LMFDB - Embedded newform 3640.2.a.ba.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3640,2,Mod(1,3640)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3640, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3640.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3640.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:	$29.0655463357$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 11x^{5} + 24x^{4} + 33x^{3} - 41x^{2} - 31x - 4 $ x^7 - 3x^6 - 11x^5 + 24x^4 + 33x^3 - 41x^2 - 31x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.04214$ of defining polynomial
Character	$\chi$	$=$	3640.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.04214 q^{3} +1.00000 q^{5} +1.00000 q^{7} -1.91394 q^{9} +O(q^{10})$	$q-1.04214 q^{3} +1.00000 q^{5} +1.00000 q^{7} -1.91394 q^{9} -3.48491 q^{11} +1.00000 q^{13} -1.04214 q^{15} +3.19618 q^{17} +8.26121 q^{19} -1.04214 q^{21} -5.09086 q^{23} +1.00000 q^{25} +5.12103 q^{27} -7.32721 q^{29} -0.550053 q^{31} +3.63178 q^{33} +1.00000 q^{35} -12.0074 q^{37} -1.04214 q^{39} -2.76255 q^{41} +9.98907 q^{43} -1.91394 q^{45} +13.4877 q^{47} +1.00000 q^{49} -3.33088 q^{51} +9.46191 q^{53} -3.48491 q^{55} -8.60937 q^{57} +11.5921 q^{59} -7.74416 q^{61} -1.91394 q^{63} +1.00000 q^{65} -14.6306 q^{67} +5.30541 q^{69} +11.2870 q^{71} +7.70231 q^{73} -1.04214 q^{75} -3.48491 q^{77} -9.89751 q^{79} +0.404962 q^{81} +1.06221 q^{83} +3.19618 q^{85} +7.63601 q^{87} +7.81931 q^{89} +1.00000 q^{91} +0.573234 q^{93} +8.26121 q^{95} -3.66441 q^{97} +6.66990 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 4 q^{3} + 7 q^{5} + 7 q^{7} + 11 q^{9}+O(q^{10}) $ 7 * q + 4 * q^3 + 7 * q^5 + 7 * q^7 + 11 * q^9	$ 7 q + 4 q^{3} + 7 q^{5} + 7 q^{7} + 11 q^{9} + 7 q^{11} + 7 q^{13} + 4 q^{15} + 3 q^{17} + 13 q^{19} + 4 q^{21} + 11 q^{23} + 7 q^{25} + 13 q^{27} + 3 q^{29} + 12 q^{31} - 11 q^{33} + 7 q^{35} - 4 q^{37} + 4 q^{39} + 6 q^{41} + 10 q^{43} + 11 q^{45} + 15 q^{47} + 7 q^{49} + 8 q^{53} + 7 q^{55} - 18 q^{57} + 13 q^{59} - q^{61} + 11 q^{63} + 7 q^{65} + 8 q^{67} - 9 q^{69} + 20 q^{71} - 17 q^{73} + 4 q^{75} + 7 q^{77} + 6 q^{79} + 39 q^{81} + 30 q^{83} + 3 q^{85} + 21 q^{87} + 5 q^{89} + 7 q^{91} - 24 q^{93} + 13 q^{95} - 9 q^{97} + 12 q^{99}+O(q^{100}) $ 7 * q + 4 * q^3 + 7 * q^5 + 7 * q^7 + 11 * q^9 + 7 * q^11 + 7 * q^13 + 4 * q^15 + 3 * q^17 + 13 * q^19 + 4 * q^21 + 11 * q^23 + 7 * q^25 + 13 * q^27 + 3 * q^29 + 12 * q^31 - 11 * q^33 + 7 * q^35 - 4 * q^37 + 4 * q^39 + 6 * q^41 + 10 * q^43 + 11 * q^45 + 15 * q^47 + 7 * q^49 + 8 * q^53 + 7 * q^55 - 18 * q^57 + 13 * q^59 - q^61 + 11 * q^63 + 7 * q^65 + 8 * q^67 - 9 * q^69 + 20 * q^71 - 17 * q^73 + 4 * q^75 + 7 * q^77 + 6 * q^79 + 39 * q^81 + 30 * q^83 + 3 * q^85 + 21 * q^87 + 5 * q^89 + 7 * q^91 - 24 * q^93 + 13 * q^95 - 9 * q^97 + 12 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	−1.04214	−0.601682	−0.300841	−	0.953674i	$-0.597267\pi$
$3$	−1.04214	−0.601682	−0.300841	+	0.953674i	$0.597267\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−1.91394	−0.637979
$10$	0	0
$11$	−3.48491	−1.05074	−0.525370	−	0.850874i	$-0.676073\pi$
$11$	−3.48491	−1.05074	−0.525370	+	0.850874i	$0.676073\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−1.04214	−0.269080
$16$	0	0
$17$	3.19618	0.775187	0.387594	−	0.921830i	$-0.373306\pi$
$17$	3.19618	0.775187	0.387594	+	0.921830i	$0.373306\pi$
$18$	0	0
$19$	8.26121	1.89525	0.947626	−	0.319381i	$-0.103475\pi$
$19$	8.26121	1.89525	0.947626	+	0.319381i	$0.103475\pi$
$20$	0	0
$21$	−1.04214	−0.227414
$22$	0	0
$23$	−5.09086	−1.06152	−0.530759	−	0.847523i	$-0.678093\pi$
$23$	−5.09086	−1.06152	−0.530759	+	0.847523i	$0.678093\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.12103	0.985542
$28$	0	0
$29$	−7.32721	−1.36063	−0.680315	−	0.732920i	$-0.738157\pi$
$29$	−7.32721	−1.36063	−0.680315	+	0.732920i	$0.738157\pi$
$30$	0	0
$31$	−0.550053	−0.0987924	−0.0493962	−	0.998779i	$-0.515730\pi$
$31$	−0.550053	−0.0987924	−0.0493962	+	0.998779i	$0.515730\pi$
$32$	0	0
$33$	3.63178	0.632212
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	−12.0074	−1.97401	−0.987006	−	0.160684i	$-0.948630\pi$
$37$	−12.0074	−1.97401	−0.987006	+	0.160684i	$0.948630\pi$
$38$	0	0
$39$	−1.04214	−0.166877
$40$	0	0
$41$	−2.76255	−0.431437	−0.215719	−	0.976456i	$-0.569209\pi$
$41$	−2.76255	−0.431437	−0.215719	+	0.976456i	$0.569209\pi$
$42$	0	0
$43$	9.98907	1.52332	0.761660	−	0.647977i	$-0.224385\pi$
$43$	9.98907	1.52332	0.761660	+	0.647977i	$0.224385\pi$
$44$	0	0
$45$	−1.91394	−0.285313
$46$	0	0
$47$	13.4877	1.96739	0.983695	−	0.179847i	$-0.0575603\pi$
$47$	13.4877	1.96739	0.983695	+	0.179847i	$0.0575603\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−3.33088	−0.466416
$52$	0	0
$53$	9.46191	1.29969	0.649847	−	0.760065i	$-0.274833\pi$
$53$	9.46191	1.29969	0.649847	+	0.760065i	$0.274833\pi$
$54$	0	0
$55$	−3.48491	−0.469905
$56$	0	0
$57$	−8.60937	−1.14034
$58$	0	0
$59$	11.5921	1.50916	0.754581	−	0.656207i	$-0.227840\pi$
$59$	11.5921	1.50916	0.754581	+	0.656207i	$0.227840\pi$
$60$	0	0
$61$	−7.74416	−0.991537	−0.495769	−	0.868455i	$-0.665114\pi$
$61$	−7.74416	−0.991537	−0.495769	+	0.868455i	$0.665114\pi$
$62$	0	0
$63$	−1.91394	−0.241133
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	−14.6306	−1.78741	−0.893704	−	0.448656i	$-0.851903\pi$
$67$	−14.6306	−1.78741	−0.893704	+	0.448656i	$0.851903\pi$
$68$	0	0
$69$	5.30541	0.638696
$70$	0	0
$71$	11.2870	1.33952	0.669758	−	0.742580i	$-0.266398\pi$
$71$	11.2870	1.33952	0.669758	+	0.742580i	$0.266398\pi$
$72$	0	0
$73$	7.70231	0.901487	0.450744	−	0.892653i	$-0.351159\pi$
$73$	7.70231	0.901487	0.450744	+	0.892653i	$0.351159\pi$
$74$	0	0
$75$	−1.04214	−0.120336
$76$	0	0
$77$	−3.48491	−0.397143
$78$	0	0
$79$	−9.89751	−1.11356	−0.556779	−	0.830661i	$-0.687963\pi$
$79$	−9.89751	−1.11356	−0.556779	+	0.830661i	$0.687963\pi$
$80$	0	0
$81$	0.404962	0.0449958
$82$	0	0
$83$	1.06221	0.116592	0.0582962	−	0.998299i	$-0.481433\pi$
$83$	1.06221	0.116592	0.0582962	+	0.998299i	$0.481433\pi$
$84$	0	0
$85$	3.19618	0.346674
$86$	0	0
$87$	7.63601	0.818666
$88$	0	0
$89$	7.81931	0.828845	0.414422	−	0.910085i	$-0.363984\pi$
$89$	7.81931	0.828845	0.414422	+	0.910085i	$0.363984\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	0.573234	0.0594416
$94$	0	0
$95$	8.26121	0.847583
$96$	0	0
$97$	−3.66441	−0.372065	−0.186032	−	0.982544i	$-0.559563\pi$
$97$	−3.66441	−0.372065	−0.186032	+	0.982544i	$0.559563\pi$
$98$	0	0
$99$	6.66990	0.670350
$100$	0	0
$101$	8.50810	0.846588	0.423294	−	0.905992i	$-0.360874\pi$
$101$	8.50810	0.846588	0.423294	+	0.905992i	$0.360874\pi$
$102$	0	0
$103$	−9.89525	−0.975008	−0.487504	−	0.873121i	$-0.662093\pi$
$103$	−9.89525	−0.975008	−0.487504	+	0.873121i	$0.662093\pi$
$104$	0	0
$105$	−1.04214	−0.101703
$106$	0	0
$107$	17.2945	1.67192	0.835960	−	0.548790i	$-0.184911\pi$
$107$	17.2945	1.67192	0.835960	+	0.548790i	$0.184911\pi$
$108$	0	0
$109$	9.23830	0.884869	0.442434	−	0.896801i	$-0.354115\pi$
$109$	9.23830	0.884869	0.442434	+	0.896801i	$0.354115\pi$
$110$	0	0
$111$	12.5135	1.18773
$112$	0	0
$113$	13.8284	1.30087	0.650435	−	0.759562i	$-0.274587\pi$
$113$	13.8284	1.30087	0.650435	+	0.759562i	$0.274587\pi$
$114$	0	0
$115$	−5.09086	−0.474725
$116$	0	0
$117$	−1.91394	−0.176943
$118$	0	0
$119$	3.19618	0.292993
$120$	0	0
$121$	1.14461	0.104055
$122$	0	0
$123$	2.87897	0.259588
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−16.6252	−1.47525	−0.737623	−	0.675213i	$-0.764052\pi$
$127$	−16.6252	−1.47525	−0.737623	+	0.675213i	$0.764052\pi$
$128$	0	0
$129$	−10.4101	−0.916554
$130$	0	0
$131$	5.11386	0.446800	0.223400	−	0.974727i	$-0.428284\pi$
$131$	5.11386	0.446800	0.223400	+	0.974727i	$0.428284\pi$
$132$	0	0
$133$	8.26121	0.716338
$134$	0	0
$135$	5.12103	0.440748
$136$	0	0
$137$	−6.97730	−0.596111	−0.298056	−	0.954549i	$-0.596338\pi$
$137$	−6.97730	−0.596111	−0.298056	+	0.954549i	$0.596338\pi$
$138$	0	0
$139$	9.02208	0.765243	0.382621	−	0.923905i	$-0.375021\pi$
$139$	9.02208	0.765243	0.382621	+	0.923905i	$0.375021\pi$
$140$	0	0
$141$	−14.0562	−1.18374
$142$	0	0
$143$	−3.48491	−0.291423
$144$	0	0
$145$	−7.32721	−0.608492
$146$	0	0
$147$	−1.04214	−0.0859546
$148$	0	0
$149$	−8.23625	−0.674740	−0.337370	−	0.941372i	$-0.609537\pi$
$149$	−8.23625	−0.674740	−0.337370	+	0.941372i	$0.609537\pi$
$150$	0	0
$151$	17.1018	1.39173	0.695864	−	0.718173i	$-0.255022\pi$
$151$	17.1018	1.39173	0.695864	+	0.718173i	$0.255022\pi$
$152$	0	0
$153$	−6.11728	−0.494553
$154$	0	0
$155$	−0.550053	−0.0441813
$156$	0	0
$157$	7.79287	0.621939	0.310970	−	0.950420i	$-0.399346\pi$
$157$	7.79287	0.621939	0.310970	+	0.950420i	$0.399346\pi$
$158$	0	0
$159$	−9.86067	−0.782002
$160$	0	0
$161$	−5.09086	−0.401216
$162$	0	0
$163$	−11.2292	−0.879537	−0.439768	−	0.898111i	$-0.644939\pi$
$163$	−11.2292	−0.879537	−0.439768	+	0.898111i	$0.644939\pi$
$164$	0	0
$165$	3.63178	0.282734
$166$	0	0
$167$	23.4504	1.81465	0.907323	−	0.420433i	$-0.138122\pi$
$167$	23.4504	1.81465	0.907323	+	0.420433i	$0.138122\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−15.8114	−1.20913
$172$	0	0
$173$	−13.5830	−1.03270	−0.516349	−	0.856378i	$-0.672709\pi$
$173$	−13.5830	−1.03270	−0.516349	+	0.856378i	$0.672709\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	−12.0806	−0.908035
$178$	0	0
$179$	0.941456	0.0703677	0.0351839	−	0.999381i	$-0.488798\pi$
$179$	0.941456	0.0703677	0.0351839	+	0.999381i	$0.488798\pi$
$180$	0	0
$181$	−2.09805	−0.155947	−0.0779735	−	0.996955i	$-0.524845\pi$
$181$	−2.09805	−0.155947	−0.0779735	+	0.996955i	$0.524845\pi$
$182$	0	0
$183$	8.07052	0.596590
$184$	0	0
$185$	−12.0074	−0.882805
$186$	0	0
$187$	−11.1384	−0.814520
$188$	0	0
$189$	5.12103	0.372500
$190$	0	0
$191$	15.4895	1.12078	0.560391	−	0.828228i	$-0.310651\pi$
$191$	15.4895	1.12078	0.560391	+	0.828228i	$0.310651\pi$
$192$	0	0
$193$	3.03558	0.218506	0.109253	−	0.994014i	$-0.465154\pi$
$193$	3.03558	0.218506	0.109253	+	0.994014i	$0.465154\pi$
$194$	0	0
$195$	−1.04214	−0.0746295
$196$	0	0
$197$	11.2017	0.798085	0.399042	−	0.916932i	$-0.369343\pi$
$197$	11.2017	0.798085	0.399042	+	0.916932i	$0.369343\pi$
$198$	0	0
$199$	−5.75280	−0.407805	−0.203902	−	0.978991i	$-0.565363\pi$
$199$	−5.75280	−0.407805	−0.203902	+	0.978991i	$0.565363\pi$
$200$	0	0
$201$	15.2472	1.07545
$202$	0	0
$203$	−7.32721	−0.514270
$204$	0	0
$205$	−2.76255	−0.192945
$206$	0	0
$207$	9.74359	0.677226
$208$	0	0
$209$	−28.7896	−1.99142
$210$	0	0
$211$	10.4666	0.720547	0.360274	−	0.932847i	$-0.382683\pi$
$211$	10.4666	0.720547	0.360274	+	0.932847i	$0.382683\pi$
$212$	0	0
$213$	−11.7626	−0.805962
$214$	0	0
$215$	9.98907	0.681249
$216$	0	0
$217$	−0.550053	−0.0373400
$218$	0	0
$219$	−8.02692	−0.542409
$220$	0	0
$221$	3.19618	0.214998
$222$	0	0
$223$	23.9219	1.60193	0.800966	−	0.598710i	$-0.204320\pi$
$223$	23.9219	1.60193	0.800966	+	0.598710i	$0.204320\pi$
$224$	0	0
$225$	−1.91394	−0.127596
$226$	0	0
$227$	−5.06787	−0.336366	−0.168183	−	0.985756i	$-0.553790\pi$
$227$	−5.06787	−0.336366	−0.168183	+	0.985756i	$0.553790\pi$
$228$	0	0
$229$	13.6710	0.903403	0.451701	−	0.892169i	$-0.350817\pi$
$229$	13.6710	0.903403	0.451701	+	0.892169i	$0.350817\pi$
$230$	0	0
$231$	3.63178	0.238953
$232$	0	0
$233$	11.1850	0.732751	0.366376	−	0.930467i	$-0.380599\pi$
$233$	11.1850	0.732751	0.366376	+	0.930467i	$0.380599\pi$
$234$	0	0
$235$	13.4877	0.879843
$236$	0	0
$237$	10.3146	0.670007
$238$	0	0
$239$	7.66616	0.495883	0.247941	−	0.968775i	$-0.420246\pi$
$239$	7.66616	0.495883	0.247941	+	0.968775i	$0.420246\pi$
$240$	0	0
$241$	9.14097	0.588822	0.294411	−	0.955679i	$-0.404877\pi$
$241$	9.14097	0.588822	0.294411	+	0.955679i	$0.404877\pi$
$242$	0	0
$243$	−15.7851	−1.01262
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	8.26121	0.525648
$248$	0	0
$249$	−1.10697	−0.0701515
$250$	0	0
$251$	−1.03014	−0.0650221	−0.0325110	−	0.999471i	$-0.510350\pi$
$251$	−1.03014	−0.0650221	−0.0325110	+	0.999471i	$0.510350\pi$
$252$	0	0
$253$	17.7412	1.11538
$254$	0	0
$255$	−3.33088	−0.208588
$256$	0	0
$257$	−6.56596	−0.409574	−0.204787	−	0.978807i	$-0.565650\pi$
$257$	−6.56596	−0.409574	−0.204787	+	0.978807i	$0.565650\pi$
$258$	0	0
$259$	−12.0074	−0.746106
$260$	0	0
$261$	14.0238	0.868053
$262$	0	0
$263$	23.3249	1.43827	0.719136	−	0.694869i	$-0.244538\pi$
$263$	23.3249	1.43827	0.719136	+	0.694869i	$0.244538\pi$
$264$	0	0
$265$	9.46191	0.581241
$266$	0	0
$267$	−8.14884	−0.498701
$268$	0	0
$269$	6.05677	0.369288	0.184644	−	0.982806i	$-0.440887\pi$
$269$	6.05677	0.369288	0.184644	+	0.982806i	$0.440887\pi$
$270$	0	0
$271$	18.8223	1.14338	0.571688	−	0.820471i	$-0.306289\pi$
$271$	18.8223	1.14338	0.571688	+	0.820471i	$0.306289\pi$
$272$	0	0
$273$	−1.04214	−0.0630734
$274$	0	0
$275$	−3.48491	−0.210148
$276$	0	0
$277$	−28.7398	−1.72681	−0.863404	−	0.504514i	$-0.831672\pi$
$277$	−28.7398	−1.72681	−0.863404	+	0.504514i	$0.831672\pi$
$278$	0	0
$279$	1.05277	0.0630275
$280$	0	0
$281$	11.2634	0.671917	0.335959	−	0.941877i	$-0.390940\pi$
$281$	11.2634	0.671917	0.335959	+	0.941877i	$0.390940\pi$
$282$	0	0
$283$	25.6051	1.52206	0.761032	−	0.648714i	$-0.224693\pi$
$283$	25.6051	1.52206	0.761032	+	0.648714i	$0.224693\pi$
$284$	0	0
$285$	−8.60937	−0.509975
$286$	0	0
$287$	−2.76255	−0.163068
$288$	0	0
$289$	−6.78445	−0.399085
$290$	0	0
$291$	3.81884	0.223865
$292$	0	0
$293$	−10.7985	−0.630853	−0.315427	−	0.948950i	$-0.602148\pi$
$293$	−10.7985	−0.630853	−0.315427	+	0.948950i	$0.602148\pi$
$294$	0	0
$295$	11.5921	0.674918
$296$	0	0
$297$	−17.8463	−1.03555
$298$	0	0
$299$	−5.09086	−0.294412
$300$	0	0
$301$	9.98907	0.575761
$302$	0	0
$303$	−8.86667	−0.509377
$304$	0	0
$305$	−7.74416	−0.443429
$306$	0	0
$307$	−1.28244	−0.0731925	−0.0365962	−	0.999330i	$-0.511652\pi$
$307$	−1.28244	−0.0731925	−0.0365962	+	0.999330i	$0.511652\pi$
$308$	0	0
$309$	10.3123	0.586645
$310$	0	0
$311$	26.2935	1.49097	0.745484	−	0.666524i	$-0.232219\pi$
$311$	26.2935	1.49097	0.745484	+	0.666524i	$0.232219\pi$
$312$	0	0
$313$	−1.06405	−0.0601438	−0.0300719	−	0.999548i	$-0.509574\pi$
$313$	−1.06405	−0.0601438	−0.0300719	+	0.999548i	$0.509574\pi$
$314$	0	0
$315$	−1.91394	−0.107838
$316$	0	0
$317$	−4.69969	−0.263961	−0.131980	−	0.991252i	$-0.542134\pi$
$317$	−4.69969	−0.263961	−0.131980	+	0.991252i	$0.542134\pi$
$318$	0	0
$319$	25.5347	1.42967
$320$	0	0
$321$	−18.0233	−1.00596
$322$	0	0
$323$	26.4043	1.46918
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	−9.62764	−0.532410
$328$	0	0
$329$	13.4877	0.743603
$330$	0	0
$331$	10.1323	0.556923	0.278462	−	0.960447i	$-0.410175\pi$
$331$	10.1323	0.556923	0.278462	+	0.960447i	$0.410175\pi$
$332$	0	0
$333$	22.9815	1.25938
$334$	0	0
$335$	−14.6306	−0.799354
$336$	0	0
$337$	17.6231	0.959992	0.479996	−	0.877271i	$-0.340638\pi$
$337$	17.6231	0.959992	0.479996	+	0.877271i	$0.340638\pi$
$338$	0	0
$339$	−14.4112	−0.782710
$340$	0	0
$341$	1.91689	0.103805
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	5.30541	0.285634
$346$	0	0
$347$	0.920630	0.0494220	0.0247110	−	0.999695i	$-0.492133\pi$
$347$	0.920630	0.0494220	0.0247110	+	0.999695i	$0.492133\pi$
$348$	0	0
$349$	−0.627561	−0.0335926	−0.0167963	−	0.999859i	$-0.505347\pi$
$349$	−0.627561	−0.0335926	−0.0167963	+	0.999859i	$0.505347\pi$
$350$	0	0
$351$	5.12103	0.273340
$352$	0	0
$353$	−12.3716	−0.658472	−0.329236	−	0.944248i	$-0.606791\pi$
$353$	−12.3716	−0.658472	−0.329236	+	0.944248i	$0.606791\pi$
$354$	0	0
$355$	11.2870	0.599049
$356$	0	0
$357$	−3.33088	−0.176289
$358$	0	0
$359$	14.9845	0.790854	0.395427	−	0.918497i	$-0.370597\pi$
$359$	14.9845	0.790854	0.395427	+	0.918497i	$0.370597\pi$
$360$	0	0
$361$	49.2477	2.59198
$362$	0	0
$363$	−1.19285	−0.0626082
$364$	0	0
$365$	7.70231	0.403157
$366$	0	0
$367$	−16.3618	−0.854079	−0.427040	−	0.904233i	$-0.640444\pi$
$367$	−16.3618	−0.854079	−0.427040	+	0.904233i	$0.640444\pi$
$368$	0	0
$369$	5.28734	0.275248
$370$	0	0
$371$	9.46191	0.491238
$372$	0	0
$373$	−23.8961	−1.23729	−0.618647	−	0.785669i	$-0.712319\pi$
$373$	−23.8961	−1.23729	−0.618647	+	0.785669i	$0.712319\pi$
$374$	0	0
$375$	−1.04214	−0.0538161
$376$	0	0
$377$	−7.32721	−0.377371
$378$	0	0
$379$	25.3283	1.30103	0.650513	−	0.759495i	$-0.274554\pi$
$379$	25.3283	1.30103	0.650513	+	0.759495i	$0.274554\pi$
$380$	0	0
$381$	17.3258	0.887628
$382$	0	0
$383$	11.5664	0.591014	0.295507	−	0.955341i	$-0.404511\pi$
$383$	11.5664	0.591014	0.295507	+	0.955341i	$0.404511\pi$
$384$	0	0
$385$	−3.48491	−0.177608
$386$	0	0
$387$	−19.1185	−0.971846
$388$	0	0
$389$	−20.9467	−1.06204	−0.531020	−	0.847359i	$-0.678191\pi$
$389$	−20.9467	−1.06204	−0.531020	+	0.847359i	$0.678191\pi$
$390$	0	0
$391$	−16.2713	−0.822875
$392$	0	0
$393$	−5.32938	−0.268832
$394$	0	0
$395$	−9.89751	−0.497998
$396$	0	0
$397$	−28.9708	−1.45400	−0.727002	−	0.686636i	$-0.759087\pi$
$397$	−28.9708	−1.45400	−0.727002	+	0.686636i	$0.759087\pi$
$398$	0	0
$399$	−8.60937	−0.431008
$400$	0	0
$401$	15.8004	0.789032	0.394516	−	0.918889i	$-0.370912\pi$
$401$	15.8004	0.789032	0.394516	+	0.918889i	$0.370912\pi$
$402$	0	0
$403$	−0.550053	−0.0274001
$404$	0	0
$405$	0.404962	0.0201227
$406$	0	0
$407$	41.8449	2.07417
$408$	0	0
$409$	14.0773	0.696077	0.348038	−	0.937480i	$-0.386848\pi$
$409$	14.0773	0.696077	0.348038	+	0.937480i	$0.386848\pi$
$410$	0	0
$411$	7.27135	0.358669
$412$	0	0
$413$	11.5921	0.570410
$414$	0	0
$415$	1.06221	0.0521417
$416$	0	0
$417$	−9.40231	−0.460433
$418$	0	0
$419$	1.99999	0.0977059	0.0488530	−	0.998806i	$-0.484443\pi$
$419$	1.99999	0.0977059	0.0488530	+	0.998806i	$0.484443\pi$
$420$	0	0
$421$	−9.10351	−0.443678	−0.221839	−	0.975083i	$-0.571206\pi$
$421$	−9.10351	−0.443678	−0.221839	+	0.975083i	$0.571206\pi$
$422$	0	0
$423$	−25.8147	−1.25515
$424$	0	0
$425$	3.19618	0.155037
$426$	0	0
$427$	−7.74416	−0.374766
$428$	0	0
$429$	3.63178	0.175344
$430$	0	0
$431$	−33.6482	−1.62078	−0.810389	−	0.585893i	$-0.800744\pi$
$431$	−33.6482	−1.62078	−0.810389	+	0.585893i	$0.800744\pi$
$432$	0	0
$433$	3.70901	0.178244	0.0891218	−	0.996021i	$-0.471594\pi$
$433$	3.70901	0.178244	0.0891218	+	0.996021i	$0.471594\pi$
$434$	0	0
$435$	7.63601	0.366119
$436$	0	0
$437$	−42.0567	−2.01185
$438$	0	0
$439$	6.70021	0.319784	0.159892	−	0.987135i	$-0.448885\pi$
$439$	6.70021	0.319784	0.159892	+	0.987135i	$0.448885\pi$
$440$	0	0
$441$	−1.91394	−0.0911398
$442$	0	0
$443$	−35.8535	−1.70345	−0.851725	−	0.523989i	$-0.824443\pi$
$443$	−35.8535	−1.70345	−0.851725	+	0.523989i	$0.824443\pi$
$444$	0	0
$445$	7.81931	0.370671
$446$	0	0
$447$	8.58335	0.405979
$448$	0	0
$449$	5.95422	0.280997	0.140498	−	0.990081i	$-0.455130\pi$
$449$	5.95422	0.280997	0.140498	+	0.990081i	$0.455130\pi$
$450$	0	0
$451$	9.62723	0.453329
$452$	0	0
$453$	−17.8226	−0.837378
$454$	0	0
$455$	1.00000	0.0468807
$456$	0	0
$457$	−16.1063	−0.753419	−0.376710	−	0.926331i	$-0.622945\pi$
$457$	−16.1063	−0.753419	−0.376710	+	0.926331i	$0.622945\pi$
$458$	0	0
$459$	16.3677	0.763980
$460$	0	0
$461$	21.0185	0.978931	0.489466	−	0.872023i	$-0.337192\pi$
$461$	21.0185	0.978931	0.489466	+	0.872023i	$0.337192\pi$
$462$	0	0
$463$	−25.9004	−1.20369	−0.601847	−	0.798612i	$-0.705568\pi$
$463$	−25.9004	−1.20369	−0.601847	+	0.798612i	$0.705568\pi$
$464$	0	0
$465$	0.573234	0.0265831
$466$	0	0
$467$	−24.8638	−1.15056	−0.575280	−	0.817956i	$-0.695107\pi$
$467$	−24.8638	−1.15056	−0.575280	+	0.817956i	$0.695107\pi$
$468$	0	0
$469$	−14.6306	−0.675577
$470$	0	0
$471$	−8.12130	−0.374210
$472$	0	0
$473$	−34.8110	−1.60061
$474$	0	0
$475$	8.26121	0.379050
$476$	0	0
$477$	−18.1095	−0.829177
$478$	0	0
$479$	−26.2987	−1.20162	−0.600810	−	0.799392i	$-0.705155\pi$
$479$	−26.2987	−1.20162	−0.600810	+	0.799392i	$0.705155\pi$
$480$	0	0
$481$	−12.0074	−0.547492
$482$	0	0
$483$	5.30541	0.241405
$484$	0	0
$485$	−3.66441	−0.166392
$486$	0	0
$487$	−16.3807	−0.742281	−0.371141	−	0.928577i	$-0.621033\pi$
$487$	−16.3807	−0.742281	−0.371141	+	0.928577i	$0.621033\pi$
$488$	0	0
$489$	11.7024	0.529201
$490$	0	0
$491$	−27.5272	−1.24229	−0.621143	−	0.783698i	$-0.713331\pi$
$491$	−27.5272	−1.24229	−0.621143	+	0.783698i	$0.713331\pi$
$492$	0	0
$493$	−23.4191	−1.05474
$494$	0	0
$495$	6.66990	0.299790
$496$	0	0
$497$	11.2870	0.506289
$498$	0	0
$499$	−15.8771	−0.710757	−0.355378	−	0.934723i	$-0.615648\pi$
$499$	−15.8771	−0.710757	−0.355378	+	0.934723i	$0.615648\pi$
$500$	0	0
$501$	−24.4387	−1.09184
$502$	0	0
$503$	−4.82750	−0.215248	−0.107624	−	0.994192i	$-0.534324\pi$
$503$	−4.82750	−0.215248	−0.107624	+	0.994192i	$0.534324\pi$
$504$	0	0
$505$	8.50810	0.378606
$506$	0	0
$507$	−1.04214	−0.0462832
$508$	0	0
$509$	25.3703	1.12452	0.562260	−	0.826960i	$-0.309932\pi$
$509$	25.3703	1.12452	0.562260	+	0.826960i	$0.309932\pi$
$510$	0	0
$511$	7.70231	0.340730
$512$	0	0
$513$	42.3059	1.86785
$514$	0	0
$515$	−9.89525	−0.436037
$516$	0	0
$517$	−47.0036	−2.06722
$518$	0	0
$519$	14.1555	0.621356
$520$	0	0
$521$	33.3816	1.46248	0.731238	−	0.682123i	$-0.238943\pi$
$521$	33.3816	1.46248	0.731238	+	0.682123i	$0.238943\pi$
$522$	0	0
$523$	−0.808146	−0.0353378	−0.0176689	−	0.999844i	$-0.505624\pi$
$523$	−0.808146	−0.0353378	−0.0176689	+	0.999844i	$0.505624\pi$
$524$	0	0
$525$	−1.04214	−0.0454829
$526$	0	0
$527$	−1.75807	−0.0765826
$528$	0	0
$529$	2.91689	0.126821
$530$	0	0
$531$	−22.1865	−0.962813
$532$	0	0
$533$	−2.76255	−0.119659
$534$	0	0
$535$	17.2945	0.747705
$536$	0	0
$537$	−0.981132	−0.0423390
$538$	0	0
$539$	−3.48491	−0.150106
$540$	0	0
$541$	−25.2147	−1.08407	−0.542033	−	0.840357i	$-0.682345\pi$
$541$	−25.2147	−1.08407	−0.542033	+	0.840357i	$0.682345\pi$
$542$	0	0
$543$	2.18647	0.0938305
$544$	0	0
$545$	9.23830	0.395725
$546$	0	0
$547$	−17.5802	−0.751674	−0.375837	−	0.926686i	$-0.622645\pi$
$547$	−17.5802	−0.751674	−0.375837	+	0.926686i	$0.622645\pi$
$548$	0	0
$549$	14.8218	0.632580
$550$	0	0
$551$	−60.5317	−2.57874
$552$	0	0
$553$	−9.89751	−0.420885
$554$	0	0
$555$	12.5135	0.531168
$556$	0	0
$557$	12.1262	0.513804	0.256902	−	0.966437i	$-0.417298\pi$
$557$	12.1262	0.513804	0.256902	+	0.966437i	$0.417298\pi$
$558$	0	0
$559$	9.98907	0.422493
$560$	0	0
$561$	11.6078	0.490082
$562$	0	0
$563$	28.3380	1.19430	0.597152	−	0.802128i	$-0.296299\pi$
$563$	28.3380	1.19430	0.597152	+	0.802128i	$0.296299\pi$
$564$	0	0
$565$	13.8284	0.581767
$566$	0	0
$567$	0.404962	0.0170068
$568$	0	0
$569$	12.1090	0.507637	0.253819	−	0.967252i	$-0.418313\pi$
$569$	12.1090	0.507637	0.253819	+	0.967252i	$0.418313\pi$
$570$	0	0
$571$	−18.0797	−0.756611	−0.378306	−	0.925681i	$-0.623493\pi$
$571$	−18.0797	−0.756611	−0.378306	+	0.925681i	$0.623493\pi$
$572$	0	0
$573$	−16.1423	−0.674355
$574$	0	0
$575$	−5.09086	−0.212304
$576$	0	0
$577$	−18.1911	−0.757307	−0.378654	−	0.925538i	$-0.623613\pi$
$577$	−18.1911	−0.757307	−0.378654	+	0.925538i	$0.623613\pi$
$578$	0	0
$579$	−3.16351	−0.131471
$580$	0	0
$581$	1.06221	0.0440678
$582$	0	0
$583$	−32.9739	−1.36564
$584$	0	0
$585$	−1.91394	−0.0791315
$586$	0	0
$587$	6.43186	0.265471	0.132736	−	0.991151i	$-0.457624\pi$
$587$	6.43186	0.265471	0.132736	+	0.991151i	$0.457624\pi$
$588$	0	0
$589$	−4.54411	−0.187237
$590$	0	0
$591$	−11.6737	−0.480193
$592$	0	0
$593$	31.8706	1.30877	0.654385	−	0.756161i	$-0.272928\pi$
$593$	31.8706	1.30877	0.654385	+	0.756161i	$0.272928\pi$
$594$	0	0
$595$	3.19618	0.131031
$596$	0	0
$597$	5.99524	0.245369
$598$	0	0
$599$	19.1209	0.781257	0.390629	−	0.920548i	$-0.372258\pi$
$599$	19.1209	0.781257	0.390629	+	0.920548i	$0.372258\pi$
$600$	0	0
$601$	40.2151	1.64041	0.820204	−	0.572071i	$-0.193860\pi$
$601$	40.2151	1.64041	0.820204	+	0.572071i	$0.193860\pi$
$602$	0	0
$603$	28.0020	1.14033
$604$	0	0
$605$	1.14461	0.0465349
$606$	0	0
$607$	−0.899788	−0.0365213	−0.0182606	−	0.999833i	$-0.505813\pi$
$607$	−0.899788	−0.0365213	−0.0182606	+	0.999833i	$0.505813\pi$
$608$	0	0
$609$	7.63601	0.309427
$610$	0	0
$611$	13.4877	0.545656
$612$	0	0
$613$	7.93043	0.320307	0.160153	−	0.987092i	$-0.448801\pi$
$613$	7.93043	0.320307	0.160153	+	0.987092i	$0.448801\pi$
$614$	0	0
$615$	2.87897	0.116091
$616$	0	0
$617$	3.20237	0.128923	0.0644613	−	0.997920i	$-0.479467\pi$
$617$	3.20237	0.128923	0.0644613	+	0.997920i	$0.479467\pi$
$618$	0	0
$619$	−19.3958	−0.779583	−0.389791	−	0.920903i	$-0.627453\pi$
$619$	−19.3958	−0.779583	−0.389791	+	0.920903i	$0.627453\pi$
$620$	0	0
$621$	−26.0705	−1.04617
$622$	0	0
$623$	7.81931	0.313274
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	30.0029	1.19820
$628$	0	0
$629$	−38.3779	−1.53023
$630$	0	0
$631$	−9.83179	−0.391397	−0.195699	−	0.980664i	$-0.562697\pi$
$631$	−9.83179	−0.391397	−0.195699	+	0.980664i	$0.562697\pi$
$632$	0	0
$633$	−10.9077	−0.433540
$634$	0	0
$635$	−16.6252	−0.659750
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−21.6025	−0.854582
$640$	0	0
$641$	−6.18129	−0.244146	−0.122073	−	0.992521i	$-0.538954\pi$
$641$	−6.18129	−0.244146	−0.122073	+	0.992521i	$0.538954\pi$
$642$	0	0
$643$	−26.1750	−1.03224	−0.516120	−	0.856516i	$-0.672624\pi$
$643$	−26.1750	−1.03224	−0.516120	+	0.856516i	$0.672624\pi$
$644$	0	0
$645$	−10.4101	−0.409895
$646$	0	0
$647$	−32.9571	−1.29568	−0.647839	−	0.761778i	$-0.724327\pi$
$647$	−32.9571	−1.29568	−0.647839	+	0.761778i	$0.724327\pi$
$648$	0	0
$649$	−40.3974	−1.58574
$650$	0	0
$651$	0.573234	0.0224668
$652$	0	0
$653$	−32.1386	−1.25768	−0.628839	−	0.777535i	$-0.716470\pi$
$653$	−32.1386	−1.25768	−0.628839	+	0.777535i	$0.716470\pi$
$654$	0	0
$655$	5.11386	0.199815
$656$	0	0
$657$	−14.7417	−0.575130
$658$	0	0
$659$	−32.1843	−1.25372	−0.626861	−	0.779131i	$-0.715661\pi$
$659$	−32.1843	−1.25372	−0.626861	+	0.779131i	$0.715661\pi$
$660$	0	0
$661$	31.4746	1.22422	0.612110	−	0.790772i	$-0.290321\pi$
$661$	31.4746	1.22422	0.612110	+	0.790772i	$0.290321\pi$
$662$	0	0
$663$	−3.33088	−0.129361
$664$	0	0
$665$	8.26121	0.320356
$666$	0	0
$667$	37.3018	1.44433
$668$	0	0
$669$	−24.9301	−0.963853
$670$	0	0
$671$	26.9877	1.04185
$672$	0	0
$673$	−44.2815	−1.70693	−0.853463	−	0.521154i	$-0.825502\pi$
$673$	−44.2815	−1.70693	−0.853463	+	0.521154i	$0.825502\pi$
$674$	0	0
$675$	5.12103	0.197108
$676$	0	0
$677$	−6.05793	−0.232825	−0.116413	−	0.993201i	$-0.537139\pi$
$677$	−6.05793	−0.232825	−0.116413	+	0.993201i	$0.537139\pi$
$678$	0	0
$679$	−3.66441	−0.140627
$680$	0	0
$681$	5.28144	0.202385
$682$	0	0
$683$	39.2137	1.50047	0.750235	−	0.661171i	$-0.229940\pi$
$683$	39.2137	1.50047	0.750235	+	0.661171i	$0.229940\pi$
$684$	0	0
$685$	−6.97730	−0.266589
$686$	0	0
$687$	−14.2471	−0.543561
$688$	0	0
$689$	9.46191	0.360470
$690$	0	0
$691$	18.5908	0.707226	0.353613	−	0.935392i	$-0.384953\pi$
$691$	18.5908	0.707226	0.353613	+	0.935392i	$0.384953\pi$
$692$	0	0
$693$	6.66990	0.253369
$694$	0	0
$695$	9.02208	0.342227
$696$	0	0
$697$	−8.82960	−0.334445
$698$	0	0
$699$	−11.6563	−0.440883
$700$	0	0
$701$	48.0493	1.81480	0.907398	−	0.420272i	$-0.138065\pi$
$701$	48.0493	1.81480	0.907398	+	0.420272i	$0.138065\pi$
$702$	0	0
$703$	−99.1961	−3.74125
$704$	0	0
$705$	−14.0562	−0.529386
$706$	0	0
$707$	8.50810	0.319980
$708$	0	0
$709$	−46.3117	−1.73927	−0.869636	−	0.493693i	$-0.835647\pi$
$709$	−46.3117	−1.73927	−0.869636	+	0.493693i	$0.835647\pi$
$710$	0	0
$711$	18.9432	0.710426
$712$	0	0
$713$	2.80024	0.104870
$714$	0	0
$715$	−3.48491	−0.130328
$716$	0	0
$717$	−7.98924	−0.298364
$718$	0	0
$719$	−3.06690	−0.114376	−0.0571880	−	0.998363i	$-0.518213\pi$
$719$	−3.06690	−0.114376	−0.0571880	+	0.998363i	$0.518213\pi$
$720$	0	0
$721$	−9.89525	−0.368519
$722$	0	0
$723$	−9.52621	−0.354283
$724$	0	0
$725$	−7.32721	−0.272126
$726$	0	0
$727$	−5.21180	−0.193295	−0.0966474	−	0.995319i	$-0.530812\pi$
$727$	−5.21180	−0.193295	−0.0966474	+	0.995319i	$0.530812\pi$
$728$	0	0
$729$	15.2355	0.564277
$730$	0	0
$731$	31.9269	1.18086
$732$	0	0
$733$	13.7602	0.508244	0.254122	−	0.967172i	$-0.418213\pi$
$733$	13.7602	0.508244	0.254122	+	0.967172i	$0.418213\pi$
$734$	0	0
$735$	−1.04214	−0.0384401
$736$	0	0
$737$	50.9862	1.87810
$738$	0	0
$739$	−3.97363	−0.146172	−0.0730861	−	0.997326i	$-0.523285\pi$
$739$	−3.97363	−0.146172	−0.0730861	+	0.997326i	$0.523285\pi$
$740$	0	0
$741$	−8.60937	−0.316273
$742$	0	0
$743$	−5.83400	−0.214029	−0.107014	−	0.994257i	$-0.534129\pi$
$743$	−5.83400	−0.214029	−0.107014	+	0.994257i	$0.534129\pi$
$744$	0	0
$745$	−8.23625	−0.301753
$746$	0	0
$747$	−2.03300	−0.0743834
$748$	0	0
$749$	17.2945	0.631926
$750$	0	0
$751$	6.79376	0.247908	0.123954	−	0.992288i	$-0.460443\pi$
$751$	6.79376	0.247908	0.123954	+	0.992288i	$0.460443\pi$
$752$	0	0
$753$	1.07356	0.0391226
$754$	0	0
$755$	17.1018	0.622400
$756$	0	0
$757$	19.7424	0.717551	0.358775	−	0.933424i	$-0.383194\pi$
$757$	19.7424	0.717551	0.358775	+	0.933424i	$0.383194\pi$
$758$	0	0
$759$	−18.4889	−0.671104
$760$	0	0
$761$	−43.9837	−1.59441	−0.797204	−	0.603710i	$-0.793688\pi$
$761$	−43.9837	−1.59441	−0.797204	+	0.603710i	$0.793688\pi$
$762$	0	0
$763$	9.23830	0.334449
$764$	0	0
$765$	−6.11728	−0.221171
$766$	0	0
$767$	11.5921	0.418566
$768$	0	0
$769$	−48.3698	−1.74426	−0.872129	−	0.489276i	$-0.837261\pi$
$769$	−48.3698	−1.74426	−0.872129	+	0.489276i	$0.837261\pi$
$770$	0	0
$771$	6.84268	0.246433
$772$	0	0
$773$	−9.71721	−0.349504	−0.174752	−	0.984612i	$-0.555912\pi$
$773$	−9.71721	−0.349504	−0.174752	+	0.984612i	$0.555912\pi$
$774$	0	0
$775$	−0.550053	−0.0197585
$776$	0	0
$777$	12.5135	0.448919
$778$	0	0
$779$	−22.8220	−0.817683
$780$	0	0
$781$	−39.3340	−1.40748
$782$	0	0
$783$	−37.5229	−1.34096
$784$	0	0
$785$	7.79287	0.278140
$786$	0	0
$787$	−35.0117	−1.24803	−0.624015	−	0.781412i	$-0.714500\pi$
$787$	−35.0117	−1.24803	−0.624015	+	0.781412i	$0.714500\pi$
$788$	0	0
$789$	−24.3079	−0.865382
$790$	0	0
$791$	13.8284	0.491683
$792$	0	0
$793$	−7.74416	−0.275003
$794$	0	0
$795$	−9.86067	−0.349722
$796$	0	0
$797$	−27.4922	−0.973825	−0.486913	−	0.873451i	$-0.661877\pi$
$797$	−27.4922	−0.973825	−0.486913	+	0.873451i	$0.661877\pi$
$798$	0	0
$799$	43.1092	1.52509
$800$	0	0
$801$	−14.9657	−0.528785
$802$	0	0
$803$	−26.8419	−0.947229
$804$	0	0
$805$	−5.09086	−0.179429
$806$	0	0
$807$	−6.31202	−0.222194
$808$	0	0
$809$	−32.7478	−1.15135	−0.575676	−	0.817678i	$-0.695261\pi$
$809$	−32.7478	−1.15135	−0.575676	+	0.817678i	$0.695261\pi$
$810$	0	0
$811$	26.4093	0.927356	0.463678	−	0.886004i	$-0.346529\pi$
$811$	26.4093	0.927356	0.463678	+	0.886004i	$0.346529\pi$
$812$	0	0
$813$	−19.6156	−0.687949
$814$	0	0
$815$	−11.2292	−0.393341
$816$	0	0
$817$	82.5219	2.88708
$818$	0	0
$819$	−1.91394	−0.0668784
$820$	0	0
$821$	−42.9213	−1.49796	−0.748982	−	0.662590i	$-0.769457\pi$
$821$	−42.9213	−1.49796	−0.748982	+	0.662590i	$0.769457\pi$
$822$	0	0
$823$	33.4985	1.16769	0.583843	−	0.811867i	$-0.301549\pi$
$823$	33.4985	1.16769	0.583843	+	0.811867i	$0.301549\pi$
$824$	0	0
$825$	3.63178	0.126442
$826$	0	0
$827$	−51.4283	−1.78834	−0.894169	−	0.447729i	$-0.852233\pi$
$827$	−51.4283	−1.78834	−0.894169	+	0.447729i	$0.852233\pi$
$828$	0	0
$829$	14.4238	0.500959	0.250479	−	0.968122i	$-0.419412\pi$
$829$	14.4238	0.500959	0.250479	+	0.968122i	$0.419412\pi$
$830$	0	0
$831$	29.9510	1.03899
$832$	0	0
$833$	3.19618	0.110741
$834$	0	0
$835$	23.4504	0.811535
$836$	0	0
$837$	−2.81684	−0.0973641
$838$	0	0
$839$	−26.0505	−0.899363	−0.449682	−	0.893189i	$-0.648463\pi$
$839$	−26.0505	−0.899363	−0.449682	+	0.893189i	$0.648463\pi$
$840$	0	0
$841$	24.6881	0.851313
$842$	0	0
$843$	−11.7381	−0.404281
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	1.14461	0.0393292
$848$	0	0
$849$	−26.6842	−0.915799
$850$	0	0
$851$	61.1283	2.09545
$852$	0	0
$853$	35.5611	1.21759	0.608795	−	0.793328i	$-0.291653\pi$
$853$	35.5611	1.21759	0.608795	+	0.793328i	$0.291653\pi$
$854$	0	0
$855$	−15.8114	−0.540740
$856$	0	0
$857$	−45.6236	−1.55847	−0.779236	−	0.626730i	$-0.784393\pi$
$857$	−45.6236	−1.55847	−0.779236	+	0.626730i	$0.784393\pi$
$858$	0	0
$859$	35.2937	1.20421	0.602104	−	0.798418i	$-0.294329\pi$
$859$	35.2937	1.20421	0.602104	+	0.798418i	$0.294329\pi$
$860$	0	0
$861$	2.87897	0.0981151
$862$	0	0
$863$	5.39804	0.183751	0.0918757	−	0.995770i	$-0.470714\pi$
$863$	5.39804	0.183751	0.0918757	+	0.995770i	$0.470714\pi$
$864$	0	0
$865$	−13.5830	−0.461837
$866$	0	0
$867$	7.07037	0.240122
$868$	0	0
$869$	34.4920	1.17006
$870$	0	0
$871$	−14.6306	−0.495738
$872$	0	0
$873$	7.01345	0.237369
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	−32.8129	−1.10801	−0.554006	−	0.832512i	$-0.686902\pi$
$877$	−32.8129	−1.10801	−0.554006	+	0.832512i	$0.686902\pi$
$878$	0	0
$879$	11.2536	0.379573
$880$	0	0
$881$	33.8708	1.14114	0.570569	−	0.821250i	$-0.306723\pi$
$881$	33.8708	1.14114	0.570569	+	0.821250i	$0.306723\pi$
$882$	0	0
$883$	−44.3429	−1.49226	−0.746130	−	0.665801i	$-0.768090\pi$
$883$	−44.3429	−1.49226	−0.746130	+	0.665801i	$0.768090\pi$
$884$	0	0
$885$	−12.0806	−0.406086
$886$	0	0
$887$	−34.0458	−1.14315	−0.571573	−	0.820551i	$-0.693667\pi$
$887$	−34.0458	−1.14315	−0.571573	+	0.820551i	$0.693667\pi$
$888$	0	0
$889$	−16.6252	−0.557590
$890$	0	0
$891$	−1.41126	−0.0472789
$892$	0	0
$893$	111.425	3.72870
$894$	0	0
$895$	0.941456	0.0314694
$896$	0	0
$897$	5.30541	0.177143
$898$	0	0
$899$	4.03036	0.134420
$900$	0	0
$901$	30.2420	1.00751
$902$	0	0
$903$	−10.4101	−0.346425
$904$	0	0
$905$	−2.09805	−0.0697416
$906$	0	0
$907$	26.0405	0.864661	0.432331	−	0.901715i	$-0.357691\pi$
$907$	26.0405	0.864661	0.432331	+	0.901715i	$0.357691\pi$
$908$	0	0
$909$	−16.2840	−0.540105
$910$	0	0
$911$	7.06587	0.234103	0.117051	−	0.993126i	$-0.462656\pi$
$911$	7.06587	0.234103	0.117051	+	0.993126i	$0.462656\pi$
$912$	0	0
$913$	−3.70170	−0.122508
$914$	0	0
$915$	8.07052	0.266803
$916$	0	0
$917$	5.11386	0.168875
$918$	0	0
$919$	−25.1604	−0.829964	−0.414982	−	0.909830i	$-0.636212\pi$
$919$	−25.1604	−0.829964	−0.414982	+	0.909830i	$0.636212\pi$
$920$	0	0
$921$	1.33648	0.0440386
$922$	0	0
$923$	11.2870	0.371515
$924$	0	0
$925$	−12.0074	−0.394802
$926$	0	0
$927$	18.9389	0.622035
$928$	0	0
$929$	10.1830	0.334094	0.167047	−	0.985949i	$-0.446577\pi$
$929$	10.1830	0.334094	0.167047	+	0.985949i	$0.446577\pi$
$930$	0	0
$931$	8.26121	0.270750
$932$	0	0
$933$	−27.4016	−0.897088
$934$	0	0
$935$	−11.1384	−0.364265
$936$	0	0
$937$	28.3041	0.924654	0.462327	−	0.886709i	$-0.347015\pi$
$937$	28.3041	0.924654	0.462327	+	0.886709i	$0.347015\pi$
$938$	0	0
$939$	1.10890	0.0361875
$940$	0	0
$941$	−5.51343	−0.179733	−0.0898664	−	0.995954i	$-0.528644\pi$
$941$	−5.51343	−0.179733	−0.0898664	+	0.995954i	$0.528644\pi$
$942$	0	0
$943$	14.0638	0.457979
$944$	0	0
$945$	5.12103	0.166587
$946$	0	0
$947$	−19.7044	−0.640306	−0.320153	−	0.947366i	$-0.603734\pi$
$947$	−19.7044	−0.640306	−0.320153	+	0.947366i	$0.603734\pi$
$948$	0	0
$949$	7.70231	0.250028
$950$	0	0
$951$	4.89775	0.158820
$952$	0	0
$953$	20.5543	0.665818	0.332909	−	0.942959i	$-0.391970\pi$
$953$	20.5543	0.665818	0.332909	+	0.942959i	$0.391970\pi$
$954$	0	0
$955$	15.4895	0.501229
$956$	0	0
$957$	−26.6108	−0.860206
$958$	0	0
$959$	−6.97730	−0.225309
$960$	0	0
$961$	−30.6974	−0.990240
$962$	0	0
$963$	−33.1005	−1.06665
$964$	0	0
$965$	3.03558	0.0977187
$966$	0	0
$967$	54.0689	1.73874	0.869369	−	0.494163i	$-0.164525\pi$
$967$	54.0689	1.73874	0.869369	+	0.494163i	$0.164525\pi$
$968$	0	0
$969$	−27.5171	−0.883976
$970$	0	0
$971$	−5.32934	−0.171027	−0.0855134	−	0.996337i	$-0.527253\pi$
$971$	−5.32934	−0.171027	−0.0855134	+	0.996337i	$0.527253\pi$
$972$	0	0
$973$	9.02208	0.289235
$974$	0	0
$975$	−1.04214	−0.0333753
$976$	0	0
$977$	−25.3709	−0.811688	−0.405844	−	0.913942i	$-0.633022\pi$
$977$	−25.3709	−0.811688	−0.405844	+	0.913942i	$0.633022\pi$
$978$	0	0
$979$	−27.2496	−0.870901
$980$	0	0
$981$	−17.6815	−0.564527
$982$	0	0
$983$	32.0565	1.02244	0.511222	−	0.859449i	$-0.329193\pi$
$983$	32.0565	1.02244	0.511222	+	0.859449i	$0.329193\pi$
$984$	0	0
$985$	11.2017	0.356914
$986$	0	0
$987$	−14.0562	−0.447413
$988$	0	0
$989$	−50.8530	−1.61703
$990$	0	0
$991$	−20.1687	−0.640678	−0.320339	−	0.947303i	$-0.603797\pi$
$991$	−20.1687	−0.640678	−0.320339	+	0.947303i	$0.603797\pi$
$992$	0	0
$993$	−10.5593	−0.335091
$994$	0	0
$995$	−5.75280	−0.182376
$996$	0	0
$997$	−16.8383	−0.533274	−0.266637	−	0.963797i	$-0.585913\pi$
$997$	−16.8383	−0.533274	−0.266637	+	0.963797i	$0.585913\pi$
$998$	0	0
$999$	−61.4905	−1.94547

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3640.2.a.ba.1.2	✓	7
4.3	odd	2		7280.2.a.cf.1.6		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3640.2.a.ba.1.2	✓	7	1.1	even	1	trivial
7280.2.a.cf.1.6		7	4.3	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:	\( N \)	\(=\)	\( 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3640.a (trivial)

\(f(q)\)	\(=\)	\(q-1.04214 q^{3} +1.00000 q^{5} +1.00000 q^{7} -1.91394 q^{9} +O(q^{10})\)	\(q-1.04214 q^{3} +1.00000 q^{5} +1.00000 q^{7} -1.91394 q^{9} -3.48491 q^{11} +1.00000 q^{13} -1.04214 q^{15} +3.19618 q^{17} +8.26121 q^{19} -1.04214 q^{21} -5.09086 q^{23} +1.00000 q^{25} +5.12103 q^{27} -7.32721 q^{29} -0.550053 q^{31} +3.63178 q^{33} +1.00000 q^{35} -12.0074 q^{37} -1.04214 q^{39} -2.76255 q^{41} +9.98907 q^{43} -1.91394 q^{45} +13.4877 q^{47} +1.00000 q^{49} -3.33088 q^{51} +9.46191 q^{53} -3.48491 q^{55} -8.60937 q^{57} +11.5921 q^{59} -7.74416 q^{61} -1.91394 q^{63} +1.00000 q^{65} -14.6306 q^{67} +5.30541 q^{69} +11.2870 q^{71} +7.70231 q^{73} -1.04214 q^{75} -3.48491 q^{77} -9.89751 q^{79} +0.404962 q^{81} +1.06221 q^{83} +3.19618 q^{85} +7.63601 q^{87} +7.81931 q^{89} +1.00000 q^{91} +0.573234 q^{93} +8.26121 q^{95} -3.66441 q^{97} +6.66990 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 4 q^{3} + 7 q^{5} + 7 q^{7} + 11 q^{9}+O(q^{10}) \) 7 * q + 4 * q^3 + 7 * q^5 + 7 * q^7 + 11 * q^9	\( 7 q + 4 q^{3} + 7 q^{5} + 7 q^{7} + 11 q^{9} + 7 q^{11} + 7 q^{13} + 4 q^{15} + 3 q^{17} + 13 q^{19} + 4 q^{21} + 11 q^{23} + 7 q^{25} + 13 q^{27} + 3 q^{29} + 12 q^{31} - 11 q^{33} + 7 q^{35} - 4 q^{37} + 4 q^{39} + 6 q^{41} + 10 q^{43} + 11 q^{45} + 15 q^{47} + 7 q^{49} + 8 q^{53} + 7 q^{55} - 18 q^{57} + 13 q^{59} - q^{61} + 11 q^{63} + 7 q^{65} + 8 q^{67} - 9 q^{69} + 20 q^{71} - 17 q^{73} + 4 q^{75} + 7 q^{77} + 6 q^{79} + 39 q^{81} + 30 q^{83} + 3 q^{85} + 21 q^{87} + 5 q^{89} + 7 q^{91} - 24 q^{93} + 13 q^{95} - 9 q^{97} + 12 q^{99}+O(q^{100}) \) 7 * q + 4 * q^3 + 7 * q^5 + 7 * q^7 + 11 * q^9 + 7 * q^11 + 7 * q^13 + 4 * q^15 + 3 * q^17 + 13 * q^19 + 4 * q^21 + 11 * q^23 + 7 * q^25 + 13 * q^27 + 3 * q^29 + 12 * q^31 - 11 * q^33 + 7 * q^35 - 4 * q^37 + 4 * q^39 + 6 * q^41 + 10 * q^43 + 11 * q^45 + 15 * q^47 + 7 * q^49 + 8 * q^53 + 7 * q^55 - 18 * q^57 + 13 * q^59 - q^61 + 11 * q^63 + 7 * q^65 + 8 * q^67 - 9 * q^69 + 20 * q^71 - 17 * q^73 + 4 * q^75 + 7 * q^77 + 6 * q^79 + 39 * q^81 + 30 * q^83 + 3 * q^85 + 21 * q^87 + 5 * q^89 + 7 * q^91 - 24 * q^93 + 13 * q^95 - 9 * q^97 + 12 * q^99

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