$	0	0
$970$	23139.4	0.765939
$971$	−10208.6	−0.337394	−0.168697	−	0.985668i	$-0.553956\pi$
$971$	−10208.6	−0.337394	−0.168697	+	0.985668i	$0.553956\pi$
$972$	0	0
$973$	0	0
$974$	−25504.4	−0.839029
$975$	0	0
$976$	−6225.27	−0.204166
$977$	35478.1	1.16177	0.580883	−	0.813987i	$-0.302707\pi$
$977$	35478.1	1.16177	0.580883	+	0.813987i	$0.302707\pi$
$978$	0	0
$979$	−6354.27	−0.207439
$980$	0	0
$981$	0	0
$982$	47800.7	1.55334
$983$	54435.8	1.76626	0.883130	−	0.469128i	$-0.155432\pi$
$983$	54435.8	1.76626	0.883130	+	0.469128i	$0.155432\pi$
$984$	0	0
$985$	22342.9	0.722746
$986$	−14420.8	−0.465773
$987$	0	0
$988$	−7353.04	−0.236773
$989$	25514.7	0.820346
$990$	0	0
$991$	−47319.6	−1.51681	−0.758404	−	0.651784i	$-0.774021\pi$
$991$	−47319.6	−1.51681	−0.758404	+	0.651784i	$0.774021\pi$
$992$	−4362.38	−0.139623
$993$	0	0
$994$	0	0
$995$	−7715.35	−0.245822
$996$	0	0
$997$	−26273.1	−0.834580	−0.417290	−	0.908773i	$-0.637020\pi$
$997$	−26273.1	−0.834580	−0.417290	+	0.908773i	$0.637020\pi$
$998$	−10436.3	−0.331017
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.4.a.y.1.1		2
3.2	odd	2		2205.4.a.ba.1.2		2
7.6	odd	2		315.4.a.h.1.1	✓	2
21.20	even	2		315.4.a.j.1.2	yes	2
35.34	odd	2		1575.4.a.u.1.2		2
105.104	even	2		1575.4.a.r.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.4.a.h.1.1	✓	2	7.6	odd	2
315.4.a.j.1.2	yes	2	21.20	even	2
1575.4.a.r.1.1		2	105.104	even	2
1575.4.a.u.1.2		2	35.34	odd	2
2205.4.a.y.1.1		2	1.1	even	1	trivial
2205.4.a.ba.1.2		2	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2205.a (trivial)

\(f(q)\)	\(=\)	\(q-2.56155 q^{2} -1.43845 q^{4} +5.00000 q^{5} +24.1771 q^{8} +O(q^{10})\)	\(q-2.56155 q^{2} -1.43845 q^{4} +5.00000 q^{5} +24.1771 q^{8} -12.8078 q^{10} -6.24621 q^{11} +56.3542 q^{13} -50.4233 q^{16} +24.6004 q^{17} +90.7083 q^{19} -7.19224 q^{20} +16.0000 q^{22} +69.8617 q^{23} +25.0000 q^{25} -144.354 q^{26} +228.847 q^{29} +67.8920 q^{31} -64.2547 q^{32} -63.0152 q^{34} -58.8466 q^{37} -232.354 q^{38} +120.885 q^{40} +19.2007 q^{41} +365.218 q^{43} +8.98485 q^{44} -178.955 q^{46} -195.153 q^{47} -64.0388 q^{50} -81.0625 q^{52} +511.201 q^{53} -31.2311 q^{55} -586.203 q^{58} -284.000 q^{59} +123.460 q^{61} -173.909 q^{62} +567.978 q^{64} +281.771 q^{65} +144.968 q^{67} -35.3863 q^{68} +73.0284 q^{71} -638.850 q^{73} +150.739 q^{74} -130.479 q^{76} +976.189 q^{79} -252.116 q^{80} -49.1837 q^{82} -484.466 q^{83} +123.002 q^{85} -935.525 q^{86} -151.015 q^{88} +1017.30 q^{89} -100.492 q^{92} +499.896 q^{94} +453.542 q^{95} -1806.67 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - 7 q^{4} + 10 q^{5} + 3 q^{8}+O(q^{10}) \) 2 * q - q^2 - 7 * q^4 + 10 * q^5 + 3 * q^8	\( 2 q - q^{2} - 7 q^{4} + 10 q^{5} + 3 q^{8} - 5 q^{10} + 4 q^{11} + 22 q^{13} - 39 q^{16} - 58 q^{17} - 35 q^{20} + 32 q^{22} + 82 q^{23} + 50 q^{25} - 198 q^{26} + 334 q^{29} + 210 q^{31} + 123 q^{32} - 192 q^{34} + 6 q^{37} - 374 q^{38} + 15 q^{40} - 176 q^{41} + 46 q^{43} - 48 q^{44} - 160 q^{46} - 514 q^{47} - 25 q^{50} + 110 q^{52} + 808 q^{53} + 20 q^{55} - 422 q^{58} - 568 q^{59} + 618 q^{61} + 48 q^{62} + 769 q^{64} + 110 q^{65} + 694 q^{67} + 424 q^{68} + 814 q^{71} - 82 q^{73} + 252 q^{74} + 374 q^{76} + 600 q^{79} - 195 q^{80} - 354 q^{82} + 268 q^{83} - 290 q^{85} - 1434 q^{86} - 368 q^{88} + 72 q^{89} - 168 q^{92} + 2 q^{94} - 1626 q^{97}+O(q^{100}) \) 2 * q - q^2 - 7 * q^4 + 10 * q^5 + 3 * q^8 - 5 * q^10 + 4 * q^11 + 22 * q^13 - 39 * q^16 - 58 * q^17 - 35 * q^20 + 32 * q^22 + 82 * q^23 + 50 * q^25 - 198 * q^26 + 334 * q^29 + 210 * q^31 + 123 * q^32 - 192 * q^34 + 6 * q^37 - 374 * q^38 + 15 * q^40 - 176 * q^41 + 46 * q^43 - 48 * q^44 - 160 * q^46 - 514 * q^47 - 25 * q^50 + 110 * q^52 + 808 * q^53 + 20 * q^55 - 422 * q^58 - 568 * q^59 + 618 * q^61 + 48 * q^62 + 769 * q^64 + 110 * q^65 + 694 * q^67 + 424 * q^68 + 814 * q^71 - 82 * q^73 + 252 * q^74 + 374 * q^76 + 600 * q^79 - 195 * q^80 - 354 * q^82 + 268 * q^83 - 290 * q^85 - 1434 * q^86 - 368 * q^88 + 72 * q^89 - 168 * q^92 + 2 * q^94 - 1626 * q^97

Introduction

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