LMFDB - Embedded newform 1960.2.a.w.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1960,2,Mod(1,1960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1960, 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("1960.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1960 = 2^{3} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1960.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:	$15.6506787962$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1944.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 9x - 6 $ x^3 - 9*x - 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 280)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.28995$ of defining polynomial
Character	$\chi$	$=$	1960.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.28995 q^{3} +1.00000 q^{5} +7.82374 q^{9} +O(q^{10})$	$q-3.28995 q^{3} +1.00000 q^{5} +7.82374 q^{9} +5.82374 q^{11} +2.75615 q^{13} -3.28995 q^{15} +2.00000 q^{17} -0.756152 q^{19} -0.533794 q^{23} +1.00000 q^{25} -15.8698 q^{27} -0.823739 q^{29} -2.57989 q^{31} -19.1598 q^{33} +4.75615 q^{37} -9.06759 q^{39} +6.06759 q^{41} +0.710055 q^{43} +7.82374 q^{45} -12.8913 q^{47} -6.57989 q^{51} -8.40363 q^{53} +5.82374 q^{55} +2.48770 q^{57} +8.00000 q^{59} +9.40363 q^{61} +2.75615 q^{65} +11.8698 q^{67} +1.75615 q^{69} +3.51230 q^{73} -3.28995 q^{75} -9.51230 q^{79} +28.7397 q^{81} -6.71005 q^{83} +2.00000 q^{85} +2.71005 q^{87} +1.75615 q^{89} +8.48770 q^{93} -0.756152 q^{95} -2.00000 q^{97} +45.5634 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{5} + 9 q^{9}+O(q^{10}) $ 3 * q + 3 * q^5 + 9 * q^9	$ 3 q + 3 q^{5} + 9 q^{9} + 3 q^{11} + 3 q^{13} + 6 q^{17} + 3 q^{19} + 3 q^{23} + 3 q^{25} - 18 q^{27} + 12 q^{29} + 12 q^{31} - 18 q^{33} + 9 q^{37} - 18 q^{39} + 9 q^{41} + 12 q^{43} + 9 q^{45} - 15 q^{47} + 9 q^{53} + 3 q^{55} + 18 q^{57} + 24 q^{59} - 6 q^{61} + 3 q^{65} + 6 q^{67} - 18 q^{79} + 27 q^{81} - 30 q^{83} + 6 q^{85} + 18 q^{87} + 36 q^{93} + 3 q^{95} - 6 q^{97} + 63 q^{99}+O(q^{100}) $ 3 * q + 3 * q^5 + 9 * q^9 + 3 * q^11 + 3 * q^13 + 6 * q^17 + 3 * q^19 + 3 * q^23 + 3 * q^25 - 18 * q^27 + 12 * q^29 + 12 * q^31 - 18 * q^33 + 9 * q^37 - 18 * q^39 + 9 * q^41 + 12 * q^43 + 9 * q^45 - 15 * q^47 + 9 * q^53 + 3 * q^55 + 18 * q^57 + 24 * q^59 - 6 * q^61 + 3 * q^65 + 6 * q^67 - 18 * q^79 + 27 * q^81 - 30 * q^83 + 6 * q^85 + 18 * q^87 + 36 * q^93 + 3 * q^95 - 6 * q^97 + 63 * 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
$3$	−3.28995	−1.89945	−0.949725	−	0.313084i	$-0.898638\pi$
$3$	−3.28995	−1.89945	−0.949725	+	0.313084i	$0.898638\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	7.82374	2.60791
$10$	0	0
$11$	5.82374	1.75592	0.877962	−	0.478731i	$-0.158903\pi$
$11$	5.82374	1.75592	0.877962	+	0.478731i	$0.158903\pi$
$12$	0	0
$13$	2.75615	0.764419	0.382209	−	0.924076i	$-0.375163\pi$
$13$	2.75615	0.764419	0.382209	+	0.924076i	$0.375163\pi$
$14$	0	0
$15$	−3.28995	−0.849460
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	−0.756152	−0.173473	−0.0867365	−	0.996231i	$-0.527644\pi$
$19$	−0.756152	−0.173473	−0.0867365	+	0.996231i	$0.527644\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.533794	−0.111304	−0.0556518	−	0.998450i	$-0.517724\pi$
$23$	−0.533794	−0.111304	−0.0556518	+	0.998450i	$0.517724\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−15.8698	−3.05415
$28$	0	0
$29$	−0.823739	−0.152964	−0.0764822	−	0.997071i	$-0.524369\pi$
$29$	−0.823739	−0.152964	−0.0764822	+	0.997071i	$0.524369\pi$
$30$	0	0
$31$	−2.57989	−0.463362	−0.231681	−	0.972792i	$-0.574423\pi$
$31$	−2.57989	−0.463362	−0.231681	+	0.972792i	$0.574423\pi$
$32$	0	0
$33$	−19.1598	−3.33529
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.75615	0.781906	0.390953	−	0.920411i	$-0.372145\pi$
$37$	4.75615	0.781906	0.390953	+	0.920411i	$0.372145\pi$
$38$	0	0
$39$	−9.06759	−1.45198
$40$	0	0
$41$	6.06759	0.947598	0.473799	−	0.880633i	$-0.342882\pi$
$41$	6.06759	0.947598	0.473799	+	0.880633i	$0.342882\pi$
$42$	0	0
$43$	0.710055	0.108282	0.0541412	−	0.998533i	$-0.482758\pi$
$43$	0.710055	0.108282	0.0541412	+	0.998533i	$0.482758\pi$
$44$	0	0
$45$	7.82374	1.16629
$46$	0	0
$47$	−12.8913	−1.88039	−0.940197	−	0.340632i	$-0.889359\pi$
$47$	−12.8913	−1.88039	−0.940197	+	0.340632i	$0.889359\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−6.57989	−0.921369
$52$	0	0
$53$	−8.40363	−1.15433	−0.577164	−	0.816629i	$-0.695841\pi$
$53$	−8.40363	−1.15433	−0.577164	+	0.816629i	$0.695841\pi$
$54$	0	0
$55$	5.82374	0.785273
$56$	0	0
$57$	2.48770	0.329504
$58$	0	0
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	9.40363	1.20401	0.602006	−	0.798492i	$-0.294368\pi$
$61$	9.40363	1.20401	0.602006	+	0.798492i	$0.294368\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.75615	0.341859
$66$	0	0
$67$	11.8698	1.45013	0.725066	−	0.688680i	$-0.241809\pi$
$67$	11.8698	1.45013	0.725066	+	0.688680i	$0.241809\pi$
$68$	0	0
$69$	1.75615	0.211416
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	3.51230	0.411084	0.205542	−	0.978648i	$-0.434104\pi$
$73$	3.51230	0.411084	0.205542	+	0.978648i	$0.434104\pi$
$74$	0	0
$75$	−3.28995	−0.379890
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−9.51230	−1.07022	−0.535109	−	0.844783i	$-0.679729\pi$
$79$	−9.51230	−1.07022	−0.535109	+	0.844783i	$0.679729\pi$
$80$	0	0
$81$	28.7397	3.19330
$82$	0	0
$83$	−6.71005	−0.736524	−0.368262	−	0.929722i	$-0.620047\pi$
$83$	−6.71005	−0.736524	−0.368262	+	0.929722i	$0.620047\pi$
$84$	0	0
$85$	2.00000	0.216930
$86$	0	0
$87$	2.71005	0.290548
$88$	0	0
$89$	1.75615	0.186152	0.0930758	−	0.995659i	$-0.470330\pi$
$89$	1.75615	0.186152	0.0930758	+	0.995659i	$0.470330\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	8.48770	0.880133
$94$	0	0
$95$	−0.756152	−0.0775795
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	45.5634	4.57929
$100$	0	0
$101$	2.33604	0.232445	0.116222	−	0.993223i	$-0.462921\pi$
$101$	2.33604	0.232445	0.116222	+	0.993223i	$0.462921\pi$
$102$	0	0
$103$	−2.22236	−0.218975	−0.109488	−	0.993988i	$-0.534921\pi$
$103$	−2.22236	−0.218975	−0.109488	+	0.993988i	$0.534921\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	15.8698	1.53419	0.767097	−	0.641531i	$-0.221700\pi$
$107$	15.8698	1.53419	0.767097	+	0.641531i	$0.221700\pi$
$108$	0	0
$109$	−3.26845	−0.313061	−0.156531	−	0.987673i	$-0.550031\pi$
$109$	−3.26845	−0.313061	−0.156531	+	0.987673i	$0.550031\pi$
$110$	0	0
$111$	−15.6475	−1.48519
$112$	0	0
$113$	−13.1598	−1.23797	−0.618984	−	0.785404i	$-0.712455\pi$
$113$	−13.1598	−1.23797	−0.618984	+	0.785404i	$0.712455\pi$
$114$	0	0
$115$	−0.533794	−0.0497765
$116$	0	0
$117$	21.5634	1.99354
$118$	0	0
$119$	0	0
$120$	0	0
$121$	22.9159	2.08327
$122$	0	0
$123$	−19.9620	−1.79992
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	11.9159	1.05737	0.528684	−	0.848819i	$-0.322686\pi$
$127$	11.9159	1.05737	0.528684	+	0.848819i	$0.322686\pi$
$128$	0	0
$129$	−2.33604	−0.205677
$130$	0	0
$131$	21.5634	1.88400	0.942002	−	0.335608i	$-0.108942\pi$
$131$	21.5634	1.88400	0.942002	+	0.335608i	$0.108942\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−15.8698	−1.36586
$136$	0	0
$137$	4.00000	0.341743	0.170872	−	0.985293i	$-0.445342\pi$
$137$	4.00000	0.341743	0.170872	+	0.985293i	$0.445342\pi$
$138$	0	0
$139$	6.57989	0.558099	0.279049	−	0.960277i	$-0.409981\pi$
$139$	6.57989	0.558099	0.279049	+	0.960277i	$0.409981\pi$
$140$	0	0
$141$	42.4118	3.57171
$142$	0	0
$143$	16.0511	1.34226
$144$	0	0
$145$	−0.823739	−0.0684078
$146$	0	0
$147$	0	0
$148$	0	0
$149$	0.108674	0.00890294	0.00445147	−	0.999990i	$-0.498583\pi$
$149$	0.108674	0.00890294	0.00445147	+	0.999990i	$0.498583\pi$
$150$	0	0
$151$	−13.0676	−1.06343	−0.531713	−	0.846925i	$-0.678451\pi$
$151$	−13.0676	−1.06343	−0.531713	+	0.846925i	$0.678451\pi$
$152$	0	0
$153$	15.6475	1.26502
$154$	0	0
$155$	−2.57989	−0.207222
$156$	0	0
$157$	8.89133	0.709605	0.354803	−	0.934941i	$-0.384548\pi$
$157$	8.89133	0.709605	0.354803	+	0.934941i	$0.384548\pi$
$158$	0	0
$159$	27.6475	2.19259
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−11.5123	−0.901713	−0.450857	−	0.892596i	$-0.648881\pi$
$163$	−11.5123	−0.901713	−0.450857	+	0.892596i	$0.648881\pi$
$164$	0	0
$165$	−19.1598	−1.49159
$166$	0	0
$167$	1.46621	0.113458	0.0567292	−	0.998390i	$-0.481933\pi$
$167$	1.46621	0.113458	0.0567292	+	0.998390i	$0.481933\pi$
$168$	0	0
$169$	−5.40363	−0.415664
$170$	0	0
$171$	−5.91593	−0.452403
$172$	0	0
$173$	−11.2438	−0.854854	−0.427427	−	0.904050i	$-0.640580\pi$
$173$	−11.2438	−0.854854	−0.427427	+	0.904050i	$0.640580\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−26.3196	−1.97830
$178$	0	0
$179$	7.33604	0.548321	0.274161	−	0.961684i	$-0.411600\pi$
$179$	7.33604	0.548321	0.274161	+	0.961684i	$0.411600\pi$
$180$	0	0
$181$	−18.4712	−1.37295	−0.686477	−	0.727151i	$-0.740844\pi$
$181$	−18.4712	−1.37295	−0.686477	+	0.727151i	$0.740844\pi$
$182$	0	0
$183$	−30.9374	−2.28696
$184$	0	0
$185$	4.75615	0.349679
$186$	0	0
$187$	11.6475	0.851748
$188$	0	0
$189$	0	0
$190$	0	0
$191$	14.5799	1.05496	0.527482	−	0.849566i	$-0.323136\pi$
$191$	14.5799	1.05496	0.527482	+	0.849566i	$0.323136\pi$
$192$	0	0
$193$	−18.8073	−1.35378	−0.676888	−	0.736086i	$-0.736672\pi$
$193$	−18.8073	−1.35378	−0.676888	+	0.736086i	$0.736672\pi$
$194$	0	0
$195$	−9.06759	−0.649343
$196$	0	0
$197$	2.75615	0.196368	0.0981838	−	0.995168i	$-0.468697\pi$
$197$	2.75615	0.196368	0.0981838	+	0.995168i	$0.468697\pi$
$198$	0	0
$199$	15.0246	1.06507	0.532533	−	0.846409i	$-0.321240\pi$
$199$	15.0246	1.06507	0.532533	+	0.846409i	$0.321240\pi$
$200$	0	0
$201$	−39.0511	−2.75445
$202$	0	0
$203$	0	0
$204$	0	0
$205$	6.06759	0.423779
$206$	0	0
$207$	−4.17626	−0.290270
$208$	0	0
$209$	−4.40363	−0.304605
$210$	0	0
$211$	−21.9159	−1.50875	−0.754377	−	0.656441i	$-0.772061\pi$
$211$	−21.9159	−1.50875	−0.754377	+	0.656441i	$0.772061\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0.710055	0.0484253
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−11.5553	−0.780834
$220$	0	0
$221$	5.51230	0.370798
$222$	0	0
$223$	18.8073	1.25943	0.629714	−	0.776827i	$-0.283172\pi$
$223$	18.8073	1.25943	0.629714	+	0.776827i	$0.283172\pi$
$224$	0	0
$225$	7.82374	0.521583
$226$	0	0
$227$	16.6721	1.10657	0.553283	−	0.832994i	$-0.313375\pi$
$227$	16.6721	1.10657	0.553283	+	0.832994i	$0.313375\pi$
$228$	0	0
$229$	19.5123	1.28941	0.644705	−	0.764432i	$-0.276980\pi$
$229$	19.5123	1.28941	0.644705	+	0.764432i	$0.276980\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	0	0
$235$	−12.8913	−0.840937
$236$	0	0
$237$	31.2950	2.03283
$238$	0	0
$239$	−16.0922	−1.04092	−0.520459	−	0.853887i	$-0.674239\pi$
$239$	−16.0922	−1.04092	−0.520459	+	0.853887i	$0.674239\pi$
$240$	0	0
$241$	0.891326	0.0574153	0.0287077	−	0.999588i	$-0.490861\pi$
$241$	0.891326	0.0574153	0.0287077	+	0.999588i	$0.490861\pi$
$242$	0	0
$243$	−46.9424	−3.01136
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−2.08407	−0.132606
$248$	0	0
$249$	22.0757	1.39899
$250$	0	0
$251$	17.4712	1.10277	0.551387	−	0.834250i	$-0.314099\pi$
$251$	17.4712	1.10277	0.551387	+	0.834250i	$0.314099\pi$
$252$	0	0
$253$	−3.10867	−0.195441
$254$	0	0
$255$	−6.57989	−0.412049
$256$	0	0
$257$	−7.02461	−0.438183	−0.219091	−	0.975704i	$-0.570309\pi$
$257$	−7.02461	−0.438183	−0.219091	+	0.975704i	$0.570309\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−6.44472	−0.398918
$262$	0	0
$263$	17.7776	1.09622	0.548108	−	0.836407i	$-0.315348\pi$
$263$	17.7776	1.09622	0.548108	+	0.836407i	$0.315348\pi$
$264$	0	0
$265$	−8.40363	−0.516231
$266$	0	0
$267$	−5.77764	−0.353586
$268$	0	0
$269$	−9.40363	−0.573349	−0.286675	−	0.958028i	$-0.592550\pi$
$269$	−9.40363	−0.573349	−0.286675	+	0.958028i	$0.592550\pi$
$270$	0	0
$271$	7.02461	0.426714	0.213357	−	0.976974i	$-0.431560\pi$
$271$	7.02461	0.426714	0.213357	+	0.976974i	$0.431560\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	5.82374	0.351185
$276$	0	0
$277$	27.1598	1.63187	0.815937	−	0.578141i	$-0.196222\pi$
$277$	27.1598	1.63187	0.815937	+	0.578141i	$0.196222\pi$
$278$	0	0
$279$	−20.1844	−1.20841
$280$	0	0
$281$	−0.620977	−0.0370444	−0.0185222	−	0.999828i	$-0.505896\pi$
$281$	−0.620977	−0.0370444	−0.0185222	+	0.999828i	$0.505896\pi$
$282$	0	0
$283$	3.15978	0.187829	0.0939147	−	0.995580i	$-0.470062\pi$
$283$	3.15978	0.187829	0.0939147	+	0.995580i	$0.470062\pi$
$284$	0	0
$285$	2.48770	0.147358
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	6.57989	0.385720
$292$	0	0
$293$	11.2438	0.656873	0.328436	−	0.944526i	$-0.393478\pi$
$293$	11.2438	0.656873	0.328436	+	0.944526i	$0.393478\pi$
$294$	0	0
$295$	8.00000	0.465778
$296$	0	0
$297$	−92.4218	−5.36286
$298$	0	0
$299$	−1.47122	−0.0850826
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−7.68545	−0.441518
$304$	0	0
$305$	9.40363	0.538450
$306$	0	0
$307$	23.5173	1.34220	0.671102	−	0.741365i	$-0.265821\pi$
$307$	23.5173	1.34220	0.671102	+	0.741365i	$0.265821\pi$
$308$	0	0
$309$	7.31144	0.415933
$310$	0	0
$311$	−29.1598	−1.65350	−0.826750	−	0.562570i	$-0.809813\pi$
$311$	−29.1598	−1.65350	−0.826750	+	0.562570i	$0.809813\pi$
$312$	0	0
$313$	−24.1844	−1.36698	−0.683491	−	0.729959i	$-0.739539\pi$
$313$	−24.1844	−1.36698	−0.683491	+	0.729959i	$0.739539\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	12.1352	0.681579	0.340790	−	0.940140i	$-0.389306\pi$
$317$	12.1352	0.681579	0.340790	+	0.940140i	$0.389306\pi$
$318$	0	0
$319$	−4.79724	−0.268594
$320$	0	0
$321$	−52.2109	−2.91413
$322$	0	0
$323$	−1.51230	−0.0841468
$324$	0	0
$325$	2.75615	0.152884
$326$	0	0
$327$	10.7530	0.594644
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−1.82374	−0.100242	−0.0501209	−	0.998743i	$-0.515961\pi$
$331$	−1.82374	−0.100242	−0.0501209	+	0.998743i	$0.515961\pi$
$332$	0	0
$333$	37.2109	2.03914
$334$	0	0
$335$	11.8698	0.648518
$336$	0	0
$337$	−17.7827	−0.968683	−0.484341	−	0.874879i	$-0.660941\pi$
$337$	−17.7827	−0.968683	−0.484341	+	0.874879i	$0.660941\pi$
$338$	0	0
$339$	43.2950	2.35146
$340$	0	0
$341$	−15.0246	−0.813628
$342$	0	0
$343$	0	0
$344$	0	0
$345$	1.75615	0.0945480
$346$	0	0
$347$	−1.28995	−0.0692479	−0.0346239	−	0.999400i	$-0.511023\pi$
$347$	−1.28995	−0.0692479	−0.0346239	+	0.999400i	$0.511023\pi$
$348$	0	0
$349$	1.31144	0.0701995	0.0350998	−	0.999384i	$-0.488825\pi$
$349$	1.31144	0.0701995	0.0350998	+	0.999384i	$0.488825\pi$
$350$	0	0
$351$	−43.7397	−2.33465
$352$	0	0
$353$	−11.0246	−0.586781	−0.293390	−	0.955993i	$-0.594784\pi$
$353$	−11.0246	−0.586781	−0.293390	+	0.955993i	$0.594784\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	10.2274	0.539780	0.269890	−	0.962891i	$-0.413013\pi$
$359$	10.2274	0.539780	0.269890	+	0.962891i	$0.413013\pi$
$360$	0	0
$361$	−18.4282	−0.969907
$362$	0	0
$363$	−75.3922	−3.95706
$364$	0	0
$365$	3.51230	0.183842
$366$	0	0
$367$	17.3411	0.905196	0.452598	−	0.891715i	$-0.350497\pi$
$367$	17.3411	0.905196	0.452598	+	0.891715i	$0.350497\pi$
$368$	0	0
$369$	47.4712	2.47125
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−17.7827	−0.920751	−0.460375	−	0.887724i	$-0.652285\pi$
$373$	−17.7827	−0.920751	−0.460375	+	0.887724i	$0.652285\pi$
$374$	0	0
$375$	−3.28995	−0.169892
$376$	0	0
$377$	−2.27035	−0.116929
$378$	0	0
$379$	−33.2109	−1.70593	−0.852964	−	0.521969i	$-0.825198\pi$
$379$	−33.2109	−1.70593	−0.852964	+	0.521969i	$0.825198\pi$
$380$	0	0
$381$	−39.2028	−2.00842
$382$	0	0
$383$	14.7612	0.754260	0.377130	−	0.926160i	$-0.376911\pi$
$383$	14.7612	0.754260	0.377130	+	0.926160i	$0.376911\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	5.55528	0.282391
$388$	0	0
$389$	−22.8073	−1.15637	−0.578187	−	0.815904i	$-0.696240\pi$
$389$	−22.8073	−1.15637	−0.578187	+	0.815904i	$0.696240\pi$
$390$	0	0
$391$	−1.06759	−0.0539902
$392$	0	0
$393$	−70.9424	−3.57857
$394$	0	0
$395$	−9.51230	−0.478616
$396$	0	0
$397$	−14.2703	−0.716208	−0.358104	−	0.933682i	$-0.616577\pi$
$397$	−14.2703	−0.716208	−0.358104	+	0.933682i	$0.616577\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−11.4877	−0.573668	−0.286834	−	0.957980i	$-0.592603\pi$
$401$	−11.4877	−0.573668	−0.286834	+	0.957980i	$0.592603\pi$
$402$	0	0
$403$	−7.11057	−0.354203
$404$	0	0
$405$	28.7397	1.42809
$406$	0	0
$407$	27.6986	1.37297
$408$	0	0
$409$	1.79913	0.0889614	0.0444807	−	0.999010i	$-0.485837\pi$
$409$	1.79913	0.0889614	0.0444807	+	0.999010i	$0.485837\pi$
$410$	0	0
$411$	−13.1598	−0.649124
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−6.71005	−0.329384
$416$	0	0
$417$	−21.6475	−1.06008
$418$	0	0
$419$	27.4282	1.33996	0.669978	−	0.742381i	$-0.266303\pi$
$419$	27.4282	1.33996	0.669978	+	0.742381i	$0.266303\pi$
$420$	0	0
$421$	2.91593	0.142114	0.0710569	−	0.997472i	$-0.477363\pi$
$421$	2.91593	0.142114	0.0710569	+	0.997472i	$0.477363\pi$
$422$	0	0
$423$	−100.858	−4.90390
$424$	0	0
$425$	2.00000	0.0970143
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−52.8073	−2.54956
$430$	0	0
$431$	14.5799	0.702289	0.351144	−	0.936321i	$-0.385793\pi$
$431$	14.5799	0.702289	0.351144	+	0.936321i	$0.385793\pi$
$432$	0	0
$433$	21.1598	1.01687	0.508437	−	0.861099i	$-0.330223\pi$
$433$	21.1598	1.01687	0.508437	+	0.861099i	$0.330223\pi$
$434$	0	0
$435$	2.71005	0.129937
$436$	0	0
$437$	0.403629	0.0193082
$438$	0	0
$439$	−30.9424	−1.47680	−0.738401	−	0.674362i	$-0.764419\pi$
$439$	−30.9424	−1.47680	−0.738401	+	0.674362i	$0.764419\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	37.5173	1.78250	0.891251	−	0.453511i	$-0.149829\pi$
$443$	37.5173	1.78250	0.891251	+	0.453511i	$0.149829\pi$
$444$	0	0
$445$	1.75615	0.0832496
$446$	0	0
$447$	−0.357532	−0.0169107
$448$	0	0
$449$	31.0922	1.46733	0.733666	−	0.679511i	$-0.237808\pi$
$449$	31.0922	1.46733	0.733666	+	0.679511i	$0.237808\pi$
$450$	0	0
$451$	35.3360	1.66391
$452$	0	0
$453$	42.9916	2.01992
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−4.35252	−0.203602	−0.101801	−	0.994805i	$-0.532461\pi$
$457$	−4.35252	−0.203602	−0.101801	+	0.994805i	$0.532461\pi$
$458$	0	0
$459$	−31.7397	−1.48148
$460$	0	0
$461$	37.2950	1.73700	0.868500	−	0.495690i	$-0.165085\pi$
$461$	37.2950	1.73700	0.868500	+	0.495690i	$0.165085\pi$
$462$	0	0
$463$	−9.81873	−0.456315	−0.228158	−	0.973624i	$-0.573270\pi$
$463$	−9.81873	−0.456315	−0.228158	+	0.973624i	$0.573270\pi$
$464$	0	0
$465$	8.48770	0.393608
$466$	0	0
$467$	−22.8022	−1.05516	−0.527581	−	0.849505i	$-0.676901\pi$
$467$	−22.8022	−1.05516	−0.527581	+	0.849505i	$0.676901\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−29.2520	−1.34786
$472$	0	0
$473$	4.13517	0.190136
$474$	0	0
$475$	−0.756152	−0.0346946
$476$	0	0
$477$	−65.7478	−3.01038
$478$	0	0
$479$	−20.4447	−0.934143	−0.467071	−	0.884220i	$-0.654691\pi$
$479$	−20.4447	−0.934143	−0.467071	+	0.884220i	$0.654691\pi$
$480$	0	0
$481$	13.1087	0.597704
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−2.00000	−0.0908153
$486$	0	0
$487$	−42.8073	−1.93978	−0.969891	−	0.243539i	$-0.921691\pi$
$487$	−42.8073	−1.93978	−0.969891	+	0.243539i	$0.921691\pi$
$488$	0	0
$489$	37.8748	1.71276
$490$	0	0
$491$	−18.8502	−0.850699	−0.425350	−	0.905029i	$-0.639849\pi$
$491$	−18.8502	−0.850699	−0.425350	+	0.905029i	$0.639849\pi$
$492$	0	0
$493$	−1.64748	−0.0741986
$494$	0	0
$495$	45.5634	2.04792
$496$	0	0
$497$	0	0
$498$	0	0
$499$	27.7397	1.24180	0.620899	−	0.783890i	$-0.286768\pi$
$499$	27.7397	1.24180	0.620899	+	0.783890i	$0.286768\pi$
$500$	0	0
$501$	−4.82374	−0.215509
$502$	0	0
$503$	14.7101	0.655889	0.327944	−	0.944697i	$-0.393644\pi$
$503$	14.7101	0.655889	0.327944	+	0.944697i	$0.393644\pi$
$504$	0	0
$505$	2.33604	0.103952
$506$	0	0
$507$	17.7776	0.789533
$508$	0	0
$509$	−13.9835	−0.619809	−0.309904	−	0.950768i	$-0.600297\pi$
$509$	−13.9835	−0.619809	−0.309904	+	0.950768i	$0.600297\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	12.0000	0.529813
$514$	0	0
$515$	−2.22236	−0.0979288
$516$	0	0
$517$	−75.0757	−3.30183
$518$	0	0
$519$	36.9916	1.62375
$520$	0	0
$521$	20.7232	0.907899	0.453950	−	0.891027i	$-0.350015\pi$
$521$	20.7232	0.907899	0.453950	+	0.891027i	$0.350015\pi$
$522$	0	0
$523$	−22.3525	−0.977408	−0.488704	−	0.872450i	$-0.662530\pi$
$523$	−22.3525	−0.977408	−0.488704	+	0.872450i	$0.662530\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−5.15978	−0.224764
$528$	0	0
$529$	−22.7151	−0.987611
$530$	0	0
$531$	62.5899	2.71617
$532$	0	0
$533$	16.7232	0.724362
$534$	0	0
$535$	15.8698	0.686113
$536$	0	0
$537$	−24.1352	−1.04151
$538$	0	0
$539$	0	0
$540$	0	0
$541$	32.6556	1.40397	0.701987	−	0.712190i	$-0.252296\pi$
$541$	32.6556	1.40397	0.701987	+	0.712190i	$0.252296\pi$
$542$	0	0
$543$	60.7693	2.60786
$544$	0	0
$545$	−3.26845	−0.140005
$546$	0	0
$547$	31.1648	1.33251	0.666255	−	0.745724i	$-0.267896\pi$
$547$	31.1648	1.33251	0.666255	+	0.745724i	$0.267896\pi$
$548$	0	0
$549$	73.5715	3.13996
$550$	0	0
$551$	0.622871	0.0265352
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−15.6475	−0.664198
$556$	0	0
$557$	−44.7232	−1.89498	−0.947491	−	0.319782i	$-0.896390\pi$
$557$	−44.7232	−1.89498	−0.947491	+	0.319782i	$0.896390\pi$
$558$	0	0
$559$	1.95702	0.0827731
$560$	0	0
$561$	−38.3196	−1.61785
$562$	0	0
$563$	−32.4927	−1.36940	−0.684702	−	0.728823i	$-0.740068\pi$
$563$	−32.4927	−1.36940	−0.684702	+	0.728823i	$0.740068\pi$
$564$	0	0
$565$	−13.1598	−0.553636
$566$	0	0
$567$	0	0
$568$	0	0
$569$	34.6740	1.45361	0.726804	−	0.686845i	$-0.241005\pi$
$569$	34.6740	1.45361	0.726804	+	0.686845i	$0.241005\pi$
$570$	0	0
$571$	3.11057	0.130173	0.0650866	−	0.997880i	$-0.479268\pi$
$571$	3.11057	0.130173	0.0650866	+	0.997880i	$0.479268\pi$
$572$	0	0
$573$	−47.9670	−2.00385
$574$	0	0
$575$	−0.533794	−0.0222607
$576$	0	0
$577$	0.672083	0.0279792	0.0139896	−	0.999902i	$-0.495547\pi$
$577$	0.672083	0.0279792	0.0139896	+	0.999902i	$0.495547\pi$
$578$	0	0
$579$	61.8748	2.57143
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−48.9405	−2.02691
$584$	0	0
$585$	21.5634	0.891537
$586$	0	0
$587$	−4.67208	−0.192838	−0.0964188	−	0.995341i	$-0.530739\pi$
$587$	−4.67208	−0.192838	−0.0964188	+	0.995341i	$0.530739\pi$
$588$	0	0
$589$	1.95079	0.0803808
$590$	0	0
$591$	−9.06759	−0.372991
$592$	0	0
$593$	20.1844	0.828873	0.414437	−	0.910078i	$-0.363979\pi$
$593$	20.1844	0.828873	0.414437	+	0.910078i	$0.363979\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−49.4301	−2.02304
$598$	0	0
$599$	21.5123	0.878969	0.439484	−	0.898250i	$-0.355161\pi$
$599$	21.5123	0.878969	0.439484	+	0.898250i	$0.355161\pi$
$600$	0	0
$601$	5.29495	0.215986	0.107993	−	0.994152i	$-0.465558\pi$
$601$	5.29495	0.215986	0.107993	+	0.994152i	$0.465558\pi$
$602$	0	0
$603$	92.8665	3.78182
$604$	0	0
$605$	22.9159	0.931665
$606$	0	0
$607$	36.7182	1.49034	0.745172	−	0.666872i	$-0.232367\pi$
$607$	36.7182	1.49034	0.745172	+	0.666872i	$0.232367\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−35.5304	−1.43741
$612$	0	0
$613$	40.7232	1.64479	0.822397	−	0.568914i	$-0.192636\pi$
$613$	40.7232	1.64479	0.822397	+	0.568914i	$0.192636\pi$
$614$	0	0
$615$	−19.9620	−0.804947
$616$	0	0
$617$	16.8073	0.676635	0.338317	−	0.941032i	$-0.390142\pi$
$617$	16.8073	0.676635	0.338317	+	0.941032i	$0.390142\pi$
$618$	0	0
$619$	−35.7908	−1.43855	−0.719276	−	0.694724i	$-0.755526\pi$
$619$	−35.7908	−1.43855	−0.719276	+	0.694724i	$0.755526\pi$
$620$	0	0
$621$	8.47122	0.339938
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	14.4877	0.578583
$628$	0	0
$629$	9.51230	0.379280
$630$	0	0
$631$	−32.4447	−1.29160	−0.645802	−	0.763505i	$-0.723477\pi$
$631$	−32.4447	−1.29160	−0.645802	+	0.763505i	$0.723477\pi$
$632$	0	0
$633$	72.1022	2.86581
$634$	0	0
$635$	11.9159	0.472869
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	16.9078	0.667818	0.333909	−	0.942605i	$-0.391632\pi$
$641$	16.9078	0.667818	0.333909	+	0.942605i	$0.391632\pi$
$642$	0	0
$643$	0.135174	0.00533075	0.00266538	−	0.999996i	$-0.499152\pi$
$643$	0.135174	0.00533075	0.00266538	+	0.999996i	$0.499152\pi$
$644$	0	0
$645$	−2.33604	−0.0919816
$646$	0	0
$647$	−4.08908	−0.160758	−0.0803791	−	0.996764i	$-0.525613\pi$
$647$	−4.08908	−0.160758	−0.0803791	+	0.996764i	$0.525613\pi$
$648$	0	0
$649$	46.5899	1.82881
$650$	0	0
$651$	0	0
$652$	0	0
$653$	31.9159	1.24897	0.624483	−	0.781038i	$-0.285310\pi$
$653$	31.9159	1.24897	0.624483	+	0.781038i	$0.285310\pi$
$654$	0	0
$655$	21.5634	0.842552
$656$	0	0
$657$	27.4793	1.07207
$658$	0	0
$659$	4.70505	0.183283	0.0916413	−	0.995792i	$-0.470789\pi$
$659$	4.70505	0.183283	0.0916413	+	0.995792i	$0.470789\pi$
$660$	0	0
$661$	4.51420	0.175582	0.0877910	−	0.996139i	$-0.472019\pi$
$661$	4.51420	0.175582	0.0877910	+	0.996139i	$0.472019\pi$
$662$	0	0
$663$	−18.1352	−0.704312
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0.439706	0.0170255
$668$	0	0
$669$	−61.8748	−2.39222
$670$	0	0
$671$	54.7643	2.11415
$672$	0	0
$673$	46.9424	1.80950	0.904749	−	0.425945i	$-0.140058\pi$
$673$	46.9424	1.80950	0.904749	+	0.425945i	$0.140058\pi$
$674$	0	0
$675$	−15.8698	−0.610830
$676$	0	0
$677$	−10.4036	−0.399844	−0.199922	−	0.979812i	$-0.564069\pi$
$677$	−10.4036	−0.399844	−0.199922	+	0.979812i	$0.564069\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−54.8502	−2.10187
$682$	0	0
$683$	5.06258	0.193714	0.0968571	−	0.995298i	$-0.469121\pi$
$683$	5.06258	0.193714	0.0968571	+	0.995298i	$0.469121\pi$
$684$	0	0
$685$	4.00000	0.152832
$686$	0	0
$687$	−64.1944	−2.44917
$688$	0	0
$689$	−23.1617	−0.882390
$690$	0	0
$691$	−4.44472	−0.169085	−0.0845425	−	0.996420i	$-0.526943\pi$
$691$	−4.44472	−0.169085	−0.0845425	+	0.996420i	$0.526943\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	6.57989	0.249589
$696$	0	0
$697$	12.1352	0.459653
$698$	0	0
$699$	−59.2190	−2.23987
$700$	0	0
$701$	21.7562	0.821719	0.410859	−	0.911699i	$-0.365229\pi$
$701$	21.7562	0.821719	0.410859	+	0.911699i	$0.365229\pi$
$702$	0	0
$703$	−3.59637	−0.135640
$704$	0	0
$705$	42.4118	1.59732
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−41.0081	−1.54009	−0.770046	−	0.637988i	$-0.779767\pi$
$709$	−41.0081	−1.54009	−0.770046	+	0.637988i	$0.779767\pi$
$710$	0	0
$711$	−74.4218	−2.79103
$712$	0	0
$713$	1.37713	0.0515739
$714$	0	0
$715$	16.0511	0.600277
$716$	0	0
$717$	52.9424	1.97717
$718$	0	0
$719$	23.7397	0.885340	0.442670	−	0.896685i	$-0.354031\pi$
$719$	23.7397	0.885340	0.442670	+	0.896685i	$0.354031\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−2.93241	−0.109058
$724$	0	0
$725$	−0.823739	−0.0305929
$726$	0	0
$727$	−28.8534	−1.07011	−0.535056	−	0.844817i	$-0.679709\pi$
$727$	−28.8534	−1.07011	−0.535056	+	0.844817i	$0.679709\pi$
$728$	0	0
$729$	68.2190	2.52663
$730$	0	0
$731$	1.42011	0.0525247
$732$	0	0
$733$	46.8584	1.73075	0.865377	−	0.501122i	$-0.167079\pi$
$733$	46.8584	1.73075	0.865377	+	0.501122i	$0.167079\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	69.1268	2.54632
$738$	0	0
$739$	16.1433	0.593841	0.296920	−	0.954902i	$-0.404040\pi$
$739$	16.1433	0.593841	0.296920	+	0.954902i	$0.404040\pi$
$740$	0	0
$741$	6.85647	0.251879
$742$	0	0
$743$	−16.2243	−0.595210	−0.297605	−	0.954689i	$-0.596188\pi$
$743$	−16.2243	−0.595210	−0.297605	+	0.954689i	$0.596188\pi$
$744$	0	0
$745$	0.108674	0.00398152
$746$	0	0
$747$	−52.4977	−1.92079
$748$	0	0
$749$	0	0
$750$	0	0
$751$	21.5123	0.784995	0.392498	−	0.919753i	$-0.371611\pi$
$751$	21.5123	0.784995	0.392498	+	0.919753i	$0.371611\pi$
$752$	0	0
$753$	−57.4793	−2.09466
$754$	0	0
$755$	−13.0676	−0.475578
$756$	0	0
$757$	0.840220	0.0305383	0.0152692	−	0.999883i	$-0.495139\pi$
$757$	0.840220	0.0305383	0.0152692	+	0.999883i	$0.495139\pi$
$758$	0	0
$759$	10.2274	0.371230
$760$	0	0
$761$	−28.8913	−1.04731	−0.523655	−	0.851930i	$-0.675432\pi$
$761$	−28.8913	−1.04731	−0.523655	+	0.851930i	$0.675432\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	15.6475	0.565736
$766$	0	0
$767$	22.0492	0.796151
$768$	0	0
$769$	−27.3790	−0.987313	−0.493656	−	0.869657i	$-0.664340\pi$
$769$	−27.3790	−0.987313	−0.493656	+	0.869657i	$0.664340\pi$
$770$	0	0
$771$	23.1106	0.832307
$772$	0	0
$773$	−38.2355	−1.37524	−0.687618	−	0.726073i	$-0.741343\pi$
$773$	−38.2355	−1.37524	−0.687618	+	0.726073i	$0.741343\pi$
$774$	0	0
$775$	−2.57989	−0.0926724
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−4.58802	−0.164383
$780$	0	0
$781$	0	0
$782$	0	0
$783$	13.0726	0.467176
$784$	0	0
$785$	8.89133	0.317345
$786$	0	0
$787$	7.33293	0.261391	0.130695	−	0.991423i	$-0.458279\pi$
$787$	7.33293	0.261391	0.130695	+	0.991423i	$0.458279\pi$
$788$	0	0
$789$	−58.4875	−2.08221
$790$	0	0
$791$	0	0
$792$	0	0
$793$	25.9178	0.920369
$794$	0	0
$795$	27.6475	0.980555
$796$	0	0
$797$	14.4877	0.513181	0.256590	−	0.966520i	$-0.417401\pi$
$797$	14.4877	0.513181	0.256590	+	0.966520i	$0.417401\pi$
$798$	0	0
$799$	−25.7827	−0.912125
$800$	0	0
$801$	13.7397	0.485467
$802$	0	0
$803$	20.4547	0.721832
$804$	0	0
$805$	0	0
$806$	0	0
$807$	30.9374	1.08905
$808$	0	0
$809$	−11.3525	−0.399133	−0.199567	−	0.979884i	$-0.563953\pi$
$809$	−11.3525	−0.399133	−0.199567	+	0.979884i	$0.563953\pi$
$810$	0	0
$811$	−28.7662	−1.01012	−0.505058	−	0.863085i	$-0.668529\pi$
$811$	−28.7662	−1.01012	−0.505058	+	0.863085i	$0.668529\pi$
$812$	0	0
$813$	−23.1106	−0.810523
$814$	0	0
$815$	−11.5123	−0.403258
$816$	0	0
$817$	−0.536909	−0.0187841
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−16.0492	−0.560121	−0.280061	−	0.959982i	$-0.590355\pi$
$821$	−16.0492	−0.560121	−0.280061	+	0.959982i	$0.590355\pi$
$822$	0	0
$823$	14.0050	0.488184	0.244092	−	0.969752i	$-0.421510\pi$
$823$	14.0050	0.488184	0.244092	+	0.969752i	$0.421510\pi$
$824$	0	0
$825$	−19.1598	−0.667058
$826$	0	0
$827$	15.2899	0.531683	0.265842	−	0.964017i	$-0.414350\pi$
$827$	15.2899	0.531683	0.265842	+	0.964017i	$0.414350\pi$
$828$	0	0
$829$	−21.2950	−0.739604	−0.369802	−	0.929111i	$-0.620575\pi$
$829$	−21.2950	−0.739604	−0.369802	+	0.929111i	$0.620575\pi$
$830$	0	0
$831$	−89.3542	−3.09966
$832$	0	0
$833$	0	0
$834$	0	0
$835$	1.46621	0.0507402
$836$	0	0
$837$	40.9424	1.41518
$838$	0	0
$839$	42.2274	1.45785	0.728925	−	0.684593i	$-0.240020\pi$
$839$	42.2274	1.45785	0.728925	+	0.684593i	$0.240020\pi$
$840$	0	0
$841$	−28.3215	−0.976602
$842$	0	0
$843$	2.04298	0.0703640
$844$	0	0
$845$	−5.40363	−0.185890
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−10.3955	−0.356773
$850$	0	0
$851$	−2.53880	−0.0870290
$852$	0	0
$853$	−13.2931	−0.455146	−0.227573	−	0.973761i	$-0.573079\pi$
$853$	−13.2931	−0.455146	−0.227573	+	0.973761i	$0.573079\pi$
$854$	0	0
$855$	−5.91593	−0.202321
$856$	0	0
$857$	38.4547	1.31359	0.656794	−	0.754070i	$-0.271912\pi$
$857$	38.4547	1.31359	0.656794	+	0.754070i	$0.271912\pi$
$858$	0	0
$859$	8.00000	0.272956	0.136478	−	0.990643i	$-0.456422\pi$
$859$	8.00000	0.272956	0.136478	+	0.990643i	$0.456422\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−47.0807	−1.60265	−0.801323	−	0.598232i	$-0.795870\pi$
$863$	−47.0807	−1.60265	−0.801323	+	0.598232i	$0.795870\pi$
$864$	0	0
$865$	−11.2438	−0.382302
$866$	0	0
$867$	42.7693	1.45252
$868$	0	0
$869$	−55.3972	−1.87922
$870$	0	0
$871$	32.7151	1.10851
$872$	0	0
$873$	−15.6475	−0.529587
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−4.57177	−0.154378	−0.0771888	−	0.997016i	$-0.524594\pi$
$877$	−4.57177	−0.154378	−0.0771888	+	0.997016i	$0.524594\pi$
$878$	0	0
$879$	−36.9916	−1.24770
$880$	0	0
$881$	−14.2849	−0.481272	−0.240636	−	0.970615i	$-0.577356\pi$
$881$	−14.2849	−0.481272	−0.240636	+	0.970615i	$0.577356\pi$
$882$	0	0
$883$	−15.9670	−0.537334	−0.268667	−	0.963233i	$-0.586583\pi$
$883$	−15.9670	−0.537334	−0.268667	+	0.963233i	$0.586583\pi$
$884$	0	0
$885$	−26.3196	−0.884722
$886$	0	0
$887$	−7.16479	−0.240570	−0.120285	−	0.992739i	$-0.538381\pi$
$887$	−7.16479	−0.240570	−0.120285	+	0.992739i	$0.538381\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	167.372	5.60718
$892$	0	0
$893$	9.74780	0.326198
$894$	0	0
$895$	7.33604	0.245217
$896$	0	0
$897$	4.84022	0.161610
$898$	0	0
$899$	2.12516	0.0708779
$900$	0	0
$901$	−16.8073	−0.559931
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−18.4712	−0.614004
$906$	0	0
$907$	8.26534	0.274446	0.137223	−	0.990540i	$-0.456182\pi$
$907$	8.26534	0.274446	0.137223	+	0.990540i	$0.456182\pi$
$908$	0	0
$909$	18.2766	0.606196
$910$	0	0
$911$	26.2274	0.868951	0.434476	−	0.900684i	$-0.356934\pi$
$911$	26.2274	0.868951	0.434476	+	0.900684i	$0.356934\pi$
$912$	0	0
$913$	−39.0776	−1.29328
$914$	0	0
$915$	−30.9374	−1.02276
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−18.7580	−0.618771	−0.309385	−	0.950937i	$-0.600123\pi$
$919$	−18.7580	−0.618771	−0.309385	+	0.950937i	$0.600123\pi$
$920$	0	0
$921$	−77.3707	−2.54945
$922$	0	0
$923$	0	0
$924$	0	0
$925$	4.75615	0.156381
$926$	0	0
$927$	−17.3871	−0.571069
$928$	0	0
$929$	−12.9078	−0.423491	−0.211746	−	0.977325i	$-0.567915\pi$
$929$	−12.9078	−0.423491	−0.211746	+	0.977325i	$0.567915\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	95.9341	3.14074
$934$	0	0
$935$	11.6475	0.380913
$936$	0	0
$937$	1.78265	0.0582367	0.0291183	−	0.999576i	$-0.490730\pi$
$937$	1.78265	0.0582367	0.0291183	+	0.999576i	$0.490730\pi$
$938$	0	0
$939$	79.5653	2.59652
$940$	0	0
$941$	−45.1268	−1.47109	−0.735546	−	0.677475i	$-0.763074\pi$
$941$	−45.1268	−1.47109	−0.735546	+	0.677475i	$0.763074\pi$
$942$	0	0
$943$	−3.23884	−0.105471
$944$	0	0
$945$	0	0
$946$	0	0
$947$	22.0872	0.717737	0.358869	−	0.933388i	$-0.383163\pi$
$947$	22.0872	0.717737	0.358869	+	0.933388i	$0.383163\pi$
$948$	0	0
$949$	9.68044	0.314240
$950$	0	0
$951$	−39.9241	−1.29463
$952$	0	0
$953$	−45.3442	−1.46884	−0.734421	−	0.678694i	$-0.762546\pi$
$953$	−45.3442	−1.46884	−0.734421	+	0.678694i	$0.762546\pi$
$954$	0	0
$955$	14.5799	0.471794
$956$	0	0
$957$	15.7827	0.510181
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−24.3442	−0.785296
$962$	0	0
$963$	124.161	4.00105
$964$	0	0
$965$	−18.8073	−0.605427
$966$	0	0
$967$	−17.0296	−0.547636	−0.273818	−	0.961782i	$-0.588287\pi$
$967$	−17.0296	−0.547636	−0.273818	+	0.961782i	$0.588287\pi$
$968$	0	0
$969$	4.97539	0.159833
$970$	0	0
$971$	−29.2109	−0.937422	−0.468711	−	0.883352i	$-0.655281\pi$
$971$	−29.2109	−0.937422	−0.468711	+	0.883352i	$0.655281\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−9.06759	−0.290395
$976$	0	0
$977$	0.840220	0.0268810	0.0134405	−	0.999910i	$-0.495722\pi$
$977$	0.840220	0.0268810	0.0134405	+	0.999910i	$0.495722\pi$
$978$	0	0
$979$	10.2274	0.326868
$980$	0	0
$981$	−25.5715	−0.816436
$982$	0	0
$983$	8.31645	0.265253	0.132627	−	0.991166i	$-0.457659\pi$
$983$	8.31645	0.265253	0.132627	+	0.991166i	$0.457659\pi$
$984$	0	0
$985$	2.75615	0.0878183
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−0.379023	−0.0120522
$990$	0	0
$991$	−20.2603	−0.643591	−0.321795	−	0.946809i	$-0.604286\pi$
$991$	−20.2603	−0.643591	−0.321795	+	0.946809i	$0.604286\pi$
$992$	0	0
$993$	6.00000	0.190404
$994$	0	0
$995$	15.0246	0.476312
$996$	0	0
$997$	−53.0776	−1.68098	−0.840492	−	0.541823i	$-0.817734\pi$
$997$	−53.0776	−1.68098	−0.840492	+	0.541823i	$0.817734\pi$
$998$	0	0
$999$	−75.4793	−2.38806

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1960.2.a.w.1.1		3
4.3	odd	2		3920.2.a.cc.1.3		3
5.4	even	2		9800.2.a.ce.1.3		3
7.2	even	3		280.2.q.e.81.3	✓	6
7.3	odd	6		1960.2.q.w.961.1		6
7.4	even	3		280.2.q.e.121.3	yes	6
7.5	odd	6		1960.2.q.w.361.1		6
7.6	odd	2		1960.2.a.v.1.3		3
21.2	odd	6		2520.2.bi.q.361.3		6
21.11	odd	6		2520.2.bi.q.1801.3		6
28.11	odd	6		560.2.q.l.401.1		6
28.23	odd	6		560.2.q.l.81.1		6
28.27	even	2		3920.2.a.cb.1.1		3
35.2	odd	12		1400.2.bh.i.249.1		12
35.4	even	6		1400.2.q.j.401.1		6
35.9	even	6		1400.2.q.j.1201.1		6
35.18	odd	12		1400.2.bh.i.849.1		12
35.23	odd	12		1400.2.bh.i.249.6		12
35.32	odd	12		1400.2.bh.i.849.6		12
35.34	odd	2		9800.2.a.cf.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.2.q.e.81.3	✓	6	7.2	even	3
280.2.q.e.121.3	yes	6	7.4	even	3
560.2.q.l.81.1		6	28.23	odd	6
560.2.q.l.401.1		6	28.11	odd	6
1400.2.q.j.401.1		6	35.4	even	6
1400.2.q.j.1201.1		6	35.9	even	6
1400.2.bh.i.249.1		12	35.2	odd	12
1400.2.bh.i.249.6		12	35.23	odd	12
1400.2.bh.i.849.1		12	35.18	odd	12
1400.2.bh.i.849.6		12	35.32	odd	12
1960.2.a.v.1.3		3	7.6	odd	2
1960.2.a.w.1.1		3	1.1	even	1	trivial
1960.2.q.w.361.1		6	7.5	odd	6
1960.2.q.w.961.1		6	7.3	odd	6
2520.2.bi.q.361.3		6	21.2	odd	6
2520.2.bi.q.1801.3		6	21.11	odd	6
3920.2.a.cb.1.1		3	28.27	even	2
3920.2.a.cc.1.3		3	4.3	odd	2
9800.2.a.ce.1.3		3	5.4	even	2
9800.2.a.cf.1.1		3	35.34	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:	\( N \)	\(=\)	\( 1960 = 2^{3} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1960.a (trivial)

\(f(q)\)	\(=\)	\(q-3.28995 q^{3} +1.00000 q^{5} +7.82374 q^{9} +O(q^{10})\)	\(q-3.28995 q^{3} +1.00000 q^{5} +7.82374 q^{9} +5.82374 q^{11} +2.75615 q^{13} -3.28995 q^{15} +2.00000 q^{17} -0.756152 q^{19} -0.533794 q^{23} +1.00000 q^{25} -15.8698 q^{27} -0.823739 q^{29} -2.57989 q^{31} -19.1598 q^{33} +4.75615 q^{37} -9.06759 q^{39} +6.06759 q^{41} +0.710055 q^{43} +7.82374 q^{45} -12.8913 q^{47} -6.57989 q^{51} -8.40363 q^{53} +5.82374 q^{55} +2.48770 q^{57} +8.00000 q^{59} +9.40363 q^{61} +2.75615 q^{65} +11.8698 q^{67} +1.75615 q^{69} +3.51230 q^{73} -3.28995 q^{75} -9.51230 q^{79} +28.7397 q^{81} -6.71005 q^{83} +2.00000 q^{85} +2.71005 q^{87} +1.75615 q^{89} +8.48770 q^{93} -0.756152 q^{95} -2.00000 q^{97} +45.5634 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{5} + 9 q^{9}+O(q^{10}) \) 3 * q + 3 * q^5 + 9 * q^9	\( 3 q + 3 q^{5} + 9 q^{9} + 3 q^{11} + 3 q^{13} + 6 q^{17} + 3 q^{19} + 3 q^{23} + 3 q^{25} - 18 q^{27} + 12 q^{29} + 12 q^{31} - 18 q^{33} + 9 q^{37} - 18 q^{39} + 9 q^{41} + 12 q^{43} + 9 q^{45} - 15 q^{47} + 9 q^{53} + 3 q^{55} + 18 q^{57} + 24 q^{59} - 6 q^{61} + 3 q^{65} + 6 q^{67} - 18 q^{79} + 27 q^{81} - 30 q^{83} + 6 q^{85} + 18 q^{87} + 36 q^{93} + 3 q^{95} - 6 q^{97} + 63 q^{99}+O(q^{100}) \) 3 * q + 3 * q^5 + 9 * q^9 + 3 * q^11 + 3 * q^13 + 6 * q^17 + 3 * q^19 + 3 * q^23 + 3 * q^25 - 18 * q^27 + 12 * q^29 + 12 * q^31 - 18 * q^33 + 9 * q^37 - 18 * q^39 + 9 * q^41 + 12 * q^43 + 9 * q^45 - 15 * q^47 + 9 * q^53 + 3 * q^55 + 18 * q^57 + 24 * q^59 - 6 * q^61 + 3 * q^65 + 6 * q^67 - 18 * q^79 + 27 * q^81 - 30 * q^83 + 6 * q^85 + 18 * q^87 + 36 * q^93 + 3 * q^95 - 6 * q^97 + 63 * q^99

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