LMFDB - Embedded newform 308.4.a.d.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [308,4,Mod(1,308)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(308, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("308.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 308 = 2^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	308.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:	$18.1725882818$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 43x^{2} - 11x + 222 $ x^4 - x^3 - 43x^2 - 11x + 222
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$6.83284$ of defining polynomial
Character	$\chi$	$=$	308.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.83284 q^{3} +12.0220 q^{5} -7.00000 q^{7} +7.02203 q^{9} +11.0000 q^{11} +31.7043 q^{13} +70.1226 q^{15} +112.106 q^{17} -6.19461 q^{19} -40.8299 q^{21} +113.822 q^{23} +19.5293 q^{25} -116.528 q^{27} +71.8038 q^{29} -88.9496 q^{31} +64.1612 q^{33} -84.1542 q^{35} -92.9024 q^{37} +184.926 q^{39} +192.683 q^{41} +124.387 q^{43} +84.4191 q^{45} -272.822 q^{47} +49.0000 q^{49} +653.896 q^{51} -122.144 q^{53} +132.242 q^{55} -36.1322 q^{57} -67.4038 q^{59} -173.205 q^{61} -49.1542 q^{63} +381.150 q^{65} +22.9068 q^{67} +663.903 q^{69} +769.223 q^{71} -57.1402 q^{73} +113.911 q^{75} -77.0000 q^{77} -381.875 q^{79} -869.286 q^{81} -177.002 q^{83} +1347.74 q^{85} +418.820 q^{87} -243.840 q^{89} -221.930 q^{91} -518.829 q^{93} -74.4719 q^{95} -1817.16 q^{97} +77.2424 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 3 q^{3} + q^{5} - 28 q^{7} - 19 q^{9} + 44 q^{11} + 98 q^{13} - 33 q^{15} + 64 q^{17} + 114 q^{19} + 21 q^{21} + 231 q^{23} + 417 q^{25} + 63 q^{27} + 268 q^{29} + 33 q^{31} - 33 q^{33} - 7 q^{35}+ \cdots - 209 q^{99}+O(q^{100}) $ 4 * q - 3 * q^3 + q^5 - 28 * q^7 - 19 * q^9 + 44 * q^11 + 98 * q^13 - 33 * q^15 + 64 * q^17 + 114 * q^19 + 21 * q^21 + 231 * q^23 + 417 * q^25 + 63 * q^27 + 268 * q^29 + 33 * q^31 - 33 * q^33 - 7 * q^35 + 357 * q^37 + 140 * q^39 + 364 * q^41 + 44 * q^43 + 912 * q^45 + 720 * q^47 + 196 * q^49 + 794 * q^51 + 740 * q^53 + 11 * q^55 + 138 * q^57 + 787 * q^59 + 1020 * q^61 + 133 * q^63 + 296 * q^65 - 995 * q^67 + 1059 * q^69 - 1011 * q^71 + 1592 * q^73 - 324 * q^75 - 308 * q^77 - 178 * q^79 - 1396 * q^81 + 324 * q^83 - 94 * q^85 - 1262 * q^87 + 19 * q^89 - 686 * q^91 + 637 * q^93 - 2418 * q^95 - 555 * q^97 - 209 * q^99

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
$3$	5.83284	1.12253	0.561265	−	0.827636i	$-0.310315\pi$
$3$	5.83284	1.12253	0.561265	+	0.827636i	$0.310315\pi$
$4$	0	0
$5$	12.0220	1.07528	0.537642	−	0.843173i	$-0.319315\pi$
$5$	12.0220	1.07528	0.537642	+	0.843173i	$0.319315\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	7.02203	0.260075
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	31.7043	0.676400	0.338200	−	0.941074i	$-0.390182\pi$
$13$	31.7043	0.676400	0.338200	+	0.941074i	$0.390182\pi$
$14$	0	0
$15$	70.1226	1.20704
$16$	0	0
$17$	112.106	1.59939	0.799697	−	0.600404i	$-0.204993\pi$
$17$	112.106	1.59939	0.799697	+	0.600404i	$0.204993\pi$
$18$	0	0
$19$	−6.19461	−0.0747969	−0.0373985	−	0.999300i	$-0.511907\pi$
$19$	−6.19461	−0.0747969	−0.0373985	+	0.999300i	$0.511907\pi$
$20$	0	0
$21$	−40.8299	−0.424277
$22$	0	0
$23$	113.822	1.03189	0.515944	−	0.856622i	$-0.327441\pi$
$23$	113.822	1.03189	0.515944	+	0.856622i	$0.327441\pi$
$24$	0	0
$25$	19.5293	0.156234
$26$	0	0
$27$	−116.528	−0.830588
$28$	0	0
$29$	71.8038	0.459780	0.229890	−	0.973217i	$-0.426163\pi$
$29$	71.8038	0.459780	0.229890	+	0.973217i	$0.426163\pi$
$30$	0	0
$31$	−88.9496	−0.515349	−0.257675	−	0.966232i	$-0.582956\pi$
$31$	−88.9496	−0.515349	−0.257675	+	0.966232i	$0.582956\pi$
$32$	0	0
$33$	64.1612	0.338456
$34$	0	0
$35$	−84.1542	−0.406419
$36$	0	0
$37$	−92.9024	−0.412785	−0.206393	−	0.978469i	$-0.566172\pi$
$37$	−92.9024	−0.412785	−0.206393	+	0.978469i	$0.566172\pi$
$38$	0	0
$39$	184.926	0.759280
$40$	0	0
$41$	192.683	0.733953	0.366977	−	0.930230i	$-0.380393\pi$
$41$	192.683	0.733953	0.366977	+	0.930230i	$0.380393\pi$
$42$	0	0
$43$	124.387	0.441137	0.220569	−	0.975371i	$-0.429209\pi$
$43$	124.387	0.441137	0.220569	+	0.975371i	$0.429209\pi$
$44$	0	0
$45$	84.4191	0.279655
$46$	0	0
$47$	−272.822	−0.846707	−0.423354	−	0.905965i	$-0.639147\pi$
$47$	−272.822	−0.846707	−0.423354	+	0.905965i	$0.639147\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	653.896	1.79537
$52$	0	0
$53$	−122.144	−0.316561	−0.158280	−	0.987394i	$-0.550595\pi$
$53$	−122.144	−0.316561	−0.158280	+	0.987394i	$0.550595\pi$
$54$	0	0
$55$	132.242	0.324210
$56$	0	0
$57$	−36.1322	−0.0839619
$58$	0	0
$59$	−67.4038	−0.148733	−0.0743663	−	0.997231i	$-0.523693\pi$
$59$	−67.4038	−0.148733	−0.0743663	+	0.997231i	$0.523693\pi$
$60$	0	0
$61$	−173.205	−0.363551	−0.181775	−	0.983340i	$-0.558184\pi$
$61$	−173.205	−0.363551	−0.181775	+	0.983340i	$0.558184\pi$
$62$	0	0
$63$	−49.1542	−0.0982992
$64$	0	0
$65$	381.150	0.727321
$66$	0	0
$67$	22.9068	0.0417687	0.0208844	−	0.999782i	$-0.493352\pi$
$67$	22.9068	0.0417687	0.0208844	+	0.999782i	$0.493352\pi$
$68$	0	0
$69$	663.903	1.15833
$70$	0	0
$71$	769.223	1.28577	0.642887	−	0.765961i	$-0.277736\pi$
$71$	769.223	1.28577	0.642887	+	0.765961i	$0.277736\pi$
$72$	0	0
$73$	−57.1402	−0.0916131	−0.0458066	−	0.998950i	$-0.514586\pi$
$73$	−57.1402	−0.0916131	−0.0458066	+	0.998950i	$0.514586\pi$
$74$	0	0
$75$	113.911	0.175378
$76$	0	0
$77$	−77.0000	−0.113961
$78$	0	0
$79$	−381.875	−0.543851	−0.271926	−	0.962318i	$-0.587661\pi$
$79$	−381.875	−0.543851	−0.271926	+	0.962318i	$0.587661\pi$
$80$	0	0
$81$	−869.286	−1.19244
$82$	0	0
$83$	−177.002	−0.234078	−0.117039	−	0.993127i	$-0.537340\pi$
$83$	−177.002	−0.234078	−0.117039	+	0.993127i	$0.537340\pi$
$84$	0	0
$85$	1347.74	1.71980
$86$	0	0
$87$	418.820	0.516118
$88$	0	0
$89$	−243.840	−0.290416	−0.145208	−	0.989401i	$-0.546385\pi$
$89$	−243.840	−0.290416	−0.145208	+	0.989401i	$0.546385\pi$
$90$	0	0
$91$	−221.930	−0.255655
$92$	0	0
$93$	−518.829	−0.578495
$94$	0	0
$95$	−74.4719	−0.0804279
$96$	0	0
$97$	−1817.16	−1.90211	−0.951057	−	0.309016i	$-0.900000\pi$
$97$	−1817.16	−1.90211	−0.951057	+	0.309016i	$0.900000\pi$
$98$	0	0
$99$	77.2424	0.0784157
$100$	0	0
$101$	925.682	0.911969	0.455984	−	0.889988i	$-0.349287\pi$
$101$	925.682	0.911969	0.455984	+	0.889988i	$0.349287\pi$
$102$	0	0
$103$	−1463.12	−1.39966	−0.699830	−	0.714309i	$-0.746741\pi$
$103$	−1463.12	−1.39966	−0.699830	+	0.714309i	$0.746741\pi$
$104$	0	0
$105$	−490.858	−0.456218
$106$	0	0
$107$	911.434	0.823474	0.411737	−	0.911303i	$-0.364922\pi$
$107$	911.434	0.823474	0.411737	+	0.911303i	$0.364922\pi$
$108$	0	0
$109$	771.888	0.678289	0.339144	−	0.940734i	$-0.389862\pi$
$109$	771.888	0.678289	0.339144	+	0.940734i	$0.389862\pi$
$110$	0	0
$111$	−541.885	−0.463364
$112$	0	0
$113$	64.8840	0.0540157	0.0270078	−	0.999635i	$-0.491402\pi$
$113$	64.8840	0.0540157	0.0270078	+	0.999635i	$0.491402\pi$
$114$	0	0
$115$	1368.37	1.10957
$116$	0	0
$117$	222.629	0.175915
$118$	0	0
$119$	−784.742	−0.604514
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	1123.89	0.823885
$124$	0	0
$125$	−1267.97	−0.907287
$126$	0	0
$127$	−369.047	−0.257856	−0.128928	−	0.991654i	$-0.541154\pi$
$127$	−369.047	−0.257856	−0.128928	+	0.991654i	$0.541154\pi$
$128$	0	0
$129$	725.532	0.495190
$130$	0	0
$131$	−2397.95	−1.59931	−0.799657	−	0.600457i	$-0.794985\pi$
$131$	−2397.95	−1.59931	−0.799657	+	0.600457i	$0.794985\pi$
$132$	0	0
$133$	43.3623	0.0282706
$134$	0	0
$135$	−1400.91	−0.893118
$136$	0	0
$137$	−1111.64	−0.693238	−0.346619	−	0.938006i	$-0.612670\pi$
$137$	−1111.64	−0.693238	−0.346619	+	0.938006i	$0.612670\pi$
$138$	0	0
$139$	−2014.94	−1.22953	−0.614765	−	0.788710i	$-0.710749\pi$
$139$	−2014.94	−1.22953	−0.614765	+	0.788710i	$0.710749\pi$
$140$	0	0
$141$	−1591.33	−0.950455
$142$	0	0
$143$	348.748	0.203942
$144$	0	0
$145$	863.228	0.494394
$146$	0	0
$147$	285.809	0.160362
$148$	0	0
$149$	1537.51	0.845354	0.422677	−	0.906280i	$-0.361090\pi$
$149$	1537.51	0.845354	0.422677	+	0.906280i	$0.361090\pi$
$150$	0	0
$151$	−2809.26	−1.51400	−0.757002	−	0.653413i	$-0.773337\pi$
$151$	−2809.26	−1.51400	−0.757002	+	0.653413i	$0.773337\pi$
$152$	0	0
$153$	787.212	0.415963
$154$	0	0
$155$	−1069.36	−0.554146
$156$	0	0
$157$	1484.81	0.754781	0.377390	−	0.926054i	$-0.376821\pi$
$157$	1484.81	0.754781	0.377390	+	0.926054i	$0.376821\pi$
$158$	0	0
$159$	−712.445	−0.355349
$160$	0	0
$161$	−796.751	−0.390017
$162$	0	0
$163$	−3129.79	−1.50395	−0.751976	−	0.659191i	$-0.770899\pi$
$163$	−3129.79	−1.50395	−0.751976	+	0.659191i	$0.770899\pi$
$164$	0	0
$165$	771.349	0.363936
$166$	0	0
$167$	−3351.63	−1.55303	−0.776517	−	0.630097i	$-0.783015\pi$
$167$	−3351.63	−1.55303	−0.776517	+	0.630097i	$0.783015\pi$
$168$	0	0
$169$	−1191.84	−0.542483
$170$	0	0
$171$	−43.4988	−0.0194528
$172$	0	0
$173$	3327.39	1.46229	0.731147	−	0.682220i	$-0.238985\pi$
$173$	3327.39	1.46229	0.731147	+	0.682220i	$0.238985\pi$
$174$	0	0
$175$	−136.705	−0.0590510
$176$	0	0
$177$	−393.156	−0.166957
$178$	0	0
$179$	−3704.49	−1.54685	−0.773426	−	0.633886i	$-0.781459\pi$
$179$	−3704.49	−1.54685	−0.773426	+	0.633886i	$0.781459\pi$
$180$	0	0
$181$	379.186	0.155716	0.0778582	−	0.996964i	$-0.475192\pi$
$181$	379.186	0.155716	0.0778582	+	0.996964i	$0.475192\pi$
$182$	0	0
$183$	−1010.28	−0.408097
$184$	0	0
$185$	−1116.88	−0.443861
$186$	0	0
$187$	1233.17	0.482235
$188$	0	0
$189$	815.698	0.313933
$190$	0	0
$191$	−3357.55	−1.27196	−0.635978	−	0.771707i	$-0.719403\pi$
$191$	−3357.55	−1.27196	−0.635978	+	0.771707i	$0.719403\pi$
$192$	0	0
$193$	2667.38	0.994830	0.497415	−	0.867513i	$-0.334283\pi$
$193$	2667.38	0.994830	0.497415	+	0.867513i	$0.334283\pi$
$194$	0	0
$195$	2223.19	0.816441
$196$	0	0
$197$	−3403.70	−1.23098	−0.615491	−	0.788144i	$-0.711042\pi$
$197$	−3403.70	−1.23098	−0.615491	+	0.788144i	$0.711042\pi$
$198$	0	0
$199$	4448.74	1.58474	0.792370	−	0.610041i	$-0.208847\pi$
$199$	4448.74	1.58474	0.792370	+	0.610041i	$0.208847\pi$
$200$	0	0
$201$	133.611	0.0468867
$202$	0	0
$203$	−502.627	−0.173781
$204$	0	0
$205$	2316.45	0.789208
$206$	0	0
$207$	799.258	0.268369
$208$	0	0
$209$	−68.1408	−0.0225521
$210$	0	0
$211$	3350.65	1.09322	0.546608	−	0.837389i	$-0.315919\pi$
$211$	3350.65	1.09322	0.546608	+	0.837389i	$0.315919\pi$
$212$	0	0
$213$	4486.75	1.44332
$214$	0	0
$215$	1495.39	0.474347
$216$	0	0
$217$	622.648	0.194784
$218$	0	0
$219$	−333.290	−0.102839
$220$	0	0
$221$	3554.24	1.08183
$222$	0	0
$223$	2775.98	0.833603	0.416802	−	0.908997i	$-0.363151\pi$
$223$	2775.98	0.833603	0.416802	+	0.908997i	$0.363151\pi$
$224$	0	0
$225$	137.135	0.0406327
$226$	0	0
$227$	−3389.95	−0.991183	−0.495592	−	0.868556i	$-0.665049\pi$
$227$	−3389.95	−0.991183	−0.495592	+	0.868556i	$0.665049\pi$
$228$	0	0
$229$	5565.87	1.60613	0.803063	−	0.595895i	$-0.203202\pi$
$229$	5565.87	1.60613	0.803063	+	0.595895i	$0.203202\pi$
$230$	0	0
$231$	−449.129	−0.127924
$232$	0	0
$233$	6088.68	1.71194	0.855971	−	0.517023i	$-0.172960\pi$
$233$	6088.68	1.71194	0.855971	+	0.517023i	$0.172960\pi$
$234$	0	0
$235$	−3279.88	−0.910450
$236$	0	0
$237$	−2227.41	−0.610490
$238$	0	0
$239$	−1284.25	−0.347578	−0.173789	−	0.984783i	$-0.555601\pi$
$239$	−1284.25	−0.347578	−0.173789	+	0.984783i	$0.555601\pi$
$240$	0	0
$241$	3508.32	0.937720	0.468860	−	0.883273i	$-0.344665\pi$
$241$	3508.32	0.937720	0.468860	+	0.883273i	$0.344665\pi$
$242$	0	0
$243$	−1924.14	−0.507958
$244$	0	0
$245$	589.080	0.153612
$246$	0	0
$247$	−196.396	−0.0505926
$248$	0	0
$249$	−1032.42	−0.262759
$250$	0	0
$251$	636.222	0.159992	0.0799959	−	0.996795i	$-0.474509\pi$
$251$	636.222	0.159992	0.0799959	+	0.996795i	$0.474509\pi$
$252$	0	0
$253$	1252.04	0.311126
$254$	0	0
$255$	7861.16	1.93053
$256$	0	0
$257$	3186.64	0.773452	0.386726	−	0.922195i	$-0.373606\pi$
$257$	3186.64	0.773452	0.386726	+	0.922195i	$0.373606\pi$
$258$	0	0
$259$	650.317	0.156018
$260$	0	0
$261$	504.209	0.119577
$262$	0	0
$263$	−4437.62	−1.04044	−0.520219	−	0.854033i	$-0.674150\pi$
$263$	−4437.62	−1.04044	−0.520219	+	0.854033i	$0.674150\pi$
$264$	0	0
$265$	−1468.42	−0.340393
$266$	0	0
$267$	−1422.28	−0.326001
$268$	0	0
$269$	1191.51	0.270065	0.135032	−	0.990841i	$-0.456886\pi$
$269$	1191.51	0.270065	0.135032	+	0.990841i	$0.456886\pi$
$270$	0	0
$271$	322.829	0.0723634	0.0361817	−	0.999345i	$-0.488480\pi$
$271$	322.829	0.0723634	0.0361817	+	0.999345i	$0.488480\pi$
$272$	0	0
$273$	−1294.48	−0.286981
$274$	0	0
$275$	214.822	0.0471064
$276$	0	0
$277$	3919.08	0.850090	0.425045	−	0.905172i	$-0.360258\pi$
$277$	3919.08	0.850090	0.425045	+	0.905172i	$0.360258\pi$
$278$	0	0
$279$	−624.607	−0.134030
$280$	0	0
$281$	−8295.86	−1.76117	−0.880587	−	0.473885i	$-0.842851\pi$
$281$	−8295.86	−1.76117	−0.880587	+	0.473885i	$0.842851\pi$
$282$	0	0
$283$	5061.76	1.06322	0.531609	−	0.846990i	$-0.321588\pi$
$283$	5061.76	1.06322	0.531609	+	0.846990i	$0.321588\pi$
$284$	0	0
$285$	−434.382	−0.0902828
$286$	0	0
$287$	−1348.78	−0.277408
$288$	0	0
$289$	7654.75	1.55806
$290$	0	0
$291$	−10599.2	−2.13518
$292$	0	0
$293$	−2908.90	−0.579998	−0.289999	−	0.957027i	$-0.593655\pi$
$293$	−2908.90	−0.579998	−0.289999	+	0.957027i	$0.593655\pi$
$294$	0	0
$295$	−810.331	−0.159930
$296$	0	0
$297$	−1281.81	−0.250432
$298$	0	0
$299$	3608.63	0.697969
$300$	0	0
$301$	−870.711	−0.166734
$302$	0	0
$303$	5399.36	1.02371
$304$	0	0
$305$	−2082.27	−0.390920
$306$	0	0
$307$	−3613.12	−0.671699	−0.335850	−	0.941916i	$-0.609023\pi$
$307$	−3613.12	−0.671699	−0.335850	+	0.941916i	$0.609023\pi$
$308$	0	0
$309$	−8534.12	−1.57116
$310$	0	0
$311$	6958.57	1.26876	0.634380	−	0.773022i	$-0.281256\pi$
$311$	6958.57	1.26876	0.634380	+	0.773022i	$0.281256\pi$
$312$	0	0
$313$	−5067.50	−0.915119	−0.457560	−	0.889179i	$-0.651276\pi$
$313$	−5067.50	−0.915119	−0.457560	+	0.889179i	$0.651276\pi$
$314$	0	0
$315$	−590.934	−0.105700
$316$	0	0
$317$	−2326.87	−0.412272	−0.206136	−	0.978523i	$-0.566089\pi$
$317$	−2326.87	−0.412272	−0.206136	+	0.978523i	$0.566089\pi$
$318$	0	0
$319$	789.842	0.138629
$320$	0	0
$321$	5316.25	0.924374
$322$	0	0
$323$	−694.453	−0.119630
$324$	0	0
$325$	619.163	0.105677
$326$	0	0
$327$	4502.30	0.761400
$328$	0	0
$329$	1909.76	0.320025
$330$	0	0
$331$	5078.45	0.843314	0.421657	−	0.906755i	$-0.361449\pi$
$331$	5078.45	0.843314	0.421657	+	0.906755i	$0.361449\pi$
$332$	0	0
$333$	−652.364	−0.107355
$334$	0	0
$335$	275.386	0.0449132
$336$	0	0
$337$	−10019.1	−1.61952	−0.809759	−	0.586763i	$-0.800402\pi$
$337$	−10019.1	−1.61952	−0.809759	+	0.586763i	$0.800402\pi$
$338$	0	0
$339$	378.458	0.0606343
$340$	0	0
$341$	−978.446	−0.155384
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	7981.46	1.24553
$346$	0	0
$347$	1205.36	0.186476	0.0932379	−	0.995644i	$-0.470278\pi$
$347$	1205.36	0.186476	0.0932379	+	0.995644i	$0.470278\pi$
$348$	0	0
$349$	6115.00	0.937904	0.468952	−	0.883224i	$-0.344632\pi$
$349$	6115.00	0.937904	0.468952	+	0.883224i	$0.344632\pi$
$350$	0	0
$351$	−3694.45	−0.561810
$352$	0	0
$353$	−11056.8	−1.66712	−0.833558	−	0.552432i	$-0.813700\pi$
$353$	−11056.8	−1.66712	−0.833558	+	0.552432i	$0.813700\pi$
$354$	0	0
$355$	9247.62	1.38257
$356$	0	0
$357$	−4577.28	−0.678586
$358$	0	0
$359$	5340.03	0.785059	0.392530	−	0.919739i	$-0.371600\pi$
$359$	5340.03	0.785059	0.392530	+	0.919739i	$0.371600\pi$
$360$	0	0
$361$	−6820.63	−0.994405
$362$	0	0
$363$	705.774	0.102048
$364$	0	0
$365$	−686.942	−0.0985101
$366$	0	0
$367$	2363.54	0.336173	0.168087	−	0.985772i	$-0.446241\pi$
$367$	2363.54	0.336173	0.168087	+	0.985772i	$0.446241\pi$
$368$	0	0
$369$	1353.03	0.190883
$370$	0	0
$371$	855.006	0.119649
$372$	0	0
$373$	6158.54	0.854899	0.427449	−	0.904039i	$-0.359412\pi$
$373$	6158.54	0.854899	0.427449	+	0.904039i	$0.359412\pi$
$374$	0	0
$375$	−7395.88	−1.01846
$376$	0	0
$377$	2276.49	0.310995
$378$	0	0
$379$	4192.82	0.568261	0.284130	−	0.958786i	$-0.408295\pi$
$379$	4192.82	0.568261	0.284130	+	0.958786i	$0.408295\pi$
$380$	0	0
$381$	−2152.60	−0.289451
$382$	0	0
$383$	4878.83	0.650905	0.325452	−	0.945558i	$-0.394483\pi$
$383$	4878.83	0.650905	0.325452	+	0.945558i	$0.394483\pi$
$384$	0	0
$385$	−925.697	−0.122540
$386$	0	0
$387$	873.452	0.114729
$388$	0	0
$389$	11150.3	1.45332	0.726659	−	0.686998i	$-0.241072\pi$
$389$	11150.3	1.45332	0.726659	+	0.686998i	$0.241072\pi$
$390$	0	0
$391$	12760.1	1.65040
$392$	0	0
$393$	−13986.9	−1.79528
$394$	0	0
$395$	−4590.91	−0.584794
$396$	0	0
$397$	4894.60	0.618774	0.309387	−	0.950936i	$-0.399876\pi$
$397$	4894.60	0.618774	0.309387	+	0.950936i	$0.399876\pi$
$398$	0	0
$399$	252.925	0.0317346
$400$	0	0
$401$	−142.340	−0.0177260	−0.00886302	−	0.999961i	$-0.502821\pi$
$401$	−142.340	−0.0177260	−0.00886302	+	0.999961i	$0.502821\pi$
$402$	0	0
$403$	−2820.09	−0.348582
$404$	0	0
$405$	−10450.6	−1.28221
$406$	0	0
$407$	−1021.93	−0.124459
$408$	0	0
$409$	6428.55	0.777192	0.388596	−	0.921408i	$-0.372960\pi$
$409$	6428.55	0.777192	0.388596	+	0.921408i	$0.372960\pi$
$410$	0	0
$411$	−6484.00	−0.778181
$412$	0	0
$413$	471.827	0.0562157
$414$	0	0
$415$	−2127.92	−0.251700
$416$	0	0
$417$	−11752.8	−1.38019
$418$	0	0
$419$	−13744.6	−1.60254	−0.801272	−	0.598301i	$-0.795843\pi$
$419$	−13744.6	−1.60254	−0.801272	+	0.598301i	$0.795843\pi$
$420$	0	0
$421$	−2826.45	−0.327204	−0.163602	−	0.986526i	$-0.552311\pi$
$421$	−2826.45	−0.327204	−0.163602	+	0.986526i	$0.552311\pi$
$422$	0	0
$423$	−1915.77	−0.220208
$424$	0	0
$425$	2189.35	0.249880
$426$	0	0
$427$	1212.43	0.137409
$428$	0	0
$429$	2034.19	0.228931
$430$	0	0
$431$	3347.48	0.374112	0.187056	−	0.982349i	$-0.440105\pi$
$431$	3347.48	0.374112	0.187056	+	0.982349i	$0.440105\pi$
$432$	0	0
$433$	6888.91	0.764572	0.382286	−	0.924044i	$-0.375137\pi$
$433$	6888.91	0.764572	0.382286	+	0.924044i	$0.375137\pi$
$434$	0	0
$435$	5035.07	0.554973
$436$	0	0
$437$	−705.080	−0.0771821
$438$	0	0
$439$	−9512.89	−1.03423	−0.517113	−	0.855917i	$-0.672993\pi$
$439$	−9512.89	−1.03423	−0.517113	+	0.855917i	$0.672993\pi$
$440$	0	0
$441$	344.080	0.0371536
$442$	0	0
$443$	2320.62	0.248885	0.124442	−	0.992227i	$-0.460286\pi$
$443$	2320.62	0.248885	0.124442	+	0.992227i	$0.460286\pi$
$444$	0	0
$445$	−2931.46	−0.312279
$446$	0	0
$447$	8968.05	0.948936
$448$	0	0
$449$	870.673	0.0915136	0.0457568	−	0.998953i	$-0.485430\pi$
$449$	870.673	0.0915136	0.0457568	+	0.998953i	$0.485430\pi$
$450$	0	0
$451$	2119.52	0.221295
$452$	0	0
$453$	−16386.0	−1.69952
$454$	0	0
$455$	−2668.05	−0.274902
$456$	0	0
$457$	−1863.00	−0.190695	−0.0953475	−	0.995444i	$-0.530396\pi$
$457$	−1863.00	−0.190695	−0.0953475	+	0.995444i	$0.530396\pi$
$458$	0	0
$459$	−13063.5	−1.32844
$460$	0	0
$461$	−10639.4	−1.07490	−0.537448	−	0.843297i	$-0.680611\pi$
$461$	−10639.4	−1.07490	−0.537448	+	0.843297i	$0.680611\pi$
$462$	0	0
$463$	−9034.34	−0.906828	−0.453414	−	0.891300i	$-0.649794\pi$
$463$	−9034.34	−0.906828	−0.453414	+	0.891300i	$0.649794\pi$
$464$	0	0
$465$	−6237.38	−0.622046
$466$	0	0
$467$	−12465.4	−1.23518	−0.617592	−	0.786498i	$-0.711892\pi$
$467$	−12465.4	−1.23518	−0.617592	+	0.786498i	$0.711892\pi$
$468$	0	0
$469$	−160.347	−0.0157871
$470$	0	0
$471$	8660.65	0.847265
$472$	0	0
$473$	1368.26	0.133008
$474$	0	0
$475$	−120.976	−0.0116858
$476$	0	0
$477$	−857.697	−0.0823296
$478$	0	0
$479$	18173.0	1.73350	0.866749	−	0.498745i	$-0.166206\pi$
$479$	18173.0	1.73350	0.866749	+	0.498745i	$0.166206\pi$
$480$	0	0
$481$	−2945.41	−0.279208
$482$	0	0
$483$	−4647.32	−0.437806
$484$	0	0
$485$	−21846.0	−2.04531
$486$	0	0
$487$	17548.4	1.63284	0.816420	−	0.577459i	$-0.195956\pi$
$487$	17548.4	1.63284	0.816420	+	0.577459i	$0.195956\pi$
$488$	0	0
$489$	−18255.6	−1.68823
$490$	0	0
$491$	−7362.51	−0.676711	−0.338356	−	0.941018i	$-0.609871\pi$
$491$	−7362.51	−0.676711	−0.338356	+	0.941018i	$0.609871\pi$
$492$	0	0
$493$	8049.64	0.735370
$494$	0	0
$495$	928.610	0.0843190
$496$	0	0
$497$	−5384.56	−0.485977
$498$	0	0
$499$	−7716.80	−0.692287	−0.346144	−	0.938182i	$-0.612509\pi$
$499$	−7716.80	−0.692287	−0.346144	+	0.938182i	$0.612509\pi$
$500$	0	0
$501$	−19549.5	−1.74333
$502$	0	0
$503$	−3600.83	−0.319191	−0.159596	−	0.987182i	$-0.551019\pi$
$503$	−3600.83	−0.319191	−0.159596	+	0.987182i	$0.551019\pi$
$504$	0	0
$505$	11128.6	0.980625
$506$	0	0
$507$	−6951.79	−0.608954
$508$	0	0
$509$	22467.0	1.95645	0.978224	−	0.207550i	$-0.0665489\pi$
$509$	22467.0	1.95645	0.978224	+	0.207550i	$0.0665489\pi$
$510$	0	0
$511$	399.982	0.0346265
$512$	0	0
$513$	721.848	0.0621255
$514$	0	0
$515$	−17589.6	−1.50503
$516$	0	0
$517$	−3001.05	−0.255292
$518$	0	0
$519$	19408.1	1.64147
$520$	0	0
$521$	6869.00	0.577613	0.288807	−	0.957387i	$-0.406742\pi$
$521$	6869.00	0.577613	0.288807	+	0.957387i	$0.406742\pi$
$522$	0	0
$523$	4906.19	0.410197	0.205098	−	0.978741i	$-0.434249\pi$
$523$	4906.19	0.410197	0.205098	+	0.978741i	$0.434249\pi$
$524$	0	0
$525$	−797.378	−0.0662865
$526$	0	0
$527$	−9971.79	−0.824247
$528$	0	0
$529$	788.336	0.0647930
$530$	0	0
$531$	−473.312	−0.0386817
$532$	0	0
$533$	6108.89	0.496446
$534$	0	0
$535$	10957.3	0.885467
$536$	0	0
$537$	−21607.7	−1.73639
$538$	0	0
$539$	539.000	0.0430730
$540$	0	0
$541$	5806.66	0.461456	0.230728	−	0.973018i	$-0.425889\pi$
$541$	5806.66	0.461456	0.230728	+	0.973018i	$0.425889\pi$
$542$	0	0
$543$	2211.73	0.174797
$544$	0	0
$545$	9279.66	0.729353
$546$	0	0
$547$	−5568.85	−0.435296	−0.217648	−	0.976027i	$-0.569838\pi$
$547$	−5568.85	−0.435296	−0.217648	+	0.976027i	$0.569838\pi$
$548$	0	0
$549$	−1216.25	−0.0945506
$550$	0	0
$551$	−444.797	−0.0343902
$552$	0	0
$553$	2673.12	0.205557
$554$	0	0
$555$	−6514.56	−0.498248
$556$	0	0
$557$	25993.9	1.97738	0.988688	−	0.149990i	$-0.0479242\pi$
$557$	25993.9	1.97738	0.988688	+	0.149990i	$0.0479242\pi$
$558$	0	0
$559$	3943.62	0.298385
$560$	0	0
$561$	7192.86	0.541324
$562$	0	0
$563$	14827.3	1.10994	0.554970	−	0.831871i	$-0.312730\pi$
$563$	14827.3	1.10994	0.554970	+	0.831871i	$0.312730\pi$
$564$	0	0
$565$	780.037	0.0580822
$566$	0	0
$567$	6085.00	0.450698
$568$	0	0
$569$	1683.23	0.124015	0.0620076	−	0.998076i	$-0.480250\pi$
$569$	1683.23	0.124015	0.0620076	+	0.998076i	$0.480250\pi$
$570$	0	0
$571$	−14211.7	−1.04158	−0.520791	−	0.853684i	$-0.674363\pi$
$571$	−14211.7	−1.04158	−0.520791	+	0.853684i	$0.674363\pi$
$572$	0	0
$573$	−19584.1	−1.42781
$574$	0	0
$575$	2222.85	0.161216
$576$	0	0
$577$	18583.7	1.34081	0.670406	−	0.741995i	$-0.266120\pi$
$577$	18583.7	1.34081	0.670406	+	0.741995i	$0.266120\pi$
$578$	0	0
$579$	15558.4	1.11673
$580$	0	0
$581$	1239.01	0.0884730
$582$	0	0
$583$	−1343.58	−0.0954467
$584$	0	0
$585$	2676.45	0.189158
$586$	0	0
$587$	−10260.5	−0.721460	−0.360730	−	0.932670i	$-0.617472\pi$
$587$	−10260.5	−0.721460	−0.360730	+	0.932670i	$0.617472\pi$
$588$	0	0
$589$	551.009	0.0385466
$590$	0	0
$591$	−19853.2	−1.38181
$592$	0	0
$593$	18587.4	1.28717	0.643587	−	0.765373i	$-0.277445\pi$
$593$	18587.4	1.28717	0.643587	+	0.765373i	$0.277445\pi$
$594$	0	0
$595$	−9434.19	−0.650024
$596$	0	0
$597$	25948.8	1.77892
$598$	0	0
$599$	20343.0	1.38764	0.693818	−	0.720151i	$-0.255927\pi$
$599$	20343.0	1.38764	0.693818	+	0.720151i	$0.255927\pi$
$600$	0	0
$601$	1169.16	0.0793531	0.0396765	−	0.999213i	$-0.487367\pi$
$601$	1169.16	0.0793531	0.0396765	+	0.999213i	$0.487367\pi$
$602$	0	0
$603$	160.852	0.0108630
$604$	0	0
$605$	1454.67	0.0977530
$606$	0	0
$607$	24247.6	1.62138	0.810691	−	0.585474i	$-0.199092\pi$
$607$	24247.6	1.62138	0.810691	+	0.585474i	$0.199092\pi$
$608$	0	0
$609$	−2931.74	−0.195074
$610$	0	0
$611$	−8649.65	−0.572713
$612$	0	0
$613$	17695.1	1.16590	0.582951	−	0.812507i	$-0.301898\pi$
$613$	17695.1	1.16590	0.582951	+	0.812507i	$0.301898\pi$
$614$	0	0
$615$	13511.5	0.885910
$616$	0	0
$617$	−20656.8	−1.34783	−0.673916	−	0.738808i	$-0.735389\pi$
$617$	−20656.8	−1.34783	−0.673916	+	0.738808i	$0.735389\pi$
$618$	0	0
$619$	1545.14	0.100330	0.0501651	−	0.998741i	$-0.484025\pi$
$619$	1545.14	0.100330	0.0501651	+	0.998741i	$0.484025\pi$
$620$	0	0
$621$	−13263.4	−0.857074
$622$	0	0
$623$	1706.88	0.109767
$624$	0	0
$625$	−17684.8	−1.13183
$626$	0	0
$627$	−397.454	−0.0253155
$628$	0	0
$629$	−10414.9	−0.660206
$630$	0	0
$631$	15573.4	0.982514	0.491257	−	0.871015i	$-0.336538\pi$
$631$	15573.4	0.982514	0.491257	+	0.871015i	$0.336538\pi$
$632$	0	0
$633$	19543.8	1.22717
$634$	0	0
$635$	−4436.70	−0.277268
$636$	0	0
$637$	1553.51	0.0966285
$638$	0	0
$639$	5401.51	0.334398
$640$	0	0
$641$	23349.3	1.43876	0.719379	−	0.694618i	$-0.244427\pi$
$641$	23349.3	1.43876	0.719379	+	0.694618i	$0.244427\pi$
$642$	0	0
$643$	5534.12	0.339416	0.169708	−	0.985494i	$-0.445718\pi$
$643$	5534.12	0.339416	0.169708	+	0.985494i	$0.445718\pi$
$644$	0	0
$645$	8722.36	0.532469
$646$	0	0
$647$	150.814	0.00916400	0.00458200	−	0.999990i	$-0.498541\pi$
$647$	150.814	0.00916400	0.00458200	+	0.999990i	$0.498541\pi$
$648$	0	0
$649$	−741.442	−0.0448446
$650$	0	0
$651$	3631.80	0.218651
$652$	0	0
$653$	3019.58	0.180958	0.0904789	−	0.995898i	$-0.471160\pi$
$653$	3019.58	0.180958	0.0904789	+	0.995898i	$0.471160\pi$
$654$	0	0
$655$	−28828.3	−1.71971
$656$	0	0
$657$	−401.241	−0.0238263
$658$	0	0
$659$	1247.67	0.0737516	0.0368758	−	0.999320i	$-0.488259\pi$
$659$	1247.67	0.0737516	0.0368758	+	0.999320i	$0.488259\pi$
$660$	0	0
$661$	−16016.3	−0.942455	−0.471227	−	0.882012i	$-0.656189\pi$
$661$	−16016.3	−0.942455	−0.471227	+	0.882012i	$0.656189\pi$
$662$	0	0
$663$	20731.3	1.21439
$664$	0	0
$665$	521.303	0.0303989
$666$	0	0
$667$	8172.82	0.474442
$668$	0	0
$669$	16191.9	0.935745
$670$	0	0
$671$	−1905.25	−0.109615
$672$	0	0
$673$	26356.3	1.50960	0.754800	−	0.655955i	$-0.227734\pi$
$673$	26356.3	1.50960	0.754800	+	0.655955i	$0.227734\pi$
$674$	0	0
$675$	−2275.71	−0.129766
$676$	0	0
$677$	−12161.0	−0.690375	−0.345188	−	0.938534i	$-0.612185\pi$
$677$	−12161.0	−0.690375	−0.345188	+	0.938534i	$0.612185\pi$
$678$	0	0
$679$	12720.1	0.718931
$680$	0	0
$681$	−19773.0	−1.11263
$682$	0	0
$683$	16582.6	0.929011	0.464505	−	0.885570i	$-0.346232\pi$
$683$	16582.6	0.929011	0.464505	+	0.885570i	$0.346232\pi$
$684$	0	0
$685$	−13364.1	−0.745427
$686$	0	0
$687$	32464.8	1.80293
$688$	0	0
$689$	−3872.48	−0.214122
$690$	0	0
$691$	−25394.6	−1.39805	−0.699026	−	0.715096i	$-0.746383\pi$
$691$	−25394.6	−1.39805	−0.699026	+	0.715096i	$0.746383\pi$
$692$	0	0
$693$	−540.697	−0.0296383
$694$	0	0
$695$	−24223.6	−1.32209
$696$	0	0
$697$	21601.0	1.17388
$698$	0	0
$699$	35514.3	1.92171
$700$	0	0
$701$	−22612.5	−1.21835	−0.609173	−	0.793037i	$-0.708499\pi$
$701$	−22612.5	−1.21835	−0.609173	+	0.793037i	$0.708499\pi$
$702$	0	0
$703$	575.494	0.0308751
$704$	0	0
$705$	−19131.0	−1.02201
$706$	0	0
$707$	−6479.78	−0.344692
$708$	0	0
$709$	−31216.8	−1.65356	−0.826779	−	0.562526i	$-0.809830\pi$
$709$	−31216.8	−1.65356	−0.826779	+	0.562526i	$0.809830\pi$
$710$	0	0
$711$	−2681.54	−0.141442
$712$	0	0
$713$	−10124.4	−0.531783
$714$	0	0
$715$	4192.65	0.219296
$716$	0	0
$717$	−7490.81	−0.390167
$718$	0	0
$719$	−3403.97	−0.176560	−0.0882800	−	0.996096i	$-0.528137\pi$
$719$	−3403.97	−0.176560	−0.0882800	+	0.996096i	$0.528137\pi$
$720$	0	0
$721$	10241.8	0.529022
$722$	0	0
$723$	20463.4	1.05262
$724$	0	0
$725$	1402.28	0.0718334
$726$	0	0
$727$	21153.4	1.07914	0.539572	−	0.841940i	$-0.318586\pi$
$727$	21153.4	1.07914	0.539572	+	0.841940i	$0.318586\pi$
$728$	0	0
$729$	12247.5	0.622238
$730$	0	0
$731$	13944.6	0.705552
$732$	0	0
$733$	−16329.7	−0.822851	−0.411425	−	0.911443i	$-0.634969\pi$
$733$	−16329.7	−0.822851	−0.411425	+	0.911443i	$0.634969\pi$
$734$	0	0
$735$	3436.01	0.172434
$736$	0	0
$737$	251.974	0.0125938
$738$	0	0
$739$	14287.3	0.711186	0.355593	−	0.934641i	$-0.384279\pi$
$739$	14287.3	0.711186	0.355593	+	0.934641i	$0.384279\pi$
$740$	0	0
$741$	−1145.55	−0.0567918
$742$	0	0
$743$	17613.0	0.869662	0.434831	−	0.900512i	$-0.356808\pi$
$743$	17613.0	0.869662	0.434831	+	0.900512i	$0.356808\pi$
$744$	0	0
$745$	18484.0	0.908995
$746$	0	0
$747$	−1242.91	−0.0608778
$748$	0	0
$749$	−6380.04	−0.311244
$750$	0	0
$751$	−13264.4	−0.644508	−0.322254	−	0.946653i	$-0.604440\pi$
$751$	−13264.4	−0.644508	−0.322254	+	0.946653i	$0.604440\pi$
$752$	0	0
$753$	3710.98	0.179596
$754$	0	0
$755$	−33773.0	−1.62798
$756$	0	0
$757$	10473.0	0.502838	0.251419	−	0.967878i	$-0.419103\pi$
$757$	10473.0	0.502838	0.251419	+	0.967878i	$0.419103\pi$
$758$	0	0
$759$	7302.93	0.349248
$760$	0	0
$761$	−8667.70	−0.412883	−0.206441	−	0.978459i	$-0.566188\pi$
$761$	−8667.70	−0.412883	−0.206441	+	0.978459i	$0.566188\pi$
$762$	0	0
$763$	−5403.22	−0.256369
$764$	0	0
$765$	9463.89	0.447278
$766$	0	0
$767$	−2136.99	−0.100603
$768$	0	0
$769$	4996.46	0.234300	0.117150	−	0.993114i	$-0.462624\pi$
$769$	4996.46	0.234300	0.117150	+	0.993114i	$0.462624\pi$
$770$	0	0
$771$	18587.2	0.868224
$772$	0	0
$773$	−36952.7	−1.71940	−0.859700	−	0.510798i	$-0.829350\pi$
$773$	−36952.7	−1.71940	−0.859700	+	0.510798i	$0.829350\pi$
$774$	0	0
$775$	−1737.12	−0.0805152
$776$	0	0
$777$	3793.19	0.175135
$778$	0	0
$779$	−1193.60	−0.0548974
$780$	0	0
$781$	8461.45	0.387675
$782$	0	0
$783$	−8367.17	−0.381888
$784$	0	0
$785$	17850.4	0.811603
$786$	0	0
$787$	−28793.6	−1.30417	−0.652085	−	0.758146i	$-0.726105\pi$
$787$	−28793.6	−1.30417	−0.652085	+	0.758146i	$0.726105\pi$
$788$	0	0
$789$	−25883.9	−1.16792
$790$	0	0
$791$	−454.188	−0.0204160
$792$	0	0
$793$	−5491.34	−0.245906
$794$	0	0
$795$	−8565.03	−0.382101
$796$	0	0
$797$	−20084.5	−0.892634	−0.446317	−	0.894875i	$-0.647265\pi$
$797$	−20084.5	−0.892634	−0.446317	+	0.894875i	$0.647265\pi$
$798$	0	0
$799$	−30585.0	−1.35422
$800$	0	0
$801$	−1712.25	−0.0755300
$802$	0	0
$803$	−628.542	−0.0276224
$804$	0	0
$805$	−9578.56	−0.419379
$806$	0	0
$807$	6949.87	0.303156
$808$	0	0
$809$	−35938.9	−1.56186	−0.780929	−	0.624620i	$-0.785254\pi$
$809$	−35938.9	−1.56186	−0.780929	+	0.624620i	$0.785254\pi$
$810$	0	0
$811$	25477.6	1.10313	0.551566	−	0.834132i	$-0.314031\pi$
$811$	25477.6	1.10313	0.551566	+	0.834132i	$0.314031\pi$
$812$	0	0
$813$	1883.01	0.0812301
$814$	0	0
$815$	−37626.4	−1.61717
$816$	0	0
$817$	−770.532	−0.0329957
$818$	0	0
$819$	−1558.40	−0.0664896
$820$	0	0
$821$	−3296.85	−0.140147	−0.0700736	−	0.997542i	$-0.522323\pi$
$821$	−3296.85	−0.140147	−0.0700736	+	0.997542i	$0.522323\pi$
$822$	0	0
$823$	−24892.7	−1.05432	−0.527160	−	0.849766i	$-0.676743\pi$
$823$	−24892.7	−1.05432	−0.527160	+	0.849766i	$0.676743\pi$
$824$	0	0
$825$	1253.02	0.0528784
$826$	0	0
$827$	28719.0	1.20756	0.603782	−	0.797149i	$-0.293660\pi$
$827$	28719.0	1.20756	0.603782	+	0.797149i	$0.293660\pi$
$828$	0	0
$829$	−9911.29	−0.415239	−0.207620	−	0.978210i	$-0.566572\pi$
$829$	−9911.29	−0.415239	−0.207620	+	0.978210i	$0.566572\pi$
$830$	0	0
$831$	22859.4	0.954252
$832$	0	0
$833$	5493.19	0.228485
$834$	0	0
$835$	−40293.4	−1.66995
$836$	0	0
$837$	10365.2	0.428043
$838$	0	0
$839$	30490.4	1.25464	0.627322	−	0.778760i	$-0.284151\pi$
$839$	30490.4	1.25464	0.627322	+	0.778760i	$0.284151\pi$
$840$	0	0
$841$	−19233.2	−0.788602
$842$	0	0
$843$	−48388.4	−1.97697
$844$	0	0
$845$	−14328.3	−0.583323
$846$	0	0
$847$	−847.000	−0.0343604
$848$	0	0
$849$	29524.5	1.19349
$850$	0	0
$851$	−10574.3	−0.425948
$852$	0	0
$853$	27614.6	1.10845	0.554225	−	0.832367i	$-0.313015\pi$
$853$	27614.6	1.10845	0.554225	+	0.832367i	$0.313015\pi$
$854$	0	0
$855$	−522.944	−0.0209173
$856$	0	0
$857$	7889.71	0.314478	0.157239	−	0.987561i	$-0.449741\pi$
$857$	7889.71	0.314478	0.157239	+	0.987561i	$0.449741\pi$
$858$	0	0
$859$	39695.8	1.57672	0.788361	−	0.615213i	$-0.210930\pi$
$859$	39695.8	1.57672	0.788361	+	0.615213i	$0.210930\pi$
$860$	0	0
$861$	−7867.24	−0.311399
$862$	0	0
$863$	−8181.18	−0.322701	−0.161350	−	0.986897i	$-0.551585\pi$
$863$	−8181.18	−0.322701	−0.161350	+	0.986897i	$0.551585\pi$
$864$	0	0
$865$	40002.0	1.57238
$866$	0	0
$867$	44649.0	1.74897
$868$	0	0
$869$	−4200.62	−0.163977
$870$	0	0
$871$	726.243	0.0282524
$872$	0	0
$873$	−12760.2	−0.494693
$874$	0	0
$875$	8875.81	0.342922
$876$	0	0
$877$	28605.9	1.10143	0.550714	−	0.834694i	$-0.314355\pi$
$877$	28605.9	1.10143	0.550714	+	0.834694i	$0.314355\pi$
$878$	0	0
$879$	−16967.1	−0.651066
$880$	0	0
$881$	−38093.6	−1.45676	−0.728380	−	0.685173i	$-0.759726\pi$
$881$	−38093.6	−1.45676	−0.728380	+	0.685173i	$0.759726\pi$
$882$	0	0
$883$	−46608.1	−1.77632	−0.888159	−	0.459537i	$-0.848015\pi$
$883$	−46608.1	−1.77632	−0.888159	+	0.459537i	$0.848015\pi$
$884$	0	0
$885$	−4726.53	−0.179526
$886$	0	0
$887$	17837.3	0.675219	0.337610	−	0.941286i	$-0.390382\pi$
$887$	17837.3	0.675219	0.337610	+	0.941286i	$0.390382\pi$
$888$	0	0
$889$	2583.33	0.0974603
$890$	0	0
$891$	−9562.15	−0.359533
$892$	0	0
$893$	1690.03	0.0633311
$894$	0	0
$895$	−44535.5	−1.66330
$896$	0	0
$897$	21048.6	0.783492
$898$	0	0
$899$	−6386.92	−0.236947
$900$	0	0
$901$	−13693.0	−0.506305
$902$	0	0
$903$	−5078.72	−0.187164
$904$	0	0
$905$	4558.59	0.167439
$906$	0	0
$907$	−32139.9	−1.17661	−0.588307	−	0.808638i	$-0.700205\pi$
$907$	−32139.9	−1.17661	−0.588307	+	0.808638i	$0.700205\pi$
$908$	0	0
$909$	6500.17	0.237181
$910$	0	0
$911$	−21179.0	−0.770243	−0.385122	−	0.922866i	$-0.625841\pi$
$911$	−21179.0	−0.770243	−0.385122	+	0.922866i	$0.625841\pi$
$912$	0	0
$913$	−1947.02	−0.0705770
$914$	0	0
$915$	−12145.6	−0.438820
$916$	0	0
$917$	16785.7	0.604484
$918$	0	0
$919$	−25214.4	−0.905057	−0.452529	−	0.891750i	$-0.649478\pi$
$919$	−25214.4	−0.905057	−0.452529	+	0.891750i	$0.649478\pi$
$920$	0	0
$921$	−21074.8	−0.754003
$922$	0	0
$923$	24387.7	0.869697
$924$	0	0
$925$	−1814.32	−0.0644912
$926$	0	0
$927$	−10274.0	−0.364017
$928$	0	0
$929$	−48511.1	−1.71324	−0.856618	−	0.515951i	$-0.827439\pi$
$929$	−48511.1	−1.71324	−0.856618	+	0.515951i	$0.827439\pi$
$930$	0	0
$931$	−303.536	−0.0106853
$932$	0	0
$933$	40588.2	1.42422
$934$	0	0
$935$	14825.2	0.518540
$936$	0	0
$937$	38413.1	1.33927	0.669637	−	0.742689i	$-0.266450\pi$
$937$	38413.1	1.33927	0.669637	+	0.742689i	$0.266450\pi$
$938$	0	0
$939$	−29558.0	−1.02725
$940$	0	0
$941$	−23920.5	−0.828677	−0.414339	−	0.910123i	$-0.635987\pi$
$941$	−23920.5	−0.828677	−0.414339	+	0.910123i	$0.635987\pi$
$942$	0	0
$943$	21931.5	0.757357
$944$	0	0
$945$	9806.35	0.337567
$946$	0	0
$947$	52682.8	1.80777	0.903887	−	0.427772i	$-0.140701\pi$
$947$	52682.8	1.80777	0.903887	+	0.427772i	$0.140701\pi$
$948$	0	0
$949$	−1811.59	−0.0619671
$950$	0	0
$951$	−13572.3	−0.462788
$952$	0	0
$953$	−30076.1	−1.02231	−0.511155	−	0.859489i	$-0.670782\pi$
$953$	−30076.1	−1.02231	−0.511155	+	0.859489i	$0.670782\pi$
$954$	0	0
$955$	−40364.6	−1.36771
$956$	0	0
$957$	4607.02	0.155615
$958$	0	0
$959$	7781.46	0.262019
$960$	0	0
$961$	−21879.0	−0.734415
$962$	0	0
$963$	6400.12	0.214165
$964$	0	0
$965$	32067.3	1.06972
$966$	0	0
$967$	33266.1	1.10627	0.553136	−	0.833091i	$-0.313431\pi$
$967$	33266.1	1.10627	0.553136	+	0.833091i	$0.313431\pi$
$968$	0	0
$969$	−4050.64	−0.134288
$970$	0	0
$971$	23611.6	0.780364	0.390182	−	0.920738i	$-0.372412\pi$
$971$	23611.6	0.780364	0.390182	+	0.920738i	$0.372412\pi$
$972$	0	0
$973$	14104.6	0.464719
$974$	0	0
$975$	3611.48	0.118625
$976$	0	0
$977$	19512.6	0.638960	0.319480	−	0.947593i	$-0.396492\pi$
$977$	19512.6	0.638960	0.319480	+	0.947593i	$0.396492\pi$
$978$	0	0
$979$	−2682.24	−0.0875637
$980$	0	0
$981$	5420.22	0.176406
$982$	0	0
$983$	−37547.1	−1.21828	−0.609138	−	0.793064i	$-0.708485\pi$
$983$	−37547.1	−1.21828	−0.609138	+	0.793064i	$0.708485\pi$
$984$	0	0
$985$	−40919.3	−1.32365
$986$	0	0
$987$	11139.3	0.359238
$988$	0	0
$989$	14158.0	0.455204
$990$	0	0
$991$	57959.2	1.85786	0.928928	−	0.370262i	$-0.120732\pi$
$991$	57959.2	1.85786	0.928928	+	0.370262i	$0.120732\pi$
$992$	0	0
$993$	29621.8	0.946646
$994$	0	0
$995$	53483.0	1.70404
$996$	0	0
$997$	−30812.9	−0.978791	−0.489395	−	0.872062i	$-0.662782\pi$
$997$	−30812.9	−0.978791	−0.489395	+	0.872062i	$0.662782\pi$
$998$	0	0
$999$	10825.8	0.342855

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	308.4.a.d.1.4	✓	4
4.3	odd	2		1232.4.a.v.1.1		4
7.6	odd	2		2156.4.a.f.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
308.4.a.d.1.4	✓	4	1.1	even	1	trivial
1232.4.a.v.1.1		4	4.3	odd	2
2156.4.a.f.1.1		4	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 308 = 2^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	308.a (trivial)

\(f(q)\)	\(=\)	\(q+5.83284 q^{3} +12.0220 q^{5} -7.00000 q^{7} +7.02203 q^{9} +11.0000 q^{11} +31.7043 q^{13} +70.1226 q^{15} +112.106 q^{17} -6.19461 q^{19} -40.8299 q^{21} +113.822 q^{23} +19.5293 q^{25} -116.528 q^{27} +71.8038 q^{29} -88.9496 q^{31} +64.1612 q^{33} -84.1542 q^{35} -92.9024 q^{37} +184.926 q^{39} +192.683 q^{41} +124.387 q^{43} +84.4191 q^{45} -272.822 q^{47} +49.0000 q^{49} +653.896 q^{51} -122.144 q^{53} +132.242 q^{55} -36.1322 q^{57} -67.4038 q^{59} -173.205 q^{61} -49.1542 q^{63} +381.150 q^{65} +22.9068 q^{67} +663.903 q^{69} +769.223 q^{71} -57.1402 q^{73} +113.911 q^{75} -77.0000 q^{77} -381.875 q^{79} -869.286 q^{81} -177.002 q^{83} +1347.74 q^{85} +418.820 q^{87} -243.840 q^{89} -221.930 q^{91} -518.829 q^{93} -74.4719 q^{95} -1817.16 q^{97} +77.2424 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 3 q^{3} + q^{5} - 28 q^{7} - 19 q^{9} + 44 q^{11} + 98 q^{13} - 33 q^{15} + 64 q^{17} + 114 q^{19} + 21 q^{21} + 231 q^{23} + 417 q^{25} + 63 q^{27} + 268 q^{29} + 33 q^{31} - 33 q^{33} - 7 q^{35}+ \cdots - 209 q^{99}+O(q^{100}) \) 4 * q - 3 * q^3 + q^5 - 28 * q^7 - 19 * q^9 + 44 * q^11 + 98 * q^13 - 33 * q^15 + 64 * q^17 + 114 * q^19 + 21 * q^21 + 231 * q^23 + 417 * q^25 + 63 * q^27 + 268 * q^29 + 33 * q^31 - 33 * q^33 - 7 * q^35 + 357 * q^37 + 140 * q^39 + 364 * q^41 + 44 * q^43 + 912 * q^45 + 720 * q^47 + 196 * q^49 + 794 * q^51 + 740 * q^53 + 11 * q^55 + 138 * q^57 + 787 * q^59 + 1020 * q^61 + 133 * q^63 + 296 * q^65 - 995 * q^67 + 1059 * q^69 - 1011 * q^71 + 1592 * q^73 - 324 * q^75 - 308 * q^77 - 178 * q^79 - 1396 * q^81 + 324 * q^83 - 94 * q^85 - 1262 * q^87 + 19 * q^89 - 686 * q^91 + 637 * q^93 - 2418 * q^95 - 555 * q^97 - 209 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more