LMFDB - Embedded newform 6975.2.a.ca.1.4

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 = [6,-1,0,7,0,0,2,3,0,0,-7,0,4,-10,0,17,0] 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:	$0$
Dimension:	$6$
Coefficient field:	6.6.75968016.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 9x^{4} + 9x^{3} + 14x^{2} - 6x - 6 $ x^6 - x^5 - 9x^4 + 9x^3 + 14x^2 - 6x - 6
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2325)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.667396$ 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)

$f(q)$	$=$	$q+0.667396 q^{2} -1.55458 q^{4} +3.64222 q^{7} -2.37231 q^{8} +1.15709 q^{11} +5.71167 q^{13} +2.43080 q^{14} +1.52589 q^{16} +1.37855 q^{17} +2.28711 q^{19} +0.772236 q^{22} -5.17025 q^{23} +3.81195 q^{26} -5.66213 q^{28} +5.51387 q^{29} -1.00000 q^{31} +5.76300 q^{32} +0.920040 q^{34} +2.77224 q^{37} +1.52641 q^{38} +0.122562 q^{41} +1.11738 q^{43} -1.79879 q^{44} -3.45060 q^{46} +8.92527 q^{47} +6.26574 q^{49} -8.87926 q^{52} -3.42456 q^{53} -8.64048 q^{56} +3.67993 q^{58} -0.542646 q^{59} +10.4883 q^{61} -0.667396 q^{62} +0.794421 q^{64} -15.8274 q^{67} -2.14307 q^{68} -10.7253 q^{71} -8.34049 q^{73} +1.85018 q^{74} -3.55550 q^{76} +4.21437 q^{77} +5.83719 q^{79} +0.0817974 q^{82} -16.4935 q^{83} +0.745734 q^{86} -2.74498 q^{88} +8.27438 q^{89} +20.8031 q^{91} +8.03757 q^{92} +5.95669 q^{94} +9.97701 q^{97} +4.18173 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{2} + 7 q^{4} + 2 q^{7} + 3 q^{8} - 7 q^{11} + 4 q^{13} - 10 q^{14} + 17 q^{16} + 17 q^{19} + 2 q^{22} + q^{23} - 2 q^{26} + 22 q^{28} + 8 q^{29} - 6 q^{31} + 35 q^{32} - 13 q^{34} + 14 q^{37}+ \cdots - 37 q^{98}+O(q^{100}) $ 6 * q - q^2 + 7 * q^4 + 2 * q^7 + 3 * q^8 - 7 * q^11 + 4 * q^13 - 10 * q^14 + 17 * q^16 + 17 * q^19 + 2 * q^22 + q^23 - 2 * q^26 + 22 * q^28 + 8 * q^29 - 6 * q^31 + 35 * q^32 - 13 * q^34 + 14 * q^37 - 10 * q^38 - 18 * q^41 + 15 * q^43 - 4 * q^44 - 10 * q^46 + 13 * q^47 + 18 * q^49 - 38 * q^52 + 13 * q^53 - 32 * q^56 + 23 * q^58 - 32 * q^59 + q^62 + 29 * q^64 + 9 * q^67 - 9 * q^68 - q^71 + 14 * q^73 - 20 * q^74 + 42 * q^76 + 14 * q^77 + 13 * q^79 - 32 * q^82 - 9 * q^83 + 11 * q^86 + 36 * q^88 - 34 * q^89 - 2 * q^91 + 46 * q^92 + 19 * q^94 + 30 * q^97 - 37 * 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 $

$2$	0.667396	0.471920	0.235960	−	0.971763i	$-0.424177\pi$
$2$	0.667396	0.471920	0.235960	+	0.971763i	$0.424177\pi$
$3$	0	0
$3$	0	0
$4$	−1.55458	−0.777291
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.64222	1.37663	0.688314	−	0.725413i	$-0.258351\pi$
$7$	3.64222	1.37663	0.688314	+	0.725413i	$0.258351\pi$
$8$	−2.37231	−0.838740
$9$	0	0
$10$	0	0
$11$	1.15709	0.348875	0.174438	−	0.984668i	$-0.444189\pi$
$11$	1.15709	0.348875	0.174438	+	0.984668i	$0.444189\pi$
$12$	0	0
$13$	5.71167	1.58413	0.792066	−	0.610435i	$-0.209005\pi$
$13$	5.71167	1.58413	0.792066	+	0.610435i	$0.209005\pi$
$14$	2.43080	0.649659
$15$	0	0
$16$	1.52589	0.381473
$17$	1.37855	0.334348	0.167174	−	0.985927i	$-0.446536\pi$
$17$	1.37855	0.334348	0.167174	+	0.985927i	$0.446536\pi$
$18$	0	0
$19$	2.28711	0.524699	0.262349	−	0.964973i	$-0.415503\pi$
$19$	2.28711	0.524699	0.262349	+	0.964973i	$0.415503\pi$
$20$	0	0
$21$	0	0
$22$	0.772236	0.164641
$23$	−5.17025	−1.07807	−0.539035	−	0.842283i	$-0.681211\pi$
$23$	−5.17025	−1.07807	−0.539035	+	0.842283i	$0.681211\pi$
$24$	0	0
$25$	0	0
$26$	3.81195	0.747584
$27$	0	0
$28$	−5.66213	−1.07004
$29$	5.51387	1.02390	0.511950	−	0.859015i	$-0.328923\pi$
$29$	5.51387	1.02390	0.511950	+	0.859015i	$0.328923\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	5.76300	1.01876
$33$	0	0
$34$	0.920040	0.157786
$35$	0	0
$36$	0	0
$37$	2.77224	0.455753	0.227876	−	0.973690i	$-0.426822\pi$
$37$	2.77224	0.455753	0.227876	+	0.973690i	$0.426822\pi$
$38$	1.52641	0.247616
$39$	0	0
$40$	0	0
$41$	0.122562	0.0191410	0.00957048	−	0.999954i	$-0.496954\pi$
$41$	0.122562	0.0191410	0.00957048	+	0.999954i	$0.496954\pi$
$42$	0	0
$43$	1.11738	0.170399	0.0851993	−	0.996364i	$-0.472847\pi$
$43$	1.11738	0.170399	0.0851993	+	0.996364i	$0.472847\pi$
$44$	−1.79879	−0.271178
$45$	0	0
$46$	−3.45060	−0.508763
$47$	8.92527	1.30189	0.650943	−	0.759127i	$-0.274374\pi$
$47$	8.92527	1.30189	0.650943	+	0.759127i	$0.274374\pi$
$48$	0	0
$49$	6.26574	0.895106
$50$	0	0
$51$	0	0
$52$	−8.87926	−1.23133
$53$	−3.42456	−0.470400	−0.235200	−	0.971947i	$-0.575574\pi$
$53$	−3.42456	−0.470400	−0.235200	+	0.971947i	$0.575574\pi$
$54$	0	0
$55$	0	0
$56$	−8.64048	−1.15463
$57$	0	0
$58$	3.67993	0.483199
$59$	−0.542646	−0.0706465	−0.0353232	−	0.999376i	$-0.511246\pi$
$59$	−0.542646	−0.0706465	−0.0353232	+	0.999376i	$0.511246\pi$
$60$	0	0
$61$	10.4883	1.34289	0.671444	−	0.741056i	$-0.265675\pi$
$61$	10.4883	1.34289	0.671444	+	0.741056i	$0.265675\pi$
$62$	−0.667396	−0.0847594
$63$	0	0
$64$	0.794421	0.0993026
$65$	0	0
$66$	0	0
$67$	−15.8274	−1.93363	−0.966815	−	0.255477i	$-0.917767\pi$
$67$	−15.8274	−1.93363	−0.966815	+	0.255477i	$0.917767\pi$
$68$	−2.14307	−0.259886
$69$	0	0
$70$	0	0
$71$	−10.7253	−1.27286	−0.636428	−	0.771336i	$-0.719589\pi$
$71$	−10.7253	−1.27286	−0.636428	+	0.771336i	$0.719589\pi$
$72$	0	0
$73$	−8.34049	−0.976181	−0.488090	−	0.872793i	$-0.662306\pi$
$73$	−8.34049	−0.976181	−0.488090	+	0.872793i	$0.662306\pi$
$74$	1.85018	0.215079
$75$	0	0
$76$	−3.55550	−0.407844
$77$	4.21437	0.480272
$78$	0	0
$79$	5.83719	0.656735	0.328368	−	0.944550i	$-0.393502\pi$
$79$	5.83719	0.656735	0.328368	+	0.944550i	$0.393502\pi$
$80$	0	0
$81$	0	0
$82$	0.0817974	0.00903301
$83$	−16.4935	−1.81039	−0.905197	−	0.424993i	$-0.860277\pi$
$83$	−16.4935	−1.81039	−0.905197	+	0.424993i	$0.860277\pi$
$84$	0	0
$85$	0	0
$86$	0.745734	0.0804146
$87$	0	0
$88$	−2.74498	−0.292616
$89$	8.27438	0.877083	0.438541	−	0.898711i	$-0.355495\pi$
$89$	8.27438	0.877083	0.438541	+	0.898711i	$0.355495\pi$
$90$	0	0
$91$	20.8031	2.18076
$92$	8.03757	0.837975
$93$	0	0
$94$	5.95669	0.614386
$95$	0	0
$96$	0	0
$97$	9.97701	1.01301	0.506506	−	0.862237i	$-0.330937\pi$
$97$	9.97701	1.01301	0.506506	+	0.862237i	$0.330937\pi$
$98$	4.18173	0.422418
$99$	0	0
$100$	0	0
$101$	7.18097	0.714533	0.357266	−	0.934003i	$-0.383709\pi$
$101$	7.18097	0.714533	0.357266	+	0.934003i	$0.383709\pi$
$102$	0	0
$103$	−1.70411	−0.167911	−0.0839555	−	0.996470i	$-0.526755\pi$
$103$	−1.70411	−0.167911	−0.0839555	+	0.996470i	$0.526755\pi$
$104$	−13.5499	−1.32867
$105$	0	0
$106$	−2.28554	−0.221991
$107$	18.1570	1.75530	0.877652	−	0.479299i	$-0.159109\pi$
$107$	18.1570	1.75530	0.877652	+	0.479299i	$0.159109\pi$
$108$	0	0
$109$	6.50614	0.623176	0.311588	−	0.950217i	$-0.399139\pi$
$109$	6.50614	0.623176	0.311588	+	0.950217i	$0.399139\pi$
$110$	0	0
$111$	0	0
$112$	5.55763	0.525147
$113$	−3.36042	−0.316122	−0.158061	−	0.987429i	$-0.550524\pi$
$113$	−3.36042	−0.316122	−0.158061	+	0.987429i	$0.550524\pi$
$114$	0	0
$115$	0	0
$116$	−8.57176	−0.795868
$117$	0	0
$118$	−0.362160	−0.0333395
$119$	5.02099	0.460273
$120$	0	0
$121$	−9.66115	−0.878286
$122$	6.99984	0.633736
$123$	0	0
$124$	1.55458	0.139606
$125$	0	0
$126$	0	0
$127$	10.6138	0.941823	0.470911	−	0.882181i	$-0.343925\pi$
$127$	10.6138	0.941823	0.470911	+	0.882181i	$0.343925\pi$
$128$	−10.9958	−0.971902
$129$	0	0
$130$	0	0
$131$	−14.5080	−1.26757	−0.633786	−	0.773508i	$-0.718500\pi$
$131$	−14.5080	−1.26757	−0.633786	+	0.773508i	$0.718500\pi$
$132$	0	0
$133$	8.33014	0.722315
$134$	−10.5632	−0.912519
$135$	0	0
$136$	−3.27036	−0.280431
$137$	−5.31941	−0.454468	−0.227234	−	0.973840i	$-0.572968\pi$
$137$	−5.31941	−0.454468	−0.227234	+	0.973840i	$0.572968\pi$
$138$	0	0
$139$	−8.27443	−0.701828	−0.350914	−	0.936408i	$-0.614129\pi$
$139$	−8.27443	−0.701828	−0.350914	+	0.936408i	$0.614129\pi$
$140$	0	0
$141$	0	0
$142$	−7.15801	−0.600687
$143$	6.60891	0.552665
$144$	0	0
$145$	0	0
$146$	−5.56641	−0.460679
$147$	0	0
$148$	−4.30967	−0.354253
$149$	−15.9731	−1.30857	−0.654284	−	0.756249i	$-0.727030\pi$
$149$	−15.9731	−1.30857	−0.654284	+	0.756249i	$0.727030\pi$
$150$	0	0
$151$	7.39770	0.602016	0.301008	−	0.953622i	$-0.402677\pi$
$151$	7.39770	0.602016	0.301008	+	0.953622i	$0.402677\pi$
$152$	−5.42574	−0.440086
$153$	0	0
$154$	2.81265	0.226650
$155$	0	0
$156$	0	0
$157$	15.7243	1.25493	0.627467	−	0.778643i	$-0.284092\pi$
$157$	15.7243	1.25493	0.627467	+	0.778643i	$0.284092\pi$
$158$	3.89572	0.309927
$159$	0	0
$160$	0	0
$161$	−18.8312	−1.48410
$162$	0	0
$163$	−9.68816	−0.758836	−0.379418	−	0.925225i	$-0.623876\pi$
$163$	−9.68816	−0.758836	−0.379418	+	0.925225i	$0.623876\pi$
$164$	−0.190533	−0.0148781
$165$	0	0
$166$	−11.0077	−0.854361
$167$	−19.1075	−1.47858	−0.739292	−	0.673385i	$-0.764840\pi$
$167$	−19.1075	−1.47858	−0.739292	+	0.673385i	$0.764840\pi$
$168$	0	0
$169$	19.6232	1.50948
$170$	0	0
$171$	0	0
$172$	−1.73706	−0.132449
$173$	21.1286	1.60638	0.803189	−	0.595724i	$-0.203135\pi$
$173$	21.1286	1.60638	0.803189	+	0.595724i	$0.203135\pi$
$174$	0	0
$175$	0	0
$176$	1.76559	0.133087
$177$	0	0
$178$	5.52229	0.413913
$179$	−13.0689	−0.976816	−0.488408	−	0.872615i	$-0.662422\pi$
$179$	−13.0689	−0.976816	−0.488408	+	0.872615i	$0.662422\pi$
$180$	0	0
$181$	9.98243	0.741988	0.370994	−	0.928635i	$-0.379017\pi$
$181$	9.98243	0.741988	0.370994	+	0.928635i	$0.379017\pi$
$182$	13.8839	1.02915
$183$	0	0
$184$	12.2654	0.904221
$185$	0	0
$186$	0	0
$187$	1.59511	0.116646
$188$	−13.8751	−1.01194
$189$	0	0
$190$	0	0
$191$	8.69024	0.628804	0.314402	−	0.949290i	$-0.398196\pi$
$191$	8.69024	0.628804	0.314402	+	0.949290i	$0.398196\pi$
$192$	0	0
$193$	18.4643	1.32909	0.664544	−	0.747249i	$-0.268626\pi$
$193$	18.4643	1.32909	0.664544	+	0.747249i	$0.268626\pi$
$194$	6.65862	0.478061
$195$	0	0
$196$	−9.74061	−0.695758
$197$	−1.73458	−0.123584	−0.0617919	−	0.998089i	$-0.519682\pi$
$197$	−1.73458	−0.123584	−0.0617919	+	0.998089i	$0.519682\pi$
$198$	0	0
$199$	13.7835	0.977084	0.488542	−	0.872540i	$-0.337529\pi$
$199$	13.7835	0.977084	0.488542	+	0.872540i	$0.337529\pi$
$200$	0	0
$201$	0	0
$202$	4.79255	0.337202
$203$	20.0827	1.40953
$204$	0	0
$205$	0	0
$206$	−1.13732	−0.0792406
$207$	0	0
$208$	8.71539	0.604304
$209$	2.64639	0.183054
$210$	0	0
$211$	−8.73784	−0.601538	−0.300769	−	0.953697i	$-0.597243\pi$
$211$	−8.73784	−0.601538	−0.300769	+	0.953697i	$0.597243\pi$
$212$	5.32377	0.365638
$213$	0	0
$214$	12.1179	0.828363
$215$	0	0
$216$	0	0
$217$	−3.64222	−0.247250
$218$	4.34217	0.294089
$219$	0	0
$220$	0	0
$221$	7.87384	0.529652
$222$	0	0
$223$	−10.4409	−0.699176	−0.349588	−	0.936903i	$-0.613679\pi$
$223$	−10.4409	−0.699176	−0.349588	+	0.936903i	$0.613679\pi$
$224$	20.9901	1.40246
$225$	0	0
$226$	−2.24273	−0.149184
$227$	25.1149	1.66693	0.833466	−	0.552571i	$-0.186353\pi$
$227$	25.1149	1.66693	0.833466	+	0.552571i	$0.186353\pi$
$228$	0	0
$229$	−3.58524	−0.236919	−0.118460	−	0.992959i	$-0.537796\pi$
$229$	−3.58524	−0.236919	−0.118460	+	0.992959i	$0.537796\pi$
$230$	0	0
$231$	0	0
$232$	−13.0806	−0.858785
$233$	18.2989	1.19880	0.599401	−	0.800449i	$-0.295406\pi$
$233$	18.2989	1.19880	0.599401	+	0.800449i	$0.295406\pi$
$234$	0	0
$235$	0	0
$236$	0.843588	0.0549129
$237$	0	0
$238$	3.35099	0.217212
$239$	9.65731	0.624680	0.312340	−	0.949970i	$-0.398887\pi$
$239$	9.65731	0.624680	0.312340	+	0.949970i	$0.398887\pi$
$240$	0	0
$241$	−5.31631	−0.342454	−0.171227	−	0.985232i	$-0.554773\pi$
$241$	−5.31631	−0.342454	−0.171227	+	0.985232i	$0.554773\pi$
$242$	−6.44781	−0.414481
$243$	0	0
$244$	−16.3049	−1.04381
$245$	0	0
$246$	0	0
$247$	13.0632	0.831192
$248$	2.37231	0.150642
$249$	0	0
$250$	0	0
$251$	9.20850	0.581235	0.290618	−	0.956839i	$-0.406139\pi$
$251$	9.20850	0.581235	0.290618	+	0.956839i	$0.406139\pi$
$252$	0	0
$253$	−5.98243	−0.376112
$254$	7.08361	0.444465
$255$	0	0
$256$	−8.92740	−0.557963
$257$	−7.04525	−0.439470	−0.219735	−	0.975560i	$-0.570519\pi$
$257$	−7.04525	−0.439470	−0.219735	+	0.975560i	$0.570519\pi$
$258$	0	0
$259$	10.0971	0.627402
$260$	0	0
$261$	0	0
$262$	−9.68260	−0.598193
$263$	4.35053	0.268265	0.134133	−	0.990963i	$-0.457175\pi$
$263$	4.35053	0.268265	0.134133	+	0.990963i	$0.457175\pi$
$264$	0	0
$265$	0	0
$266$	5.55951	0.340875
$267$	0	0
$268$	24.6051	1.50299
$269$	20.1710	1.22985	0.614925	−	0.788586i	$-0.289186\pi$
$269$	20.1710	1.22985	0.614925	+	0.788586i	$0.289186\pi$
$270$	0	0
$271$	13.0303	0.791533	0.395767	−	0.918351i	$-0.370479\pi$
$271$	13.0303	0.791533	0.395767	+	0.918351i	$0.370479\pi$
$272$	2.10352	0.127545
$273$	0	0
$274$	−3.55015	−0.214473
$275$	0	0
$276$	0	0
$277$	−5.93472	−0.356583	−0.178292	−	0.983978i	$-0.557057\pi$
$277$	−5.93472	−0.356583	−0.178292	+	0.983978i	$0.557057\pi$
$278$	−5.52232	−0.331207
$279$	0	0
$280$	0	0
$281$	2.56333	0.152915	0.0764577	−	0.997073i	$-0.475639\pi$
$281$	2.56333	0.152915	0.0764577	+	0.997073i	$0.475639\pi$
$282$	0	0
$283$	22.5046	1.33776	0.668879	−	0.743371i	$-0.266774\pi$
$283$	22.5046	1.33776	0.668879	+	0.743371i	$0.266774\pi$
$284$	16.6733	0.989380
$285$	0	0
$286$	4.41076	0.260814
$287$	0.446397	0.0263500
$288$	0	0
$289$	−15.0996	−0.888211
$290$	0	0
$291$	0	0
$292$	12.9660	0.758777
$293$	9.86771	0.576478	0.288239	−	0.957559i	$-0.406930\pi$
$293$	9.86771	0.576478	0.288239	+	0.957559i	$0.406930\pi$
$294$	0	0
$295$	0	0
$296$	−6.57662	−0.382258
$297$	0	0
$298$	−10.6604	−0.617540
$299$	−29.5307	−1.70781
$300$	0	0
$301$	4.06973	0.234576
$302$	4.93719	0.284104
$303$	0	0
$304$	3.48988	0.200158
$305$	0	0
$306$	0	0
$307$	13.4887	0.769843	0.384922	−	0.922949i	$-0.374229\pi$
$307$	13.4887	0.769843	0.384922	+	0.922949i	$0.374229\pi$
$308$	−6.55158	−0.373311
$309$	0	0
$310$	0	0
$311$	−8.33830	−0.472822	−0.236411	−	0.971653i	$-0.575971\pi$
$311$	−8.33830	−0.472822	−0.236411	+	0.971653i	$0.575971\pi$
$312$	0	0
$313$	7.03489	0.397636	0.198818	−	0.980036i	$-0.436290\pi$
$313$	7.03489	0.397636	0.198818	+	0.980036i	$0.436290\pi$
$314$	10.4943	0.592229
$315$	0	0
$316$	−9.07439	−0.510474
$317$	−28.9660	−1.62689	−0.813446	−	0.581640i	$-0.802411\pi$
$317$	−28.9660	−1.62689	−0.813446	+	0.581640i	$0.802411\pi$
$318$	0	0
$319$	6.38004	0.357213
$320$	0	0
$321$	0	0
$322$	−12.5678	−0.700378
$323$	3.15290	0.175432
$324$	0	0
$325$	0	0
$326$	−6.46584	−0.358110
$327$	0	0
$328$	−0.290756	−0.0160543
$329$	32.5078	1.79221
$330$	0	0
$331$	4.97223	0.273299	0.136649	−	0.990619i	$-0.456367\pi$
$331$	4.97223	0.273299	0.136649	+	0.990619i	$0.456367\pi$
$332$	25.6405	1.40720
$333$	0	0
$334$	−12.7523	−0.697773
$335$	0	0
$336$	0	0
$337$	3.96964	0.216240	0.108120	−	0.994138i	$-0.465517\pi$
$337$	3.96964	0.216240	0.108120	+	0.994138i	$0.465517\pi$
$338$	13.0964	0.712352
$339$	0	0
$340$	0	0
$341$	−1.15709	−0.0626599
$342$	0	0
$343$	−2.67434	−0.144401
$344$	−2.65077	−0.142920
$345$	0	0
$346$	14.1011	0.758082
$347$	32.8011	1.76085	0.880427	−	0.474182i	$-0.157256\pi$
$347$	32.8011	1.76085	0.880427	+	0.474182i	$0.157256\pi$
$348$	0	0
$349$	36.2630	1.94111	0.970557	−	0.240872i	$-0.0774332\pi$
$349$	36.2630	1.94111	0.970557	+	0.240872i	$0.0774332\pi$
$350$	0	0
$351$	0	0
$352$	6.66830	0.355422
$353$	30.9661	1.64816	0.824080	−	0.566473i	$-0.191693\pi$
$353$	30.9661	1.64816	0.824080	+	0.566473i	$0.191693\pi$
$354$	0	0
$355$	0	0
$356$	−12.8632	−0.681749
$357$	0	0
$358$	−8.72213	−0.460979
$359$	11.6777	0.616324	0.308162	−	0.951334i	$-0.400286\pi$
$359$	11.6777	0.616324	0.308162	+	0.951334i	$0.400286\pi$
$360$	0	0
$361$	−13.7691	−0.724691
$362$	6.66223	0.350159
$363$	0	0
$364$	−32.3402	−1.69509
$365$	0	0
$366$	0	0
$367$	2.52502	0.131805	0.0659026	−	0.997826i	$-0.479007\pi$
$367$	2.52502	0.131805	0.0659026	+	0.997826i	$0.479007\pi$
$368$	−7.88924	−0.411255
$369$	0	0
$370$	0	0
$371$	−12.4730	−0.647566
$372$	0	0
$373$	15.0961	0.781646	0.390823	−	0.920466i	$-0.372191\pi$
$373$	15.0961	0.781646	0.390823	+	0.920466i	$0.372191\pi$
$374$	1.06457	0.0550475
$375$	0	0
$376$	−21.1736	−1.09194
$377$	31.4934	1.62199
$378$	0	0
$379$	−19.1633	−0.984354	−0.492177	−	0.870495i	$-0.663799\pi$
$379$	−19.1633	−0.984354	−0.492177	+	0.870495i	$0.663799\pi$
$380$	0	0
$381$	0	0
$382$	5.79983	0.296745
$383$	2.27917	0.116460	0.0582301	−	0.998303i	$-0.481454\pi$
$383$	2.27917	0.116460	0.0582301	+	0.998303i	$0.481454\pi$
$384$	0	0
$385$	0	0
$386$	12.3230	0.627223
$387$	0	0
$388$	−15.5101	−0.787405
$389$	14.5865	0.739563	0.369782	−	0.929119i	$-0.379432\pi$
$389$	14.5865	0.739563	0.369782	+	0.929119i	$0.379432\pi$
$390$	0	0
$391$	−7.12745	−0.360451
$392$	−14.8643	−0.750761
$393$	0	0
$394$	−1.15765	−0.0583217
$395$	0	0
$396$	0	0
$397$	33.0931	1.66089	0.830447	−	0.557098i	$-0.188085\pi$
$397$	33.0931	1.66089	0.830447	+	0.557098i	$0.188085\pi$
$398$	9.19903	0.461106
$399$	0	0
$400$	0	0
$401$	−10.6123	−0.529953	−0.264977	−	0.964255i	$-0.585364\pi$
$401$	−10.6123	−0.529953	−0.264977	+	0.964255i	$0.585364\pi$
$402$	0	0
$403$	−5.71167	−0.284519
$404$	−11.1634	−0.555400
$405$	0	0
$406$	13.4031	0.665185
$407$	3.20772	0.159001
$408$	0	0
$409$	−28.1971	−1.39426	−0.697128	−	0.716947i	$-0.745539\pi$
$409$	−28.1971	−1.39426	−0.697128	+	0.716947i	$0.745539\pi$
$410$	0	0
$411$	0	0
$412$	2.64918	0.130516
$413$	−1.97643	−0.0972540
$414$	0	0
$415$	0	0
$416$	32.9164	1.61386
$417$	0	0
$418$	1.76619	0.0863871
$419$	−30.2128	−1.47599	−0.737996	−	0.674805i	$-0.764228\pi$
$419$	−30.2128	−1.47599	−0.737996	+	0.674805i	$0.764228\pi$
$420$	0	0
$421$	−18.4246	−0.897959	−0.448980	−	0.893542i	$-0.648212\pi$
$421$	−18.4246	−0.897959	−0.448980	+	0.893542i	$0.648212\pi$
$422$	−5.83160	−0.283878
$423$	0	0
$424$	8.12414	0.394543
$425$	0	0
$426$	0	0
$427$	38.2006	1.84866
$428$	−28.2266	−1.36438
$429$	0	0
$430$	0	0
$431$	13.6428	0.657152	0.328576	−	0.944478i	$-0.393431\pi$
$431$	13.6428	0.657152	0.328576	+	0.944478i	$0.393431\pi$
$432$	0	0
$433$	−33.6141	−1.61539	−0.807696	−	0.589599i	$-0.799286\pi$
$433$	−33.6141	−1.61539	−0.807696	+	0.589599i	$0.799286\pi$
$434$	−2.43080	−0.116682
$435$	0	0
$436$	−10.1143	−0.484389
$437$	−11.8249	−0.565662
$438$	0	0
$439$	−21.9603	−1.04811	−0.524054	−	0.851685i	$-0.675581\pi$
$439$	−21.9603	−1.04811	−0.524054	+	0.851685i	$0.675581\pi$
$440$	0	0
$441$	0	0
$442$	5.25497	0.249953
$443$	5.44404	0.258654	0.129327	−	0.991602i	$-0.458718\pi$
$443$	5.44404	0.258654	0.129327	+	0.991602i	$0.458718\pi$
$444$	0	0
$445$	0	0
$446$	−6.96824	−0.329956
$447$	0	0
$448$	2.89345	0.136703
$449$	−36.8483	−1.73898	−0.869489	−	0.493952i	$-0.835552\pi$
$449$	−36.8483	−1.73898	−0.869489	+	0.493952i	$0.835552\pi$
$450$	0	0
$451$	0.141815	0.00667781
$452$	5.22406	0.245719
$453$	0	0
$454$	16.7616	0.786659
$455$	0	0
$456$	0	0
$457$	−5.03603	−0.235575	−0.117788	−	0.993039i	$-0.537580\pi$
$457$	−5.03603	−0.235575	−0.117788	+	0.993039i	$0.537580\pi$
$458$	−2.39277	−0.111807
$459$	0	0
$460$	0	0
$461$	22.5649	1.05095	0.525476	−	0.850808i	$-0.323887\pi$
$461$	22.5649	1.05095	0.525476	+	0.850808i	$0.323887\pi$
$462$	0	0
$463$	7.92517	0.368314	0.184157	−	0.982897i	$-0.441045\pi$
$463$	7.92517	0.368314	0.184157	+	0.982897i	$0.441045\pi$
$464$	8.41357	0.390590
$465$	0	0
$466$	12.2126	0.565739
$467$	10.7488	0.497397	0.248698	−	0.968581i	$-0.419997\pi$
$467$	10.7488	0.497397	0.248698	+	0.968581i	$0.419997\pi$
$468$	0	0
$469$	−57.6470	−2.66189
$470$	0	0
$471$	0	0
$472$	1.28733	0.0592540
$473$	1.29291	0.0594479
$474$	0	0
$475$	0	0
$476$	−7.80554	−0.357766
$477$	0	0
$478$	6.44525	0.294799
$479$	31.1215	1.42198	0.710989	−	0.703203i	$-0.248248\pi$
$479$	31.1215	1.42198	0.710989	+	0.703203i	$0.248248\pi$
$480$	0	0
$481$	15.8341	0.721973
$482$	−3.54809	−0.161611
$483$	0	0
$484$	15.0190	0.682684
$485$	0	0
$486$	0	0
$487$	22.1648	1.00438	0.502192	−	0.864756i	$-0.332527\pi$
$487$	22.1648	1.00438	0.502192	+	0.864756i	$0.332527\pi$
$488$	−24.8815	−1.12633
$489$	0	0
$490$	0	0
$491$	−38.3744	−1.73181	−0.865905	−	0.500208i	$-0.833257\pi$
$491$	−38.3744	−1.73181	−0.865905	+	0.500208i	$0.833257\pi$
$492$	0	0
$493$	7.60116	0.342339
$494$	8.71834	0.392256
$495$	0	0
$496$	−1.52589	−0.0685146
$497$	−39.0638	−1.75225
$498$	0	0
$499$	−3.14013	−0.140572	−0.0702858	−	0.997527i	$-0.522391\pi$
$499$	−3.14013	−0.140572	−0.0702858	+	0.997527i	$0.522391\pi$
$500$	0	0
$501$	0	0
$502$	6.14572	0.274297
$503$	−26.4210	−1.17805	−0.589027	−	0.808114i	$-0.700489\pi$
$503$	−26.4210	−1.17805	−0.589027	+	0.808114i	$0.700489\pi$
$504$	0	0
$505$	0	0
$506$	−3.99265	−0.177495
$507$	0	0
$508$	−16.5000	−0.732070
$509$	27.5147	1.21957	0.609785	−	0.792567i	$-0.291256\pi$
$509$	27.5147	1.21957	0.609785	+	0.792567i	$0.291256\pi$
$510$	0	0
$511$	−30.3779	−1.34384
$512$	16.0335	0.708588
$513$	0	0
$514$	−4.70197	−0.207395
$515$	0	0
$516$	0	0
$517$	10.3273	0.454196
$518$	6.73875	0.296084
$519$	0	0
$520$	0	0
$521$	−30.2824	−1.32670	−0.663349	−	0.748310i	$-0.730865\pi$
$521$	−30.2824	−1.32670	−0.663349	+	0.748310i	$0.730865\pi$
$522$	0	0
$523$	−42.8200	−1.87239	−0.936193	−	0.351486i	$-0.885676\pi$
$523$	−42.8200	−1.87239	−0.936193	+	0.351486i	$0.885676\pi$
$524$	22.5539	0.985273
$525$	0	0
$526$	2.90353	0.126600
$527$	−1.37855	−0.0600507
$528$	0	0
$529$	3.73144	0.162236
$530$	0	0
$531$	0	0
$532$	−12.9499	−0.561449
$533$	0.700034	0.0303218
$534$	0	0
$535$	0	0
$536$	37.5477	1.62181
$537$	0	0
$538$	13.4621	0.580391
$539$	7.25001	0.312280
$540$	0	0
$541$	−10.9585	−0.471142	−0.235571	−	0.971857i	$-0.575696\pi$
$541$	−10.9585	−0.471142	−0.235571	+	0.971857i	$0.575696\pi$
$542$	8.69636	0.373541
$543$	0	0
$544$	7.94460	0.340622
$545$	0	0
$546$	0	0
$547$	11.8536	0.506822	0.253411	−	0.967359i	$-0.418448\pi$
$547$	11.8536	0.506822	0.253411	+	0.967359i	$0.418448\pi$
$548$	8.26947	0.353254
$549$	0	0
$550$	0	0
$551$	12.6108	0.537239
$552$	0	0
$553$	21.2603	0.904080
$554$	−3.96081	−0.168279
$555$	0	0
$556$	12.8633	0.545525
$557$	31.2361	1.32351	0.661757	−	0.749718i	$-0.269811\pi$
$557$	31.2361	1.32351	0.661757	+	0.749718i	$0.269811\pi$
$558$	0	0
$559$	6.38210	0.269934
$560$	0	0
$561$	0	0
$562$	1.71075	0.0721638
$563$	16.1583	0.680990	0.340495	−	0.940246i	$-0.389405\pi$
$563$	16.1583	0.680990	0.340495	+	0.940246i	$0.389405\pi$
$564$	0	0
$565$	0	0
$566$	15.0195	0.631315
$567$	0	0
$568$	25.4437	1.06760
$569$	8.39847	0.352082	0.176041	−	0.984383i	$-0.443671\pi$
$569$	8.39847	0.352082	0.176041	+	0.984383i	$0.443671\pi$
$570$	0	0
$571$	−5.39179	−0.225640	−0.112820	−	0.993615i	$-0.535988\pi$
$571$	−5.39179	−0.225640	−0.112820	+	0.993615i	$0.535988\pi$
$572$	−10.2741	−0.429582
$573$	0	0
$574$	0.297924	0.0124351
$575$	0	0
$576$	0	0
$577$	−14.7206	−0.612828	−0.306414	−	0.951898i	$-0.599129\pi$
$577$	−14.7206	−0.612828	−0.306414	+	0.951898i	$0.599129\pi$
$578$	−10.0774	−0.419165
$579$	0	0
$580$	0	0
$581$	−60.0728	−2.49224
$582$	0	0
$583$	−3.96252	−0.164111
$584$	19.7863	0.818761
$585$	0	0
$586$	6.58567	0.272052
$587$	41.7243	1.72215	0.861073	−	0.508481i	$-0.169793\pi$
$587$	41.7243	1.72215	0.861073	+	0.508481i	$0.169793\pi$
$588$	0	0
$589$	−2.28711	−0.0942387
$590$	0	0
$591$	0	0
$592$	4.23013	0.173857
$593$	40.4139	1.65960	0.829801	−	0.558059i	$-0.188454\pi$
$593$	40.4139	1.65960	0.829801	+	0.558059i	$0.188454\pi$
$594$	0	0
$595$	0	0
$596$	24.8315	1.01714
$597$	0	0
$598$	−19.7087	−0.805949
$599$	−38.8888	−1.58895	−0.794477	−	0.607294i	$-0.792255\pi$
$599$	−38.8888	−1.58895	−0.794477	+	0.607294i	$0.792255\pi$
$600$	0	0
$601$	13.9675	0.569747	0.284873	−	0.958565i	$-0.408048\pi$
$601$	13.9675	0.569747	0.284873	+	0.958565i	$0.408048\pi$
$602$	2.71612	0.110701
$603$	0	0
$604$	−11.5003	−0.467942
$605$	0	0
$606$	0	0
$607$	−40.8060	−1.65626	−0.828132	−	0.560534i	$-0.810596\pi$
$607$	−40.8060	−1.65626	−0.828132	+	0.560534i	$0.810596\pi$
$608$	13.1806	0.534544
$609$	0	0
$610$	0	0
$611$	50.9782	2.06236
$612$	0	0
$613$	−26.4962	−1.07017	−0.535086	−	0.844798i	$-0.679721\pi$
$613$	−26.4962	−1.07017	−0.535086	+	0.844798i	$0.679721\pi$
$614$	9.00233	0.363304
$615$	0	0
$616$	−9.99780	−0.402823
$617$	−28.7271	−1.15651	−0.578255	−	0.815856i	$-0.696266\pi$
$617$	−28.7271	−1.15651	−0.578255	+	0.815856i	$0.696266\pi$
$618$	0	0
$619$	−44.0519	−1.77060	−0.885299	−	0.465023i	$-0.846046\pi$
$619$	−44.0519	−1.77060	−0.885299	+	0.465023i	$0.846046\pi$
$620$	0	0
$621$	0	0
$622$	−5.56495	−0.223134
$623$	30.1371	1.20742
$624$	0	0
$625$	0	0
$626$	4.69506	0.187652
$627$	0	0
$628$	−24.4447	−0.975449
$629$	3.82167	0.152380
$630$	0	0
$631$	−0.777435	−0.0309492	−0.0154746	−	0.999880i	$-0.504926\pi$
$631$	−0.777435	−0.0309492	−0.0154746	+	0.999880i	$0.504926\pi$
$632$	−13.8476	−0.550830
$633$	0	0
$634$	−19.3318	−0.767764
$635$	0	0
$636$	0	0
$637$	35.7878	1.41797
$638$	4.25801	0.168576
$639$	0	0
$640$	0	0
$641$	−12.6585	−0.499981	−0.249991	−	0.968248i	$-0.580428\pi$
$641$	−12.6585	−0.499981	−0.249991	+	0.968248i	$0.580428\pi$
$642$	0	0
$643$	−46.7414	−1.84330	−0.921651	−	0.388021i	$-0.873159\pi$
$643$	−46.7414	−1.84330	−0.921651	+	0.388021i	$0.873159\pi$
$644$	29.2746	1.15358
$645$	0	0
$646$	2.10423	0.0827899
$647$	19.0171	0.747639	0.373820	−	0.927501i	$-0.378048\pi$
$647$	19.0171	0.747639	0.373820	+	0.927501i	$0.378048\pi$
$648$	0	0
$649$	−0.627890	−0.0246468
$650$	0	0
$651$	0	0
$652$	15.0611	0.589836
$653$	−23.6344	−0.924885	−0.462442	−	0.886649i	$-0.653027\pi$
$653$	−23.6344	−0.924885	−0.462442	+	0.886649i	$0.653027\pi$
$654$	0	0
$655$	0	0
$656$	0.187016	0.00730176
$657$	0	0
$658$	21.6956	0.845781
$659$	−0.736021	−0.0286713	−0.0143356	−	0.999897i	$-0.504563\pi$
$659$	−0.736021	−0.0286713	−0.0143356	+	0.999897i	$0.504563\pi$
$660$	0	0
$661$	23.1277	0.899561	0.449781	−	0.893139i	$-0.351502\pi$
$661$	23.1277	0.899561	0.449781	+	0.893139i	$0.351502\pi$
$662$	3.31845	0.128975
$663$	0	0
$664$	39.1277	1.51845
$665$	0	0
$666$	0	0
$667$	−28.5081	−1.10384
$668$	29.7042	1.14929
$669$	0	0
$670$	0	0
$671$	12.1359	0.468500
$672$	0	0
$673$	−32.9126	−1.26869	−0.634344	−	0.773051i	$-0.718730\pi$
$673$	−32.9126	−1.26869	−0.634344	+	0.773051i	$0.718730\pi$
$674$	2.64932	0.102048
$675$	0	0
$676$	−30.5059	−1.17330
$677$	−4.83090	−0.185667	−0.0928333	−	0.995682i	$-0.529592\pi$
$677$	−4.83090	−0.185667	−0.0928333	+	0.995682i	$0.529592\pi$
$678$	0	0
$679$	36.3384	1.39454
$680$	0	0
$681$	0	0
$682$	−0.772236	−0.0295705
$683$	36.8203	1.40889	0.704444	−	0.709759i	$-0.251196\pi$
$683$	36.8203	1.40889	0.704444	+	0.709759i	$0.251196\pi$
$684$	0	0
$685$	0	0
$686$	−1.78484	−0.0681456
$687$	0	0
$688$	1.70500	0.0650025
$689$	−19.5600	−0.745176
$690$	0	0
$691$	25.2787	0.961646	0.480823	−	0.876818i	$-0.340338\pi$
$691$	25.2787	0.961646	0.480823	+	0.876818i	$0.340338\pi$
$692$	−32.8462	−1.24862
$693$	0	0
$694$	21.8913	0.830983
$695$	0	0
$696$	0	0
$697$	0.168958	0.00639975
$698$	24.2018	0.916051
$699$	0	0
$700$	0	0
$701$	−50.8414	−1.92025	−0.960127	−	0.279565i	$-0.909810\pi$
$701$	−50.8414	−1.92025	−0.960127	+	0.279565i	$0.909810\pi$
$702$	0	0
$703$	6.34041	0.239133
$704$	0.919215	0.0346442
$705$	0	0
$706$	20.6667	0.777800
$707$	26.1546	0.983646
$708$	0	0
$709$	−37.1976	−1.39698	−0.698492	−	0.715618i	$-0.746145\pi$
$709$	−37.1976	−1.39698	−0.698492	+	0.715618i	$0.746145\pi$
$710$	0	0
$711$	0	0
$712$	−19.6294	−0.735644
$713$	5.17025	0.193627
$714$	0	0
$715$	0	0
$716$	20.3167	0.759270
$717$	0	0
$718$	7.79364	0.290856
$719$	−35.3863	−1.31969	−0.659843	−	0.751403i	$-0.729377\pi$
$719$	−35.3863	−1.31969	−0.659843	+	0.751403i	$0.729377\pi$
$720$	0	0
$721$	−6.20674	−0.231151
$722$	−9.18946	−0.341996
$723$	0	0
$724$	−15.5185	−0.576741
$725$	0	0
$726$	0	0
$727$	−40.7826	−1.51254	−0.756272	−	0.654257i	$-0.772981\pi$
$727$	−40.7826	−1.51254	−0.756272	+	0.654257i	$0.772981\pi$
$728$	−49.3516	−1.82909
$729$	0	0
$730$	0	0
$731$	1.54036	0.0569724
$732$	0	0
$733$	26.8011	0.989922	0.494961	−	0.868915i	$-0.335182\pi$
$733$	26.8011	0.989922	0.494961	+	0.868915i	$0.335182\pi$
$734$	1.68519	0.0622015
$735$	0	0
$736$	−29.7961	−1.09830
$737$	−18.3138	−0.674596
$738$	0	0
$739$	−8.28484	−0.304763	−0.152381	−	0.988322i	$-0.548694\pi$
$739$	−8.28484	−0.304763	−0.152381	+	0.988322i	$0.548694\pi$
$740$	0	0
$741$	0	0
$742$	−8.32443	−0.305599
$743$	24.8614	0.912077	0.456039	−	0.889960i	$-0.349268\pi$
$743$	24.8614	0.912077	0.456039	+	0.889960i	$0.349268\pi$
$744$	0	0
$745$	0	0
$746$	10.0751	0.368874
$747$	0	0
$748$	−2.47973	−0.0906677
$749$	66.1317	2.41640
$750$	0	0
$751$	17.0776	0.623172	0.311586	−	0.950218i	$-0.399140\pi$
$751$	17.0776	0.623172	0.311586	+	0.950218i	$0.399140\pi$
$752$	13.6190	0.496634
$753$	0	0
$754$	21.0186	0.765451
$755$	0	0
$756$	0	0
$757$	−22.6984	−0.824988	−0.412494	−	0.910960i	$-0.635342\pi$
$757$	−22.6984	−0.824988	−0.412494	+	0.910960i	$0.635342\pi$
$758$	−12.7895	−0.464536
$759$	0	0
$760$	0	0
$761$	−4.34342	−0.157449	−0.0787244	−	0.996896i	$-0.525085\pi$
$761$	−4.34342	−0.157449	−0.0787244	+	0.996896i	$0.525085\pi$
$762$	0	0
$763$	23.6968	0.857881
$764$	−13.5097	−0.488764
$765$	0	0
$766$	1.52111	0.0549599
$767$	−3.09942	−0.111913
$768$	0	0
$769$	2.57132	0.0927242	0.0463621	−	0.998925i	$-0.485237\pi$
$769$	2.57132	0.0927242	0.0463621	+	0.998925i	$0.485237\pi$
$770$	0	0
$771$	0	0
$772$	−28.7042	−1.03309
$773$	−12.4442	−0.447588	−0.223794	−	0.974636i	$-0.571844\pi$
$773$	−12.4442	−0.447588	−0.223794	+	0.974636i	$0.571844\pi$
$774$	0	0
$775$	0	0
$776$	−23.6686	−0.849653
$777$	0	0
$778$	9.73495	0.349015
$779$	0.280313	0.0100432
$780$	0	0
$781$	−12.4101	−0.444068
$782$	−4.75683	−0.170104
$783$	0	0
$784$	9.56084	0.341459
$785$	0	0
$786$	0	0
$787$	37.3565	1.33161	0.665807	−	0.746124i	$-0.268087\pi$
$787$	37.3565	1.33161	0.665807	+	0.746124i	$0.268087\pi$
$788$	2.69655	0.0960606
$789$	0	0
$790$	0	0
$791$	−12.2394	−0.435183
$792$	0	0
$793$	59.9056	2.12731
$794$	22.0862	0.783809
$795$	0	0
$796$	−21.4275	−0.759479
$797$	42.1378	1.49260	0.746299	−	0.665611i	$-0.231829\pi$
$797$	42.1378	1.49260	0.746299	+	0.665611i	$0.231829\pi$
$798$	0	0
$799$	12.3040	0.435283
$800$	0	0
$801$	0	0
$802$	−7.08261	−0.250096
$803$	−9.65069	−0.340565
$804$	0	0
$805$	0	0
$806$	−3.81195	−0.134270
$807$	0	0
$808$	−17.0355	−0.599307
$809$	−2.15676	−0.0758277	−0.0379138	−	0.999281i	$-0.512071\pi$
$809$	−2.15676	−0.0758277	−0.0379138	+	0.999281i	$0.512071\pi$
$810$	0	0
$811$	12.8658	0.451780	0.225890	−	0.974153i	$-0.427471\pi$
$811$	12.8658	0.451780	0.225890	+	0.974153i	$0.427471\pi$
$812$	−31.2202	−1.09561
$813$	0	0
$814$	2.14082	0.0750358
$815$	0	0
$816$	0	0
$817$	2.55557	0.0894079
$818$	−18.8186	−0.657977
$819$	0	0
$820$	0	0
$821$	−13.9740	−0.487696	−0.243848	−	0.969813i	$-0.578410\pi$
$821$	−13.9740	−0.487696	−0.243848	+	0.969813i	$0.578410\pi$
$822$	0	0
$823$	45.9522	1.60179	0.800897	−	0.598802i	$-0.204357\pi$
$823$	45.9522	1.60179	0.800897	+	0.598802i	$0.204357\pi$
$824$	4.04269	0.140834
$825$	0	0
$826$	−1.31906	−0.0458961
$827$	−6.37379	−0.221638	−0.110819	−	0.993841i	$-0.535347\pi$
$827$	−6.37379	−0.221638	−0.110819	+	0.993841i	$0.535347\pi$
$828$	0	0
$829$	−21.0176	−0.729970	−0.364985	−	0.931013i	$-0.618926\pi$
$829$	−21.0176	−0.729970	−0.364985	+	0.931013i	$0.618926\pi$
$830$	0	0
$831$	0	0
$832$	4.53747	0.157308
$833$	8.63765	0.299277
$834$	0	0
$835$	0	0
$836$	−4.11403	−0.142287
$837$	0	0
$838$	−20.1639	−0.696551
$839$	−20.6247	−0.712045	−0.356022	−	0.934477i	$-0.615867\pi$
$839$	−20.6247	−0.712045	−0.356022	+	0.934477i	$0.615867\pi$
$840$	0	0
$841$	1.40275	0.0483708
$842$	−12.2965	−0.423765
$843$	0	0
$844$	13.5837	0.467570
$845$	0	0
$846$	0	0
$847$	−35.1880	−1.20907
$848$	−5.22551	−0.179445
$849$	0	0
$850$	0	0
$851$	−14.3331	−0.491334
$852$	0	0
$853$	6.65175	0.227752	0.113876	−	0.993495i	$-0.463673\pi$
$853$	6.65175	0.227752	0.113876	+	0.993495i	$0.463673\pi$
$854$	25.4949	0.872418
$855$	0	0
$856$	−43.0741	−1.47224
$857$	26.3351	0.899590	0.449795	−	0.893132i	$-0.351497\pi$
$857$	26.3351	0.899590	0.449795	+	0.893132i	$0.351497\pi$
$858$	0	0
$859$	26.8919	0.917541	0.458770	−	0.888555i	$-0.348290\pi$
$859$	26.8919	0.917541	0.458770	+	0.888555i	$0.348290\pi$
$860$	0	0
$861$	0	0
$862$	9.10517	0.310123
$863$	5.04588	0.171764	0.0858818	−	0.996305i	$-0.472629\pi$
$863$	5.04588	0.171764	0.0858818	+	0.996305i	$0.472629\pi$
$864$	0	0
$865$	0	0
$866$	−22.4339	−0.762336
$867$	0	0
$868$	5.66213	0.192185
$869$	6.75414	0.229119
$870$	0	0
$871$	−90.4012	−3.06313
$872$	−15.4346	−0.522682
$873$	0	0
$874$	−7.89190	−0.266947
$875$	0	0
$876$	0	0
$877$	1.50014	0.0506560	0.0253280	−	0.999679i	$-0.491937\pi$
$877$	1.50014	0.0506560	0.0253280	+	0.999679i	$0.491937\pi$
$878$	−14.6562	−0.494623
$879$	0	0
$880$	0	0
$881$	52.2328	1.75977	0.879884	−	0.475188i	$-0.157620\pi$
$881$	52.2328	1.75977	0.879884	+	0.475188i	$0.157620\pi$
$882$	0	0
$883$	39.4859	1.32881	0.664403	−	0.747375i	$-0.268686\pi$
$883$	39.4859	1.32881	0.664403	+	0.747375i	$0.268686\pi$
$884$	−12.2405	−0.411694
$885$	0	0
$886$	3.63333	0.122064
$887$	−48.5387	−1.62977	−0.814885	−	0.579623i	$-0.803200\pi$
$887$	−48.5387	−1.62977	−0.814885	+	0.579623i	$0.803200\pi$
$888$	0	0
$889$	38.6578	1.29654
$890$	0	0
$891$	0	0
$892$	16.2313	0.543464
$893$	20.4131	0.683097
$894$	0	0
$895$	0	0
$896$	−40.0491	−1.33795
$897$	0	0
$898$	−24.5924	−0.820659
$899$	−5.51387	−0.183898
$900$	0	0
$901$	−4.72094	−0.157277
$902$	0.0946468	0.00315139
$903$	0	0
$904$	7.97198	0.265144
$905$	0	0
$906$	0	0
$907$	−47.3682	−1.57284	−0.786418	−	0.617694i	$-0.788067\pi$
$907$	−47.3682	−1.57284	−0.786418	+	0.617694i	$0.788067\pi$
$908$	−39.0431	−1.29569
$909$	0	0
$910$	0	0
$911$	28.7118	0.951264	0.475632	−	0.879644i	$-0.342219\pi$
$911$	28.7118	0.951264	0.475632	+	0.879644i	$0.342219\pi$
$912$	0	0
$913$	−19.0844	−0.631602
$914$	−3.36102	−0.111173
$915$	0	0
$916$	5.57355	0.184155
$917$	−52.8414	−1.74498
$918$	0	0
$919$	−3.19601	−0.105427	−0.0527133	−	0.998610i	$-0.516787\pi$
$919$	−3.19601	−0.105427	−0.0527133	+	0.998610i	$0.516787\pi$
$920$	0	0
$921$	0	0
$922$	15.0597	0.495966
$923$	−61.2593	−2.01637
$924$	0	0
$925$	0	0
$926$	5.28923	0.173815
$927$	0	0
$928$	31.7764	1.04311
$929$	−2.28411	−0.0749392	−0.0374696	−	0.999298i	$-0.511930\pi$
$929$	−2.28411	−0.0749392	−0.0374696	+	0.999298i	$0.511930\pi$
$930$	0	0
$931$	14.3304	0.469661
$932$	−28.4472	−0.931818
$933$	0	0
$934$	7.17373	0.234732
$935$	0	0
$936$	0	0
$937$	−9.90513	−0.323587	−0.161793	−	0.986825i	$-0.551728\pi$
$937$	−9.90513	−0.323587	−0.161793	+	0.986825i	$0.551728\pi$
$938$	−38.4734	−1.25620
$939$	0	0
$940$	0	0
$941$	8.74045	0.284930	0.142465	−	0.989800i	$-0.454497\pi$
$941$	8.74045	0.284930	0.142465	+	0.989800i	$0.454497\pi$
$942$	0	0
$943$	−0.633676	−0.0206353
$944$	−0.828019	−0.0269497
$945$	0	0
$946$	0.862880	0.0280547
$947$	−55.8934	−1.81629	−0.908146	−	0.418654i	$-0.862502\pi$
$947$	−55.8934	−1.81629	−0.908146	+	0.418654i	$0.862502\pi$
$948$	0	0
$949$	−47.6381	−1.54640
$950$	0	0
$951$	0	0
$952$	−11.9114	−0.386049
$953$	36.3033	1.17598	0.587989	−	0.808869i	$-0.299920\pi$
$953$	36.3033	1.17598	0.587989	+	0.808869i	$0.299920\pi$
$954$	0	0
$955$	0	0
$956$	−15.0131	−0.485558
$957$	0	0
$958$	20.7704	0.671060
$959$	−19.3744	−0.625634
$960$	0	0
$961$	1.00000	0.0322581
$962$	10.5676	0.340714
$963$	0	0
$964$	8.26465	0.266186
$965$	0	0
$966$	0	0
$967$	−25.7135	−0.826890	−0.413445	−	0.910529i	$-0.635675\pi$
$967$	−25.7135	−0.826890	−0.413445	+	0.910529i	$0.635675\pi$
$968$	22.9193	0.736653
$969$	0	0
$970$	0	0
$971$	−33.1226	−1.06296	−0.531478	−	0.847072i	$-0.678363\pi$
$971$	−33.1226	−1.06296	−0.531478	+	0.847072i	$0.678363\pi$
$972$	0	0
$973$	−30.1373	−0.966156
$974$	14.7927	0.473989
$975$	0	0
$976$	16.0040	0.512275
$977$	22.6735	0.725388	0.362694	−	0.931908i	$-0.381857\pi$
$977$	22.6735	0.725388	0.362694	+	0.931908i	$0.381857\pi$
$978$	0	0
$979$	9.57419	0.305992
$980$	0	0
$981$	0	0
$982$	−25.6109	−0.817276
$983$	−9.35725	−0.298450	−0.149225	−	0.988803i	$-0.547678\pi$
$983$	−9.35725	−0.298450	−0.149225	+	0.988803i	$0.547678\pi$
$984$	0	0
$985$	0	0
$986$	5.07298	0.161557
$987$	0	0
$988$	−20.3078	−0.646079
$989$	−5.77712	−0.183702
$990$	0	0
$991$	12.9858	0.412506	0.206253	−	0.978499i	$-0.433873\pi$
$991$	12.9858	0.412506	0.206253	+	0.978499i	$0.433873\pi$
$992$	−5.76300	−0.182976
$993$	0	0
$994$	−26.0710	−0.826922
$995$	0	0
$996$	0	0
$997$	3.74266	0.118531	0.0592656	−	0.998242i	$-0.481124\pi$
$997$	3.74266	0.118531	0.0592656	+	0.998242i	$0.481124\pi$
$998$	−2.09571	−0.0663385
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6975.2.a.ca.1.4		6
3.2	odd	2		2325.2.a.bb.1.3	yes	6
5.4	even	2		6975.2.a.cc.1.3		6
15.2	even	4		2325.2.c.r.1024.6		12
15.8	even	4		2325.2.c.r.1024.7		12
15.14	odd	2		2325.2.a.y.1.4	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2325.2.a.y.1.4	✓	6	15.14	odd	2
2325.2.a.bb.1.3	yes	6	3.2	odd	2
2325.2.c.r.1024.6		12	15.2	even	4
2325.2.c.r.1024.7		12	15.8	even	4
6975.2.a.ca.1.4		6	1.1	even	1	trivial
6975.2.a.cc.1.3		6	5.4	even	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)

\(f(q)\)	\(=\)	\(q+0.667396 q^{2} -1.55458 q^{4} +3.64222 q^{7} -2.37231 q^{8} +1.15709 q^{11} +5.71167 q^{13} +2.43080 q^{14} +1.52589 q^{16} +1.37855 q^{17} +2.28711 q^{19} +0.772236 q^{22} -5.17025 q^{23} +3.81195 q^{26} -5.66213 q^{28} +5.51387 q^{29} -1.00000 q^{31} +5.76300 q^{32} +0.920040 q^{34} +2.77224 q^{37} +1.52641 q^{38} +0.122562 q^{41} +1.11738 q^{43} -1.79879 q^{44} -3.45060 q^{46} +8.92527 q^{47} +6.26574 q^{49} -8.87926 q^{52} -3.42456 q^{53} -8.64048 q^{56} +3.67993 q^{58} -0.542646 q^{59} +10.4883 q^{61} -0.667396 q^{62} +0.794421 q^{64} -15.8274 q^{67} -2.14307 q^{68} -10.7253 q^{71} -8.34049 q^{73} +1.85018 q^{74} -3.55550 q^{76} +4.21437 q^{77} +5.83719 q^{79} +0.0817974 q^{82} -16.4935 q^{83} +0.745734 q^{86} -2.74498 q^{88} +8.27438 q^{89} +20.8031 q^{91} +8.03757 q^{92} +5.95669 q^{94} +9.97701 q^{97} +4.18173 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{2} + 7 q^{4} + 2 q^{7} + 3 q^{8} - 7 q^{11} + 4 q^{13} - 10 q^{14} + 17 q^{16} + 17 q^{19} + 2 q^{22} + q^{23} - 2 q^{26} + 22 q^{28} + 8 q^{29} - 6 q^{31} + 35 q^{32} - 13 q^{34} + 14 q^{37}+ \cdots - 37 q^{98}+O(q^{100}) \) 6 * q - q^2 + 7 * q^4 + 2 * q^7 + 3 * q^8 - 7 * q^11 + 4 * q^13 - 10 * q^14 + 17 * q^16 + 17 * q^19 + 2 * q^22 + q^23 - 2 * q^26 + 22 * q^28 + 8 * q^29 - 6 * q^31 + 35 * q^32 - 13 * q^34 + 14 * q^37 - 10 * q^38 - 18 * q^41 + 15 * q^43 - 4 * q^44 - 10 * q^46 + 13 * q^47 + 18 * q^49 - 38 * q^52 + 13 * q^53 - 32 * q^56 + 23 * q^58 - 32 * q^59 + q^62 + 29 * q^64 + 9 * q^67 - 9 * q^68 - q^71 + 14 * q^73 - 20 * q^74 + 42 * q^76 + 14 * q^77 + 13 * q^79 - 32 * q^82 - 9 * q^83 + 11 * q^86 + 36 * q^88 - 34 * q^89 - 2 * q^91 + 46 * q^92 + 19 * q^94 + 30 * q^97 - 37 * q^98

Introduction

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