LMFDB - Embedded newform 6975.2.a.bg.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6975,2,Mod(1,6975)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6975.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6975, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6975 = 3^{2} \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6975.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,2,0,2,0,0,-8,6,0,0,-3,0,-4,-10,0,4,7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$55.6956554098$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.311108$ of defining polynomial
Character	$\chi$	$=$	6975.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.688892 q^{2} -1.52543 q^{4} -4.90321 q^{7} -2.42864 q^{8} -3.21432 q^{11} +0.836535 q^{13} -3.37778 q^{14} +1.37778 q^{16} +6.73975 q^{17} +3.00000 q^{19} -2.21432 q^{22} +5.21432 q^{23} +0.576283 q^{26} +7.47949 q^{28} +1.49532 q^{29} +1.00000 q^{31} +5.80642 q^{32} +4.64296 q^{34} -0.407896 q^{37} +2.06668 q^{38} -9.54617 q^{41} +7.61285 q^{43} +4.90321 q^{44} +3.59210 q^{46} -6.62222 q^{47} +17.0415 q^{49} -1.27607 q^{52} -12.8731 q^{53} +11.9081 q^{56} +1.03011 q^{58} +8.16839 q^{59} +0.755569 q^{61} +0.688892 q^{62} +1.24443 q^{64} -2.85236 q^{67} -10.2810 q^{68} +8.29529 q^{71} -14.0415 q^{73} -0.280996 q^{74} -4.57628 q^{76} +15.7605 q^{77} -2.77631 q^{79} -6.57628 q^{82} +10.8874 q^{83} +5.24443 q^{86} +7.80642 q^{88} +2.78568 q^{89} -4.10171 q^{91} -7.95407 q^{92} -4.56199 q^{94} -8.95407 q^{97} +11.7397 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{2} + 2 q^{4} - 8 q^{7} + 6 q^{8} - 3 q^{11} - 4 q^{13} - 10 q^{14} + 4 q^{16} + 7 q^{17} + 9 q^{19} + 9 q^{23} - 18 q^{26} - 4 q^{28} - 9 q^{29} + 3 q^{31} + 4 q^{32} - 6 q^{34} - 8 q^{37} + 6 q^{38}+ \cdots + 22 q^{98}+O(q^{100}) $ 3 * q + 2 * q^2 + 2 * q^4 - 8 * q^7 + 6 * q^8 - 3 * q^11 - 4 * q^13 - 10 * q^14 + 4 * q^16 + 7 * q^17 + 9 * q^19 + 9 * q^23 - 18 * q^26 - 4 * q^28 - 9 * q^29 + 3 * q^31 + 4 * q^32 - 6 * q^34 - 8 * q^37 + 6 * q^38 - 2 * q^41 - 4 * q^43 + 8 * q^44 + 4 * q^46 - 20 * q^47 + 11 * q^49 - 30 * q^52 - 25 * q^53 - 4 * q^56 + 10 * q^58 - 2 * q^59 + 2 * q^61 + 2 * q^62 + 4 * q^64 - 15 * q^67 - 24 * q^68 + 12 * q^71 - 2 * q^73 + 6 * q^74 + 6 * q^76 + 14 * q^77 + 12 * q^79 + 13 * q^83 + 16 * q^86 + 10 * q^88 + 15 * q^89 + 14 * q^91 - 4 * q^92 - 7 * q^97 + 22 * q^98

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 $

$2$	0.688892	0.487120	0.243560	−	0.969886i	$-0.421685\pi$
$2$	0.688892	0.487120	0.243560	+	0.969886i	$0.421685\pi$
$3$	0	0
$3$	0	0
$4$	−1.52543	−0.762714
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.90321	−1.85324	−0.926620	−	0.375999i	$-0.877300\pi$
$7$	−4.90321	−1.85324	−0.926620	+	0.375999i	$0.877300\pi$
$8$	−2.42864	−0.858654
$9$	0	0
$10$	0	0
$11$	−3.21432	−0.969154	−0.484577	−	0.874749i	$-0.661026\pi$
$11$	−3.21432	−0.969154	−0.484577	+	0.874749i	$0.661026\pi$
$12$	0	0
$13$	0.836535	0.232013	0.116007	−	0.993248i	$-0.462991\pi$
$13$	0.836535	0.232013	0.116007	+	0.993248i	$0.462991\pi$
$14$	−3.37778	−0.902751
$15$	0	0
$16$	1.37778	0.344446
$17$	6.73975	1.63463	0.817314	−	0.576192i	$-0.195462\pi$
$17$	6.73975	1.63463	0.817314	+	0.576192i	$0.195462\pi$
$18$	0	0
$19$	3.00000	0.688247	0.344124	−	0.938924i	$-0.388176\pi$
$19$	3.00000	0.688247	0.344124	+	0.938924i	$0.388176\pi$
$20$	0	0
$21$	0	0
$22$	−2.21432	−0.472095
$23$	5.21432	1.08726	0.543630	−	0.839325i	$-0.317049\pi$
$23$	5.21432	1.08726	0.543630	+	0.839325i	$0.317049\pi$
$24$	0	0
$25$	0	0
$26$	0.576283	0.113018
$27$	0	0
$28$	7.47949	1.41349
$29$	1.49532	0.277673	0.138837	−	0.990315i	$-0.455664\pi$
$29$	1.49532	0.277673	0.138837	+	0.990315i	$0.455664\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	5.80642	1.02644
$33$	0	0
$34$	4.64296	0.796261
$35$	0	0
$36$	0	0
$37$	−0.407896	−0.0670577	−0.0335288	−	0.999438i	$-0.510675\pi$
$37$	−0.407896	−0.0670577	−0.0335288	+	0.999438i	$0.510675\pi$
$38$	2.06668	0.335259
$39$	0	0
$40$	0	0
$41$	−9.54617	−1.49086	−0.745431	−	0.666583i	$-0.767756\pi$
$41$	−9.54617	−1.49086	−0.745431	+	0.666583i	$0.767756\pi$
$42$	0	0
$43$	7.61285	1.16095	0.580474	−	0.814279i	$-0.302867\pi$
$43$	7.61285	1.16095	0.580474	+	0.814279i	$0.302867\pi$
$44$	4.90321	0.739187
$45$	0	0
$46$	3.59210	0.529627
$47$	−6.62222	−0.965949	−0.482975	−	0.875634i	$-0.660444\pi$
$47$	−6.62222	−0.965949	−0.482975	+	0.875634i	$0.660444\pi$
$48$	0	0
$49$	17.0415	2.43450
$50$	0	0
$51$	0	0
$52$	−1.27607	−0.176960
$53$	−12.8731	−1.76826	−0.884128	−	0.467244i	$-0.845247\pi$
$53$	−12.8731	−1.76826	−0.884128	+	0.467244i	$0.845247\pi$
$54$	0	0
$55$	0	0
$56$	11.9081	1.59129
$57$	0	0
$58$	1.03011	0.135260
$59$	8.16839	1.06343	0.531717	−	0.846922i	$-0.321547\pi$
$59$	8.16839	1.06343	0.531717	+	0.846922i	$0.321547\pi$
$60$	0	0
$61$	0.755569	0.0967407	0.0483703	−	0.998829i	$-0.484597\pi$
$61$	0.755569	0.0967407	0.0483703	+	0.998829i	$0.484597\pi$
$62$	0.688892	0.0874894
$63$	0	0
$64$	1.24443	0.155554
$65$	0	0
$66$	0	0
$67$	−2.85236	−0.348471	−0.174235	−	0.984704i	$-0.555745\pi$
$67$	−2.85236	−0.348471	−0.174235	+	0.984704i	$0.555745\pi$
$68$	−10.2810	−1.24675
$69$	0	0
$70$	0	0
$71$	8.29529	0.984469	0.492235	−	0.870463i	$-0.336180\pi$
$71$	8.29529	0.984469	0.492235	+	0.870463i	$0.336180\pi$
$72$	0	0
$73$	−14.0415	−1.64343	−0.821716	−	0.569897i	$-0.806983\pi$
$73$	−14.0415	−1.64343	−0.821716	+	0.569897i	$0.806983\pi$
$74$	−0.280996	−0.0326652
$75$	0	0
$76$	−4.57628	−0.524936
$77$	15.7605	1.79607
$78$	0	0
$79$	−2.77631	−0.312360	−0.156180	−	0.987729i	$-0.549918\pi$
$79$	−2.77631	−0.312360	−0.156180	+	0.987729i	$0.549918\pi$
$80$	0	0
$81$	0	0
$82$	−6.57628	−0.726229
$83$	10.8874	1.19505	0.597523	−	0.801852i	$-0.296152\pi$
$83$	10.8874	1.19505	0.597523	+	0.801852i	$0.296152\pi$
$84$	0	0
$85$	0	0
$86$	5.24443	0.565522
$87$	0	0
$88$	7.80642	0.832168
$89$	2.78568	0.295282	0.147641	−	0.989041i	$-0.452832\pi$
$89$	2.78568	0.295282	0.147641	+	0.989041i	$0.452832\pi$
$90$	0	0
$91$	−4.10171	−0.429976
$92$	−7.95407	−0.829269
$93$	0	0
$94$	−4.56199	−0.470534
$95$	0	0
$96$	0	0
$97$	−8.95407	−0.909148	−0.454574	−	0.890709i	$-0.650208\pi$
$97$	−8.95407	−0.909148	−0.454574	+	0.890709i	$0.650208\pi$
$98$	11.7397	1.18589
$99$	0	0
$100$	0	0
$101$	−3.57136	−0.355364	−0.177682	−	0.984088i	$-0.556860\pi$
$101$	−3.57136	−0.355364	−0.177682	+	0.984088i	$0.556860\pi$
$102$	0	0
$103$	14.8479	1.46301	0.731504	−	0.681837i	$-0.238819\pi$
$103$	14.8479	1.46301	0.731504	+	0.681837i	$0.238819\pi$
$104$	−2.03164	−0.199219
$105$	0	0
$106$	−8.86818	−0.861354
$107$	−20.1684	−1.94975	−0.974876	−	0.222749i	$-0.928497\pi$
$107$	−20.1684	−1.94975	−0.974876	+	0.222749i	$0.928497\pi$
$108$	0	0
$109$	6.76986	0.648435	0.324217	−	0.945983i	$-0.394899\pi$
$109$	6.76986	0.648435	0.324217	+	0.945983i	$0.394899\pi$
$110$	0	0
$111$	0	0
$112$	−6.75557	−0.638341
$113$	−9.93978	−0.935056	−0.467528	−	0.883978i	$-0.654855\pi$
$113$	−9.93978	−0.935056	−0.467528	+	0.883978i	$0.654855\pi$
$114$	0	0
$115$	0	0
$116$	−2.28100	−0.211785
$117$	0	0
$118$	5.62714	0.518020
$119$	−33.0464	−3.02936
$120$	0	0
$121$	−0.668149	−0.0607408
$122$	0.520505	0.0471243
$123$	0	0
$124$	−1.52543	−0.136987
$125$	0	0
$126$	0	0
$127$	4.56199	0.404811	0.202406	−	0.979302i	$-0.435124\pi$
$127$	4.56199	0.404811	0.202406	+	0.979302i	$0.435124\pi$
$128$	−10.7556	−0.950667
$129$	0	0
$130$	0	0
$131$	−19.1590	−1.67393	−0.836966	−	0.547255i	$-0.815673\pi$
$131$	−19.1590	−1.67393	−0.836966	+	0.547255i	$0.815673\pi$
$132$	0	0
$133$	−14.7096	−1.27549
$134$	−1.96497	−0.169747
$135$	0	0
$136$	−16.3684	−1.40358
$137$	6.44446	0.550587	0.275294	−	0.961360i	$-0.411225\pi$
$137$	6.44446	0.550587	0.275294	+	0.961360i	$0.411225\pi$
$138$	0	0
$139$	18.6844	1.58479	0.792397	−	0.610006i	$-0.208833\pi$
$139$	18.6844	1.58479	0.792397	+	0.610006i	$0.208833\pi$
$140$	0	0
$141$	0	0
$142$	5.71456	0.479555
$143$	−2.68889	−0.224856
$144$	0	0
$145$	0	0
$146$	−9.67307	−0.800549
$147$	0	0
$148$	0.622216	0.0511458
$149$	0.647405	0.0530375	0.0265187	−	0.999648i	$-0.491558\pi$
$149$	0.647405	0.0530375	0.0265187	+	0.999648i	$0.491558\pi$
$150$	0	0
$151$	−10.7239	−0.872701	−0.436350	−	0.899777i	$-0.643729\pi$
$151$	−10.7239	−0.872701	−0.436350	+	0.899777i	$0.643729\pi$
$152$	−7.28592	−0.590966
$153$	0	0
$154$	10.8573	0.874904
$155$	0	0
$156$	0	0
$157$	7.99063	0.637722	0.318861	−	0.947802i	$-0.396700\pi$
$157$	7.99063	0.637722	0.318861	+	0.947802i	$0.396700\pi$
$158$	−1.91258	−0.152157
$159$	0	0
$160$	0	0
$161$	−25.5669	−2.01496
$162$	0	0
$163$	−23.2351	−1.81991	−0.909955	−	0.414706i	$-0.863884\pi$
$163$	−23.2351	−1.81991	−0.909955	+	0.414706i	$0.863884\pi$
$164$	14.5620	1.13710
$165$	0	0
$166$	7.50024	0.582131
$167$	−0.295286	−0.0228499	−0.0114250	−	0.999935i	$-0.503637\pi$
$167$	−0.295286	−0.0228499	−0.0114250	+	0.999935i	$0.503637\pi$
$168$	0	0
$169$	−12.3002	−0.946170
$170$	0	0
$171$	0	0
$172$	−11.6128	−0.885471
$173$	−6.73038	−0.511701	−0.255851	−	0.966716i	$-0.582356\pi$
$173$	−6.73038	−0.511701	−0.255851	+	0.966716i	$0.582356\pi$
$174$	0	0
$175$	0	0
$176$	−4.42864	−0.333821
$177$	0	0
$178$	1.91903	0.143838
$179$	15.8938	1.18796	0.593981	−	0.804479i	$-0.297555\pi$
$179$	15.8938	1.18796	0.593981	+	0.804479i	$0.297555\pi$
$180$	0	0
$181$	−12.9382	−0.961692	−0.480846	−	0.876805i	$-0.659670\pi$
$181$	−12.9382	−0.961692	−0.480846	+	0.876805i	$0.659670\pi$
$182$	−2.82564	−0.209450
$183$	0	0
$184$	−12.6637	−0.933581
$185$	0	0
$186$	0	0
$187$	−21.6637	−1.58421
$188$	10.1017	0.736743
$189$	0	0
$190$	0	0
$191$	11.4128	0.825803	0.412901	−	0.910776i	$-0.364515\pi$
$191$	11.4128	0.825803	0.412901	+	0.910776i	$0.364515\pi$
$192$	0	0
$193$	−13.2810	−0.955987	−0.477994	−	0.878363i	$-0.658636\pi$
$193$	−13.2810	−0.955987	−0.477994	+	0.878363i	$0.658636\pi$
$194$	−6.16839	−0.442864
$195$	0	0
$196$	−25.9956	−1.85683
$197$	−21.0923	−1.50277	−0.751384	−	0.659866i	$-0.770613\pi$
$197$	−21.0923	−1.50277	−0.751384	+	0.659866i	$0.770613\pi$
$198$	0	0
$199$	22.3575	1.58488	0.792441	−	0.609948i	$-0.208810\pi$
$199$	22.3575	1.58488	0.792441	+	0.609948i	$0.208810\pi$
$200$	0	0
$201$	0	0
$202$	−2.46028	−0.173105
$203$	−7.33185	−0.514595
$204$	0	0
$205$	0	0
$206$	10.2286	0.712661
$207$	0	0
$208$	1.15257	0.0799160
$209$	−9.64296	−0.667017
$210$	0	0
$211$	10.0509	0.691929	0.345965	−	0.938248i	$-0.387552\pi$
$211$	10.0509	0.691929	0.345965	+	0.938248i	$0.387552\pi$
$212$	19.6370	1.34867
$213$	0	0
$214$	−13.8938	−0.949764
$215$	0	0
$216$	0	0
$217$	−4.90321	−0.332852
$218$	4.66370	0.315866
$219$	0	0
$220$	0	0
$221$	5.63804	0.379255
$222$	0	0
$223$	−19.5812	−1.31125	−0.655627	−	0.755085i	$-0.727596\pi$
$223$	−19.5812	−1.31125	−0.655627	+	0.755085i	$0.727596\pi$
$224$	−28.4701	−1.90224
$225$	0	0
$226$	−6.84743	−0.455485
$227$	7.08250	0.470082	0.235041	−	0.971985i	$-0.424478\pi$
$227$	7.08250	0.470082	0.235041	+	0.971985i	$0.424478\pi$
$228$	0	0
$229$	−4.25581	−0.281232	−0.140616	−	0.990064i	$-0.544908\pi$
$229$	−4.25581	−0.281232	−0.140616	+	0.990064i	$0.544908\pi$
$230$	0	0
$231$	0	0
$232$	−3.63158	−0.238425
$233$	23.0321	1.50888	0.754442	−	0.656367i	$-0.227908\pi$
$233$	23.0321	1.50888	0.754442	+	0.656367i	$0.227908\pi$
$234$	0	0
$235$	0	0
$236$	−12.4603	−0.811095
$237$	0	0
$238$	−22.7654	−1.47566
$239$	−2.41282	−0.156072	−0.0780361	−	0.996951i	$-0.524865\pi$
$239$	−2.41282	−0.156072	−0.0780361	+	0.996951i	$0.524865\pi$
$240$	0	0
$241$	−25.7669	−1.65979	−0.829897	−	0.557916i	$-0.811601\pi$
$241$	−25.7669	−1.65979	−0.829897	+	0.557916i	$0.811601\pi$
$242$	−0.460282	−0.0295881
$243$	0	0
$244$	−1.15257	−0.0737854
$245$	0	0
$246$	0	0
$247$	2.50961	0.159682
$248$	−2.42864	−0.154219
$249$	0	0
$250$	0	0
$251$	−22.3526	−1.41088	−0.705442	−	0.708768i	$-0.749251\pi$
$251$	−22.3526	−1.41088	−0.705442	+	0.708768i	$0.749251\pi$
$252$	0	0
$253$	−16.7605	−1.05372
$254$	3.14272	0.197192
$255$	0	0
$256$	−9.89829	−0.618643
$257$	18.7719	1.17096	0.585478	−	0.810688i	$-0.300907\pi$
$257$	18.7719	1.17096	0.585478	+	0.810688i	$0.300907\pi$
$258$	0	0
$259$	2.00000	0.124274
$260$	0	0
$261$	0	0
$262$	−13.1985	−0.815406
$263$	−14.6222	−0.901644	−0.450822	−	0.892614i	$-0.648869\pi$
$263$	−14.6222	−0.901644	−0.450822	+	0.892614i	$0.648869\pi$
$264$	0	0
$265$	0	0
$266$	−10.1334	−0.621316
$267$	0	0
$268$	4.35106	0.265784
$269$	6.52051	0.397562	0.198781	−	0.980044i	$-0.436302\pi$
$269$	6.52051	0.397562	0.198781	+	0.980044i	$0.436302\pi$
$270$	0	0
$271$	14.8780	0.903776	0.451888	−	0.892075i	$-0.350751\pi$
$271$	14.8780	0.903776	0.451888	+	0.892075i	$0.350751\pi$
$272$	9.28592	0.563042
$273$	0	0
$274$	4.43954	0.268202
$275$	0	0
$276$	0	0
$277$	0.133353	0.00801241	0.00400621	−	0.999992i	$-0.498725\pi$
$277$	0.133353	0.00801241	0.00400621	+	0.999992i	$0.498725\pi$
$278$	12.8716	0.771985
$279$	0	0
$280$	0	0
$281$	2.85728	0.170451	0.0852255	−	0.996362i	$-0.472839\pi$
$281$	2.85728	0.170451	0.0852255	+	0.996362i	$0.472839\pi$
$282$	0	0
$283$	−5.04149	−0.299685	−0.149843	−	0.988710i	$-0.547877\pi$
$283$	−5.04149	−0.299685	−0.149843	+	0.988710i	$0.547877\pi$
$284$	−12.6539	−0.750868
$285$	0	0
$286$	−1.85236	−0.109532
$287$	46.8069	2.76292
$288$	0	0
$289$	28.4242	1.67201
$290$	0	0
$291$	0	0
$292$	21.4193	1.25347
$293$	12.8222	0.749084	0.374542	−	0.927210i	$-0.377800\pi$
$293$	12.8222	0.749084	0.374542	+	0.927210i	$0.377800\pi$
$294$	0	0
$295$	0	0
$296$	0.990632	0.0575793
$297$	0	0
$298$	0.445992	0.0258356
$299$	4.36196	0.252259
$300$	0	0
$301$	−37.3274	−2.15152
$302$	−7.38763	−0.425110
$303$	0	0
$304$	4.13335	0.237064
$305$	0	0
$306$	0	0
$307$	−21.1066	−1.20462	−0.602310	−	0.798263i	$-0.705753\pi$
$307$	−21.1066	−1.20462	−0.602310	+	0.798263i	$0.705753\pi$
$308$	−24.0415	−1.36989
$309$	0	0
$310$	0	0
$311$	31.8415	1.80556	0.902782	−	0.430099i	$-0.141521\pi$
$311$	31.8415	1.80556	0.902782	+	0.430099i	$0.141521\pi$
$312$	0	0
$313$	−1.35704	−0.0767045	−0.0383522	−	0.999264i	$-0.512211\pi$
$313$	−1.35704	−0.0767045	−0.0383522	+	0.999264i	$0.512211\pi$
$314$	5.50468	0.310647
$315$	0	0
$316$	4.23506	0.238241
$317$	−8.40345	−0.471985	−0.235992	−	0.971755i	$-0.575834\pi$
$317$	−8.40345	−0.471985	−0.235992	+	0.971755i	$0.575834\pi$
$318$	0	0
$319$	−4.80642	−0.269108
$320$	0	0
$321$	0	0
$322$	−17.6128	−0.981526
$323$	20.2192	1.12503
$324$	0	0
$325$	0	0
$326$	−16.0065	−0.886515
$327$	0	0
$328$	23.1842	1.28013
$329$	32.4701	1.79014
$330$	0	0
$331$	2.08097	0.114380	0.0571901	−	0.998363i	$-0.481786\pi$
$331$	2.08097	0.114380	0.0571901	+	0.998363i	$0.481786\pi$
$332$	−16.6079	−0.911478
$333$	0	0
$334$	−0.203420	−0.0111307
$335$	0	0
$336$	0	0
$337$	−1.51897	−0.0827438	−0.0413719	−	0.999144i	$-0.513173\pi$
$337$	−1.51897	−0.0827438	−0.0413719	+	0.999144i	$0.513173\pi$
$338$	−8.47352	−0.460899
$339$	0	0
$340$	0	0
$341$	−3.21432	−0.174065
$342$	0	0
$343$	−49.2355	−2.65847
$344$	−18.4889	−0.996853
$345$	0	0
$346$	−4.63651	−0.249260
$347$	7.71900	0.414378	0.207189	−	0.978301i	$-0.433569\pi$
$347$	7.71900	0.414378	0.207189	+	0.978301i	$0.433569\pi$
$348$	0	0
$349$	−29.1432	−1.56000	−0.780000	−	0.625780i	$-0.784781\pi$
$349$	−29.1432	−1.56000	−0.780000	+	0.625780i	$0.784781\pi$
$350$	0	0
$351$	0	0
$352$	−18.6637	−0.994779
$353$	−17.8637	−0.950791	−0.475395	−	0.879772i	$-0.657695\pi$
$353$	−17.8637	−0.950791	−0.475395	+	0.879772i	$0.657695\pi$
$354$	0	0
$355$	0	0
$356$	−4.24935	−0.225215
$357$	0	0
$358$	10.9491	0.578680
$359$	15.5812	0.822345	0.411172	−	0.911558i	$-0.365119\pi$
$359$	15.5812	0.822345	0.411172	+	0.911558i	$0.365119\pi$
$360$	0	0
$361$	−10.0000	−0.526316
$362$	−8.91306	−0.468460
$363$	0	0
$364$	6.25686	0.327949
$365$	0	0
$366$	0	0
$367$	−18.7239	−0.977381	−0.488690	−	0.872457i	$-0.662525\pi$
$367$	−18.7239	−0.977381	−0.488690	+	0.872457i	$0.662525\pi$
$368$	7.18421	0.374503
$369$	0	0
$370$	0	0
$371$	63.1195	3.27700
$372$	0	0
$373$	8.89384	0.460506	0.230253	−	0.973131i	$-0.426045\pi$
$373$	8.89384	0.460506	0.230253	+	0.973131i	$0.426045\pi$
$374$	−14.9240	−0.771699
$375$	0	0
$376$	16.0830	0.829416
$377$	1.25088	0.0644238
$378$	0	0
$379$	4.76986	0.245011	0.122506	−	0.992468i	$-0.460907\pi$
$379$	4.76986	0.245011	0.122506	+	0.992468i	$0.460907\pi$
$380$	0	0
$381$	0	0
$382$	7.86220	0.402265
$383$	−8.29237	−0.423720	−0.211860	−	0.977300i	$-0.567952\pi$
$383$	−8.29237	−0.423720	−0.211860	+	0.977300i	$0.567952\pi$
$384$	0	0
$385$	0	0
$386$	−9.14917	−0.465681
$387$	0	0
$388$	13.6588	0.693420
$389$	15.3274	0.777130	0.388565	−	0.921421i	$-0.372971\pi$
$389$	15.3274	0.777130	0.388565	+	0.921421i	$0.372971\pi$
$390$	0	0
$391$	35.1432	1.77727
$392$	−41.3876	−2.09039
$393$	0	0
$394$	−14.5303	−0.732028
$395$	0	0
$396$	0	0
$397$	−17.5763	−0.882128	−0.441064	−	0.897476i	$-0.645399\pi$
$397$	−17.5763	−0.882128	−0.441064	+	0.897476i	$0.645399\pi$
$398$	15.4019	0.772028
$399$	0	0
$400$	0	0
$401$	13.2874	0.663544	0.331772	−	0.943360i	$-0.392354\pi$
$401$	13.2874	0.663544	0.331772	+	0.943360i	$0.392354\pi$
$402$	0	0
$403$	0.836535	0.0416708
$404$	5.44785	0.271041
$405$	0	0
$406$	−5.05086	−0.250670
$407$	1.31111	0.0649892
$408$	0	0
$409$	5.11108	0.252727	0.126363	−	0.991984i	$-0.459669\pi$
$409$	5.11108	0.252727	0.126363	+	0.991984i	$0.459669\pi$
$410$	0	0
$411$	0	0
$412$	−22.6494	−1.11586
$413$	−40.0513	−1.97080
$414$	0	0
$415$	0	0
$416$	4.85728	0.238148
$417$	0	0
$418$	−6.64296	−0.324918
$419$	6.88892	0.336546	0.168273	−	0.985740i	$-0.446181\pi$
$419$	6.88892	0.336546	0.168273	+	0.985740i	$0.446181\pi$
$420$	0	0
$421$	−5.00492	−0.243925	−0.121962	−	0.992535i	$-0.538919\pi$
$421$	−5.00492	−0.243925	−0.121962	+	0.992535i	$0.538919\pi$
$422$	6.92396	0.337053
$423$	0	0
$424$	31.2641	1.51832
$425$	0	0
$426$	0	0
$427$	−3.70471	−0.179284
$428$	30.7654	1.48710
$429$	0	0
$430$	0	0
$431$	−28.7239	−1.38358	−0.691791	−	0.722097i	$-0.743178\pi$
$431$	−28.7239	−1.38358	−0.691791	+	0.722097i	$0.743178\pi$
$432$	0	0
$433$	−8.38715	−0.403061	−0.201530	−	0.979482i	$-0.564591\pi$
$433$	−8.38715	−0.403061	−0.201530	+	0.979482i	$0.564591\pi$
$434$	−3.37778	−0.162139
$435$	0	0
$436$	−10.3269	−0.494570
$437$	15.6430	0.748304
$438$	0	0
$439$	4.38271	0.209175	0.104588	−	0.994516i	$-0.466648\pi$
$439$	4.38271	0.209175	0.104588	+	0.994516i	$0.466648\pi$
$440$	0	0
$441$	0	0
$442$	3.88400	0.184743
$443$	27.7496	1.31842	0.659211	−	0.751958i	$-0.270890\pi$
$443$	27.7496	1.31842	0.659211	+	0.751958i	$0.270890\pi$
$444$	0	0
$445$	0	0
$446$	−13.4893	−0.638739
$447$	0	0
$448$	−6.10171	−0.288279
$449$	−7.92840	−0.374164	−0.187082	−	0.982344i	$-0.559903\pi$
$449$	−7.92840	−0.374164	−0.187082	+	0.982344i	$0.559903\pi$
$450$	0	0
$451$	30.6844	1.44487
$452$	15.1624	0.713180
$453$	0	0
$454$	4.87908	0.228986
$455$	0	0
$456$	0	0
$457$	−39.9704	−1.86973	−0.934867	−	0.354997i	$-0.884482\pi$
$457$	−39.9704	−1.86973	−0.934867	+	0.354997i	$0.884482\pi$
$458$	−2.93179	−0.136994
$459$	0	0
$460$	0	0
$461$	39.0207	1.81738	0.908689	−	0.417475i	$-0.137085\pi$
$461$	39.0207	1.81738	0.908689	+	0.417475i	$0.137085\pi$
$462$	0	0
$463$	−5.43801	−0.252726	−0.126363	−	0.991984i	$-0.540330\pi$
$463$	−5.43801	−0.252726	−0.126363	+	0.991984i	$0.540330\pi$
$464$	2.06022	0.0956435
$465$	0	0
$466$	15.8666	0.735008
$467$	0.235063	0.0108774	0.00543872	−	0.999985i	$-0.498269\pi$
$467$	0.235063	0.0108774	0.00543872	+	0.999985i	$0.498269\pi$
$468$	0	0
$469$	13.9857	0.645800
$470$	0	0
$471$	0	0
$472$	−19.8381	−0.913121
$473$	−24.4701	−1.12514
$474$	0	0
$475$	0	0
$476$	50.4099	2.31053
$477$	0	0
$478$	−1.66217	−0.0760260
$479$	22.7368	1.03887	0.519436	−	0.854509i	$-0.326142\pi$
$479$	22.7368	1.03887	0.519436	+	0.854509i	$0.326142\pi$
$480$	0	0
$481$	−0.341219	−0.0155583
$482$	−17.7506	−0.808520
$483$	0	0
$484$	1.01921	0.0463278
$485$	0	0
$486$	0	0
$487$	6.77631	0.307064	0.153532	−	0.988144i	$-0.450935\pi$
$487$	6.77631	0.307064	0.153532	+	0.988144i	$0.450935\pi$
$488$	−1.83500	−0.0830667
$489$	0	0
$490$	0	0
$491$	9.87802	0.445789	0.222894	−	0.974843i	$-0.428449\pi$
$491$	9.87802	0.445789	0.222894	+	0.974843i	$0.428449\pi$
$492$	0	0
$493$	10.0781	0.453893
$494$	1.72885	0.0777846
$495$	0	0
$496$	1.37778	0.0618643
$497$	−40.6735	−1.82446
$498$	0	0
$499$	−26.9512	−1.20650	−0.603250	−	0.797552i	$-0.706128\pi$
$499$	−26.9512	−1.20650	−0.603250	+	0.797552i	$0.706128\pi$
$500$	0	0
$501$	0	0
$502$	−15.3985	−0.687270
$503$	23.2672	1.03743	0.518716	−	0.854946i	$-0.326410\pi$
$503$	23.2672	1.03743	0.518716	+	0.854946i	$0.326410\pi$
$504$	0	0
$505$	0	0
$506$	−11.5462	−0.513290
$507$	0	0
$508$	−6.95899	−0.308755
$509$	−40.4543	−1.79311	−0.896553	−	0.442937i	$-0.853937\pi$
$509$	−40.4543	−1.79311	−0.896553	+	0.442937i	$0.853937\pi$
$510$	0	0
$511$	68.8484	3.04567
$512$	14.6923	0.649313
$513$	0	0
$514$	12.9318	0.570397
$515$	0	0
$516$	0	0
$517$	21.2859	0.936154
$518$	1.37778	0.0605364
$519$	0	0
$520$	0	0
$521$	−18.0667	−0.791515	−0.395758	−	0.918355i	$-0.629518\pi$
$521$	−18.0667	−0.791515	−0.395758	+	0.918355i	$0.629518\pi$
$522$	0	0
$523$	13.3876	0.585400	0.292700	−	0.956204i	$-0.405446\pi$
$523$	13.3876	0.585400	0.292700	+	0.956204i	$0.405446\pi$
$524$	29.2257	1.27673
$525$	0	0
$526$	−10.0731	−0.439209
$527$	6.73975	0.293588
$528$	0	0
$529$	4.18913	0.182136
$530$	0	0
$531$	0	0
$532$	22.4385	0.972832
$533$	−7.98571	−0.345899
$534$	0	0
$535$	0	0
$536$	6.92735	0.299216
$537$	0	0
$538$	4.49193	0.193661
$539$	−54.7768	−2.35940
$540$	0	0
$541$	13.9733	0.600758	0.300379	−	0.953820i	$-0.402887\pi$
$541$	13.9733	0.600758	0.300379	+	0.953820i	$0.402887\pi$
$542$	10.2494	0.440247
$543$	0	0
$544$	39.1338	1.67785
$545$	0	0
$546$	0	0
$547$	−31.2810	−1.33748	−0.668739	−	0.743497i	$-0.733166\pi$
$547$	−31.2810	−1.33748	−0.668739	+	0.743497i	$0.733166\pi$
$548$	−9.83056	−0.419941
$549$	0	0
$550$	0	0
$551$	4.48595	0.191108
$552$	0	0
$553$	13.6128	0.578877
$554$	0.0918659	0.00390301
$555$	0	0
$556$	−28.5018	−1.20874
$557$	−13.0939	−0.554805	−0.277403	−	0.960754i	$-0.589474\pi$
$557$	−13.0939	−0.554805	−0.277403	+	0.960754i	$0.589474\pi$
$558$	0	0
$559$	6.36842	0.269355
$560$	0	0
$561$	0	0
$562$	1.96836	0.0830301
$563$	−31.5560	−1.32993	−0.664964	−	0.746876i	$-0.731553\pi$
$563$	−31.5560	−1.32993	−0.664964	+	0.746876i	$0.731553\pi$
$564$	0	0
$565$	0	0
$566$	−3.47304	−0.145983
$567$	0	0
$568$	−20.1463	−0.845318
$569$	6.30666	0.264389	0.132195	−	0.991224i	$-0.457798\pi$
$569$	6.30666	0.264389	0.132195	+	0.991224i	$0.457798\pi$
$570$	0	0
$571$	−22.3368	−0.934765	−0.467382	−	0.884055i	$-0.654803\pi$
$571$	−22.3368	−0.934765	−0.467382	+	0.884055i	$0.654803\pi$
$572$	4.10171	0.171501
$573$	0	0
$574$	32.2449	1.34588
$575$	0	0
$576$	0	0
$577$	−29.4621	−1.22653	−0.613263	−	0.789879i	$-0.710143\pi$
$577$	−29.4621	−1.22653	−0.613263	+	0.789879i	$0.710143\pi$
$578$	19.5812	0.814471
$579$	0	0
$580$	0	0
$581$	−53.3832	−2.21471
$582$	0	0
$583$	41.3783	1.71371
$584$	34.1017	1.41114
$585$	0	0
$586$	8.83314	0.364894
$587$	26.4499	1.09170	0.545851	−	0.837882i	$-0.316206\pi$
$587$	26.4499	1.09170	0.545851	+	0.837882i	$0.316206\pi$
$588$	0	0
$589$	3.00000	0.123613
$590$	0	0
$591$	0	0
$592$	−0.561993	−0.0230978
$593$	−35.5047	−1.45800	−0.729001	−	0.684512i	$-0.760015\pi$
$593$	−35.5047	−1.45800	−0.729001	+	0.684512i	$0.760015\pi$
$594$	0	0
$595$	0	0
$596$	−0.987569	−0.0404524
$597$	0	0
$598$	3.00492	0.122880
$599$	28.5368	1.16598	0.582991	−	0.812479i	$-0.301882\pi$
$599$	28.5368	1.16598	0.582991	+	0.812479i	$0.301882\pi$
$600$	0	0
$601$	20.1639	0.822504	0.411252	−	0.911522i	$-0.365092\pi$
$601$	20.1639	0.822504	0.411252	+	0.911522i	$0.365092\pi$
$602$	−25.7146	−1.04805
$603$	0	0
$604$	16.3586	0.665621
$605$	0	0
$606$	0	0
$607$	−43.4371	−1.76306	−0.881529	−	0.472130i	$-0.843485\pi$
$607$	−43.4371	−1.76306	−0.881529	+	0.472130i	$0.843485\pi$
$608$	17.4193	0.706445
$609$	0	0
$610$	0	0
$611$	−5.53972	−0.224113
$612$	0	0
$613$	−45.7560	−1.84807	−0.924035	−	0.382309i	$-0.875129\pi$
$613$	−45.7560	−1.84807	−0.924035	+	0.382309i	$0.875129\pi$
$614$	−14.5402	−0.586794
$615$	0	0
$616$	−38.2766	−1.54221
$617$	−1.93332	−0.0778327	−0.0389163	−	0.999242i	$-0.512391\pi$
$617$	−1.93332	−0.0778327	−0.0389163	+	0.999242i	$0.512391\pi$
$618$	0	0
$619$	−29.9684	−1.20453	−0.602265	−	0.798296i	$-0.705735\pi$
$619$	−29.9684	−1.20453	−0.602265	+	0.798296i	$0.705735\pi$
$620$	0	0
$621$	0	0
$622$	21.9353	0.879527
$623$	−13.6588	−0.547227
$624$	0	0
$625$	0	0
$626$	−0.934855	−0.0373643
$627$	0	0
$628$	−12.1891	−0.486399
$629$	−2.74912	−0.109614
$630$	0	0
$631$	31.2652	1.24465	0.622323	−	0.782760i	$-0.286189\pi$
$631$	31.2652	1.24465	0.622323	+	0.782760i	$0.286189\pi$
$632$	6.74266	0.268209
$633$	0	0
$634$	−5.78907	−0.229913
$635$	0	0
$636$	0	0
$637$	14.2558	0.564836
$638$	−3.31111	−0.131088
$639$	0	0
$640$	0	0
$641$	−34.3210	−1.35560	−0.677798	−	0.735248i	$-0.737066\pi$
$641$	−34.3210	−1.35560	−0.677798	+	0.735248i	$0.737066\pi$
$642$	0	0
$643$	8.17283	0.322305	0.161153	−	0.986930i	$-0.448479\pi$
$643$	8.17283	0.322305	0.161153	+	0.986930i	$0.448479\pi$
$644$	39.0005	1.53683
$645$	0	0
$646$	13.9289	0.548024
$647$	−12.7768	−0.502307	−0.251154	−	0.967947i	$-0.580810\pi$
$647$	−12.7768	−0.502307	−0.251154	+	0.967947i	$0.580810\pi$
$648$	0	0
$649$	−26.2558	−1.03063
$650$	0	0
$651$	0	0
$652$	35.4434	1.38807
$653$	0.914111	0.0357719	0.0178860	−	0.999840i	$-0.494306\pi$
$653$	0.914111	0.0357719	0.0178860	+	0.999840i	$0.494306\pi$
$654$	0	0
$655$	0	0
$656$	−13.1526	−0.513521
$657$	0	0
$658$	22.3684	0.872012
$659$	33.0736	1.28836	0.644182	−	0.764872i	$-0.277198\pi$
$659$	33.0736	1.28836	0.644182	+	0.764872i	$0.277198\pi$
$660$	0	0
$661$	25.4973	0.991731	0.495865	−	0.868399i	$-0.334851\pi$
$661$	25.4973	0.991731	0.495865	+	0.868399i	$0.334851\pi$
$662$	1.43356	0.0557170
$663$	0	0
$664$	−26.4415	−1.02613
$665$	0	0
$666$	0	0
$667$	7.79706	0.301903
$668$	0.450438	0.0174280
$669$	0	0
$670$	0	0
$671$	−2.42864	−0.0937566
$672$	0	0
$673$	−43.7669	−1.68709	−0.843546	−	0.537057i	$-0.819536\pi$
$673$	−43.7669	−1.68709	−0.843546	+	0.537057i	$0.819536\pi$
$674$	−1.04641	−0.0403062
$675$	0	0
$676$	18.7631	0.721657
$677$	−29.9639	−1.15161	−0.575803	−	0.817588i	$-0.695311\pi$
$677$	−29.9639	−1.15161	−0.575803	+	0.817588i	$0.695311\pi$
$678$	0	0
$679$	43.9037	1.68487
$680$	0	0
$681$	0	0
$682$	−2.21432	−0.0847907
$683$	11.4064	0.436452	0.218226	−	0.975898i	$-0.429973\pi$
$683$	11.4064	0.436452	0.218226	+	0.975898i	$0.429973\pi$
$684$	0	0
$685$	0	0
$686$	−33.9180	−1.29499
$687$	0	0
$688$	10.4889	0.399884
$689$	−10.7688	−0.410259
$690$	0	0
$691$	−37.9403	−1.44332	−0.721658	−	0.692250i	$-0.756619\pi$
$691$	−37.9403	−1.44332	−0.721658	+	0.692250i	$0.756619\pi$
$692$	10.2667	0.390282
$693$	0	0
$694$	5.31756	0.201852
$695$	0	0
$696$	0	0
$697$	−64.3388	−2.43701
$698$	−20.0765	−0.759908
$699$	0	0
$700$	0	0
$701$	42.5654	1.60767	0.803836	−	0.594851i	$-0.202789\pi$
$701$	42.5654	1.60767	0.803836	+	0.594851i	$0.202789\pi$
$702$	0	0
$703$	−1.22369	−0.0461523
$704$	−4.00000	−0.150756
$705$	0	0
$706$	−12.3062	−0.463149
$707$	17.5111	0.658574
$708$	0	0
$709$	−21.2050	−0.796369	−0.398184	−	0.917305i	$-0.630360\pi$
$709$	−21.2050	−0.796369	−0.398184	+	0.917305i	$0.630360\pi$
$710$	0	0
$711$	0	0
$712$	−6.76541	−0.253545
$713$	5.21432	0.195278
$714$	0	0
$715$	0	0
$716$	−24.2449	−0.906075
$717$	0	0
$718$	10.7338	0.400581
$719$	−39.8992	−1.48799	−0.743995	−	0.668185i	$-0.767072\pi$
$719$	−39.8992	−1.48799	−0.743995	+	0.668185i	$0.767072\pi$
$720$	0	0
$721$	−72.8025	−2.71131
$722$	−6.88892	−0.256379
$723$	0	0
$724$	19.7364	0.733496
$725$	0	0
$726$	0	0
$727$	3.48394	0.129212	0.0646061	−	0.997911i	$-0.479421\pi$
$727$	3.48394	0.129212	0.0646061	+	0.997911i	$0.479421\pi$
$728$	9.96158	0.369201
$729$	0	0
$730$	0	0
$731$	51.3087	1.89772
$732$	0	0
$733$	15.2034	0.561551	0.280776	−	0.959773i	$-0.409408\pi$
$733$	15.2034	0.561551	0.280776	+	0.959773i	$0.409408\pi$
$734$	−12.8988	−0.476102
$735$	0	0
$736$	30.2766	1.11601
$737$	9.16839	0.337722
$738$	0	0
$739$	−27.7067	−1.01921	−0.509604	−	0.860409i	$-0.670208\pi$
$739$	−27.7067	−1.01921	−0.509604	+	0.860409i	$0.670208\pi$
$740$	0	0
$741$	0	0
$742$	43.4826	1.59629
$743$	1.97973	0.0726294	0.0363147	−	0.999340i	$-0.488438\pi$
$743$	1.97973	0.0726294	0.0363147	+	0.999340i	$0.488438\pi$
$744$	0	0
$745$	0	0
$746$	6.12690	0.224322
$747$	0	0
$748$	33.0464	1.20830
$749$	98.8899	3.61336
$750$	0	0
$751$	7.50622	0.273906	0.136953	−	0.990578i	$-0.456269\pi$
$751$	7.50622	0.273906	0.136953	+	0.990578i	$0.456269\pi$
$752$	−9.12399	−0.332718
$753$	0	0
$754$	0.861725	0.0313822
$755$	0	0
$756$	0	0
$757$	21.2938	0.773935	0.386967	−	0.922093i	$-0.373523\pi$
$757$	21.2938	0.773935	0.386967	+	0.922093i	$0.373523\pi$
$758$	3.28592	0.119350
$759$	0	0
$760$	0	0
$761$	−1.88094	−0.0681839	−0.0340920	−	0.999419i	$-0.510854\pi$
$761$	−1.88094	−0.0681839	−0.0340920	+	0.999419i	$0.510854\pi$
$762$	0	0
$763$	−33.1941	−1.20171
$764$	−17.4094	−0.629851
$765$	0	0
$766$	−5.71255	−0.206403
$767$	6.83314	0.246731
$768$	0	0
$769$	5.51606	0.198914	0.0994571	−	0.995042i	$-0.468289\pi$
$769$	5.51606	0.198914	0.0994571	+	0.995042i	$0.468289\pi$
$770$	0	0
$771$	0	0
$772$	20.2592	0.729144
$773$	29.8780	1.07464	0.537319	−	0.843379i	$-0.319437\pi$
$773$	29.8780	1.07464	0.537319	+	0.843379i	$0.319437\pi$
$774$	0	0
$775$	0	0
$776$	21.7462	0.780643
$777$	0	0
$778$	10.5589	0.378556
$779$	−28.6385	−1.02608
$780$	0	0
$781$	−26.6637	−0.954102
$782$	24.2099	0.865743
$783$	0	0
$784$	23.4795	0.838553
$785$	0	0
$786$	0	0
$787$	−24.0000	−0.855508	−0.427754	−	0.903895i	$-0.640695\pi$
$787$	−24.0000	−0.855508	−0.427754	+	0.903895i	$0.640695\pi$
$788$	32.1748	1.14618
$789$	0	0
$790$	0	0
$791$	48.7368	1.73288
$792$	0	0
$793$	0.632060	0.0224451
$794$	−12.1082	−0.429703
$795$	0	0
$796$	−34.1048	−1.20881
$797$	−42.7640	−1.51478	−0.757390	−	0.652963i	$-0.773526\pi$
$797$	−42.7640	−1.51478	−0.757390	+	0.652963i	$0.773526\pi$
$798$	0	0
$799$	−44.6321	−1.57897
$800$	0	0
$801$	0	0
$802$	9.15362	0.323226
$803$	45.1338	1.59274
$804$	0	0
$805$	0	0
$806$	0.576283	0.0202987
$807$	0	0
$808$	8.67355	0.305134
$809$	−38.2420	−1.34452	−0.672258	−	0.740317i	$-0.734676\pi$
$809$	−38.2420	−1.34452	−0.672258	+	0.740317i	$0.734676\pi$
$810$	0	0
$811$	−46.5388	−1.63420	−0.817099	−	0.576497i	$-0.804419\pi$
$811$	−46.5388	−1.63420	−0.817099	+	0.576497i	$0.804419\pi$
$812$	11.1842	0.392489
$813$	0	0
$814$	0.903212	0.0316576
$815$	0	0
$816$	0	0
$817$	22.8385	0.799019
$818$	3.52098	0.123108
$819$	0	0
$820$	0	0
$821$	3.80796	0.132899	0.0664493	−	0.997790i	$-0.478833\pi$
$821$	3.80796	0.132899	0.0664493	+	0.997790i	$0.478833\pi$
$822$	0	0
$823$	51.5910	1.79835	0.899175	−	0.437588i	$-0.144167\pi$
$823$	51.5910	1.79835	0.899175	+	0.437588i	$0.144167\pi$
$824$	−36.0602	−1.25622
$825$	0	0
$826$	−27.5910	−0.960015
$827$	−55.1595	−1.91808	−0.959042	−	0.283265i	$-0.908583\pi$
$827$	−55.1595	−1.91808	−0.959042	+	0.283265i	$0.908583\pi$
$828$	0	0
$829$	−8.60147	−0.298741	−0.149371	−	0.988781i	$-0.547725\pi$
$829$	−8.60147	−0.298741	−0.149371	+	0.988781i	$0.547725\pi$
$830$	0	0
$831$	0	0
$832$	1.04101	0.0360906
$833$	114.855	3.97950
$834$	0	0
$835$	0	0
$836$	14.7096	0.508743
$837$	0	0
$838$	4.74572	0.163938
$839$	17.2159	0.594357	0.297179	−	0.954822i	$-0.403954\pi$
$839$	17.2159	0.594357	0.297179	+	0.954822i	$0.403954\pi$
$840$	0	0
$841$	−26.7640	−0.922898
$842$	−3.44785	−0.118821
$843$	0	0
$844$	−15.3319	−0.527744
$845$	0	0
$846$	0	0
$847$	3.27607	0.112567
$848$	−17.7364	−0.609069
$849$	0	0
$850$	0	0
$851$	−2.12690	−0.0729092
$852$	0	0
$853$	42.4943	1.45498	0.727488	−	0.686121i	$-0.240688\pi$
$853$	42.4943	1.45498	0.727488	+	0.686121i	$0.240688\pi$
$854$	−2.55215	−0.0873327
$855$	0	0
$856$	48.9817	1.67416
$857$	−5.37133	−0.183481	−0.0917406	−	0.995783i	$-0.529243\pi$
$857$	−5.37133	−0.183481	−0.0917406	+	0.995783i	$0.529243\pi$
$858$	0	0
$859$	−10.6539	−0.363505	−0.181752	−	0.983344i	$-0.558177\pi$
$859$	−10.6539	−0.363505	−0.181752	+	0.983344i	$0.558177\pi$
$860$	0	0
$861$	0	0
$862$	−19.7877	−0.673971
$863$	−38.7753	−1.31993	−0.659963	−	0.751298i	$-0.729428\pi$
$863$	−38.7753	−1.31993	−0.659963	+	0.751298i	$0.729428\pi$
$864$	0	0
$865$	0	0
$866$	−5.77784	−0.196339
$867$	0	0
$868$	7.47949	0.253871
$869$	8.92396	0.302724
$870$	0	0
$871$	−2.38610	−0.0808498
$872$	−16.4415	−0.556781
$873$	0	0
$874$	10.7763	0.364514
$875$	0	0
$876$	0	0
$877$	42.7150	1.44238	0.721192	−	0.692735i	$-0.243594\pi$
$877$	42.7150	1.44238	0.721192	+	0.692735i	$0.243594\pi$
$878$	3.01921	0.101893
$879$	0	0
$880$	0	0
$881$	2.54909	0.0858809	0.0429404	−	0.999078i	$-0.486327\pi$
$881$	2.54909	0.0858809	0.0429404	+	0.999078i	$0.486327\pi$
$882$	0	0
$883$	37.7799	1.27139	0.635697	−	0.771939i	$-0.280713\pi$
$883$	37.7799	1.27139	0.635697	+	0.771939i	$0.280713\pi$
$884$	−8.60042	−0.289263
$885$	0	0
$886$	19.1165	0.642231
$887$	−46.6287	−1.56564	−0.782819	−	0.622250i	$-0.786219\pi$
$887$	−46.6287	−1.56564	−0.782819	+	0.622250i	$0.786219\pi$
$888$	0	0
$889$	−22.3684	−0.750213
$890$	0	0
$891$	0	0
$892$	29.8697	1.00011
$893$	−19.8666	−0.664812
$894$	0	0
$895$	0	0
$896$	52.7368	1.76181
$897$	0	0
$898$	−5.46181	−0.182263
$899$	1.49532	0.0498716
$900$	0	0
$901$	−86.7614	−2.89044
$902$	21.1383	0.703828
$903$	0	0
$904$	24.1401	0.802889
$905$	0	0
$906$	0	0
$907$	38.2958	1.27159	0.635795	−	0.771858i	$-0.280672\pi$
$907$	38.2958	1.27159	0.635795	+	0.771858i	$0.280672\pi$
$908$	−10.8038	−0.358538
$909$	0	0
$910$	0	0
$911$	−25.7175	−0.852058	−0.426029	−	0.904710i	$-0.640088\pi$
$911$	−25.7175	−0.852058	−0.426029	+	0.904710i	$0.640088\pi$
$912$	0	0
$913$	−34.9956	−1.15818
$914$	−27.5353	−0.910786
$915$	0	0
$916$	6.49193	0.214499
$917$	93.9407	3.10220
$918$	0	0
$919$	18.4148	0.607449	0.303725	−	0.952760i	$-0.401770\pi$
$919$	18.4148	0.607449	0.303725	+	0.952760i	$0.401770\pi$
$920$	0	0
$921$	0	0
$922$	26.8811	0.885281
$923$	6.93930	0.228410
$924$	0	0
$925$	0	0
$926$	−3.74620	−0.123108
$927$	0	0
$928$	8.68244	0.285015
$929$	−54.5659	−1.79025	−0.895124	−	0.445817i	$-0.852913\pi$
$929$	−54.5659	−1.79025	−0.895124	+	0.445817i	$0.852913\pi$
$930$	0	0
$931$	51.1245	1.67554
$932$	−35.1338	−1.15085
$933$	0	0
$934$	0.161933	0.00529862
$935$	0	0
$936$	0	0
$937$	−15.1334	−0.494385	−0.247193	−	0.968966i	$-0.579508\pi$
$937$	−15.1334	−0.494385	−0.247193	+	0.968966i	$0.579508\pi$
$938$	9.63465	0.314582
$939$	0	0
$940$	0	0
$941$	28.9704	0.944407	0.472203	−	0.881490i	$-0.343459\pi$
$941$	28.9704	0.944407	0.472203	+	0.881490i	$0.343459\pi$
$942$	0	0
$943$	−49.7768	−1.62096
$944$	11.2543	0.366295
$945$	0	0
$946$	−16.8573	−0.548077
$947$	−16.8272	−0.546809	−0.273405	−	0.961899i	$-0.588150\pi$
$947$	−16.8272	−0.546809	−0.273405	+	0.961899i	$0.588150\pi$
$948$	0	0
$949$	−11.7462	−0.381298
$950$	0	0
$951$	0	0
$952$	80.2578	2.60117
$953$	58.5817	1.89765	0.948823	−	0.315807i	$-0.102275\pi$
$953$	58.5817	1.89765	0.948823	+	0.315807i	$0.102275\pi$
$954$	0	0
$955$	0	0
$956$	3.68058	0.119038
$957$	0	0
$958$	15.6632	0.506056
$959$	−31.5986	−1.02037
$960$	0	0
$961$	1.00000	0.0322581
$962$	−0.235063	−0.00757875
$963$	0	0
$964$	39.3056	1.26595
$965$	0	0
$966$	0	0
$967$	49.4815	1.59122	0.795609	−	0.605811i	$-0.207151\pi$
$967$	49.4815	1.59122	0.795609	+	0.605811i	$0.207151\pi$
$968$	1.62269	0.0521553
$969$	0	0
$970$	0	0
$971$	−24.7620	−0.794651	−0.397326	−	0.917678i	$-0.630062\pi$
$971$	−24.7620	−0.794651	−0.397326	+	0.917678i	$0.630062\pi$
$972$	0	0
$973$	−91.6138	−2.93700
$974$	4.66815	0.149577
$975$	0	0
$976$	1.04101	0.0333219
$977$	−17.3210	−0.554146	−0.277073	−	0.960849i	$-0.589364\pi$
$977$	−17.3210	−0.554146	−0.277073	+	0.960849i	$0.589364\pi$
$978$	0	0
$979$	−8.95407	−0.286173
$980$	0	0
$981$	0	0
$982$	6.80489	0.217153
$983$	−36.8731	−1.17607	−0.588035	−	0.808836i	$-0.700098\pi$
$983$	−36.8731	−1.17607	−0.588035	+	0.808836i	$0.700098\pi$
$984$	0	0
$985$	0	0
$986$	6.94269	0.221100
$987$	0	0
$988$	−3.82822	−0.121792
$989$	39.6958	1.26225
$990$	0	0
$991$	40.8780	1.29853	0.649267	−	0.760561i	$-0.275076\pi$
$991$	40.8780	1.29853	0.649267	+	0.760561i	$0.275076\pi$
$992$	5.80642	0.184354
$993$	0	0
$994$	−28.0197	−0.888731
$995$	0	0
$996$	0	0
$997$	−28.7003	−0.908947	−0.454473	−	0.890760i	$-0.650173\pi$
$997$	−28.7003	−0.908947	−0.454473	+	0.890760i	$0.650173\pi$
$998$	−18.5664	−0.587710
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6975.2.a.bg.1.2	yes	3
3.2	odd	2		6975.2.a.z.1.2	✓	3
5.4	even	2		6975.2.a.ba.1.2	yes	3
15.14	odd	2		6975.2.a.bh.1.2	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6975.2.a.z.1.2	✓	3	3.2	odd	2
6975.2.a.ba.1.2	yes	3	5.4	even	2
6975.2.a.bg.1.2	yes	3	1.1	even	1	trivial
6975.2.a.bh.1.2	yes	3	15.14	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 6975 = 3^{2} \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6975.a (trivial)

\(f(q)\)	\(=\)	\(q+0.688892 q^{2} -1.52543 q^{4} -4.90321 q^{7} -2.42864 q^{8} -3.21432 q^{11} +0.836535 q^{13} -3.37778 q^{14} +1.37778 q^{16} +6.73975 q^{17} +3.00000 q^{19} -2.21432 q^{22} +5.21432 q^{23} +0.576283 q^{26} +7.47949 q^{28} +1.49532 q^{29} +1.00000 q^{31} +5.80642 q^{32} +4.64296 q^{34} -0.407896 q^{37} +2.06668 q^{38} -9.54617 q^{41} +7.61285 q^{43} +4.90321 q^{44} +3.59210 q^{46} -6.62222 q^{47} +17.0415 q^{49} -1.27607 q^{52} -12.8731 q^{53} +11.9081 q^{56} +1.03011 q^{58} +8.16839 q^{59} +0.755569 q^{61} +0.688892 q^{62} +1.24443 q^{64} -2.85236 q^{67} -10.2810 q^{68} +8.29529 q^{71} -14.0415 q^{73} -0.280996 q^{74} -4.57628 q^{76} +15.7605 q^{77} -2.77631 q^{79} -6.57628 q^{82} +10.8874 q^{83} +5.24443 q^{86} +7.80642 q^{88} +2.78568 q^{89} -4.10171 q^{91} -7.95407 q^{92} -4.56199 q^{94} -8.95407 q^{97} +11.7397 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{2} + 2 q^{4} - 8 q^{7} + 6 q^{8} - 3 q^{11} - 4 q^{13} - 10 q^{14} + 4 q^{16} + 7 q^{17} + 9 q^{19} + 9 q^{23} - 18 q^{26} - 4 q^{28} - 9 q^{29} + 3 q^{31} + 4 q^{32} - 6 q^{34} - 8 q^{37} + 6 q^{38}+ \cdots + 22 q^{98}+O(q^{100}) \) 3 * q + 2 * q^2 + 2 * q^4 - 8 * q^7 + 6 * q^8 - 3 * q^11 - 4 * q^13 - 10 * q^14 + 4 * q^16 + 7 * q^17 + 9 * q^19 + 9 * q^23 - 18 * q^26 - 4 * q^28 - 9 * q^29 + 3 * q^31 + 4 * q^32 - 6 * q^34 - 8 * q^37 + 6 * q^38 - 2 * q^41 - 4 * q^43 + 8 * q^44 + 4 * q^46 - 20 * q^47 + 11 * q^49 - 30 * q^52 - 25 * q^53 - 4 * q^56 + 10 * q^58 - 2 * q^59 + 2 * q^61 + 2 * q^62 + 4 * q^64 - 15 * q^67 - 24 * q^68 + 12 * q^71 - 2 * q^73 + 6 * q^74 + 6 * q^76 + 14 * q^77 + 12 * q^79 + 13 * q^83 + 16 * q^86 + 10 * q^88 + 15 * q^89 + 14 * q^91 - 4 * q^92 - 7 * q^97 + 22 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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