LMFDB - Embedded newform 7616.2.a.cc.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7616,2,Mod(1,7616)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7616, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("7616.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7616 = 2^{6} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7616.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:	$60.8140661794$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.109859312.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 7x^{4} + 15x^{3} + 13x^{2} - 9x - 2 $ x^6 - 3x^5 - 7x^4 + 15x^3 + 13x^2 - 9*x - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 3808)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.10086$ of defining polynomial
Character	$\chi$	$=$	7616.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.07243 q^{3} -3.18060 q^{5} -1.00000 q^{7} -1.84989 q^{9} +O(q^{10})$	$q+1.07243 q^{3} -3.18060 q^{5} -1.00000 q^{7} -1.84989 q^{9} -0.565788 q^{11} -6.01588 q^{13} -3.41098 q^{15} -1.00000 q^{17} +5.60956 q^{19} -1.07243 q^{21} -7.60956 q^{23} +5.11621 q^{25} -5.20118 q^{27} +8.65334 q^{29} -7.61740 q^{31} -0.606769 q^{33} +3.18060 q^{35} +0.737022 q^{37} -6.45162 q^{39} -6.04166 q^{41} -9.59679 q^{43} +5.88376 q^{45} +2.91237 q^{47} +1.00000 q^{49} -1.07243 q^{51} -7.85269 q^{53} +1.79954 q^{55} +6.01588 q^{57} +9.84464 q^{59} +11.6846 q^{61} +1.84989 q^{63} +19.1341 q^{65} -8.79143 q^{67} -8.16074 q^{69} -15.2894 q^{71} -5.75194 q^{73} +5.48679 q^{75} +0.565788 q^{77} +5.13920 q^{79} -0.0282405 q^{81} +3.50982 q^{83} +3.18060 q^{85} +9.28012 q^{87} -7.43833 q^{89} +6.01588 q^{91} -8.16914 q^{93} -17.8418 q^{95} +6.51390 q^{97} +1.04664 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{3} - 4 q^{5} - 6 q^{7} + 8 q^{9}+O(q^{10}) $ 6 * q + 4 * q^3 - 4 * q^5 - 6 * q^7 + 8 * q^9	$ 6 q + 4 q^{3} - 4 q^{5} - 6 q^{7} + 8 q^{9} + 8 q^{11} - 8 q^{13} - 6 q^{17} + 6 q^{19} - 4 q^{21} - 18 q^{23} + 12 q^{25} + 22 q^{27} + 8 q^{29} + 4 q^{31} - 6 q^{33} + 4 q^{35} + 8 q^{37} - 14 q^{39} + 16 q^{41} + 12 q^{43} - 4 q^{45} - 18 q^{47} + 6 q^{49} - 4 q^{51} + 2 q^{53} + 8 q^{55} + 8 q^{57} + 16 q^{59} - 6 q^{61} - 8 q^{63} - 22 q^{65} + 12 q^{67} - 16 q^{69} + 2 q^{71} + 8 q^{73} + 24 q^{75} - 8 q^{77} + 18 q^{79} + 18 q^{81} + 28 q^{83} + 4 q^{85} + 2 q^{87} - 2 q^{89} + 8 q^{91} + 32 q^{93} - 26 q^{95} - 18 q^{97} + 28 q^{99}+O(q^{100}) $ 6 * q + 4 * q^3 - 4 * q^5 - 6 * q^7 + 8 * q^9 + 8 * q^11 - 8 * q^13 - 6 * q^17 + 6 * q^19 - 4 * q^21 - 18 * q^23 + 12 * q^25 + 22 * q^27 + 8 * q^29 + 4 * q^31 - 6 * q^33 + 4 * q^35 + 8 * q^37 - 14 * q^39 + 16 * q^41 + 12 * q^43 - 4 * q^45 - 18 * q^47 + 6 * q^49 - 4 * q^51 + 2 * q^53 + 8 * q^55 + 8 * q^57 + 16 * q^59 - 6 * q^61 - 8 * q^63 - 22 * q^65 + 12 * q^67 - 16 * q^69 + 2 * q^71 + 8 * q^73 + 24 * q^75 - 8 * q^77 + 18 * q^79 + 18 * q^81 + 28 * q^83 + 4 * q^85 + 2 * q^87 - 2 * q^89 + 8 * q^91 + 32 * q^93 - 26 * q^95 - 18 * q^97 + 28 * 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
$3$	1.07243	0.619169	0.309584	−	0.950872i	$-0.399810\pi$
$3$	1.07243	0.619169	0.309584	+	0.950872i	$0.399810\pi$
$4$	0	0
$5$	−3.18060	−1.42241	−0.711204	−	0.702986i	$-0.751850\pi$
$5$	−3.18060	−1.42241	−0.711204	+	0.702986i	$0.751850\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−1.84989	−0.616630
$10$	0	0
$11$	−0.565788	−0.170591	−0.0852957	−	0.996356i	$-0.527183\pi$
$11$	−0.565788	−0.170591	−0.0852957	+	0.996356i	$0.527183\pi$
$12$	0	0
$13$	−6.01588	−1.66850	−0.834252	−	0.551383i	$-0.814100\pi$
$13$	−6.01588	−1.66850	−0.834252	+	0.551383i	$0.814100\pi$
$14$	0	0
$15$	−3.41098	−0.880710
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	5.60956	1.28692	0.643461	−	0.765479i	$-0.277498\pi$
$19$	5.60956	1.28692	0.643461	+	0.765479i	$0.277498\pi$
$20$	0	0
$21$	−1.07243	−0.234024
$22$	0	0
$23$	−7.60956	−1.58670	−0.793352	−	0.608763i	$-0.791666\pi$
$23$	−7.60956	−1.58670	−0.793352	+	0.608763i	$0.791666\pi$
$24$	0	0
$25$	5.11621	1.02324
$26$	0	0
$27$	−5.20118	−1.00097
$28$	0	0
$29$	8.65334	1.60689	0.803443	−	0.595382i	$-0.202999\pi$
$29$	8.65334	1.60689	0.803443	+	0.595382i	$0.202999\pi$
$30$	0	0
$31$	−7.61740	−1.36813	−0.684063	−	0.729423i	$-0.739789\pi$
$31$	−7.61740	−1.36813	−0.684063	+	0.729423i	$0.739789\pi$
$32$	0	0
$33$	−0.606769	−0.105625
$34$	0	0
$35$	3.18060	0.537619
$36$	0	0
$37$	0.737022	0.121166	0.0605828	−	0.998163i	$-0.480704\pi$
$37$	0.737022	0.121166	0.0605828	+	0.998163i	$0.480704\pi$
$38$	0	0
$39$	−6.45162	−1.03309
$40$	0	0
$41$	−6.04166	−0.943549	−0.471775	−	0.881719i	$-0.656386\pi$
$41$	−6.04166	−0.943549	−0.471775	+	0.881719i	$0.656386\pi$
$42$	0	0
$43$	−9.59679	−1.46350	−0.731748	−	0.681575i	$-0.761295\pi$
$43$	−9.59679	−1.46350	−0.731748	+	0.681575i	$0.761295\pi$
$44$	0	0
$45$	5.88376	0.877099
$46$	0	0
$47$	2.91237	0.424813	0.212407	−	0.977181i	$-0.431870\pi$
$47$	2.91237	0.424813	0.212407	+	0.977181i	$0.431870\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.07243	−0.150171
$52$	0	0
$53$	−7.85269	−1.07865	−0.539325	−	0.842098i	$-0.681320\pi$
$53$	−7.85269	−1.07865	−0.539325	+	0.842098i	$0.681320\pi$
$54$	0	0
$55$	1.79954	0.242650
$56$	0	0
$57$	6.01588	0.796822
$58$	0	0
$59$	9.84464	1.28166	0.640832	−	0.767681i	$-0.278590\pi$
$59$	9.84464	1.28166	0.640832	+	0.767681i	$0.278590\pi$
$60$	0	0
$61$	11.6846	1.49606	0.748029	−	0.663666i	$-0.231000\pi$
$61$	11.6846	1.49606	0.748029	+	0.663666i	$0.231000\pi$
$62$	0	0
$63$	1.84989	0.233064
$64$	0	0
$65$	19.1341	2.37329
$66$	0	0
$67$	−8.79143	−1.07404	−0.537022	−	0.843568i	$-0.680451\pi$
$67$	−8.79143	−1.07404	−0.537022	+	0.843568i	$0.680451\pi$
$68$	0	0
$69$	−8.16074	−0.982438
$70$	0	0
$71$	−15.2894	−1.81452	−0.907262	−	0.420566i	$-0.861831\pi$
$71$	−15.2894	−1.81452	−0.907262	+	0.420566i	$0.861831\pi$
$72$	0	0
$73$	−5.75194	−0.673213	−0.336607	−	0.941645i	$-0.609279\pi$
$73$	−5.75194	−0.673213	−0.336607	+	0.941645i	$0.609279\pi$
$74$	0	0
$75$	5.48679	0.633560
$76$	0	0
$77$	0.565788	0.0644775
$78$	0	0
$79$	5.13920	0.578205	0.289103	−	0.957298i	$-0.406643\pi$
$79$	5.13920	0.578205	0.289103	+	0.957298i	$0.406643\pi$
$80$	0	0
$81$	−0.0282405	−0.00313783
$82$	0	0
$83$	3.50982	0.385252	0.192626	−	0.981272i	$-0.438300\pi$
$83$	3.50982	0.385252	0.192626	+	0.981272i	$0.438300\pi$
$84$	0	0
$85$	3.18060	0.344984
$86$	0	0
$87$	9.28012	0.994933
$88$	0	0
$89$	−7.43833	−0.788461	−0.394231	−	0.919011i	$-0.628989\pi$
$89$	−7.43833	−0.788461	−0.394231	+	0.919011i	$0.628989\pi$
$90$	0	0
$91$	6.01588	0.630635
$92$	0	0
$93$	−8.16914	−0.847101
$94$	0	0
$95$	−17.8418	−1.83053
$96$	0	0
$97$	6.51390	0.661386	0.330693	−	0.943738i	$-0.392718\pi$
$97$	6.51390	0.661386	0.330693	+	0.943738i	$0.392718\pi$
$98$	0	0
$99$	1.04664	0.105192
$100$	0	0
$101$	−4.43574	−0.441373	−0.220686	−	0.975345i	$-0.570830\pi$
$101$	−4.43574	−0.441373	−0.220686	+	0.975345i	$0.570830\pi$
$102$	0	0
$103$	−2.38757	−0.235254	−0.117627	−	0.993058i	$-0.537529\pi$
$103$	−2.38757	−0.235254	−0.117627	+	0.993058i	$0.537529\pi$
$104$	0	0
$105$	3.41098	0.332877
$106$	0	0
$107$	16.9307	1.63675	0.818375	−	0.574685i	$-0.194875\pi$
$107$	16.9307	1.63675	0.818375	+	0.574685i	$0.194875\pi$
$108$	0	0
$109$	−2.76993	−0.265311	−0.132655	−	0.991162i	$-0.542350\pi$
$109$	−2.76993	−0.265311	−0.132655	+	0.991162i	$0.542350\pi$
$110$	0	0
$111$	0.790406	0.0750220
$112$	0	0
$113$	17.6149	1.65708	0.828538	−	0.559934i	$-0.189173\pi$
$113$	17.6149	1.65708	0.828538	+	0.559934i	$0.189173\pi$
$114$	0	0
$115$	24.2030	2.25694
$116$	0	0
$117$	11.1287	1.02885
$118$	0	0
$119$	1.00000	0.0916698
$120$	0	0
$121$	−10.6799	−0.970899
$122$	0	0
$123$	−6.47927	−0.584217
$124$	0	0
$125$	−0.369615	−0.0330594
$126$	0	0
$127$	7.65985	0.679702	0.339851	−	0.940479i	$-0.389623\pi$
$127$	7.65985	0.679702	0.339851	+	0.940479i	$0.389623\pi$
$128$	0	0
$129$	−10.2919	−0.906151
$130$	0	0
$131$	−17.6689	−1.54374	−0.771871	−	0.635779i	$-0.780679\pi$
$131$	−17.6689	−1.54374	−0.771871	+	0.635779i	$0.780679\pi$
$132$	0	0
$133$	−5.60956	−0.486411
$134$	0	0
$135$	16.5429	1.42378
$136$	0	0
$137$	0.948655	0.0810490	0.0405245	−	0.999179i	$-0.487097\pi$
$137$	0.948655	0.0810490	0.0405245	+	0.999179i	$0.487097\pi$
$138$	0	0
$139$	7.82961	0.664099	0.332050	−	0.943262i	$-0.392260\pi$
$139$	7.82961	0.664099	0.332050	+	0.943262i	$0.392260\pi$
$140$	0	0
$141$	3.12332	0.263031
$142$	0	0
$143$	3.40371	0.284632
$144$	0	0
$145$	−27.5228	−2.28564
$146$	0	0
$147$	1.07243	0.0884527
$148$	0	0
$149$	4.30332	0.352542	0.176271	−	0.984342i	$-0.443597\pi$
$149$	4.30332	0.352542	0.176271	+	0.984342i	$0.443597\pi$
$150$	0	0
$151$	17.2449	1.40337	0.701686	−	0.712487i	$-0.252431\pi$
$151$	17.2449	1.40337	0.701686	+	0.712487i	$0.252431\pi$
$152$	0	0
$153$	1.84989	0.149555
$154$	0	0
$155$	24.2279	1.94603
$156$	0	0
$157$	−2.72069	−0.217135	−0.108567	−	0.994089i	$-0.534626\pi$
$157$	−2.72069	−0.217135	−0.108567	+	0.994089i	$0.534626\pi$
$158$	0	0
$159$	−8.42147	−0.667866
$160$	0	0
$161$	7.60956	0.599718
$162$	0	0
$163$	10.4756	0.820510	0.410255	−	0.911971i	$-0.365440\pi$
$163$	10.4756	0.820510	0.410255	+	0.911971i	$0.365440\pi$
$164$	0	0
$165$	1.92989	0.150242
$166$	0	0
$167$	−0.251538	−0.0194646	−0.00973230	−	0.999953i	$-0.503098\pi$
$167$	−0.251538	−0.0194646	−0.00973230	+	0.999953i	$0.503098\pi$
$168$	0	0
$169$	23.1908	1.78391
$170$	0	0
$171$	−10.3771	−0.793555
$172$	0	0
$173$	−7.41094	−0.563443	−0.281722	−	0.959496i	$-0.590906\pi$
$173$	−7.41094	−0.563443	−0.281722	+	0.959496i	$0.590906\pi$
$174$	0	0
$175$	−5.11621	−0.386749
$176$	0	0
$177$	10.5577	0.793566
$178$	0	0
$179$	18.2380	1.36317	0.681585	−	0.731739i	$-0.261291\pi$
$179$	18.2380	1.36317	0.681585	+	0.731739i	$0.261291\pi$
$180$	0	0
$181$	4.71719	0.350626	0.175313	−	0.984513i	$-0.443906\pi$
$181$	4.71719	0.350626	0.175313	+	0.984513i	$0.443906\pi$
$182$	0	0
$183$	12.5309	0.926313
$184$	0	0
$185$	−2.34417	−0.172347
$186$	0	0
$187$	0.565788	0.0413745
$188$	0	0
$189$	5.20118	0.378330
$190$	0	0
$191$	4.31459	0.312193	0.156096	−	0.987742i	$-0.450109\pi$
$191$	4.31459	0.312193	0.156096	+	0.987742i	$0.450109\pi$
$192$	0	0
$193$	−7.64419	−0.550241	−0.275120	−	0.961410i	$-0.588718\pi$
$193$	−7.64419	−0.550241	−0.275120	+	0.961410i	$0.588718\pi$
$194$	0	0
$195$	20.5200	1.46947
$196$	0	0
$197$	20.9446	1.49224	0.746119	−	0.665812i	$-0.231915\pi$
$197$	20.9446	1.49224	0.746119	+	0.665812i	$0.231915\pi$
$198$	0	0
$199$	0.0936490	0.00663860	0.00331930	−	0.999994i	$-0.498943\pi$
$199$	0.0936490	0.00663860	0.00331930	+	0.999994i	$0.498943\pi$
$200$	0	0
$201$	−9.42821	−0.665015
$202$	0	0
$203$	−8.65334	−0.607346
$204$	0	0
$205$	19.2161	1.34211
$206$	0	0
$207$	14.0769	0.978409
$208$	0	0
$209$	−3.17382	−0.219538
$210$	0	0
$211$	26.4236	1.81908	0.909539	−	0.415618i	$-0.136435\pi$
$211$	26.4236	1.81908	0.909539	+	0.415618i	$0.136435\pi$
$212$	0	0
$213$	−16.3969	−1.12350
$214$	0	0
$215$	30.5235	2.08169
$216$	0	0
$217$	7.61740	0.517103
$218$	0	0
$219$	−6.16856	−0.416833
$220$	0	0
$221$	6.01588	0.404672
$222$	0	0
$223$	−6.69432	−0.448285	−0.224143	−	0.974556i	$-0.571958\pi$
$223$	−6.69432	−0.448285	−0.224143	+	0.974556i	$0.571958\pi$
$224$	0	0
$225$	−9.46442	−0.630961
$226$	0	0
$227$	−10.1542	−0.673955	−0.336978	−	0.941513i	$-0.609405\pi$
$227$	−10.1542	−0.673955	−0.336978	+	0.941513i	$0.609405\pi$
$228$	0	0
$229$	24.2389	1.60175	0.800875	−	0.598831i	$-0.204368\pi$
$229$	24.2389	1.60175	0.800875	+	0.598831i	$0.204368\pi$
$230$	0	0
$231$	0.606769	0.0399225
$232$	0	0
$233$	−19.4343	−1.27318	−0.636592	−	0.771201i	$-0.719656\pi$
$233$	−19.4343	−1.27318	−0.636592	+	0.771201i	$0.719656\pi$
$234$	0	0
$235$	−9.26309	−0.604258
$236$	0	0
$237$	5.51144	0.358007
$238$	0	0
$239$	−7.98354	−0.516412	−0.258206	−	0.966090i	$-0.583131\pi$
$239$	−7.98354	−0.516412	−0.258206	+	0.966090i	$0.583131\pi$
$240$	0	0
$241$	22.8402	1.47127	0.735634	−	0.677380i	$-0.236885\pi$
$241$	22.8402	1.47127	0.735634	+	0.677380i	$0.236885\pi$
$242$	0	0
$243$	15.5732	0.999024
$244$	0	0
$245$	−3.18060	−0.203201
$246$	0	0
$247$	−33.7465	−2.14724
$248$	0	0
$249$	3.76404	0.238536
$250$	0	0
$251$	−21.7023	−1.36984	−0.684918	−	0.728620i	$-0.740162\pi$
$251$	−21.7023	−1.36984	−0.684918	+	0.728620i	$0.740162\pi$
$252$	0	0
$253$	4.30540	0.270678
$254$	0	0
$255$	3.41098	0.213604
$256$	0	0
$257$	−23.3082	−1.45392	−0.726962	−	0.686678i	$-0.759068\pi$
$257$	−23.3082	−1.45392	−0.726962	+	0.686678i	$0.759068\pi$
$258$	0	0
$259$	−0.737022	−0.0457963
$260$	0	0
$261$	−16.0077	−0.990853
$262$	0	0
$263$	−8.21633	−0.506641	−0.253320	−	0.967382i	$-0.581523\pi$
$263$	−8.21633	−0.506641	−0.253320	+	0.967382i	$0.581523\pi$
$264$	0	0
$265$	24.9762	1.53428
$266$	0	0
$267$	−7.97710	−0.488191
$268$	0	0
$269$	−8.59407	−0.523990	−0.261995	−	0.965069i	$-0.584380\pi$
$269$	−8.59407	−0.523990	−0.261995	+	0.965069i	$0.584380\pi$
$270$	0	0
$271$	0.00265971	0.000161566 0	8.07829e−5	−	1.00000i	$-0.499974\pi$
$271$	0.00265971	0.000161566 0	8.07829e−5		1.00000i	$0.499974\pi$
$272$	0	0
$273$	6.45162	0.390470
$274$	0	0
$275$	−2.89469	−0.174556
$276$	0	0
$277$	16.3080	0.979854	0.489927	−	0.871763i	$-0.337023\pi$
$277$	16.3080	0.979854	0.489927	+	0.871763i	$0.337023\pi$
$278$	0	0
$279$	14.0913	0.843627
$280$	0	0
$281$	16.0529	0.957638	0.478819	−	0.877914i	$-0.341065\pi$
$281$	16.0529	0.957638	0.478819	+	0.877914i	$0.341065\pi$
$282$	0	0
$283$	−20.8537	−1.23962	−0.619811	−	0.784751i	$-0.712791\pi$
$283$	−20.8537	−1.23962	−0.619811	+	0.784751i	$0.712791\pi$
$284$	0	0
$285$	−19.1341	−1.13341
$286$	0	0
$287$	6.04166	0.356628
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	6.98571	0.409510
$292$	0	0
$293$	5.48705	0.320557	0.160279	−	0.987072i	$-0.448761\pi$
$293$	5.48705	0.320557	0.160279	+	0.987072i	$0.448761\pi$
$294$	0	0
$295$	−31.3119	−1.82305
$296$	0	0
$297$	2.94276	0.170756
$298$	0	0
$299$	45.7782	2.64742
$300$	0	0
$301$	9.59679	0.553150
$302$	0	0
$303$	−4.75703	−0.273284
$304$	0	0
$305$	−37.1640	−2.12800
$306$	0	0
$307$	9.38674	0.535730	0.267865	−	0.963457i	$-0.413682\pi$
$307$	9.38674	0.535730	0.267865	+	0.963457i	$0.413682\pi$
$308$	0	0
$309$	−2.56050	−0.145662
$310$	0	0
$311$	21.2513	1.20505	0.602525	−	0.798100i	$-0.294161\pi$
$311$	21.2513	1.20505	0.602525	+	0.798100i	$0.294161\pi$
$312$	0	0
$313$	−14.3710	−0.812295	−0.406147	−	0.913808i	$-0.633128\pi$
$313$	−14.3710	−0.812295	−0.406147	+	0.913808i	$0.633128\pi$
$314$	0	0
$315$	−5.88376	−0.331512
$316$	0	0
$317$	−24.3225	−1.36609	−0.683045	−	0.730377i	$-0.739345\pi$
$317$	−24.3225	−1.36609	−0.683045	+	0.730377i	$0.739345\pi$
$318$	0	0
$319$	−4.89595	−0.274121
$320$	0	0
$321$	18.1570	1.01342
$322$	0	0
$323$	−5.60956	−0.312125
$324$	0	0
$325$	−30.7785	−1.70728
$326$	0	0
$327$	−2.97056	−0.164272
$328$	0	0
$329$	−2.91237	−0.160564
$330$	0	0
$331$	17.3559	0.953969	0.476984	−	0.878912i	$-0.341730\pi$
$331$	17.3559	0.953969	0.476984	+	0.878912i	$0.341730\pi$
$332$	0	0
$333$	−1.36341	−0.0747143
$334$	0	0
$335$	27.9620	1.52773
$336$	0	0
$337$	−19.1221	−1.04165	−0.520824	−	0.853664i	$-0.674375\pi$
$337$	−19.1221	−1.04165	−0.520824	+	0.853664i	$0.674375\pi$
$338$	0	0
$339$	18.8908	1.02601
$340$	0	0
$341$	4.30983	0.233390
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	25.9560	1.39743
$346$	0	0
$347$	−16.5618	−0.889086	−0.444543	−	0.895758i	$-0.646634\pi$
$347$	−16.5618	−0.889086	−0.444543	+	0.895758i	$0.646634\pi$
$348$	0	0
$349$	−15.5311	−0.831359	−0.415680	−	0.909511i	$-0.636456\pi$
$349$	−15.5311	−0.831359	−0.415680	+	0.909511i	$0.636456\pi$
$350$	0	0
$351$	31.2896	1.67012
$352$	0	0
$353$	14.4021	0.766546	0.383273	−	0.923635i	$-0.374797\pi$
$353$	14.4021	0.766546	0.383273	+	0.923635i	$0.374797\pi$
$354$	0	0
$355$	48.6296	2.58099
$356$	0	0
$357$	1.07243	0.0567591
$358$	0	0
$359$	−1.13178	−0.0597333	−0.0298667	−	0.999554i	$-0.509508\pi$
$359$	−1.13178	−0.0597333	−0.0298667	+	0.999554i	$0.509508\pi$
$360$	0	0
$361$	12.4672	0.656169
$362$	0	0
$363$	−11.4535	−0.601150
$364$	0	0
$365$	18.2946	0.957583
$366$	0	0
$367$	15.9100	0.830496	0.415248	−	0.909708i	$-0.363695\pi$
$367$	15.9100	0.830496	0.415248	+	0.909708i	$0.363695\pi$
$368$	0	0
$369$	11.1764	0.581821
$370$	0	0
$371$	7.85269	0.407691
$372$	0	0
$373$	−13.0538	−0.675900	−0.337950	−	0.941164i	$-0.609734\pi$
$373$	−13.0538	−0.675900	−0.337950	+	0.941164i	$0.609734\pi$
$374$	0	0
$375$	−0.396387	−0.0204693
$376$	0	0
$377$	−52.0574	−2.68109
$378$	0	0
$379$	30.4198	1.56256	0.781279	−	0.624182i	$-0.214568\pi$
$379$	30.4198	1.56256	0.781279	+	0.624182i	$0.214568\pi$
$380$	0	0
$381$	8.21467	0.420850
$382$	0	0
$383$	−35.4221	−1.80998	−0.904992	−	0.425428i	$-0.860124\pi$
$383$	−35.4221	−1.80998	−0.904992	+	0.425428i	$0.860124\pi$
$384$	0	0
$385$	−1.79954	−0.0917132
$386$	0	0
$387$	17.7530	0.902435
$388$	0	0
$389$	12.7473	0.646312	0.323156	−	0.946346i	$-0.395256\pi$
$389$	12.7473	0.646312	0.323156	+	0.946346i	$0.395256\pi$
$390$	0	0
$391$	7.60956	0.384832
$392$	0	0
$393$	−18.9487	−0.955837
$394$	0	0
$395$	−16.3457	−0.822443
$396$	0	0
$397$	33.9721	1.70501	0.852505	−	0.522719i	$-0.175082\pi$
$397$	33.9721	1.70501	0.852505	+	0.522719i	$0.175082\pi$
$398$	0	0
$399$	−6.01588	−0.301171
$400$	0	0
$401$	16.2997	0.813967	0.406984	−	0.913435i	$-0.366581\pi$
$401$	16.2997	0.813967	0.406984	+	0.913435i	$0.366581\pi$
$402$	0	0
$403$	45.8253	2.28272
$404$	0	0
$405$	0.0898217	0.00446328
$406$	0	0
$407$	−0.416998	−0.0206698
$408$	0	0
$409$	−18.2156	−0.900703	−0.450351	−	0.892851i	$-0.648701\pi$
$409$	−18.2156	−0.900703	−0.450351	+	0.892851i	$0.648701\pi$
$410$	0	0
$411$	1.01737	0.0501830
$412$	0	0
$413$	−9.84464	−0.484423
$414$	0	0
$415$	−11.1633	−0.547986
$416$	0	0
$417$	8.39673	0.411190
$418$	0	0
$419$	21.7259	1.06138	0.530690	−	0.847566i	$-0.321933\pi$
$419$	21.7259	1.06138	0.530690	+	0.847566i	$0.321933\pi$
$420$	0	0
$421$	−10.7609	−0.524453	−0.262226	−	0.965006i	$-0.584457\pi$
$421$	−10.7609	−0.524453	−0.262226	+	0.965006i	$0.584457\pi$
$422$	0	0
$423$	−5.38757	−0.261953
$424$	0	0
$425$	−5.11621	−0.248173
$426$	0	0
$427$	−11.6846	−0.565457
$428$	0	0
$429$	3.65025	0.176236
$430$	0	0
$431$	−6.71437	−0.323420	−0.161710	−	0.986838i	$-0.551701\pi$
$431$	−6.71437	−0.323420	−0.161710	+	0.986838i	$0.551701\pi$
$432$	0	0
$433$	−32.1887	−1.54689	−0.773446	−	0.633862i	$-0.781469\pi$
$433$	−32.1887	−1.54689	−0.773446	+	0.633862i	$0.781469\pi$
$434$	0	0
$435$	−29.5163	−1.41520
$436$	0	0
$437$	−42.6863	−2.04196
$438$	0	0
$439$	25.8839	1.23537	0.617685	−	0.786426i	$-0.288071\pi$
$439$	25.8839	1.23537	0.617685	+	0.786426i	$0.288071\pi$
$440$	0	0
$441$	−1.84989	−0.0880900
$442$	0	0
$443$	3.01136	0.143074	0.0715371	−	0.997438i	$-0.477210\pi$
$443$	3.01136	0.143074	0.0715371	+	0.997438i	$0.477210\pi$
$444$	0	0
$445$	23.6583	1.12151
$446$	0	0
$447$	4.61502	0.218283
$448$	0	0
$449$	−1.06964	−0.0504795	−0.0252397	−	0.999681i	$-0.508035\pi$
$449$	−1.06964	−0.0504795	−0.0252397	+	0.999681i	$0.508035\pi$
$450$	0	0
$451$	3.41830	0.160961
$452$	0	0
$453$	18.4940	0.868924
$454$	0	0
$455$	−19.1341	−0.897020
$456$	0	0
$457$	−37.6134	−1.75948	−0.879741	−	0.475453i	$-0.842284\pi$
$457$	−37.6134	−1.75948	−0.879741	+	0.475453i	$0.842284\pi$
$458$	0	0
$459$	5.20118	0.242770
$460$	0	0
$461$	−6.64704	−0.309583	−0.154792	−	0.987947i	$-0.549471\pi$
$461$	−6.64704	−0.309583	−0.154792	+	0.987947i	$0.549471\pi$
$462$	0	0
$463$	35.8158	1.66450	0.832250	−	0.554401i	$-0.187052\pi$
$463$	35.8158	1.66450	0.832250	+	0.554401i	$0.187052\pi$
$464$	0	0
$465$	25.9828	1.20492
$466$	0	0
$467$	25.8823	1.19769	0.598846	−	0.800865i	$-0.295626\pi$
$467$	25.8823	1.19769	0.598846	+	0.800865i	$0.295626\pi$
$468$	0	0
$469$	8.79143	0.405951
$470$	0	0
$471$	−2.91776	−0.134443
$472$	0	0
$473$	5.42974	0.249660
$474$	0	0
$475$	28.6997	1.31683
$476$	0	0
$477$	14.5266	0.665127
$478$	0	0
$479$	−28.2837	−1.29231	−0.646157	−	0.763204i	$-0.723625\pi$
$479$	−28.2837	−1.29231	−0.646157	+	0.763204i	$0.723625\pi$
$480$	0	0
$481$	−4.43383	−0.202165
$482$	0	0
$483$	8.16074	0.371327
$484$	0	0
$485$	−20.7181	−0.940760
$486$	0	0
$487$	35.2167	1.59582	0.797910	−	0.602777i	$-0.205939\pi$
$487$	35.2167	1.59582	0.797910	+	0.602777i	$0.205939\pi$
$488$	0	0
$489$	11.2343	0.508034
$490$	0	0
$491$	2.01739	0.0910433	0.0455217	−	0.998963i	$-0.485505\pi$
$491$	2.01739	0.0910433	0.0455217	+	0.998963i	$0.485505\pi$
$492$	0	0
$493$	−8.65334	−0.389727
$494$	0	0
$495$	−3.32896	−0.149625
$496$	0	0
$497$	15.2894	0.685826
$498$	0	0
$499$	−10.8472	−0.485587	−0.242794	−	0.970078i	$-0.578064\pi$
$499$	−10.8472	−0.485587	−0.242794	+	0.970078i	$0.578064\pi$
$500$	0	0
$501$	−0.269757	−0.0120519
$502$	0	0
$503$	−11.9677	−0.533613	−0.266806	−	0.963750i	$-0.585968\pi$
$503$	−11.9677	−0.533613	−0.266806	+	0.963750i	$0.585968\pi$
$504$	0	0
$505$	14.1083	0.627812
$506$	0	0
$507$	24.8705	1.10454
$508$	0	0
$509$	−32.4608	−1.43880	−0.719400	−	0.694596i	$-0.755583\pi$
$509$	−32.4608	−1.43880	−0.719400	+	0.694596i	$0.755583\pi$
$510$	0	0
$511$	5.75194	0.254451
$512$	0	0
$513$	−29.1763	−1.28817
$514$	0	0
$515$	7.59390	0.334627
$516$	0	0
$517$	−1.64779	−0.0724695
$518$	0	0
$519$	−7.94773	−0.348867
$520$	0	0
$521$	−16.8689	−0.739038	−0.369519	−	0.929223i	$-0.620478\pi$
$521$	−16.8689	−0.739038	−0.369519	+	0.929223i	$0.620478\pi$
$522$	0	0
$523$	−36.4122	−1.59220	−0.796098	−	0.605168i	$-0.793106\pi$
$523$	−36.4122	−1.59220	−0.796098	+	0.605168i	$0.793106\pi$
$524$	0	0
$525$	−5.48679	−0.239463
$526$	0	0
$527$	7.61740	0.331819
$528$	0	0
$529$	34.9055	1.51763
$530$	0	0
$531$	−18.2115	−0.790312
$532$	0	0
$533$	36.3459	1.57432
$534$	0	0
$535$	−53.8497	−2.32812
$536$	0	0
$537$	19.5590	0.844032
$538$	0	0
$539$	−0.565788	−0.0243702
$540$	0	0
$541$	−21.5010	−0.924400	−0.462200	−	0.886776i	$-0.652940\pi$
$541$	−21.5010	−0.924400	−0.462200	+	0.886776i	$0.652940\pi$
$542$	0	0
$543$	5.05886	0.217097
$544$	0	0
$545$	8.81002	0.377380
$546$	0	0
$547$	27.2708	1.16602	0.583008	−	0.812466i	$-0.301876\pi$
$547$	27.2708	1.16602	0.583008	+	0.812466i	$0.301876\pi$
$548$	0	0
$549$	−21.6152	−0.922514
$550$	0	0
$551$	48.5415	2.06794
$552$	0	0
$553$	−5.13920	−0.218541
$554$	0	0
$555$	−2.51396	−0.106712
$556$	0	0
$557$	10.3029	0.436546	0.218273	−	0.975888i	$-0.429958\pi$
$557$	10.3029	0.436546	0.218273	+	0.975888i	$0.429958\pi$
$558$	0	0
$559$	57.7331	2.44185
$560$	0	0
$561$	0.606769	0.0256178
$562$	0	0
$563$	−20.2550	−0.853648	−0.426824	−	0.904335i	$-0.640368\pi$
$563$	−20.2550	−0.853648	−0.426824	+	0.904335i	$0.640368\pi$
$564$	0	0
$565$	−56.0261	−2.35704
$566$	0	0
$567$	0.0282405	0.00118599
$568$	0	0
$569$	−16.2918	−0.682988	−0.341494	−	0.939884i	$-0.610933\pi$
$569$	−16.2918	−0.682988	−0.341494	+	0.939884i	$0.610933\pi$
$570$	0	0
$571$	−36.5486	−1.52951	−0.764756	−	0.644320i	$-0.777140\pi$
$571$	−36.5486	−1.52951	−0.764756	+	0.644320i	$0.777140\pi$
$572$	0	0
$573$	4.62710	0.193300
$574$	0	0
$575$	−38.9321	−1.62358
$576$	0	0
$577$	−16.0398	−0.667745	−0.333872	−	0.942618i	$-0.608355\pi$
$577$	−16.0398	−0.667745	−0.333872	+	0.942618i	$0.608355\pi$
$578$	0	0
$579$	−8.19787	−0.340692
$580$	0	0
$581$	−3.50982	−0.145612
$582$	0	0
$583$	4.44295	0.184008
$584$	0	0
$585$	−35.3960	−1.46344
$586$	0	0
$587$	−23.8841	−0.985802	−0.492901	−	0.870085i	$-0.664064\pi$
$587$	−23.8841	−0.985802	−0.492901	+	0.870085i	$0.664064\pi$
$588$	0	0
$589$	−42.7303	−1.76067
$590$	0	0
$591$	22.4616	0.923948
$592$	0	0
$593$	14.9443	0.613690	0.306845	−	0.951759i	$-0.400727\pi$
$593$	14.9443	0.613690	0.306845	+	0.951759i	$0.400727\pi$
$594$	0	0
$595$	−3.18060	−0.130392
$596$	0	0
$597$	0.100432	0.00411042
$598$	0	0
$599$	11.1611	0.456029	0.228014	−	0.973658i	$-0.426777\pi$
$599$	11.1611	0.456029	0.228014	+	0.973658i	$0.426777\pi$
$600$	0	0
$601$	23.9318	0.976198	0.488099	−	0.872788i	$-0.337691\pi$
$601$	23.9318	0.976198	0.488099	+	0.872788i	$0.337691\pi$
$602$	0	0
$603$	16.2632	0.662288
$604$	0	0
$605$	33.9684	1.38101
$606$	0	0
$607$	10.9074	0.442717	0.221359	−	0.975192i	$-0.428951\pi$
$607$	10.9074	0.442717	0.221359	+	0.975192i	$0.428951\pi$
$608$	0	0
$609$	−9.28012	−0.376049
$610$	0	0
$611$	−17.5205	−0.708803
$612$	0	0
$613$	15.6985	0.634056	0.317028	−	0.948416i	$-0.397315\pi$
$613$	15.6985	0.634056	0.317028	+	0.948416i	$0.397315\pi$
$614$	0	0
$615$	20.6080	0.830994
$616$	0	0
$617$	32.8872	1.32399	0.661995	−	0.749508i	$-0.269710\pi$
$617$	32.8872	1.32399	0.661995	+	0.749508i	$0.269710\pi$
$618$	0	0
$619$	−19.8331	−0.797159	−0.398580	−	0.917134i	$-0.630497\pi$
$619$	−19.8331	−0.797159	−0.398580	+	0.917134i	$0.630497\pi$
$620$	0	0
$621$	39.5787	1.58824
$622$	0	0
$623$	7.43833	0.298010
$624$	0	0
$625$	−24.4054	−0.976218
$626$	0	0
$627$	−3.40371	−0.135931
$628$	0	0
$629$	−0.737022	−0.0293870
$630$	0	0
$631$	29.8673	1.18900	0.594499	−	0.804096i	$-0.297350\pi$
$631$	29.8673	1.18900	0.594499	+	0.804096i	$0.297350\pi$
$632$	0	0
$633$	28.3376	1.12632
$634$	0	0
$635$	−24.3629	−0.966813
$636$	0	0
$637$	−6.01588	−0.238358
$638$	0	0
$639$	28.2838	1.11889
$640$	0	0
$641$	−41.4059	−1.63543	−0.817717	−	0.575620i	$-0.804761\pi$
$641$	−41.4059	−1.63543	−0.817717	+	0.575620i	$0.804761\pi$
$642$	0	0
$643$	−43.7843	−1.72668	−0.863341	−	0.504620i	$-0.831633\pi$
$643$	−43.7843	−1.72668	−0.863341	+	0.504620i	$0.831633\pi$
$644$	0	0
$645$	32.7344	1.28892
$646$	0	0
$647$	0.708087	0.0278378	0.0139189	−	0.999903i	$-0.495569\pi$
$647$	0.708087	0.0278378	0.0139189	+	0.999903i	$0.495569\pi$
$648$	0	0
$649$	−5.56998	−0.218641
$650$	0	0
$651$	8.16914	0.320174
$652$	0	0
$653$	20.1332	0.787872	0.393936	−	0.919138i	$-0.371113\pi$
$653$	20.1332	0.787872	0.393936	+	0.919138i	$0.371113\pi$
$654$	0	0
$655$	56.1978	2.19583
$656$	0	0
$657$	10.6404	0.415123
$658$	0	0
$659$	−4.34634	−0.169310	−0.0846548	−	0.996410i	$-0.526979\pi$
$659$	−4.34634	−0.169310	−0.0846548	+	0.996410i	$0.526979\pi$
$660$	0	0
$661$	−9.79150	−0.380845	−0.190423	−	0.981702i	$-0.560986\pi$
$661$	−9.79150	−0.380845	−0.190423	+	0.981702i	$0.560986\pi$
$662$	0	0
$663$	6.45162	0.250560
$664$	0	0
$665$	17.8418	0.691874
$666$	0	0
$667$	−65.8482	−2.54965
$668$	0	0
$669$	−7.17921	−0.277564
$670$	0	0
$671$	−6.61099	−0.255215
$672$	0	0
$673$	15.4496	0.595538	0.297769	−	0.954638i	$-0.403758\pi$
$673$	15.4496	0.595538	0.297769	+	0.954638i	$0.403758\pi$
$674$	0	0
$675$	−26.6103	−1.02423
$676$	0	0
$677$	6.37751	0.245108	0.122554	−	0.992462i	$-0.460892\pi$
$677$	6.37751	0.245108	0.122554	+	0.992462i	$0.460892\pi$
$678$	0	0
$679$	−6.51390	−0.249980
$680$	0	0
$681$	−10.8896	−0.417292
$682$	0	0
$683$	−33.4023	−1.27810	−0.639051	−	0.769164i	$-0.720673\pi$
$683$	−33.4023	−1.27810	−0.639051	+	0.769164i	$0.720673\pi$
$684$	0	0
$685$	−3.01729	−0.115285
$686$	0	0
$687$	25.9946	0.991754
$688$	0	0
$689$	47.2408	1.79973
$690$	0	0
$691$	6.17478	0.234900	0.117450	−	0.993079i	$-0.462528\pi$
$691$	6.17478	0.234900	0.117450	+	0.993079i	$0.462528\pi$
$692$	0	0
$693$	−1.04664	−0.0397587
$694$	0	0
$695$	−24.9029	−0.944619
$696$	0	0
$697$	6.04166	0.228844
$698$	0	0
$699$	−20.8420	−0.788316
$700$	0	0
$701$	33.2071	1.25422	0.627108	−	0.778932i	$-0.284238\pi$
$701$	33.2071	1.25422	0.627108	+	0.778932i	$0.284238\pi$
$702$	0	0
$703$	4.13437	0.155931
$704$	0	0
$705$	−9.93404	−0.374138
$706$	0	0
$707$	4.43574	0.166823
$708$	0	0
$709$	17.3087	0.650042	0.325021	−	0.945707i	$-0.394629\pi$
$709$	17.3087	0.650042	0.325021	+	0.945707i	$0.394629\pi$
$710$	0	0
$711$	−9.50695	−0.356539
$712$	0	0
$713$	57.9651	2.17081
$714$	0	0
$715$	−10.8258	−0.404863
$716$	0	0
$717$	−8.56180	−0.319746
$718$	0	0
$719$	22.6523	0.844789	0.422394	−	0.906412i	$-0.361190\pi$
$719$	22.6523	0.844789	0.422394	+	0.906412i	$0.361190\pi$
$720$	0	0
$721$	2.38757	0.0889177
$722$	0	0
$723$	24.4946	0.910963
$724$	0	0
$725$	44.2723	1.64423
$726$	0	0
$727$	20.1743	0.748223	0.374112	−	0.927384i	$-0.377948\pi$
$727$	20.1743	0.748223	0.374112	+	0.927384i	$0.377948\pi$
$728$	0	0
$729$	16.7860	0.621703
$730$	0	0
$731$	9.59679	0.354950
$732$	0	0
$733$	−32.0124	−1.18240	−0.591202	−	0.806523i	$-0.701347\pi$
$733$	−32.0124	−1.18240	−0.591202	+	0.806523i	$0.701347\pi$
$734$	0	0
$735$	−3.41098	−0.125816
$736$	0	0
$737$	4.97408	0.183223
$738$	0	0
$739$	33.2350	1.22257	0.611285	−	0.791410i	$-0.290653\pi$
$739$	33.2350	1.22257	0.611285	+	0.791410i	$0.290653\pi$
$740$	0	0
$741$	−36.1908	−1.32950
$742$	0	0
$743$	39.5480	1.45088	0.725438	−	0.688288i	$-0.241637\pi$
$743$	39.5480	1.45088	0.725438	+	0.688288i	$0.241637\pi$
$744$	0	0
$745$	−13.6871	−0.501458
$746$	0	0
$747$	−6.49277	−0.237558
$748$	0	0
$749$	−16.9307	−0.618633
$750$	0	0
$751$	−13.9905	−0.510521	−0.255261	−	0.966872i	$-0.582161\pi$
$751$	−13.9905	−0.510521	−0.255261	+	0.966872i	$0.582161\pi$
$752$	0	0
$753$	−23.2742	−0.848160
$754$	0	0
$755$	−54.8492	−1.99617
$756$	0	0
$757$	−43.9379	−1.59695	−0.798474	−	0.602029i	$-0.794359\pi$
$757$	−43.9379	−1.59695	−0.798474	+	0.602029i	$0.794359\pi$
$758$	0	0
$759$	4.61725	0.167595
$760$	0	0
$761$	−7.93480	−0.287636	−0.143818	−	0.989604i	$-0.545938\pi$
$761$	−7.93480	−0.287636	−0.143818	+	0.989604i	$0.545938\pi$
$762$	0	0
$763$	2.76993	0.100278
$764$	0	0
$765$	−5.88376	−0.212728
$766$	0	0
$767$	−59.2242	−2.13846
$768$	0	0
$769$	3.46389	0.124911	0.0624555	−	0.998048i	$-0.480107\pi$
$769$	3.46389	0.124911	0.0624555	+	0.998048i	$0.480107\pi$
$770$	0	0
$771$	−24.9964	−0.900225
$772$	0	0
$773$	−36.9208	−1.32795	−0.663974	−	0.747755i	$-0.731132\pi$
$773$	−36.9208	−1.32795	−0.663974	+	0.747755i	$0.731132\pi$
$774$	0	0
$775$	−38.9722	−1.39992
$776$	0	0
$777$	−0.790406	−0.0283556
$778$	0	0
$779$	−33.8911	−1.21428
$780$	0	0
$781$	8.65058	0.309542
$782$	0	0
$783$	−45.0076	−1.60844
$784$	0	0
$785$	8.65343	0.308854
$786$	0	0
$787$	27.6999	0.987396	0.493698	−	0.869633i	$-0.335645\pi$
$787$	27.6999	0.987396	0.493698	+	0.869633i	$0.335645\pi$
$788$	0	0
$789$	−8.81146	−0.313696
$790$	0	0
$791$	−17.6149	−0.626315
$792$	0	0
$793$	−70.2930	−2.49618
$794$	0	0
$795$	26.7853	0.949978
$796$	0	0
$797$	0.871164	0.0308582	0.0154291	−	0.999881i	$-0.495089\pi$
$797$	0.871164	0.0308582	0.0154291	+	0.999881i	$0.495089\pi$
$798$	0	0
$799$	−2.91237	−0.103032
$800$	0	0
$801$	13.7601	0.486189
$802$	0	0
$803$	3.25437	0.114844
$804$	0	0
$805$	−24.2030	−0.853043
$806$	0	0
$807$	−9.21655	−0.324438
$808$	0	0
$809$	−44.9196	−1.57929	−0.789645	−	0.613564i	$-0.789735\pi$
$809$	−44.9196	−1.57929	−0.789645	+	0.613564i	$0.789735\pi$
$810$	0	0
$811$	37.1509	1.30454	0.652272	−	0.757985i	$-0.273816\pi$
$811$	37.1509	1.30454	0.652272	+	0.757985i	$0.273816\pi$
$812$	0	0
$813$	0.00285236	0.000100037 0
$814$	0	0
$815$	−33.3186	−1.16710
$816$	0	0
$817$	−53.8338	−1.88341
$818$	0	0
$819$	−11.1287	−0.388869
$820$	0	0
$821$	−40.4231	−1.41078	−0.705388	−	0.708821i	$-0.749227\pi$
$821$	−40.4231	−1.41078	−0.705388	+	0.708821i	$0.749227\pi$
$822$	0	0
$823$	17.5691	0.612422	0.306211	−	0.951964i	$-0.400939\pi$
$823$	17.5691	0.612422	0.306211	+	0.951964i	$0.400939\pi$
$824$	0	0
$825$	−3.10436	−0.108080
$826$	0	0
$827$	−36.8149	−1.28018	−0.640089	−	0.768300i	$-0.721103\pi$
$827$	−36.8149	−1.28018	−0.640089	+	0.768300i	$0.721103\pi$
$828$	0	0
$829$	37.8449	1.31441	0.657204	−	0.753713i	$-0.271739\pi$
$829$	37.8449	1.31441	0.657204	+	0.753713i	$0.271739\pi$
$830$	0	0
$831$	17.4892	0.606695
$832$	0	0
$833$	−1.00000	−0.0346479
$834$	0	0
$835$	0.800041	0.0276866
$836$	0	0
$837$	39.6194	1.36945
$838$	0	0
$839$	10.3934	0.358819	0.179409	−	0.983774i	$-0.442581\pi$
$839$	10.3934	0.358819	0.179409	+	0.983774i	$0.442581\pi$
$840$	0	0
$841$	45.8803	1.58208
$842$	0	0
$843$	17.2157	0.592940
$844$	0	0
$845$	−73.7606	−2.53744
$846$	0	0
$847$	10.6799	0.366965
$848$	0	0
$849$	−22.3641	−0.767536
$850$	0	0
$851$	−5.60841	−0.192254
$852$	0	0
$853$	−34.9766	−1.19758	−0.598788	−	0.800908i	$-0.704351\pi$
$853$	−34.9766	−1.19758	−0.598788	+	0.800908i	$0.704351\pi$
$854$	0	0
$855$	33.0053	1.12876
$856$	0	0
$857$	−27.3992	−0.935938	−0.467969	−	0.883745i	$-0.655014\pi$
$857$	−27.3992	−0.935938	−0.467969	+	0.883745i	$0.655014\pi$
$858$	0	0
$859$	−19.7022	−0.672229	−0.336114	−	0.941821i	$-0.609113\pi$
$859$	−19.7022	−0.672229	−0.336114	+	0.941821i	$0.609113\pi$
$860$	0	0
$861$	6.47927	0.220813
$862$	0	0
$863$	40.5207	1.37934	0.689670	−	0.724123i	$-0.257755\pi$
$863$	40.5207	1.37934	0.689670	+	0.724123i	$0.257755\pi$
$864$	0	0
$865$	23.5712	0.801446
$866$	0	0
$867$	1.07243	0.0364217
$868$	0	0
$869$	−2.90770	−0.0986368
$870$	0	0
$871$	52.8882	1.79205
$872$	0	0
$873$	−12.0500	−0.407830
$874$	0	0
$875$	0.369615	0.0124953
$876$	0	0
$877$	−39.1528	−1.32210	−0.661048	−	0.750344i	$-0.729888\pi$
$877$	−39.1528	−1.32210	−0.661048	+	0.750344i	$0.729888\pi$
$878$	0	0
$879$	5.88449	0.198479
$880$	0	0
$881$	10.6853	0.359998	0.179999	−	0.983667i	$-0.442391\pi$
$881$	10.6853	0.359998	0.179999	+	0.983667i	$0.442391\pi$
$882$	0	0
$883$	23.0187	0.774640	0.387320	−	0.921945i	$-0.373401\pi$
$883$	23.0187	0.774640	0.387320	+	0.921945i	$0.373401\pi$
$884$	0	0
$885$	−33.5798	−1.12877
$886$	0	0
$887$	−27.5360	−0.924567	−0.462284	−	0.886732i	$-0.652970\pi$
$887$	−27.5360	−0.924567	−0.462284	+	0.886732i	$0.652970\pi$
$888$	0	0
$889$	−7.65985	−0.256903
$890$	0	0
$891$	0.0159781	0.000535288 0
$892$	0	0
$893$	16.3372	0.546702
$894$	0	0
$895$	−58.0077	−1.93898
$896$	0	0
$897$	49.0940	1.63920
$898$	0	0
$899$	−65.9160	−2.19842
$900$	0	0
$901$	7.85269	0.261611
$902$	0	0
$903$	10.2919	0.342493
$904$	0	0
$905$	−15.0035	−0.498732
$906$	0	0
$907$	−20.3518	−0.675769	−0.337885	−	0.941188i	$-0.609711\pi$
$907$	−20.3518	−0.675769	−0.337885	+	0.941188i	$0.609711\pi$
$908$	0	0
$909$	8.20563	0.272164
$910$	0	0
$911$	−57.8477	−1.91658	−0.958290	−	0.285798i	$-0.907741\pi$
$911$	−57.8477	−1.91658	−0.958290	+	0.285798i	$0.907741\pi$
$912$	0	0
$913$	−1.98581	−0.0657207
$914$	0	0
$915$	−39.8558	−1.31759
$916$	0	0
$917$	17.6689	0.583480
$918$	0	0
$919$	21.3389	0.703905	0.351952	−	0.936018i	$-0.385518\pi$
$919$	21.3389	0.703905	0.351952	+	0.936018i	$0.385518\pi$
$920$	0	0
$921$	10.0666	0.331707
$922$	0	0
$923$	91.9794	3.02754
$924$	0	0
$925$	3.77076	0.123982
$926$	0	0
$927$	4.41674	0.145065
$928$	0	0
$929$	8.02916	0.263428	0.131714	−	0.991288i	$-0.457952\pi$
$929$	8.02916	0.263428	0.131714	+	0.991288i	$0.457952\pi$
$930$	0	0
$931$	5.60956	0.183846
$932$	0	0
$933$	22.7906	0.746130
$934$	0	0
$935$	−1.79954	−0.0588514
$936$	0	0
$937$	59.4541	1.94228	0.971141	−	0.238507i	$-0.0766580\pi$
$937$	59.4541	1.94228	0.971141	+	0.238507i	$0.0766580\pi$
$938$	0	0
$939$	−15.4119	−0.502948
$940$	0	0
$941$	−8.40071	−0.273855	−0.136928	−	0.990581i	$-0.543723\pi$
$941$	−8.40071	−0.273855	−0.136928	+	0.990581i	$0.543723\pi$
$942$	0	0
$943$	45.9744	1.49713
$944$	0	0
$945$	−16.5429	−0.538139
$946$	0	0
$947$	30.3153	0.985115	0.492557	−	0.870280i	$-0.336062\pi$
$947$	30.3153	0.985115	0.492557	+	0.870280i	$0.336062\pi$
$948$	0	0
$949$	34.6029	1.12326
$950$	0	0
$951$	−26.0843	−0.845840
$952$	0	0
$953$	12.9838	0.420588	0.210294	−	0.977638i	$-0.432558\pi$
$953$	12.9838	0.420588	0.210294	+	0.977638i	$0.432558\pi$
$954$	0	0
$955$	−13.7230	−0.444065
$956$	0	0
$957$	−5.25058	−0.169727
$958$	0	0
$959$	−0.948655	−0.0306337
$960$	0	0
$961$	27.0248	0.871767
$962$	0	0
$963$	−31.3199	−1.00927
$964$	0	0
$965$	24.3131	0.782666
$966$	0	0
$967$	10.8013	0.347348	0.173674	−	0.984803i	$-0.444436\pi$
$967$	10.8013	0.347348	0.173674	+	0.984803i	$0.444436\pi$
$968$	0	0
$969$	−6.01588	−0.193258
$970$	0	0
$971$	−34.2280	−1.09843	−0.549215	−	0.835681i	$-0.685073\pi$
$971$	−34.2280	−1.09843	−0.549215	+	0.835681i	$0.685073\pi$
$972$	0	0
$973$	−7.82961	−0.251006
$974$	0	0
$975$	−33.0078	−1.05710
$976$	0	0
$977$	27.9190	0.893209	0.446604	−	0.894732i	$-0.352633\pi$
$977$	27.9190	0.893209	0.446604	+	0.894732i	$0.352633\pi$
$978$	0	0
$979$	4.20852	0.134505
$980$	0	0
$981$	5.12406	0.163599
$982$	0	0
$983$	31.5704	1.00694	0.503470	−	0.864013i	$-0.332056\pi$
$983$	31.5704	1.00694	0.503470	+	0.864013i	$0.332056\pi$
$984$	0	0
$985$	−66.6163	−2.12257
$986$	0	0
$987$	−3.12332	−0.0994165
$988$	0	0
$989$	73.0274	2.32214
$990$	0	0
$991$	−13.5078	−0.429088	−0.214544	−	0.976714i	$-0.568827\pi$
$991$	−13.5078	−0.429088	−0.214544	+	0.976714i	$0.568827\pi$
$992$	0	0
$993$	18.6131	0.590668
$994$	0	0
$995$	−0.297860	−0.00944279
$996$	0	0
$997$	13.5349	0.428654	0.214327	−	0.976762i	$-0.431244\pi$
$997$	13.5349	0.428654	0.214327	+	0.976762i	$0.431244\pi$
$998$	0	0
$999$	−3.83338	−0.121283

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7616.2.a.cc.1.4		6
4.3	odd	2		7616.2.a.bu.1.3		6
8.3	odd	2		3808.2.a.p.1.4	yes	6
8.5	even	2		3808.2.a.h.1.3	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3808.2.a.h.1.3	✓	6	8.5	even	2
3808.2.a.p.1.4	yes	6	8.3	odd	2
7616.2.a.bu.1.3		6	4.3	odd	2
7616.2.a.cc.1.4		6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7616 = 2^{6} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7616.a (trivial)

\(f(q)\)	\(=\)	\(q+1.07243 q^{3} -3.18060 q^{5} -1.00000 q^{7} -1.84989 q^{9} +O(q^{10})\)	\(q+1.07243 q^{3} -3.18060 q^{5} -1.00000 q^{7} -1.84989 q^{9} -0.565788 q^{11} -6.01588 q^{13} -3.41098 q^{15} -1.00000 q^{17} +5.60956 q^{19} -1.07243 q^{21} -7.60956 q^{23} +5.11621 q^{25} -5.20118 q^{27} +8.65334 q^{29} -7.61740 q^{31} -0.606769 q^{33} +3.18060 q^{35} +0.737022 q^{37} -6.45162 q^{39} -6.04166 q^{41} -9.59679 q^{43} +5.88376 q^{45} +2.91237 q^{47} +1.00000 q^{49} -1.07243 q^{51} -7.85269 q^{53} +1.79954 q^{55} +6.01588 q^{57} +9.84464 q^{59} +11.6846 q^{61} +1.84989 q^{63} +19.1341 q^{65} -8.79143 q^{67} -8.16074 q^{69} -15.2894 q^{71} -5.75194 q^{73} +5.48679 q^{75} +0.565788 q^{77} +5.13920 q^{79} -0.0282405 q^{81} +3.50982 q^{83} +3.18060 q^{85} +9.28012 q^{87} -7.43833 q^{89} +6.01588 q^{91} -8.16914 q^{93} -17.8418 q^{95} +6.51390 q^{97} +1.04664 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{3} - 4 q^{5} - 6 q^{7} + 8 q^{9}+O(q^{10}) \) 6 * q + 4 * q^3 - 4 * q^5 - 6 * q^7 + 8 * q^9	\( 6 q + 4 q^{3} - 4 q^{5} - 6 q^{7} + 8 q^{9} + 8 q^{11} - 8 q^{13} - 6 q^{17} + 6 q^{19} - 4 q^{21} - 18 q^{23} + 12 q^{25} + 22 q^{27} + 8 q^{29} + 4 q^{31} - 6 q^{33} + 4 q^{35} + 8 q^{37} - 14 q^{39} + 16 q^{41} + 12 q^{43} - 4 q^{45} - 18 q^{47} + 6 q^{49} - 4 q^{51} + 2 q^{53} + 8 q^{55} + 8 q^{57} + 16 q^{59} - 6 q^{61} - 8 q^{63} - 22 q^{65} + 12 q^{67} - 16 q^{69} + 2 q^{71} + 8 q^{73} + 24 q^{75} - 8 q^{77} + 18 q^{79} + 18 q^{81} + 28 q^{83} + 4 q^{85} + 2 q^{87} - 2 q^{89} + 8 q^{91} + 32 q^{93} - 26 q^{95} - 18 q^{97} + 28 q^{99}+O(q^{100}) \) 6 * q + 4 * q^3 - 4 * q^5 - 6 * q^7 + 8 * q^9 + 8 * q^11 - 8 * q^13 - 6 * q^17 + 6 * q^19 - 4 * q^21 - 18 * q^23 + 12 * q^25 + 22 * q^27 + 8 * q^29 + 4 * q^31 - 6 * q^33 + 4 * q^35 + 8 * q^37 - 14 * q^39 + 16 * q^41 + 12 * q^43 - 4 * q^45 - 18 * q^47 + 6 * q^49 - 4 * q^51 + 2 * q^53 + 8 * q^55 + 8 * q^57 + 16 * q^59 - 6 * q^61 - 8 * q^63 - 22 * q^65 + 12 * q^67 - 16 * q^69 + 2 * q^71 + 8 * q^73 + 24 * q^75 - 8 * q^77 + 18 * q^79 + 18 * q^81 + 28 * q^83 + 4 * q^85 + 2 * q^87 - 2 * q^89 + 8 * q^91 + 32 * q^93 - 26 * q^95 - 18 * q^97 + 28 * q^99

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