LMFDB - Embedded newform 6900.2.a.ba.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6900,2,Mod(1,6900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6900, 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("6900.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6900.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:	$55.0967773947$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.175557.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 8x^{2} + 3x + 2 $ x^4 - 2x^3 - 8x^2 + 3*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.15613$ of defining polynomial
Character	$\chi$	$=$	6900.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +3.22855 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +3.22855 q^{7} +1.00000 q^{9} +2.15613 q^{11} +3.08372 q^{13} -5.11730 q^{17} -1.80504 q^{19} -3.22855 q^{21} +1.00000 q^{23} -1.00000 q^{27} +7.18972 q^{29} -1.50408 q^{31} -2.15613 q^{33} +11.1173 q^{37} -3.08372 q^{39} -6.38468 q^{41} +5.16744 q^{43} +1.19496 q^{47} +3.42351 q^{49} +5.11730 q^{51} +2.57649 q^{53} +1.80504 q^{57} +9.34585 q^{59} -7.96117 q^{61} +3.22855 q^{63} +5.30096 q^{67} -1.00000 q^{69} -0.966415 q^{71} -4.23985 q^{73} +6.96117 q^{77} +7.53767 q^{79} +1.00000 q^{81} -14.1622 q^{83} -7.18972 q^{87} +9.81949 q^{89} +9.95593 q^{91} +1.50408 q^{93} +16.4183 q^{97} +2.15613 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + 3 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 + 3 * q^7 + 4 * q^9	$ 4 q - 4 q^{3} + 3 q^{7} + 4 q^{9} - 2 q^{11} + q^{13} + 10 q^{17} + 2 q^{19} - 3 q^{21} + 4 q^{23} - 4 q^{27} - q^{29} - 6 q^{31} + 2 q^{33} + 14 q^{37} - q^{39} - 5 q^{41} - 2 q^{43} + 14 q^{47} + 13 q^{49} - 10 q^{51} + 11 q^{53} - 2 q^{57} - 3 q^{59} - 12 q^{61} + 3 q^{63} + 12 q^{67} - 4 q^{69} - 23 q^{71} + 5 q^{73} + 8 q^{77} + 11 q^{79} + 4 q^{81} + 5 q^{83} + q^{87} + 6 q^{89} - 19 q^{91} + 6 q^{93} + 26 q^{97} - 2 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 + 3 * q^7 + 4 * q^9 - 2 * q^11 + q^13 + 10 * q^17 + 2 * q^19 - 3 * q^21 + 4 * q^23 - 4 * q^27 - q^29 - 6 * q^31 + 2 * q^33 + 14 * q^37 - q^39 - 5 * q^41 - 2 * q^43 + 14 * q^47 + 13 * q^49 - 10 * q^51 + 11 * q^53 - 2 * q^57 - 3 * q^59 - 12 * q^61 + 3 * q^63 + 12 * q^67 - 4 * q^69 - 23 * q^71 + 5 * q^73 + 8 * q^77 + 11 * q^79 + 4 * q^81 + 5 * q^83 + q^87 + 6 * q^89 - 19 * q^91 + 6 * q^93 + 26 * q^97 - 2 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.22855	1.22028	0.610138	−	0.792295i	$-0.291114\pi$
$7$	3.22855	1.22028	0.610138	+	0.792295i	$0.291114\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.15613	0.650098	0.325049	−	0.945697i	$-0.394619\pi$
$11$	2.15613	0.650098	0.325049	+	0.945697i	$0.394619\pi$
$12$	0	0
$13$	3.08372	0.855270	0.427635	−	0.903952i	$-0.359347\pi$
$13$	3.08372	0.855270	0.427635	+	0.903952i	$0.359347\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.11730	−1.24113	−0.620564	−	0.784156i	$-0.713096\pi$
$17$	−5.11730	−1.24113	−0.620564	+	0.784156i	$0.713096\pi$
$18$	0	0
$19$	−1.80504	−0.414104	−0.207052	−	0.978330i	$-0.566387\pi$
$19$	−1.80504	−0.414104	−0.207052	+	0.978330i	$0.566387\pi$
$20$	0	0
$21$	−3.22855	−0.704526
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	7.18972	1.33510	0.667548	−	0.744566i	$-0.267344\pi$
$29$	7.18972	1.33510	0.667548	+	0.744566i	$0.267344\pi$
$30$	0	0
$31$	−1.50408	−0.270141	−0.135070	−	0.990836i	$-0.543126\pi$
$31$	−1.50408	−0.270141	−0.135070	+	0.990836i	$0.543126\pi$
$32$	0	0
$33$	−2.15613	−0.375334
$34$	0	0
$35$	0	0
$36$	0	0
$37$	11.1173	1.82767	0.913837	−	0.406082i	$-0.133105\pi$
$37$	11.1173	1.82767	0.913837	+	0.406082i	$0.133105\pi$
$38$	0	0
$39$	−3.08372	−0.493790
$40$	0	0
$41$	−6.38468	−0.997119	−0.498560	−	0.866855i	$-0.666137\pi$
$41$	−6.38468	−0.997119	−0.498560	+	0.866855i	$0.666137\pi$
$42$	0	0
$43$	5.16744	0.788027	0.394013	−	0.919105i	$-0.371086\pi$
$43$	5.16744	0.788027	0.394013	+	0.919105i	$0.371086\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.19496	0.174303	0.0871515	−	0.996195i	$-0.472224\pi$
$47$	1.19496	0.174303	0.0871515	+	0.996195i	$0.472224\pi$
$48$	0	0
$49$	3.42351	0.489072
$50$	0	0
$51$	5.11730	0.716566
$52$	0	0
$53$	2.57649	0.353909	0.176954	−	0.984219i	$-0.443376\pi$
$53$	2.57649	0.353909	0.176954	+	0.984219i	$0.443376\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.80504	0.239083
$58$	0	0
$59$	9.34585	1.21673	0.608363	−	0.793659i	$-0.291827\pi$
$59$	9.34585	1.21673	0.608363	+	0.793659i	$0.291827\pi$
$60$	0	0
$61$	−7.96117	−1.01932	−0.509662	−	0.860375i	$-0.670229\pi$
$61$	−7.96117	−1.01932	−0.509662	+	0.860375i	$0.670229\pi$
$62$	0	0
$63$	3.22855	0.406759
$64$	0	0
$65$	0	0
$66$	0	0
$67$	5.30096	0.647615	0.323808	−	0.946123i	$-0.395037\pi$
$67$	5.30096	0.647615	0.323808	+	0.946123i	$0.395037\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−0.966415	−0.114692	−0.0573462	−	0.998354i	$-0.518264\pi$
$71$	−0.966415	−0.114692	−0.0573462	+	0.998354i	$0.518264\pi$
$72$	0	0
$73$	−4.23985	−0.496237	−0.248119	−	0.968730i	$-0.579812\pi$
$73$	−4.23985	−0.496237	−0.248119	+	0.968730i	$0.579812\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	6.96117	0.793299
$78$	0	0
$79$	7.53767	0.848054	0.424027	−	0.905650i	$-0.360616\pi$
$79$	7.53767	0.848054	0.424027	+	0.905650i	$0.360616\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−14.1622	−1.55450	−0.777251	−	0.629190i	$-0.783387\pi$
$83$	−14.1622	−1.55450	−0.777251	+	0.629190i	$0.783387\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−7.18972	−0.770819
$88$	0	0
$89$	9.81949	1.04086	0.520432	−	0.853903i	$-0.325771\pi$
$89$	9.81949	1.04086	0.520432	+	0.853903i	$0.325771\pi$
$90$	0	0
$91$	9.95593	1.04366
$92$	0	0
$93$	1.50408	0.155966
$94$	0	0
$95$	0	0
$96$	0	0
$97$	16.4183	1.66702	0.833511	−	0.552503i	$-0.186327\pi$
$97$	16.4183	1.66702	0.833511	+	0.552503i	$0.186327\pi$
$98$	0	0
$99$	2.15613	0.216699
$100$	0	0
$101$	−11.8386	−1.17799	−0.588994	−	0.808138i	$-0.700476\pi$
$101$	−11.8386	−1.17799	−0.588994	+	0.808138i	$0.700476\pi$
$102$	0	0
$103$	5.57964	0.549778	0.274889	−	0.961476i	$-0.411359\pi$
$103$	5.57964	0.549778	0.274889	+	0.961476i	$0.411359\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.19496	−0.115521	−0.0577606	−	0.998330i	$-0.518396\pi$
$107$	−1.19496	−0.115521	−0.0577606	+	0.998330i	$0.518396\pi$
$108$	0	0
$109$	1.27553	0.122174	0.0610870	−	0.998132i	$-0.480543\pi$
$109$	1.27553	0.122174	0.0610870	+	0.998132i	$0.480543\pi$
$110$	0	0
$111$	−11.1173	−1.05521
$112$	0	0
$113$	−2.39598	−0.225395	−0.112698	−	0.993629i	$-0.535949\pi$
$113$	−2.39598	−0.225395	−0.112698	+	0.993629i	$0.535949\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	3.08372	0.285090
$118$	0	0
$119$	−16.5214	−1.51452
$120$	0	0
$121$	−6.35109	−0.577372
$122$	0	0
$123$	6.38468	0.575687
$124$	0	0
$125$	0	0
$126$	0	0
$127$	3.85517	0.342091	0.171046	−	0.985263i	$-0.445285\pi$
$127$	3.85517	0.342091	0.171046	+	0.985263i	$0.445285\pi$
$128$	0	0
$129$	−5.16744	−0.454968
$130$	0	0
$131$	−7.19496	−0.628627	−0.314313	−	0.949319i	$-0.601774\pi$
$131$	−7.19496	−0.628627	−0.314313	+	0.949319i	$0.601774\pi$
$132$	0	0
$133$	−5.82765	−0.505321
$134$	0	0
$135$	0	0
$136$	0	0
$137$	16.0898	1.37464	0.687321	−	0.726353i	$-0.258786\pi$
$137$	16.0898	1.37464	0.687321	+	0.726353i	$0.258786\pi$
$138$	0	0
$139$	2.26423	0.192049	0.0960247	−	0.995379i	$-0.469387\pi$
$139$	2.26423	0.192049	0.0960247	+	0.995379i	$0.469387\pi$
$140$	0	0
$141$	−1.19496	−0.100634
$142$	0	0
$143$	6.64891	0.556010
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−3.42351	−0.282366
$148$	0	0
$149$	−3.79688	−0.311052	−0.155526	−	0.987832i	$-0.549707\pi$
$149$	−3.79688	−0.311052	−0.155526	+	0.987832i	$0.549707\pi$
$150$	0	0
$151$	−9.90006	−0.805656	−0.402828	−	0.915276i	$-0.631973\pi$
$151$	−9.90006	−0.805656	−0.402828	+	0.915276i	$0.631973\pi$
$152$	0	0
$153$	−5.11730	−0.413710
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−2.84993	−0.227449	−0.113725	−	0.993512i	$-0.536278\pi$
$157$	−2.84993	−0.227449	−0.113725	+	0.993512i	$0.536278\pi$
$158$	0	0
$159$	−2.57649	−0.204329
$160$	0	0
$161$	3.22855	0.254445
$162$	0	0
$163$	13.7998	1.08088	0.540442	−	0.841381i	$-0.318257\pi$
$163$	13.7998	1.08088	0.540442	+	0.841381i	$0.318257\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−22.4632	−1.73825	−0.869126	−	0.494592i	$-0.835318\pi$
$167$	−22.4632	−1.73825	−0.869126	+	0.494592i	$0.835318\pi$
$168$	0	0
$169$	−3.49068	−0.268514
$170$	0	0
$171$	−1.80504	−0.138035
$172$	0	0
$173$	6.71349	0.510417	0.255209	−	0.966886i	$-0.417856\pi$
$173$	6.71349	0.510417	0.255209	+	0.966886i	$0.417856\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−9.34585	−0.702477
$178$	0	0
$179$	0.576494	0.0430892	0.0215446	−	0.999768i	$-0.493142\pi$
$179$	0.576494	0.0430892	0.0215446	+	0.999768i	$0.493142\pi$
$180$	0	0
$181$	−8.53767	−0.634600	−0.317300	−	0.948325i	$-0.602776\pi$
$181$	−8.53767	−0.634600	−0.317300	+	0.948325i	$0.602776\pi$
$182$	0	0
$183$	7.96117	0.588507
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−11.0336	−0.806856
$188$	0	0
$189$	−3.22855	−0.234842
$190$	0	0
$191$	−10.0724	−0.728814	−0.364407	−	0.931240i	$-0.618728\pi$
$191$	−10.0724	−0.728814	−0.364407	+	0.931240i	$0.618728\pi$
$192$	0	0
$193$	0.267374	0.0192460	0.00962300	−	0.999954i	$-0.496937\pi$
$193$	0.267374	0.0192460	0.00962300	+	0.999954i	$0.496937\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2.79980	−0.199477	−0.0997386	−	0.995014i	$-0.531801\pi$
$197$	−2.79980	−0.199477	−0.0997386	+	0.995014i	$0.531801\pi$
$198$	0	0
$199$	−15.1989	−1.07742	−0.538712	−	0.842490i	$-0.681089\pi$
$199$	−15.1989	−1.07742	−0.538712	+	0.842490i	$0.681089\pi$
$200$	0	0
$201$	−5.30096	−0.373901
$202$	0	0
$203$	23.2123	1.62919
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−3.89190	−0.269209
$210$	0	0
$211$	24.1766	1.66439	0.832194	−	0.554484i	$-0.187084\pi$
$211$	24.1766	1.66439	0.832194	+	0.554484i	$0.187084\pi$
$212$	0	0
$213$	0.966415	0.0662177
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−4.85599	−0.329646
$218$	0	0
$219$	4.23985	0.286503
$220$	0	0
$221$	−15.7803	−1.06150
$222$	0	0
$223$	29.5582	1.97936	0.989681	−	0.143288i	$-0.0457675\pi$
$223$	29.5582	1.97936	0.989681	+	0.143288i	$0.0457675\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	8.58256	0.569644	0.284822	−	0.958580i	$-0.408065\pi$
$227$	8.58256	0.569644	0.284822	+	0.958580i	$0.408065\pi$
$228$	0	0
$229$	7.97772	0.527183	0.263591	−	0.964634i	$-0.415093\pi$
$229$	7.97772	0.527183	0.263591	+	0.964634i	$0.415093\pi$
$230$	0	0
$231$	−6.96117	−0.458011
$232$	0	0
$233$	10.9192	0.715340	0.357670	−	0.933848i	$-0.383571\pi$
$233$	10.9192	0.715340	0.357670	+	0.933848i	$0.383571\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−7.53767	−0.489624
$238$	0	0
$239$	5.54291	0.358541	0.179270	−	0.983800i	$-0.442626\pi$
$239$	5.54291	0.358541	0.179270	+	0.983800i	$0.442626\pi$
$240$	0	0
$241$	−9.92549	−0.639357	−0.319678	−	0.947526i	$-0.603575\pi$
$241$	−9.92549	−0.639357	−0.319678	+	0.947526i	$0.603575\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−5.56623	−0.354171
$248$	0	0
$249$	14.1622	0.897493
$250$	0	0
$251$	−24.2764	−1.53231	−0.766155	−	0.642656i	$-0.777832\pi$
$251$	−24.2764	−1.53231	−0.766155	+	0.642656i	$0.777832\pi$
$252$	0	0
$253$	2.15613	0.135555
$254$	0	0
$255$	0	0
$256$	0	0
$257$	9.98345	0.622751	0.311375	−	0.950287i	$-0.399210\pi$
$257$	9.98345	0.622751	0.311375	+	0.950287i	$0.399210\pi$
$258$	0	0
$259$	35.8927	2.23027
$260$	0	0
$261$	7.18972	0.445032
$262$	0	0
$263$	6.66302	0.410860	0.205430	−	0.978672i	$-0.434141\pi$
$263$	6.66302	0.410860	0.205430	+	0.978672i	$0.434141\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−9.81949	−0.600943
$268$	0	0
$269$	−23.7140	−1.44587	−0.722933	−	0.690918i	$-0.757207\pi$
$269$	−23.7140	−1.44587	−0.722933	+	0.690918i	$0.757207\pi$
$270$	0	0
$271$	−14.0315	−0.852352	−0.426176	−	0.904640i	$-0.640140\pi$
$271$	−14.0315	−0.852352	−0.426176	+	0.904640i	$0.640140\pi$
$272$	0	0
$273$	−9.95593	−0.602560
$274$	0	0
$275$	0	0
$276$	0	0
$277$	24.6355	1.48020	0.740102	−	0.672495i	$-0.234777\pi$
$277$	24.6355	1.48020	0.740102	+	0.672495i	$0.234777\pi$
$278$	0	0
$279$	−1.50408	−0.0900469
$280$	0	0
$281$	21.8090	1.30102	0.650508	−	0.759499i	$-0.274556\pi$
$281$	21.8090	1.30102	0.650508	+	0.759499i	$0.274556\pi$
$282$	0	0
$283$	−18.2795	−1.08660	−0.543302	−	0.839538i	$-0.682826\pi$
$283$	−18.2795	−1.08660	−0.543302	+	0.839538i	$0.682826\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−20.6132	−1.21676
$288$	0	0
$289$	9.18680	0.540400
$290$	0	0
$291$	−16.4183	−0.962456
$292$	0	0
$293$	26.5888	1.55334	0.776668	−	0.629910i	$-0.216908\pi$
$293$	26.5888	1.55334	0.776668	+	0.629910i	$0.216908\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−2.15613	−0.125111
$298$	0	0
$299$	3.08372	0.178336
$300$	0	0
$301$	16.6833	0.961610
$302$	0	0
$303$	11.8386	0.680111
$304$	0	0
$305$	0	0
$306$	0	0
$307$	22.3886	1.27779	0.638894	−	0.769295i	$-0.279392\pi$
$307$	22.3886	1.27779	0.638894	+	0.769295i	$0.279392\pi$
$308$	0	0
$309$	−5.57964	−0.317415
$310$	0	0
$311$	−6.35424	−0.360316	−0.180158	−	0.983638i	$-0.557661\pi$
$311$	−6.35424	−0.360316	−0.180158	+	0.983638i	$0.557661\pi$
$312$	0	0
$313$	8.04489	0.454724	0.227362	−	0.973810i	$-0.426990\pi$
$313$	8.04489	0.454724	0.227362	+	0.973810i	$0.426990\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−17.8397	−1.00198	−0.500988	−	0.865454i	$-0.667030\pi$
$317$	−17.8397	−1.00198	−0.500988	+	0.865454i	$0.667030\pi$
$318$	0	0
$319$	15.5020	0.867944
$320$	0	0
$321$	1.19496	0.0666962
$322$	0	0
$323$	9.23693	0.513957
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−1.27553	−0.0705372
$328$	0	0
$329$	3.85799	0.212698
$330$	0	0
$331$	32.9429	1.81070	0.905352	−	0.424663i	$-0.139607\pi$
$331$	32.9429	1.81070	0.905352	+	0.424663i	$0.139607\pi$
$332$	0	0
$333$	11.1173	0.609225
$334$	0	0
$335$	0	0
$336$	0	0
$337$	28.7140	1.56415	0.782075	−	0.623184i	$-0.214161\pi$
$337$	28.7140	1.56415	0.782075	+	0.623184i	$0.214161\pi$
$338$	0	0
$339$	2.39598	0.130132
$340$	0	0
$341$	−3.24300	−0.175618
$342$	0	0
$343$	−11.5469	−0.623473
$344$	0	0
$345$	0	0
$346$	0	0
$347$	11.5387	0.619430	0.309715	−	0.950829i	$-0.399766\pi$
$347$	11.5387	0.619430	0.309715	+	0.950829i	$0.399766\pi$
$348$	0	0
$349$	22.5967	1.20957	0.604786	−	0.796388i	$-0.293259\pi$
$349$	22.5967	1.20957	0.604786	+	0.796388i	$0.293259\pi$
$350$	0	0
$351$	−3.08372	−0.164597
$352$	0	0
$353$	20.8033	1.10725	0.553623	−	0.832767i	$-0.313245\pi$
$353$	20.8033	1.10725	0.553623	+	0.832767i	$0.313245\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	16.5214	0.874408
$358$	0	0
$359$	17.2543	0.910647	0.455324	−	0.890326i	$-0.349524\pi$
$359$	17.2543	0.910647	0.455324	+	0.890326i	$0.349524\pi$
$360$	0	0
$361$	−15.7418	−0.828518
$362$	0	0
$363$	6.35109	0.333346
$364$	0	0
$365$	0	0
$366$	0	0
$367$	21.4020	1.11718	0.558589	−	0.829445i	$-0.311343\pi$
$367$	21.4020	1.11718	0.558589	+	0.829445i	$0.311343\pi$
$368$	0	0
$369$	−6.38468	−0.332373
$370$	0	0
$371$	8.31833	0.431866
$372$	0	0
$373$	26.4017	1.36703	0.683514	−	0.729937i	$-0.260451\pi$
$373$	26.4017	1.36703	0.683514	+	0.729937i	$0.260451\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	22.1711	1.14187
$378$	0	0
$379$	−28.0990	−1.44335	−0.721674	−	0.692233i	$-0.756627\pi$
$379$	−28.0990	−1.44335	−0.721674	+	0.692233i	$0.756627\pi$
$380$	0	0
$381$	−3.85517	−0.197507
$382$	0	0
$383$	3.82126	0.195257	0.0976285	−	0.995223i	$-0.468874\pi$
$383$	3.82126	0.195257	0.0976285	+	0.995223i	$0.468874\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	5.16744	0.262676
$388$	0	0
$389$	−5.50116	−0.278920	−0.139460	−	0.990228i	$-0.544537\pi$
$389$	−5.50116	−0.278920	−0.139460	+	0.990228i	$0.544537\pi$
$390$	0	0
$391$	−5.11730	−0.258793
$392$	0	0
$393$	7.19496	0.362938
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−19.9987	−1.00371	−0.501853	−	0.864953i	$-0.667348\pi$
$397$	−19.9987	−1.00371	−0.501853	+	0.864953i	$0.667348\pi$
$398$	0	0
$399$	5.82765	0.291747
$400$	0	0
$401$	27.2682	1.36171	0.680854	−	0.732419i	$-0.261609\pi$
$401$	27.2682	1.36171	0.680854	+	0.732419i	$0.261609\pi$
$402$	0	0
$403$	−4.63816	−0.231043
$404$	0	0
$405$	0	0
$406$	0	0
$407$	23.9704	1.18817
$408$	0	0
$409$	36.8090	1.82009	0.910044	−	0.414512i	$-0.136048\pi$
$409$	36.8090	1.82009	0.910044	+	0.414512i	$0.136048\pi$
$410$	0	0
$411$	−16.0898	−0.793650
$412$	0	0
$413$	30.1735	1.48474
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−2.26423	−0.110880
$418$	0	0
$419$	−9.25082	−0.451932	−0.225966	−	0.974135i	$-0.572554\pi$
$419$	−9.25082	−0.451932	−0.225966	+	0.974135i	$0.572554\pi$
$420$	0	0
$421$	−7.90579	−0.385305	−0.192652	−	0.981267i	$-0.561709\pi$
$421$	−7.90579	−0.385305	−0.192652	+	0.981267i	$0.561709\pi$
$422$	0	0
$423$	1.19496	0.0581010
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−25.7030	−1.24386
$428$	0	0
$429$	−6.64891	−0.321012
$430$	0	0
$431$	−11.6684	−0.562046	−0.281023	−	0.959701i	$-0.590674\pi$
$431$	−11.6684	−0.562046	−0.281023	+	0.959701i	$0.590674\pi$
$432$	0	0
$433$	−8.88142	−0.426814	−0.213407	−	0.976963i	$-0.568456\pi$
$433$	−8.88142	−0.426814	−0.213407	+	0.976963i	$0.568456\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.80504	−0.0863467
$438$	0	0
$439$	12.4122	0.592402	0.296201	−	0.955126i	$-0.404280\pi$
$439$	12.4122	0.592402	0.296201	+	0.955126i	$0.404280\pi$
$440$	0	0
$441$	3.42351	0.163024
$442$	0	0
$443$	−1.05901	−0.0503150	−0.0251575	−	0.999683i	$-0.508009\pi$
$443$	−1.05901	−0.0503150	−0.0251575	+	0.999683i	$0.508009\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	3.79688	0.179586
$448$	0	0
$449$	8.69871	0.410517	0.205259	−	0.978708i	$-0.434196\pi$
$449$	8.69871	0.410517	0.205259	+	0.978708i	$0.434196\pi$
$450$	0	0
$451$	−13.7662	−0.648226
$452$	0	0
$453$	9.90006	0.465146
$454$	0	0
$455$	0	0
$456$	0	0
$457$	21.1593	0.989789	0.494895	−	0.868953i	$-0.335207\pi$
$457$	21.1593	0.989789	0.494895	+	0.868953i	$0.335207\pi$
$458$	0	0
$459$	5.11730	0.238855
$460$	0	0
$461$	1.96608	0.0915696	0.0457848	−	0.998951i	$-0.485421\pi$
$461$	1.96608	0.0915696	0.0457848	+	0.998951i	$0.485421\pi$
$462$	0	0
$463$	−34.0375	−1.58186	−0.790930	−	0.611907i	$-0.790403\pi$
$463$	−34.0375	−1.58186	−0.790930	+	0.611907i	$0.790403\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−21.1184	−0.977241	−0.488620	−	0.872497i	$-0.662500\pi$
$467$	−21.1184	−0.977241	−0.488620	+	0.872497i	$0.662500\pi$
$468$	0	0
$469$	17.1144	0.790269
$470$	0	0
$471$	2.84993	0.131318
$472$	0	0
$473$	11.1417	0.512295
$474$	0	0
$475$	0	0
$476$	0	0
$477$	2.57649	0.117970
$478$	0	0
$479$	12.9446	0.591455	0.295727	−	0.955272i	$-0.404438\pi$
$479$	12.9446	0.591455	0.295727	+	0.955272i	$0.404438\pi$
$480$	0	0
$481$	34.2826	1.56315
$482$	0	0
$483$	−3.22855	−0.146904
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−30.4663	−1.38056	−0.690280	−	0.723542i	$-0.742513\pi$
$487$	−30.4663	−1.38056	−0.690280	+	0.723542i	$0.742513\pi$
$488$	0	0
$489$	−13.7998	−0.624048
$490$	0	0
$491$	−20.2155	−0.912312	−0.456156	−	0.889900i	$-0.650774\pi$
$491$	−20.2155	−0.912312	−0.456156	+	0.889900i	$0.650774\pi$
$492$	0	0
$493$	−36.7920	−1.65703
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−3.12012	−0.139956
$498$	0	0
$499$	32.6468	1.46147	0.730736	−	0.682660i	$-0.239177\pi$
$499$	32.6468	1.46147	0.730736	+	0.682660i	$0.239177\pi$
$500$	0	0
$501$	22.4632	1.00358
$502$	0	0
$503$	−9.75595	−0.434996	−0.217498	−	0.976061i	$-0.569790\pi$
$503$	−9.75595	−0.434996	−0.217498	+	0.976061i	$0.569790\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	3.49068	0.155026
$508$	0	0
$509$	14.0375	0.622203	0.311102	−	0.950377i	$-0.399302\pi$
$509$	14.0375	0.622203	0.311102	+	0.950377i	$0.399302\pi$
$510$	0	0
$511$	−13.6886	−0.605546
$512$	0	0
$513$	1.80504	0.0796944
$514$	0	0
$515$	0	0
$516$	0	0
$517$	2.57649	0.113314
$518$	0	0
$519$	−6.71349	−0.294690
$520$	0	0
$521$	39.0103	1.70907	0.854535	−	0.519394i	$-0.173842\pi$
$521$	39.0103	1.70907	0.854535	+	0.519394i	$0.173842\pi$
$522$	0	0
$523$	−39.9935	−1.74879	−0.874396	−	0.485212i	$-0.838742\pi$
$523$	−39.9935	−1.74879	−0.874396	+	0.485212i	$0.838742\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	7.69684	0.335279
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	9.34585	0.405575
$532$	0	0
$533$	−19.6886	−0.852806
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−0.576494	−0.0248775
$538$	0	0
$539$	7.38153	0.317945
$540$	0	0
$541$	5.51539	0.237125	0.118562	−	0.992947i	$-0.462171\pi$
$541$	5.51539	0.237125	0.118562	+	0.992947i	$0.462171\pi$
$542$	0	0
$543$	8.53767	0.366386
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−26.3211	−1.12541	−0.562705	−	0.826658i	$-0.690239\pi$
$547$	−26.3211	−1.12541	−0.562705	+	0.826658i	$0.690239\pi$
$548$	0	0
$549$	−7.96117	−0.339775
$550$	0	0
$551$	−12.9777	−0.552870
$552$	0	0
$553$	24.3357	1.03486
$554$	0	0
$555$	0	0
$556$	0	0
$557$	23.3267	0.988385	0.494192	−	0.869353i	$-0.335464\pi$
$557$	23.3267	0.988385	0.494192	+	0.869353i	$0.335464\pi$
$558$	0	0
$559$	15.9349	0.673976
$560$	0	0
$561$	11.0336	0.465838
$562$	0	0
$563$	11.3180	0.476997	0.238498	−	0.971143i	$-0.423345\pi$
$563$	11.3180	0.476997	0.238498	+	0.971143i	$0.423345\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	3.22855	0.135586
$568$	0	0
$569$	12.8722	0.539631	0.269816	−	0.962912i	$-0.413037\pi$
$569$	12.8722	0.539631	0.269816	+	0.962912i	$0.413037\pi$
$570$	0	0
$571$	−14.9754	−0.626701	−0.313350	−	0.949638i	$-0.601451\pi$
$571$	−14.9754	−0.626701	−0.313350	+	0.949638i	$0.601451\pi$
$572$	0	0
$573$	10.0724	0.420781
$574$	0	0
$575$	0	0
$576$	0	0
$577$	6.21409	0.258696	0.129348	−	0.991599i	$-0.458712\pi$
$577$	6.21409	0.258696	0.129348	+	0.991599i	$0.458712\pi$
$578$	0	0
$579$	−0.267374	−0.0111117
$580$	0	0
$581$	−45.7233	−1.89692
$582$	0	0
$583$	5.55526	0.230075
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−37.1040	−1.53145	−0.765723	−	0.643170i	$-0.777619\pi$
$587$	−37.1040	−1.53145	−0.765723	+	0.643170i	$0.777619\pi$
$588$	0	0
$589$	2.71492	0.111867
$590$	0	0
$591$	2.79980	0.115168
$592$	0	0
$593$	−22.2357	−0.913109	−0.456554	−	0.889695i	$-0.650917\pi$
$593$	−22.2357	−0.913109	−0.456554	+	0.889695i	$0.650917\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	15.1989	0.622051
$598$	0	0
$599$	3.66254	0.149647	0.0748236	−	0.997197i	$-0.476161\pi$
$599$	3.66254	0.149647	0.0748236	+	0.997197i	$0.476161\pi$
$600$	0	0
$601$	−10.5440	−0.430097	−0.215048	−	0.976603i	$-0.568991\pi$
$601$	−10.5440	−0.430097	−0.215048	+	0.976603i	$0.568991\pi$
$602$	0	0
$603$	5.30096	0.215872
$604$	0	0
$605$	0	0
$606$	0	0
$607$	32.7980	1.33123	0.665615	−	0.746295i	$-0.268169\pi$
$607$	32.7980	1.33123	0.665615	+	0.746295i	$0.268169\pi$
$608$	0	0
$609$	−23.2123	−0.940611
$610$	0	0
$611$	3.68492	0.149076
$612$	0	0
$613$	3.96713	0.160231	0.0801154	−	0.996786i	$-0.474471\pi$
$613$	3.96713	0.160231	0.0801154	+	0.996786i	$0.474471\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−35.7705	−1.44007	−0.720033	−	0.693940i	$-0.755873\pi$
$617$	−35.7705	−1.44007	−0.720033	+	0.693940i	$0.755873\pi$
$618$	0	0
$619$	10.9982	0.442056	0.221028	−	0.975267i	$-0.429059\pi$
$619$	10.9982	0.442056	0.221028	+	0.975267i	$0.429059\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	31.7027	1.27014
$624$	0	0
$625$	0	0
$626$	0	0
$627$	3.89190	0.155428
$628$	0	0
$629$	−56.8906	−2.26838
$630$	0	0
$631$	23.7108	0.943913	0.471957	−	0.881622i	$-0.343548\pi$
$631$	23.7108	0.943913	0.471957	+	0.881622i	$0.343548\pi$
$632$	0	0
$633$	−24.1766	−0.960935
$634$	0	0
$635$	0	0
$636$	0	0
$637$	10.5571	0.418289
$638$	0	0
$639$	−0.966415	−0.0382308
$640$	0	0
$641$	4.67676	0.184721	0.0923605	−	0.995726i	$-0.470559\pi$
$641$	4.67676	0.184721	0.0923605	+	0.995726i	$0.470559\pi$
$642$	0	0
$643$	−10.6578	−0.420302	−0.210151	−	0.977669i	$-0.567396\pi$
$643$	−10.6578	−0.420302	−0.210151	+	0.977669i	$0.567396\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	2.04522	0.0804061	0.0402030	−	0.999192i	$-0.487200\pi$
$647$	2.04522	0.0804061	0.0402030	+	0.999192i	$0.487200\pi$
$648$	0	0
$649$	20.1509	0.790992
$650$	0	0
$651$	4.85599	0.190321
$652$	0	0
$653$	−4.26246	−0.166803	−0.0834015	−	0.996516i	$-0.526578\pi$
$653$	−4.26246	−0.166803	−0.0834015	+	0.996516i	$0.526578\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−4.23985	−0.165412
$658$	0	0
$659$	−35.6083	−1.38710	−0.693551	−	0.720407i	$-0.743955\pi$
$659$	−35.6083	−1.38710	−0.693551	+	0.720407i	$0.743955\pi$
$660$	0	0
$661$	−36.9517	−1.43726	−0.718628	−	0.695395i	$-0.755230\pi$
$661$	−36.9517	−1.43726	−0.718628	+	0.695395i	$0.755230\pi$
$662$	0	0
$663$	15.7803	0.612857
$664$	0	0
$665$	0	0
$666$	0	0
$667$	7.18972	0.278387
$668$	0	0
$669$	−29.5582	−1.14279
$670$	0	0
$671$	−17.1653	−0.662661
$672$	0	0
$673$	43.9266	1.69325	0.846624	−	0.532192i	$-0.178632\pi$
$673$	43.9266	1.69325	0.846624	+	0.532192i	$0.178632\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	29.5998	1.13761	0.568807	−	0.822471i	$-0.307405\pi$
$677$	29.5998	1.13761	0.568807	+	0.822471i	$0.307405\pi$
$678$	0	0
$679$	53.0071	2.03423
$680$	0	0
$681$	−8.58256	−0.328884
$682$	0	0
$683$	−4.39251	−0.168075	−0.0840373	−	0.996463i	$-0.526781\pi$
$683$	−4.39251	−0.168075	−0.0840373	+	0.996463i	$0.526781\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−7.97772	−0.304369
$688$	0	0
$689$	7.94518	0.302687
$690$	0	0
$691$	19.0170	0.723442	0.361721	−	0.932286i	$-0.382189\pi$
$691$	19.0170	0.723442	0.361721	+	0.932286i	$0.382189\pi$
$692$	0	0
$693$	6.96117	0.264433
$694$	0	0
$695$	0	0
$696$	0	0
$697$	32.6723	1.23755
$698$	0	0
$699$	−10.9192	−0.413002
$700$	0	0
$701$	31.3745	1.18500	0.592500	−	0.805571i	$-0.298141\pi$
$701$	31.3745	1.18500	0.592500	+	0.805571i	$0.298141\pi$
$702$	0	0
$703$	−20.0672	−0.756848
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−38.2215	−1.43747
$708$	0	0
$709$	12.7828	0.480067	0.240033	−	0.970765i	$-0.422842\pi$
$709$	12.7828	0.480067	0.240033	+	0.970765i	$0.422842\pi$
$710$	0	0
$711$	7.53767	0.282685
$712$	0	0
$713$	−1.50408	−0.0563283
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−5.54291	−0.207004
$718$	0	0
$719$	−23.6521	−0.882073	−0.441036	−	0.897489i	$-0.645389\pi$
$719$	−23.6521	−0.882073	−0.441036	+	0.897489i	$0.645389\pi$
$720$	0	0
$721$	18.0141	0.670881
$722$	0	0
$723$	9.92549	0.369133
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−35.4031	−1.31303	−0.656514	−	0.754314i	$-0.727970\pi$
$727$	−35.4031	−1.31303	−0.656514	+	0.754314i	$0.727970\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−26.4434	−0.978043
$732$	0	0
$733$	8.65602	0.319717	0.159859	−	0.987140i	$-0.448896\pi$
$733$	8.65602	0.319717	0.159859	+	0.987140i	$0.448896\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	11.4296	0.421014
$738$	0	0
$739$	15.3458	0.564506	0.282253	−	0.959340i	$-0.408918\pi$
$739$	15.3458	0.564506	0.282253	+	0.959340i	$0.408918\pi$
$740$	0	0
$741$	5.56623	0.204481
$742$	0	0
$743$	54.1997	1.98840	0.994198	−	0.107567i	$-0.0343059\pi$
$743$	54.1997	1.98840	0.994198	+	0.107567i	$0.0343059\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−14.1622	−0.518168
$748$	0	0
$749$	−3.85799	−0.140968
$750$	0	0
$751$	18.4360	0.672738	0.336369	−	0.941730i	$-0.390801\pi$
$751$	18.4360	0.672738	0.336369	+	0.941730i	$0.390801\pi$
$752$	0	0
$753$	24.2764	0.884680
$754$	0	0
$755$	0	0
$756$	0	0
$757$	11.9003	0.432523	0.216262	−	0.976335i	$-0.430614\pi$
$757$	11.9003	0.432523	0.216262	+	0.976335i	$0.430614\pi$
$758$	0	0
$759$	−2.15613	−0.0782626
$760$	0	0
$761$	28.3359	1.02718	0.513588	−	0.858037i	$-0.328316\pi$
$761$	28.3359	1.02718	0.513588	+	0.858037i	$0.328316\pi$
$762$	0	0
$763$	4.11812	0.149086
$764$	0	0
$765$	0	0
$766$	0	0
$767$	28.8200	1.04063
$768$	0	0
$769$	−3.30096	−0.119036	−0.0595178	−	0.998227i	$-0.518956\pi$
$769$	−3.30096	−0.119036	−0.0595178	+	0.998227i	$0.518956\pi$
$770$	0	0
$771$	−9.98345	−0.359545
$772$	0	0
$773$	−24.8591	−0.894121	−0.447061	−	0.894504i	$-0.647529\pi$
$773$	−24.8591	−0.894121	−0.447061	+	0.894504i	$0.647529\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−35.8927	−1.28764
$778$	0	0
$779$	11.5246	0.412911
$780$	0	0
$781$	−2.08372	−0.0745613
$782$	0	0
$783$	−7.18972	−0.256940
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−9.40624	−0.335296	−0.167648	−	0.985847i	$-0.553617\pi$
$787$	−9.40624	−0.335296	−0.167648	+	0.985847i	$0.553617\pi$
$788$	0	0
$789$	−6.66302	−0.237210
$790$	0	0
$791$	−7.73554	−0.275044
$792$	0	0
$793$	−24.5500	−0.871797
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−43.9319	−1.55615	−0.778074	−	0.628173i	$-0.783803\pi$
$797$	−43.9319	−1.55615	−0.778074	+	0.628173i	$0.783803\pi$
$798$	0	0
$799$	−6.11498	−0.216332
$800$	0	0
$801$	9.81949	0.346955
$802$	0	0
$803$	−9.14168	−0.322603
$804$	0	0
$805$	0	0
$806$	0	0
$807$	23.7140	0.834772
$808$	0	0
$809$	−41.0092	−1.44181	−0.720903	−	0.693035i	$-0.756273\pi$
$809$	−41.0092	−1.44181	−0.720903	+	0.693035i	$0.756273\pi$
$810$	0	0
$811$	6.03860	0.212044	0.106022	−	0.994364i	$-0.466189\pi$
$811$	6.03860	0.212044	0.106022	+	0.994364i	$0.466189\pi$
$812$	0	0
$813$	14.0315	0.492106
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−9.32743	−0.326325
$818$	0	0
$819$	9.95593	0.347888
$820$	0	0
$821$	−26.4073	−0.921621	−0.460810	−	0.887499i	$-0.652441\pi$
$821$	−26.4073	−0.921621	−0.460810	+	0.887499i	$0.652441\pi$
$822$	0	0
$823$	22.4572	0.782809	0.391404	−	0.920219i	$-0.371989\pi$
$823$	22.4572	0.782809	0.391404	+	0.920219i	$0.371989\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−29.7601	−1.03486	−0.517431	−	0.855725i	$-0.673112\pi$
$827$	−29.7601	−1.03486	−0.517431	+	0.855725i	$0.673112\pi$
$828$	0	0
$829$	22.9164	0.795919	0.397959	−	0.917403i	$-0.369718\pi$
$829$	22.9164	0.795919	0.397959	+	0.917403i	$0.369718\pi$
$830$	0	0
$831$	−24.6355	−0.854596
$832$	0	0
$833$	−17.5191	−0.607002
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.50408	0.0519886
$838$	0	0
$839$	−25.5962	−0.883679	−0.441839	−	0.897094i	$-0.645674\pi$
$839$	−25.5962	−0.883679	−0.441839	+	0.897094i	$0.645674\pi$
$840$	0	0
$841$	22.6920	0.782484
$842$	0	0
$843$	−21.8090	−0.751142
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−20.5048	−0.704553
$848$	0	0
$849$	18.2795	0.627351
$850$	0	0
$851$	11.1173	0.381096
$852$	0	0
$853$	−3.24768	−0.111198	−0.0555992	−	0.998453i	$-0.517707\pi$
$853$	−3.24768	−0.111198	−0.0555992	+	0.998453i	$0.517707\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	30.6112	1.04566	0.522830	−	0.852437i	$-0.324876\pi$
$857$	30.6112	1.04566	0.522830	+	0.852437i	$0.324876\pi$
$858$	0	0
$859$	−35.4219	−1.20858	−0.604290	−	0.796765i	$-0.706543\pi$
$859$	−35.4219	−1.20858	−0.604290	+	0.796765i	$0.706543\pi$
$860$	0	0
$861$	20.6132	0.702497
$862$	0	0
$863$	−27.8640	−0.948503	−0.474252	−	0.880389i	$-0.657281\pi$
$863$	−27.8640	−0.948503	−0.474252	+	0.880389i	$0.657281\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−9.18680	−0.312000
$868$	0	0
$869$	16.2522	0.551318
$870$	0	0
$871$	16.3467	0.553886
$872$	0	0
$873$	16.4183	0.555674
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−4.42455	−0.149407	−0.0747033	−	0.997206i	$-0.523801\pi$
$877$	−4.42455	−0.149407	−0.0747033	+	0.997206i	$0.523801\pi$
$878$	0	0
$879$	−26.5888	−0.896820
$880$	0	0
$881$	38.0506	1.28196	0.640979	−	0.767558i	$-0.278529\pi$
$881$	38.0506	1.28196	0.640979	+	0.767558i	$0.278529\pi$
$882$	0	0
$883$	8.79489	0.295971	0.147986	−	0.988989i	$-0.452721\pi$
$883$	8.79489	0.295971	0.147986	+	0.988989i	$0.452721\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	39.5637	1.32842	0.664209	−	0.747547i	$-0.268768\pi$
$887$	39.5637	1.32842	0.664209	+	0.747547i	$0.268768\pi$
$888$	0	0
$889$	12.4466	0.417446
$890$	0	0
$891$	2.15613	0.0722332
$892$	0	0
$893$	−2.15695	−0.0721796
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−3.08372	−0.102962
$898$	0	0
$899$	−10.8139	−0.360664
$900$	0	0
$901$	−13.1847	−0.439246
$902$	0	0
$903$	−16.6833	−0.555186
$904$	0	0
$905$	0	0
$906$	0	0
$907$	15.9138	0.528411	0.264205	−	0.964466i	$-0.414890\pi$
$907$	15.9138	0.528411	0.264205	+	0.964466i	$0.414890\pi$
$908$	0	0
$909$	−11.8386	−0.392662
$910$	0	0
$911$	−27.6216	−0.915145	−0.457572	−	0.889172i	$-0.651281\pi$
$911$	−27.6216	−0.915145	−0.457572	+	0.889172i	$0.651281\pi$
$912$	0	0
$913$	−30.5356	−1.01058
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−23.2293	−0.767098
$918$	0	0
$919$	29.2520	0.964934	0.482467	−	0.875914i	$-0.339741\pi$
$919$	29.2520	0.964934	0.482467	+	0.875914i	$0.339741\pi$
$920$	0	0
$921$	−22.3886	−0.737731
$922$	0	0
$923$	−2.98015	−0.0980929
$924$	0	0
$925$	0	0
$926$	0	0
$927$	5.57964	0.183259
$928$	0	0
$929$	13.9336	0.457148	0.228574	−	0.973527i	$-0.426594\pi$
$929$	13.9336	0.457148	0.228574	+	0.973527i	$0.426594\pi$
$930$	0	0
$931$	−6.17956	−0.202527
$932$	0	0
$933$	6.35424	0.208028
$934$	0	0
$935$	0	0
$936$	0	0
$937$	53.4257	1.74534	0.872671	−	0.488309i	$-0.162386\pi$
$937$	53.4257	1.74534	0.872671	+	0.488309i	$0.162386\pi$
$938$	0	0
$939$	−8.04489	−0.262535
$940$	0	0
$941$	−8.10633	−0.264259	−0.132129	−	0.991232i	$-0.542181\pi$
$941$	−8.10633	−0.264259	−0.132129	+	0.991232i	$0.542181\pi$
$942$	0	0
$943$	−6.38468	−0.207914
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−44.1241	−1.43384	−0.716920	−	0.697155i	$-0.754449\pi$
$947$	−44.1241	−1.43384	−0.716920	+	0.697155i	$0.754449\pi$
$948$	0	0
$949$	−13.0745	−0.424417
$950$	0	0
$951$	17.8397	0.578491
$952$	0	0
$953$	−34.2540	−1.10959	−0.554797	−	0.831985i	$-0.687204\pi$
$953$	−34.2540	−1.10959	−0.554797	+	0.831985i	$0.687204\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−15.5020	−0.501108
$958$	0	0
$959$	51.9466	1.67744
$960$	0	0
$961$	−28.7377	−0.927024
$962$	0	0
$963$	−1.19496	−0.0385071
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−27.0518	−0.869926	−0.434963	−	0.900448i	$-0.643239\pi$
$967$	−27.0518	−0.869926	−0.434963	+	0.900448i	$0.643239\pi$
$968$	0	0
$969$	−9.23693	−0.296733
$970$	0	0
$971$	10.1202	0.324773	0.162387	−	0.986727i	$-0.448081\pi$
$971$	10.1202	0.324773	0.162387	+	0.986727i	$0.448081\pi$
$972$	0	0
$973$	7.31017	0.234353
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−29.9387	−0.957823	−0.478911	−	0.877863i	$-0.658968\pi$
$977$	−29.9387	−0.957823	−0.478911	+	0.877863i	$0.658968\pi$
$978$	0	0
$979$	21.1721	0.676664
$980$	0	0
$981$	1.27553	0.0407247
$982$	0	0
$983$	5.54373	0.176817	0.0884087	−	0.996084i	$-0.471822\pi$
$983$	5.54373	0.176817	0.0884087	+	0.996084i	$0.471822\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−3.85799	−0.122801
$988$	0	0
$989$	5.16744	0.164315
$990$	0	0
$991$	−29.1517	−0.926035	−0.463017	−	0.886349i	$-0.653233\pi$
$991$	−29.1517	−0.926035	−0.463017	+	0.886349i	$0.653233\pi$
$992$	0	0
$993$	−32.9429	−1.04541
$994$	0	0
$995$	0	0
$996$	0	0
$997$	46.5419	1.47400	0.736998	−	0.675895i	$-0.236243\pi$
$997$	46.5419	1.47400	0.736998	+	0.675895i	$0.236243\pi$
$998$	0	0
$999$	−11.1173	−0.351736

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6900.2.a.ba.1.3	✓	4
5.2	odd	4		6900.2.f.s.6349.7		8
5.3	odd	4		6900.2.f.s.6349.2		8
5.4	even	2		6900.2.a.bb.1.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6900.2.a.ba.1.3	✓	4	1.1	even	1	trivial
6900.2.a.bb.1.2	yes	4	5.4	even	2
6900.2.f.s.6349.2		8	5.3	odd	4
6900.2.f.s.6349.7		8	5.2	odd	4

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:	\( N \)	\(=\)	\( 6900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6900.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +3.22855 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +3.22855 q^{7} +1.00000 q^{9} +2.15613 q^{11} +3.08372 q^{13} -5.11730 q^{17} -1.80504 q^{19} -3.22855 q^{21} +1.00000 q^{23} -1.00000 q^{27} +7.18972 q^{29} -1.50408 q^{31} -2.15613 q^{33} +11.1173 q^{37} -3.08372 q^{39} -6.38468 q^{41} +5.16744 q^{43} +1.19496 q^{47} +3.42351 q^{49} +5.11730 q^{51} +2.57649 q^{53} +1.80504 q^{57} +9.34585 q^{59} -7.96117 q^{61} +3.22855 q^{63} +5.30096 q^{67} -1.00000 q^{69} -0.966415 q^{71} -4.23985 q^{73} +6.96117 q^{77} +7.53767 q^{79} +1.00000 q^{81} -14.1622 q^{83} -7.18972 q^{87} +9.81949 q^{89} +9.95593 q^{91} +1.50408 q^{93} +16.4183 q^{97} +2.15613 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 3 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 + 3 * q^7 + 4 * q^9	\( 4 q - 4 q^{3} + 3 q^{7} + 4 q^{9} - 2 q^{11} + q^{13} + 10 q^{17} + 2 q^{19} - 3 q^{21} + 4 q^{23} - 4 q^{27} - q^{29} - 6 q^{31} + 2 q^{33} + 14 q^{37} - q^{39} - 5 q^{41} - 2 q^{43} + 14 q^{47} + 13 q^{49} - 10 q^{51} + 11 q^{53} - 2 q^{57} - 3 q^{59} - 12 q^{61} + 3 q^{63} + 12 q^{67} - 4 q^{69} - 23 q^{71} + 5 q^{73} + 8 q^{77} + 11 q^{79} + 4 q^{81} + 5 q^{83} + q^{87} + 6 q^{89} - 19 q^{91} + 6 q^{93} + 26 q^{97} - 2 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 + 3 * q^7 + 4 * q^9 - 2 * q^11 + q^13 + 10 * q^17 + 2 * q^19 - 3 * q^21 + 4 * q^23 - 4 * q^27 - q^29 - 6 * q^31 + 2 * q^33 + 14 * q^37 - q^39 - 5 * q^41 - 2 * q^43 + 14 * q^47 + 13 * q^49 - 10 * q^51 + 11 * q^53 - 2 * q^57 - 3 * q^59 - 12 * q^61 + 3 * q^63 + 12 * q^67 - 4 * q^69 - 23 * q^71 + 5 * q^73 + 8 * q^77 + 11 * q^79 + 4 * q^81 + 5 * q^83 + q^87 + 6 * q^89 - 19 * q^91 + 6 * q^93 + 26 * q^97 - 2 * q^99

Introduction

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