LMFDB - Embedded newform 234.6.a.i.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [234,6,Mod(1,234)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(234, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("234.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 234 = 2 \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	234.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:	$37.5298138362$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{3241}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 810 $ x^2 - x - 810
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 78)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-27.9649$ of defining polynomial
Character	$\chi$	$=$	234.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.00000 q^{2} +16.0000 q^{4} +51.9298 q^{5} +125.930 q^{7} -64.0000 q^{8} -207.719 q^{10} -727.438 q^{11} +169.000 q^{13} -503.719 q^{14} +256.000 q^{16} -1752.60 q^{17} -1911.96 q^{19} +830.877 q^{20} +2909.75 q^{22} +3548.63 q^{23} -428.298 q^{25} -676.000 q^{26} +2014.88 q^{28} -4861.37 q^{29} +2315.68 q^{31} -1024.00 q^{32} +7010.38 q^{34} +6539.51 q^{35} +855.613 q^{37} +7647.85 q^{38} -3323.51 q^{40} +15940.1 q^{41} -18216.5 q^{43} -11639.0 q^{44} -14194.5 q^{46} -25491.2 q^{47} -948.690 q^{49} +1713.19 q^{50} +2704.00 q^{52} +6920.88 q^{53} -37775.7 q^{55} -8059.51 q^{56} +19445.5 q^{58} +28226.2 q^{59} +17085.5 q^{61} -9262.73 q^{62} +4096.00 q^{64} +8776.13 q^{65} -61370.4 q^{67} -28041.5 q^{68} -26158.0 q^{70} +34755.8 q^{71} -51413.4 q^{73} -3422.45 q^{74} -30591.4 q^{76} -91606.1 q^{77} +1024.40 q^{79} +13294.0 q^{80} -63760.5 q^{82} -117792. q^{83} -91011.9 q^{85} +72865.9 q^{86} +46556.0 q^{88} -24413.8 q^{89} +21282.1 q^{91} +56778.1 q^{92} +101965. q^{94} -99287.9 q^{95} +32393.1 q^{97} +3794.76 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{2} + 32 q^{4} - 10 q^{5} + 138 q^{7} - 128 q^{8} + 40 q^{10} - 544 q^{11} + 338 q^{13} - 552 q^{14} + 512 q^{16} - 1228 q^{17} - 522 q^{19} - 160 q^{20} + 2176 q^{22} + 1632 q^{23} + 282 q^{25}+ \cdots + 70440 q^{98}+O(q^{100}) $ 2 * q - 8 * q^2 + 32 * q^4 - 10 * q^5 + 138 * q^7 - 128 * q^8 + 40 * q^10 - 544 * q^11 + 338 * q^13 - 552 * q^14 + 512 * q^16 - 1228 * q^17 - 522 * q^19 - 160 * q^20 + 2176 * q^22 + 1632 * q^23 + 282 * q^25 - 1352 * q^26 + 2208 * q^28 - 2208 * q^29 + 874 * q^31 - 2048 * q^32 + 4912 * q^34 + 5792 * q^35 - 2160 * q^37 + 2088 * q^38 + 640 * q^40 + 12638 * q^41 - 17760 * q^43 - 8704 * q^44 - 6528 * q^46 - 4300 * q^47 - 17610 * q^49 - 1128 * q^50 + 5408 * q^52 + 12020 * q^53 - 49136 * q^55 - 8832 * q^56 + 8832 * q^58 + 27532 * q^59 - 38016 * q^61 - 3496 * q^62 + 8192 * q^64 - 1690 * q^65 - 90974 * q^67 - 19648 * q^68 - 23168 * q^70 + 4384 * q^71 - 95312 * q^73 + 8640 * q^74 - 8352 * q^76 - 89392 * q^77 - 62168 * q^79 - 2560 * q^80 - 50552 * q^82 - 137892 * q^83 - 123500 * q^85 + 71040 * q^86 + 34816 * q^88 - 142534 * q^89 + 23322 * q^91 + 26112 * q^92 + 17200 * q^94 - 185368 * q^95 + 4896 * q^97 + 70440 * 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 $

$2$	−4.00000	−0.707107
$2$	−4.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	16.0000	0.500000
$5$	51.9298	0.928948	0.464474	−	0.885587i	$-0.346243\pi$
$5$	51.9298	0.928948	0.464474	+	0.885587i	$0.346243\pi$
$6$	0	0
$7$	125.930	0.971367	0.485684	−	0.874135i	$-0.338571\pi$
$7$	125.930	0.971367	0.485684	+	0.874135i	$0.338571\pi$
$8$	−64.0000	−0.353553
$9$	0	0
$10$	−207.719	−0.656866
$11$	−727.438	−1.81265	−0.906326	−	0.422579i	$-0.861125\pi$
$11$	−727.438	−1.81265	−0.906326	+	0.422579i	$0.861125\pi$
$12$	0	0
$13$	169.000	0.277350
$13$	169.000	0.277350
$14$	−503.719	−0.686860
$15$	0	0
$16$	256.000	0.250000
$17$	−1752.60	−1.47082	−0.735410	−	0.677623i	$-0.763010\pi$
$17$	−1752.60	−1.47082	−0.735410	+	0.677623i	$0.763010\pi$
$18$	0	0
$19$	−1911.96	−1.21505	−0.607527	−	0.794299i	$-0.707838\pi$
$19$	−1911.96	−1.21505	−0.607527	+	0.794299i	$0.707838\pi$
$20$	830.877	0.464474
$21$	0	0
$22$	2909.75	1.28174
$23$	3548.63	1.39875	0.699377	−	0.714753i	$-0.253461\pi$
$23$	3548.63	1.39875	0.699377	+	0.714753i	$0.253461\pi$
$24$	0	0
$25$	−428.298	−0.137055
$26$	−676.000	−0.196116
$27$	0	0
$28$	2014.88	0.485684
$29$	−4861.37	−1.07340	−0.536702	−	0.843772i	$-0.680330\pi$
$29$	−4861.37	−1.07340	−0.536702	+	0.843772i	$0.680330\pi$
$30$	0	0
$31$	2315.68	0.432788	0.216394	−	0.976306i	$-0.430570\pi$
$31$	2315.68	0.432788	0.216394	+	0.976306i	$0.430570\pi$
$32$	−1024.00	−0.176777
$33$	0	0
$34$	7010.38	1.04003
$35$	6539.51	0.902350
$36$	0	0
$37$	855.613	0.102748	0.0513739	−	0.998679i	$-0.483640\pi$
$37$	855.613	0.102748	0.0513739	+	0.998679i	$0.483640\pi$
$38$	7647.85	0.859173
$39$	0	0
$40$	−3323.51	−0.328433
$41$	15940.1	1.48092	0.740461	−	0.672099i	$-0.234607\pi$
$41$	15940.1	1.48092	0.740461	+	0.672099i	$0.234607\pi$
$42$	0	0
$43$	−18216.5	−1.50243	−0.751214	−	0.660059i	$-0.770531\pi$
$43$	−18216.5	−1.50243	−0.751214	+	0.660059i	$0.770531\pi$
$44$	−11639.0	−0.906326
$45$	0	0
$46$	−14194.5	−0.989068
$47$	−25491.2	−1.68324	−0.841620	−	0.540071i	$-0.818397\pi$
$47$	−25491.2	−1.68324	−0.841620	+	0.540071i	$0.818397\pi$
$48$	0	0
$49$	−948.690	−0.0564461
$50$	1713.19	0.0969127
$51$	0	0
$52$	2704.00	0.138675
$53$	6920.88	0.338432	0.169216	−	0.985579i	$-0.445876\pi$
$53$	6920.88	0.338432	0.169216	+	0.985579i	$0.445876\pi$
$54$	0	0
$55$	−37775.7	−1.68386
$56$	−8059.51	−0.343430
$57$	0	0
$58$	19445.5	0.759011
$59$	28226.2	1.05565	0.527827	−	0.849352i	$-0.323007\pi$
$59$	28226.2	1.05565	0.527827	+	0.849352i	$0.323007\pi$
$60$	0	0
$61$	17085.5	0.587899	0.293949	−	0.955821i	$-0.405030\pi$
$61$	17085.5	0.587899	0.293949	+	0.955821i	$0.405030\pi$
$62$	−9262.73	−0.306027
$63$	0	0
$64$	4096.00	0.125000
$65$	8776.13	0.257644
$66$	0	0
$67$	−61370.4	−1.67021	−0.835107	−	0.550088i	$-0.814594\pi$
$67$	−61370.4	−1.67021	−0.835107	+	0.550088i	$0.814594\pi$
$68$	−28041.5	−0.735410
$69$	0	0
$70$	−26158.0	−0.638058
$71$	34755.8	0.818242	0.409121	−	0.912480i	$-0.365835\pi$
$71$	34755.8	0.818242	0.409121	+	0.912480i	$0.365835\pi$
$72$	0	0
$73$	−51413.4	−1.12919	−0.564597	−	0.825366i	$-0.690969\pi$
$73$	−51413.4	−1.12919	−0.564597	+	0.825366i	$0.690969\pi$
$74$	−3422.45	−0.0726537
$75$	0	0
$76$	−30591.4	−0.607527
$77$	−91606.1	−1.76075
$78$	0	0
$79$	1024.40	0.0184672	0.00923359	−	0.999957i	$-0.497061\pi$
$79$	1024.40	0.0184672	0.00923359	+	0.999957i	$0.497061\pi$
$80$	13294.0	0.232237
$81$	0	0
$82$	−63760.5	−1.04717
$83$	−117792.	−1.87681	−0.938404	−	0.345541i	$-0.887695\pi$
$83$	−117792.	−1.87681	−0.938404	+	0.345541i	$0.887695\pi$
$84$	0	0
$85$	−91011.9	−1.36632
$86$	72865.9	1.06238
$87$	0	0
$88$	46556.0	0.640869
$89$	−24413.8	−0.326708	−0.163354	−	0.986567i	$-0.552231\pi$
$89$	−24413.8	−0.326708	−0.163354	+	0.986567i	$0.552231\pi$
$90$	0	0
$91$	21282.1	0.269409
$92$	56778.1	0.699377
$93$	0	0
$94$	101965.	1.19023
$95$	−99287.9	−1.12872
$96$	0	0
$97$	32393.1	0.349561	0.174780	−	0.984607i	$-0.444078\pi$
$97$	32393.1	0.349561	0.174780	+	0.984607i	$0.444078\pi$
$98$	3794.76	0.0399134
$99$	0	0
$100$	−6852.77	−0.0685277
$101$	−136702.	−1.33343	−0.666714	−	0.745313i	$-0.732300\pi$
$101$	−136702.	−1.33343	−0.666714	+	0.745313i	$0.732300\pi$
$102$	0	0
$103$	18853.6	0.175106	0.0875532	−	0.996160i	$-0.472095\pi$
$103$	18853.6	0.175106	0.0875532	+	0.996160i	$0.472095\pi$
$104$	−10816.0	−0.0980581
$105$	0	0
$106$	−27683.5	−0.239308
$107$	−59464.2	−0.502107	−0.251054	−	0.967973i	$-0.580777\pi$
$107$	−59464.2	−0.502107	−0.251054	+	0.967973i	$0.580777\pi$
$108$	0	0
$109$	−144082.	−1.16157	−0.580783	−	0.814058i	$-0.697254\pi$
$109$	−144082.	−1.16157	−0.580783	+	0.814058i	$0.697254\pi$
$110$	151103.	1.19067
$111$	0	0
$112$	32238.0	0.242842
$113$	163868.	1.20726	0.603628	−	0.797266i	$-0.293721\pi$
$113$	163868.	1.20726	0.603628	+	0.797266i	$0.293721\pi$
$114$	0	0
$115$	184280.	1.29937
$116$	−77781.8	−0.536702
$117$	0	0
$118$	−112905.	−0.746461
$119$	−220704.	−1.42871
$120$	0	0
$121$	368115.	2.28571
$122$	−68341.9	−0.415707
$123$	0	0
$124$	37050.9	0.216394
$125$	−184522.	−1.05627
$126$	0	0
$127$	14594.3	0.0802921	0.0401461	−	0.999194i	$-0.487218\pi$
$127$	14594.3	0.0802921	0.0401461	+	0.999194i	$0.487218\pi$
$128$	−16384.0	−0.0883883
$129$	0	0
$130$	−35104.5	−0.182182
$131$	90631.6	0.461426	0.230713	−	0.973022i	$-0.425894\pi$
$131$	90631.6	0.461426	0.230713	+	0.973022i	$0.425894\pi$
$132$	0	0
$133$	−240773.	−1.18026
$134$	245482.	1.18102
$135$	0	0
$136$	112166.	0.520013
$137$	234615.	1.06796	0.533978	−	0.845498i	$-0.320696\pi$
$137$	234615.	1.06796	0.533978	+	0.845498i	$0.320696\pi$
$138$	0	0
$139$	187258.	0.822061	0.411031	−	0.911622i	$-0.365169\pi$
$139$	187258.	0.822061	0.411031	+	0.911622i	$0.365169\pi$
$140$	104632.	0.451175
$141$	0	0
$142$	−139023.	−0.578585
$143$	−122937.	−0.502739
$144$	0	0
$145$	−252450.	−0.997137
$146$	205653.	0.798461
$147$	0	0
$148$	13689.8	0.0513739
$149$	81845.0	0.302014	0.151007	−	0.988533i	$-0.451748\pi$
$149$	81845.0	0.302014	0.151007	+	0.988533i	$0.451748\pi$
$150$	0	0
$151$	−436038.	−1.55626	−0.778131	−	0.628103i	$-0.783832\pi$
$151$	−436038.	−1.55626	−0.778131	+	0.628103i	$0.783832\pi$
$152$	122366.	0.429587
$153$	0	0
$154$	366425.	1.24504
$155$	120253.	0.402037
$156$	0	0
$157$	−128820.	−0.417094	−0.208547	−	0.978012i	$-0.566873\pi$
$157$	−128820.	−0.417094	−0.208547	+	0.978012i	$0.566873\pi$
$158$	−4097.59	−0.0130583
$159$	0	0
$160$	−53176.1	−0.164216
$161$	446878.	1.35870
$162$	0	0
$163$	212197.	0.625562	0.312781	−	0.949825i	$-0.398739\pi$
$163$	212197.	0.625562	0.312781	+	0.949825i	$0.398739\pi$
$164$	255042.	0.740461
$165$	0	0
$166$	471167.	1.32710
$167$	443647.	1.23097	0.615483	−	0.788150i	$-0.288961\pi$
$167$	443647.	1.23097	0.615483	+	0.788150i	$0.288961\pi$
$168$	0	0
$169$	28561.0	0.0769231
$170$	364048.	0.966131
$171$	0	0
$172$	−291464.	−0.751214
$173$	−19751.6	−0.0501751	−0.0250875	−	0.999685i	$-0.507986\pi$
$173$	−19751.6	−0.0501751	−0.0250875	+	0.999685i	$0.507986\pi$
$174$	0	0
$175$	−53935.4	−0.133131
$176$	−186224.	−0.453163
$177$	0	0
$178$	97655.2	0.231018
$179$	−470674.	−1.09796	−0.548981	−	0.835835i	$-0.684984\pi$
$179$	−470674.	−1.09796	−0.548981	+	0.835835i	$0.684984\pi$
$180$	0	0
$181$	253684.	0.575569	0.287784	−	0.957695i	$-0.407081\pi$
$181$	253684.	0.575569	0.287784	+	0.957695i	$0.407081\pi$
$182$	−85128.5	−0.190501
$183$	0	0
$184$	−227112.	−0.494534
$185$	44431.8	0.0954474
$186$	0	0
$187$	1.27491e6	2.66608
$188$	−407859.	−0.841620
$189$	0	0
$190$	397151.	0.798127
$191$	−691969.	−1.37247	−0.686236	−	0.727379i	$-0.740738\pi$
$191$	−691969.	−1.37247	−0.686236	+	0.727379i	$0.740738\pi$
$192$	0	0
$193$	−187938.	−0.363180	−0.181590	−	0.983374i	$-0.558124\pi$
$193$	−187938.	−0.363180	−0.181590	+	0.983374i	$0.558124\pi$
$194$	−129572.	−0.247177
$195$	0	0
$196$	−15179.0	−0.0282231
$197$	392212.	0.720037	0.360019	−	0.932945i	$-0.382770\pi$
$197$	392212.	0.720037	0.360019	+	0.932945i	$0.382770\pi$
$198$	0	0
$199$	−143624.	−0.257095	−0.128547	−	0.991703i	$-0.541031\pi$
$199$	−143624.	−0.257095	−0.128547	+	0.991703i	$0.541031\pi$
$200$	27411.1	0.0484564
$201$	0	0
$202$	546806.	0.942877
$203$	−612191.	−1.04267
$204$	0	0
$205$	827768.	1.37570
$206$	−75414.5	−0.123819
$207$	0	0
$208$	43264.0	0.0693375
$209$	1.39084e6	2.20247
$210$	0	0
$211$	−663846.	−1.02651	−0.513253	−	0.858238i	$-0.671560\pi$
$211$	−663846.	−1.02651	−0.513253	+	0.858238i	$0.671560\pi$
$212$	110734.	0.169216
$213$	0	0
$214$	237857.	0.355043
$215$	−945978.	−1.39568
$216$	0	0
$217$	291613.	0.420396
$218$	576329.	0.821351
$219$	0	0
$220$	−604411.	−0.841930
$221$	−296189.	−0.407932
$222$	0	0
$223$	768220.	1.03448	0.517242	−	0.855839i	$-0.326959\pi$
$223$	768220.	1.03448	0.517242	+	0.855839i	$0.326959\pi$
$224$	−128952.	−0.171715
$225$	0	0
$226$	−655474.	−0.853658
$227$	389973.	0.502308	0.251154	−	0.967947i	$-0.419190\pi$
$227$	389973.	0.502308	0.251154	+	0.967947i	$0.419190\pi$
$228$	0	0
$229$	−93168.7	−0.117404	−0.0587018	−	0.998276i	$-0.518696\pi$
$229$	−93168.7	−0.117404	−0.0587018	+	0.998276i	$0.518696\pi$
$230$	−737118.	−0.918793
$231$	0	0
$232$	311127.	0.379506
$233$	−45435.7	−0.0548287	−0.0274143	−	0.999624i	$-0.508727\pi$
$233$	−45435.7	−0.0548287	−0.0274143	+	0.999624i	$0.508727\pi$
$234$	0	0
$235$	−1.32375e6	−1.56364
$236$	451619.	0.527827
$237$	0	0
$238$	882816.	1.01025
$239$	−640583.	−0.725405	−0.362703	−	0.931905i	$-0.618146\pi$
$239$	−640583.	−0.725405	−0.362703	+	0.931905i	$0.618146\pi$
$240$	0	0
$241$	−469774.	−0.521010	−0.260505	−	0.965472i	$-0.583889\pi$
$241$	−469774.	−0.521010	−0.260505	+	0.965472i	$0.583889\pi$
$242$	−1.47246e6	−1.61624
$243$	0	0
$244$	273368.	0.293949
$245$	−49265.3	−0.0524355
$246$	0	0
$247$	−323122.	−0.336995
$248$	−148204.	−0.153014
$249$	0	0
$250$	738088.	0.746892
$251$	382711.	0.383431	0.191715	−	0.981451i	$-0.438595\pi$
$251$	382711.	0.383431	0.191715	+	0.981451i	$0.438595\pi$
$252$	0	0
$253$	−2.58141e6	−2.53545
$254$	−58377.1	−0.0567751
$255$	0	0
$256$	65536.0	0.0625000
$257$	−132518.	−0.125154	−0.0625768	−	0.998040i	$-0.519932\pi$
$257$	−132518.	−0.125154	−0.0625768	+	0.998040i	$0.519932\pi$
$258$	0	0
$259$	107747.	0.0998059
$260$	140418.	0.128822
$261$	0	0
$262$	−362527.	−0.326277
$263$	−1.30367e6	−1.16219	−0.581095	−	0.813836i	$-0.697375\pi$
$263$	−1.30367e6	−1.16219	−0.581095	+	0.813836i	$0.697375\pi$
$264$	0	0
$265$	359400.	0.314386
$266$	963093.	0.834572
$267$	0	0
$268$	−981927.	−0.835107
$269$	−1.05757e6	−0.891105	−0.445553	−	0.895256i	$-0.646993\pi$
$269$	−1.05757e6	−0.891105	−0.445553	+	0.895256i	$0.646993\pi$
$270$	0	0
$271$	794875.	0.657470	0.328735	−	0.944422i	$-0.393378\pi$
$271$	794875.	0.657470	0.328735	+	0.944422i	$0.393378\pi$
$272$	−448664.	−0.367705
$273$	0	0
$274$	−938459.	−0.755160
$275$	311560.	0.248434
$276$	0	0
$277$	898297.	0.703430	0.351715	−	0.936107i	$-0.385599\pi$
$277$	898297.	0.703430	0.351715	+	0.936107i	$0.385599\pi$
$278$	−749033.	−0.581285
$279$	0	0
$280$	−418528.	−0.319029
$281$	711429.	0.537484	0.268742	−	0.963212i	$-0.413392\pi$
$281$	711429.	0.537484	0.268742	+	0.963212i	$0.413392\pi$
$282$	0	0
$283$	−2.42172e6	−1.79746	−0.898728	−	0.438506i	$-0.855508\pi$
$283$	−2.42172e6	−1.79746	−0.898728	+	0.438506i	$0.855508\pi$
$284$	556093.	0.409121
$285$	0	0
$286$	491748.	0.355490
$287$	2.00734e6	1.43852
$288$	0	0
$289$	1.65173e6	1.16331
$290$	1.00980e6	0.705082
$291$	0	0
$292$	−822614.	−0.564597
$293$	1.14948e6	0.782226	0.391113	−	0.920343i	$-0.372090\pi$
$293$	1.14948e6	0.782226	0.391113	+	0.920343i	$0.372090\pi$
$294$	0	0
$295$	1.46578e6	0.980648
$296$	−54759.2	−0.0363269
$297$	0	0
$298$	−327380.	−0.213556
$299$	599718.	0.387944
$300$	0	0
$301$	−2.29400e6	−1.45941
$302$	1.74415e6	1.10044
$303$	0	0
$304$	−489463.	−0.303764
$305$	887245.	0.546128
$306$	0	0
$307$	−265112.	−0.160540	−0.0802700	−	0.996773i	$-0.525578\pi$
$307$	−265112.	−0.160540	−0.0802700	+	0.996773i	$0.525578\pi$
$308$	−1.46570e6	−0.880375
$309$	0	0
$310$	−481012.	−0.284283
$311$	−1.89541e6	−1.11122	−0.555612	−	0.831441i	$-0.687516\pi$
$311$	−1.89541e6	−1.11122	−0.555612	+	0.831441i	$0.687516\pi$
$312$	0	0
$313$	−1.28051e6	−0.738791	−0.369395	−	0.929272i	$-0.620435\pi$
$313$	−1.28051e6	−0.738791	−0.369395	+	0.929272i	$0.620435\pi$
$314$	515280.	0.294930
$315$	0	0
$316$	16390.3	0.00923359
$317$	577699.	0.322889	0.161445	−	0.986882i	$-0.448385\pi$
$317$	577699.	0.322889	0.161445	+	0.986882i	$0.448385\pi$
$318$	0	0
$319$	3.53634e6	1.94571
$320$	212704.	0.116119
$321$	0	0
$322$	−1.78751e6	−0.960748
$323$	3.35090e6	1.78713
$324$	0	0
$325$	−72382.3	−0.0380123
$326$	−848788.	−0.442339
$327$	0	0
$328$	−1.02017e6	−0.523585
$329$	−3.21010e6	−1.63504
$330$	0	0
$331$	50901.8	0.0255366	0.0127683	−	0.999918i	$-0.495936\pi$
$331$	50901.8	0.0255366	0.0127683	+	0.999918i	$0.495936\pi$
$332$	−1.88467e6	−0.938404
$333$	0	0
$334$	−1.77459e6	−0.870425
$335$	−3.18695e6	−1.55154
$336$	0	0
$337$	−1.02644e6	−0.492333	−0.246166	−	0.969228i	$-0.579171\pi$
$337$	−1.02644e6	−0.492333	−0.246166	+	0.969228i	$0.579171\pi$
$338$	−114244.	−0.0543928
$339$	0	0
$340$	−1.45619e6	−0.683158
$341$	−1.68452e6	−0.784493
$342$	0	0
$343$	−2.23597e6	−1.02620
$344$	1.16585e6	0.531188
$345$	0	0
$346$	79006.6	0.0354791
$347$	−2.62322e6	−1.16953	−0.584764	−	0.811204i	$-0.698813\pi$
$347$	−2.62322e6	−1.16953	−0.584764	+	0.811204i	$0.698813\pi$
$348$	0	0
$349$	−1.22375e6	−0.537811	−0.268906	−	0.963167i	$-0.586662\pi$
$349$	−1.22375e6	−0.537811	−0.268906	+	0.963167i	$0.586662\pi$
$350$	215742.	0.0941378
$351$	0	0
$352$	744897.	0.320435
$353$	4.01591e6	1.71533	0.857663	−	0.514211i	$-0.171915\pi$
$353$	4.01591e6	1.71533	0.857663	+	0.514211i	$0.171915\pi$
$354$	0	0
$355$	1.80486e6	0.760105
$356$	−390621.	−0.163354
$357$	0	0
$358$	1.88270e6	0.776377
$359$	3.39210e6	1.38910	0.694548	−	0.719446i	$-0.255604\pi$
$359$	3.39210e6	1.38910	0.694548	+	0.719446i	$0.255604\pi$
$360$	0	0
$361$	1.17951e6	0.476357
$362$	−1.01474e6	−0.406989
$363$	0	0
$364$	340514.	0.134704
$365$	−2.66988e6	−1.04896
$366$	0	0
$367$	2.09854e6	0.813302	0.406651	−	0.913584i	$-0.366697\pi$
$367$	2.09854e6	0.813302	0.406651	+	0.913584i	$0.366697\pi$
$368$	908449.	0.349688
$369$	0	0
$370$	−177727.	−0.0674915
$371$	871544.	0.328742
$372$	0	0
$373$	4.49072e6	1.67126	0.835629	−	0.549294i	$-0.185103\pi$
$373$	4.49072e6	1.67126	0.835629	+	0.549294i	$0.185103\pi$
$374$	−5.09962e6	−1.88521
$375$	0	0
$376$	1.63144e6	0.595115
$377$	−821571.	−0.297709
$378$	0	0
$379$	4.18356e6	1.49606	0.748028	−	0.663667i	$-0.231001\pi$
$379$	4.18356e6	1.49606	0.748028	+	0.663667i	$0.231001\pi$
$380$	−1.58861e6	−0.564361
$381$	0	0
$382$	2.76788e6	0.970484
$383$	4.16386e6	1.45044	0.725220	−	0.688517i	$-0.241738\pi$
$383$	4.16386e6	1.45044	0.725220	+	0.688517i	$0.241738\pi$
$384$	0	0
$385$	−4.75709e6	−1.63565
$386$	751753.	0.256807
$387$	0	0
$388$	518289.	0.174780
$389$	2.82001e6	0.944880	0.472440	−	0.881363i	$-0.343373\pi$
$389$	2.82001e6	0.944880	0.472440	+	0.881363i	$0.343373\pi$
$390$	0	0
$391$	−6.21931e6	−2.05731
$392$	60716.2	0.0199567
$393$	0	0
$394$	−1.56885e6	−0.509143
$395$	53196.7	0.0171550
$396$	0	0
$397$	5.18346e6	1.65061	0.825304	−	0.564689i	$-0.191004\pi$
$397$	5.18346e6	1.65061	0.825304	+	0.564689i	$0.191004\pi$
$398$	574495.	0.181793
$399$	0	0
$400$	−109644.	−0.0342638
$401$	−182869.	−0.0567910	−0.0283955	−	0.999597i	$-0.509040\pi$
$401$	−182869.	−0.0567910	−0.0283955	+	0.999597i	$0.509040\pi$
$402$	0	0
$403$	391350.	0.120034
$404$	−2.18722e6	−0.666714
$405$	0	0
$406$	2.44876e6	0.737279
$407$	−622405.	−0.186246
$408$	0	0
$409$	4.16750e6	1.23188	0.615938	−	0.787795i	$-0.288777\pi$
$409$	4.16750e6	1.23188	0.615938	+	0.787795i	$0.288777\pi$
$410$	−3.31107e6	−0.972767
$411$	0	0
$412$	301658.	0.0875532
$413$	3.55451e6	1.02543
$414$	0	0
$415$	−6.11690e6	−1.74346
$416$	−173056.	−0.0490290
$417$	0	0
$418$	−5.56334e6	−1.55738
$419$	−3.40253e6	−0.946819	−0.473409	−	0.880843i	$-0.656977\pi$
$419$	−3.40253e6	−0.946819	−0.473409	+	0.880843i	$0.656977\pi$
$420$	0	0
$421$	2.83837e6	0.780485	0.390242	−	0.920712i	$-0.372391\pi$
$421$	2.83837e6	0.780485	0.390242	+	0.920712i	$0.372391\pi$
$422$	2.65538e6	0.725849
$423$	0	0
$424$	−442936.	−0.119654
$425$	750633.	0.201584
$426$	0	0
$427$	2.15157e6	0.571066
$428$	−951428.	−0.251054
$429$	0	0
$430$	3.78391e6	0.986893
$431$	3.43537e6	0.890801	0.445400	−	0.895331i	$-0.353061\pi$
$431$	3.43537e6	0.890801	0.445400	+	0.895331i	$0.353061\pi$
$432$	0	0
$433$	5.38615e6	1.38057	0.690286	−	0.723537i	$-0.257485\pi$
$433$	5.38615e6	1.38057	0.690286	+	0.723537i	$0.257485\pi$
$434$	−1.16645e6	−0.297265
$435$	0	0
$436$	−2.30531e6	−0.580783
$437$	−6.78485e6	−1.69956
$438$	0	0
$439$	−2.76940e6	−0.685842	−0.342921	−	0.939364i	$-0.611416\pi$
$439$	−2.76940e6	−0.685842	−0.342921	+	0.939364i	$0.611416\pi$
$440$	2.41765e6	0.595334
$441$	0	0
$442$	1.18475e6	0.288451
$443$	292670.	0.0708548	0.0354274	−	0.999372i	$-0.488721\pi$
$443$	292670.	0.0708548	0.0354274	+	0.999372i	$0.488721\pi$
$444$	0	0
$445$	−1.26780e6	−0.303495
$446$	−3.07288e6	−0.731491
$447$	0	0
$448$	515808.	0.121421
$449$	3.99957e6	0.936263	0.468131	−	0.883659i	$-0.344927\pi$
$449$	3.99957e6	0.936263	0.468131	+	0.883659i	$0.344927\pi$
$450$	0	0
$451$	−1.15955e7	−2.68440
$452$	2.62189e6	0.603628
$453$	0	0
$454$	−1.55989e6	−0.355185
$455$	1.10518e6	0.250267
$456$	0	0
$457$	3.90828e6	0.875378	0.437689	−	0.899127i	$-0.355797\pi$
$457$	3.90828e6	0.875378	0.437689	+	0.899127i	$0.355797\pi$
$458$	372675.	0.0830169
$459$	0	0
$460$	2.94847e6	0.649685
$461$	6.76633e6	1.48286	0.741432	−	0.671028i	$-0.234147\pi$
$461$	6.76633e6	1.48286	0.741432	+	0.671028i	$0.234147\pi$
$462$	0	0
$463$	−511273.	−0.110841	−0.0554205	−	0.998463i	$-0.517650\pi$
$463$	−511273.	−0.110841	−0.0554205	+	0.998463i	$0.517650\pi$
$464$	−1.24451e6	−0.268351
$465$	0	0
$466$	181743.	0.0387697
$467$	−8.36149e6	−1.77416	−0.887078	−	0.461620i	$-0.847268\pi$
$467$	−8.36149e6	−1.77416	−0.887078	+	0.461620i	$0.847268\pi$
$468$	0	0
$469$	−7.72836e6	−1.62239
$470$	5.29501e6	1.10566
$471$	0	0
$472$	−1.80647e6	−0.373230
$473$	1.32514e7	2.72338
$474$	0	0
$475$	818890.	0.166530
$476$	−3.53126e6	−0.714353
$477$	0	0
$478$	2.56233e6	0.512939
$479$	−9.08825e6	−1.80984	−0.904922	−	0.425576i	$-0.860071\pi$
$479$	−9.08825e6	−1.80984	−0.904922	+	0.425576i	$0.860071\pi$
$480$	0	0
$481$	144599.	0.0284971
$482$	1.87910e6	0.368410
$483$	0	0
$484$	5.88985e6	1.14285
$485$	1.68216e6	0.324724
$486$	0	0
$487$	−248040.	−0.0473913	−0.0236957	−	0.999719i	$-0.507543\pi$
$487$	−248040.	−0.0473913	−0.0236957	+	0.999719i	$0.507543\pi$
$488$	−1.09347e6	−0.207854
$489$	0	0
$490$	197061.	0.0370775
$491$	2.76666e6	0.517908	0.258954	−	0.965890i	$-0.416622\pi$
$491$	2.76666e6	0.517908	0.258954	+	0.965890i	$0.416622\pi$
$492$	0	0
$493$	8.52001e6	1.57878
$494$	1.29249e6	0.238292
$495$	0	0
$496$	592815.	0.108197
$497$	4.37679e6	0.794813
$498$	0	0
$499$	−3.50606e6	−0.630331	−0.315165	−	0.949037i	$-0.602060\pi$
$499$	−3.50606e6	−0.630331	−0.315165	+	0.949037i	$0.602060\pi$
$500$	−2.95235e6	−0.528133
$501$	0	0
$502$	−1.53084e6	−0.271126
$503$	−4.73444e6	−0.834351	−0.417176	−	0.908826i	$-0.636980\pi$
$503$	−4.73444e6	−0.834351	−0.417176	+	0.908826i	$0.636980\pi$
$504$	0	0
$505$	−7.09888e6	−1.23869
$506$	1.03256e7	1.79284
$507$	0	0
$508$	233508.	0.0401461
$509$	−8.03933e6	−1.37539	−0.687694	−	0.726000i	$-0.741377\pi$
$509$	−8.03933e6	−1.37539	−0.687694	+	0.726000i	$0.741377\pi$
$510$	0	0
$511$	−6.47447e6	−1.09686
$512$	−262144.	−0.0441942
$513$	0	0
$514$	530073.	0.0884969
$515$	979065.	0.162665
$516$	0	0
$517$	1.85433e7	3.05113
$518$	−430988.	−0.0705734
$519$	0	0
$520$	−561673.	−0.0910909
$521$	4.25006e6	0.685962	0.342981	−	0.939342i	$-0.388563\pi$
$521$	4.25006e6	0.685962	0.342981	+	0.939342i	$0.388563\pi$
$522$	0	0
$523$	9.27413e6	1.48258	0.741291	−	0.671184i	$-0.234214\pi$
$523$	9.27413e6	1.48258	0.741291	+	0.671184i	$0.234214\pi$
$524$	1.45011e6	0.230713
$525$	0	0
$526$	5.21466e6	0.821792
$527$	−4.05846e6	−0.636553
$528$	0	0
$529$	6.15643e6	0.956510
$530$	−1.43760e6	−0.222304
$531$	0	0
$532$	−3.85237e6	−0.590132
$533$	2.69388e6	0.410734
$534$	0	0
$535$	−3.08797e6	−0.466432
$536$	3.92771e6	0.590510
$537$	0	0
$538$	4.23029e6	0.630107
$539$	690114.	0.102317
$540$	0	0
$541$	−2.04934e6	−0.301038	−0.150519	−	0.988607i	$-0.548094\pi$
$541$	−2.04934e6	−0.301038	−0.150519	+	0.988607i	$0.548094\pi$
$542$	−3.17950e6	−0.464901
$543$	0	0
$544$	1.79466e6	0.260007
$545$	−7.48216e6	−1.07903
$546$	0	0
$547$	8.79085e6	1.25621	0.628105	−	0.778128i	$-0.283831\pi$
$547$	8.79085e6	1.25621	0.628105	+	0.778128i	$0.283831\pi$
$548$	3.75383e6	0.533978
$549$	0	0
$550$	−1.24624e6	−0.175669
$551$	9.29475e6	1.30424
$552$	0	0
$553$	129002.	0.0179384
$554$	−3.59319e6	−0.497400
$555$	0	0
$556$	2.99613e6	0.411031
$557$	−4.85096e6	−0.662506	−0.331253	−	0.943542i	$-0.607471\pi$
$557$	−4.85096e6	−0.662506	−0.331253	+	0.943542i	$0.607471\pi$
$558$	0	0
$559$	−3.07859e6	−0.416698
$560$	1.67411e6	0.225587
$561$	0	0
$562$	−2.84572e6	−0.380059
$563$	2.25029e6	0.299204	0.149602	−	0.988746i	$-0.452201\pi$
$563$	2.25029e6	0.299204	0.149602	+	0.988746i	$0.452201\pi$
$564$	0	0
$565$	8.50965e6	1.12148
$566$	9.68689e6	1.27099
$567$	0	0
$568$	−2.22437e6	−0.289292
$569$	1.12398e6	0.145538	0.0727691	−	0.997349i	$-0.476816\pi$
$569$	1.12398e6	0.145538	0.0727691	+	0.997349i	$0.476816\pi$
$570$	0	0
$571$	1.22111e7	1.56735	0.783674	−	0.621172i	$-0.213343\pi$
$571$	1.22111e7	1.56735	0.783674	+	0.621172i	$0.213343\pi$
$572$	−1.96699e6	−0.251370
$573$	0	0
$574$	−8.02935e6	−1.01719
$575$	−1.51987e6	−0.191707
$576$	0	0
$577$	4.65423e6	0.581981	0.290990	−	0.956726i	$-0.406015\pi$
$577$	4.65423e6	0.581981	0.290990	+	0.956726i	$0.406015\pi$
$578$	−6.60694e6	−0.822585
$579$	0	0
$580$	−4.03919e6	−0.498568
$581$	−1.48335e7	−1.82307
$582$	0	0
$583$	−5.03451e6	−0.613459
$584$	3.29046e6	0.399231
$585$	0	0
$586$	−4.59792e6	−0.553117
$587$	−6.71171e6	−0.803966	−0.401983	−	0.915647i	$-0.631679\pi$
$587$	−6.71171e6	−0.803966	−0.401983	+	0.915647i	$0.631679\pi$
$588$	0	0
$589$	−4.42750e6	−0.525860
$590$	−5.86311e6	−0.693423
$591$	0	0
$592$	219037.	0.0256870
$593$	−9.19702e6	−1.07401	−0.537007	−	0.843578i	$-0.680445\pi$
$593$	−9.19702e6	−1.07401	−0.537007	+	0.843578i	$0.680445\pi$
$594$	0	0
$595$	−1.14611e7	−1.32719
$596$	1.30952e6	0.151007
$597$	0	0
$598$	−2.39887e6	−0.274318
$599$	−3.75863e6	−0.428019	−0.214009	−	0.976832i	$-0.568652\pi$
$599$	−3.75863e6	−0.428019	−0.214009	+	0.976832i	$0.568652\pi$
$600$	0	0
$601$	−1.03306e7	−1.16665	−0.583324	−	0.812240i	$-0.698248\pi$
$601$	−1.03306e7	−1.16665	−0.583324	+	0.812240i	$0.698248\pi$
$602$	9.17599e6	1.03196
$603$	0	0
$604$	−6.97662e6	−0.778131
$605$	1.91162e7	2.12330
$606$	0	0
$607$	−6.32243e6	−0.696486	−0.348243	−	0.937404i	$-0.613222\pi$
$607$	−6.32243e6	−0.696486	−0.348243	+	0.937404i	$0.613222\pi$
$608$	1.95785e6	0.214793
$609$	0	0
$610$	−3.54898e6	−0.386171
$611$	−4.30801e6	−0.466847
$612$	0	0
$613$	1.80304e7	1.93801	0.969003	−	0.247050i	$-0.0794611\pi$
$613$	1.80304e7	1.93801	0.969003	+	0.247050i	$0.0794611\pi$
$614$	1.06045e6	0.113519
$615$	0	0
$616$	5.86279e6	0.622519
$617$	−3.30022e6	−0.349003	−0.174502	−	0.984657i	$-0.555831\pi$
$617$	−3.30022e6	−0.349003	−0.174502	+	0.984657i	$0.555831\pi$
$618$	0	0
$619$	−1.48988e7	−1.56288	−0.781438	−	0.623982i	$-0.785514\pi$
$619$	−1.48988e7	−1.56288	−0.781438	+	0.623982i	$0.785514\pi$
$620$	1.92405e6	0.201019
$621$	0	0
$622$	7.58163e6	0.785755
$623$	−3.07442e6	−0.317354
$624$	0	0
$625$	−8.24376e6	−0.844161
$626$	5.12203e6	0.522404
$627$	0	0
$628$	−2.06112e6	−0.208547
$629$	−1.49954e6	−0.151124
$630$	0	0
$631$	−1.35013e7	−1.34990	−0.674952	−	0.737862i	$-0.735836\pi$
$631$	−1.35013e7	−1.34990	−0.674952	+	0.737862i	$0.735836\pi$
$632$	−65561.4	−0.00652913
$633$	0	0
$634$	−2.31080e6	−0.228317
$635$	757877.	0.0745872
$636$	0	0
$637$	−160329.	−0.0156553
$638$	−1.41454e7	−1.37582
$639$	0	0
$640$	−850818.	−0.0821082
$641$	5.57243e6	0.535672	0.267836	−	0.963464i	$-0.413691\pi$
$641$	5.57243e6	0.535672	0.267836	+	0.963464i	$0.413691\pi$
$642$	0	0
$643$	1.04401e7	0.995810	0.497905	−	0.867232i	$-0.334103\pi$
$643$	1.04401e7	0.995810	0.497905	+	0.867232i	$0.334103\pi$
$644$	7.15005e6	0.679351
$645$	0	0
$646$	−1.34036e7	−1.26369
$647$	−7.80548e6	−0.733059	−0.366529	−	0.930406i	$-0.619454\pi$
$647$	−7.80548e6	−0.733059	−0.366529	+	0.930406i	$0.619454\pi$
$648$	0	0
$649$	−2.05328e7	−1.91353
$650$	289529.	0.0268788
$651$	0	0
$652$	3.39515e6	0.312781
$653$	−1.41064e7	−1.29459	−0.647297	−	0.762238i	$-0.724101\pi$
$653$	−1.41064e7	−1.29459	−0.647297	+	0.762238i	$0.724101\pi$
$654$	0	0
$655$	4.70648e6	0.428640
$656$	4.08067e6	0.370231
$657$	0	0
$658$	1.28404e7	1.15615
$659$	−1.60341e6	−0.143824	−0.0719121	−	0.997411i	$-0.522910\pi$
$659$	−1.60341e6	−0.143824	−0.0719121	+	0.997411i	$0.522910\pi$
$660$	0	0
$661$	1.09386e6	0.0973777	0.0486888	−	0.998814i	$-0.484496\pi$
$661$	1.09386e6	0.0973777	0.0486888	+	0.998814i	$0.484496\pi$
$662$	−203607.	−0.0180571
$663$	0	0
$664$	7.53867e6	0.663552
$665$	−1.25033e7	−1.09640
$666$	0	0
$667$	−1.72512e7	−1.50143
$668$	7.09835e6	0.615483
$669$	0	0
$670$	1.27478e7	1.09711
$671$	−1.24286e7	−1.06566
$672$	0	0
$673$	−2.24267e7	−1.90866	−0.954330	−	0.298756i	$-0.903428\pi$
$673$	−2.24267e7	−1.90866	−0.954330	+	0.298756i	$0.903428\pi$
$674$	4.10576e6	0.348132
$675$	0	0
$676$	456976.	0.0384615
$677$	3.24049e6	0.271731	0.135866	−	0.990727i	$-0.456618\pi$
$677$	3.24049e6	0.271731	0.135866	+	0.990727i	$0.456618\pi$
$678$	0	0
$679$	4.07925e6	0.339552
$680$	5.82476e6	0.483065
$681$	0	0
$682$	6.73806e6	0.554721
$683$	−7.98624e6	−0.655074	−0.327537	−	0.944838i	$-0.606219\pi$
$683$	−7.98624e6	−0.655074	−0.327537	+	0.944838i	$0.606219\pi$
$684$	0	0
$685$	1.21835e7	0.992077
$686$	8.94388e6	0.725631
$687$	0	0
$688$	−4.66342e6	−0.375607
$689$	1.16963e6	0.0938641
$690$	0	0
$691$	6.65845e6	0.530491	0.265245	−	0.964181i	$-0.414547\pi$
$691$	6.65845e6	0.530491	0.265245	+	0.964181i	$0.414547\pi$
$692$	−316026.	−0.0250875
$693$	0	0
$694$	1.04929e7	0.826981
$695$	9.72428e6	0.763652
$696$	0	0
$697$	−2.79366e7	−2.17817
$698$	4.89501e6	0.380290
$699$	0	0
$700$	−862967.	−0.0665655
$701$	1.32522e7	1.01858	0.509289	−	0.860596i	$-0.329909\pi$
$701$	1.32522e7	1.01858	0.509289	+	0.860596i	$0.329909\pi$
$702$	0	0
$703$	−1.63590e6	−0.124844
$704$	−2.97959e6	−0.226581
$705$	0	0
$706$	−1.60636e7	−1.21292
$707$	−1.72148e7	−1.29525
$708$	0	0
$709$	−2.40403e7	−1.79607	−0.898035	−	0.439923i	$-0.855006\pi$
$709$	−2.40403e7	−1.79607	−0.898035	+	0.439923i	$0.855006\pi$
$710$	−7.21945e6	−0.537475
$711$	0	0
$712$	1.56248e6	0.115509
$713$	8.21750e6	0.605363
$714$	0	0
$715$	−6.38409e6	−0.467019
$716$	−7.53078e6	−0.548981
$717$	0	0
$718$	−1.35684e7	−0.982239
$719$	−1.19668e7	−0.863287	−0.431643	−	0.902044i	$-0.642066\pi$
$719$	−1.19668e7	−0.863287	−0.431643	+	0.902044i	$0.642066\pi$
$720$	0	0
$721$	2.37423e6	0.170093
$722$	−4.71802e6	−0.336835
$723$	0	0
$724$	4.05895e6	0.287784
$725$	2.08211e6	0.147116
$726$	0	0
$727$	−9.20221e6	−0.645738	−0.322869	−	0.946444i	$-0.604647\pi$
$727$	−9.20221e6	−0.645738	−0.322869	+	0.946444i	$0.604647\pi$
$728$	−1.36206e6	−0.0952504
$729$	0	0
$730$	1.06795e7	0.741729
$731$	3.19261e7	2.20980
$732$	0	0
$733$	1.07156e7	0.736640	0.368320	−	0.929699i	$-0.379933\pi$
$733$	1.07156e7	0.736640	0.368320	+	0.929699i	$0.379933\pi$
$734$	−8.39415e6	−0.575091
$735$	0	0
$736$	−3.63380e6	−0.247267
$737$	4.46432e7	3.02752
$738$	0	0
$739$	2.91758e7	1.96522	0.982610	−	0.185681i	$-0.0594490\pi$
$739$	2.91758e7	1.96522	0.982610	+	0.185681i	$0.0594490\pi$
$740$	710908.	0.0477237
$741$	0	0
$742$	−3.48618e6	−0.232455
$743$	−2.47758e7	−1.64648	−0.823238	−	0.567697i	$-0.807835\pi$
$743$	−2.47758e7	−1.64648	−0.823238	+	0.567697i	$0.807835\pi$
$744$	0	0
$745$	4.25019e6	0.280555
$746$	−1.79629e7	−1.18176
$747$	0	0
$748$	2.03985e7	1.33304
$749$	−7.48832e6	−0.487730
$750$	0	0
$751$	2.18230e7	1.41194	0.705968	−	0.708243i	$-0.250512\pi$
$751$	2.18230e7	1.41194	0.705968	+	0.708243i	$0.250512\pi$
$752$	−6.52575e6	−0.420810
$753$	0	0
$754$	3.28628e6	0.210512
$755$	−2.26434e7	−1.44569
$756$	0	0
$757$	2.80854e7	1.78132	0.890659	−	0.454672i	$-0.150244\pi$
$757$	2.80854e7	1.78132	0.890659	+	0.454672i	$0.150244\pi$
$758$	−1.67342e7	−1.05787
$759$	0	0
$760$	6.35442e6	0.399064
$761$	1.95604e7	1.22438	0.612189	−	0.790711i	$-0.290289\pi$
$761$	1.95604e7	1.22438	0.612189	+	0.790711i	$0.290289\pi$
$762$	0	0
$763$	−1.81442e7	−1.12831
$764$	−1.10715e7	−0.686236
$765$	0	0
$766$	−1.66555e7	−1.02562
$767$	4.77022e6	0.292786
$768$	0	0
$769$	−2.29412e7	−1.39895	−0.699473	−	0.714659i	$-0.746582\pi$
$769$	−2.29412e7	−1.39895	−0.699473	+	0.714659i	$0.746582\pi$
$770$	1.90283e7	1.15658
$771$	0	0
$772$	−3.00701e6	−0.181590
$773$	−1.04250e7	−0.627522	−0.313761	−	0.949502i	$-0.601589\pi$
$773$	−1.04250e7	−0.627522	−0.313761	+	0.949502i	$0.601589\pi$
$774$	0	0
$775$	−991802.	−0.0593158
$776$	−2.07316e6	−0.123588
$777$	0	0
$778$	−1.12800e7	−0.668131
$779$	−3.04770e7	−1.79940
$780$	0	0
$781$	−2.52827e7	−1.48319
$782$	2.48773e7	1.45474
$783$	0	0
$784$	−242865.	−0.0141115
$785$	−6.68960e6	−0.387459
$786$	0	0
$787$	−2.62266e7	−1.50940	−0.754700	−	0.656070i	$-0.772218\pi$
$787$	−2.62266e7	−1.50940	−0.754700	+	0.656070i	$0.772218\pi$
$788$	6.27539e6	0.360019
$789$	0	0
$790$	−212787.	−0.0121304
$791$	2.06359e7	1.17269
$792$	0	0
$793$	2.88745e6	0.163054
$794$	−2.07339e7	−1.16716
$795$	0	0
$796$	−2.29798e6	−0.128547
$797$	−6.62000e6	−0.369158	−0.184579	−	0.982818i	$-0.559092\pi$
$797$	−6.62000e6	−0.369158	−0.184579	+	0.982818i	$0.559092\pi$
$798$	0	0
$799$	4.46758e7	2.47574
$800$	438577.	0.0242282
$801$	0	0
$802$	731477.	0.0401573
$803$	3.74000e7	2.04684
$804$	0	0
$805$	2.32063e7	1.26216
$806$	−1.56540e6	−0.0848766
$807$	0	0
$808$	8.74890e6	0.471438
$809$	3.20775e7	1.72317	0.861587	−	0.507610i	$-0.169471\pi$
$809$	3.20775e7	1.72317	0.861587	+	0.507610i	$0.169471\pi$
$810$	0	0
$811$	9.65080e6	0.515242	0.257621	−	0.966246i	$-0.417061\pi$
$811$	9.65080e6	0.515242	0.257621	+	0.966246i	$0.417061\pi$
$812$	−9.79505e6	−0.521335
$813$	0	0
$814$	2.48962e6	0.131696
$815$	1.10193e7	0.581115
$816$	0	0
$817$	3.48293e7	1.82553
$818$	−1.66700e7	−0.871068
$819$	0	0
$820$	1.32443e7	0.687850
$821$	4.08393e6	0.211456	0.105728	−	0.994395i	$-0.466283\pi$
$821$	4.08393e6	0.211456	0.105728	+	0.994395i	$0.466283\pi$
$822$	0	0
$823$	5.52729e6	0.284454	0.142227	−	0.989834i	$-0.454574\pi$
$823$	5.52729e6	0.284454	0.142227	+	0.989834i	$0.454574\pi$
$824$	−1.20663e6	−0.0619095
$825$	0	0
$826$	−1.42181e7	−0.725087
$827$	−1.20691e7	−0.613636	−0.306818	−	0.951768i	$-0.599264\pi$
$827$	−1.20691e7	−0.613636	−0.306818	+	0.951768i	$0.599264\pi$
$828$	0	0
$829$	−2.41656e7	−1.22127	−0.610634	−	0.791913i	$-0.709085\pi$
$829$	−2.41656e7	−1.22127	−0.610634	+	0.791913i	$0.709085\pi$
$830$	2.44676e7	1.23281
$831$	0	0
$832$	692224.	0.0346688
$833$	1.66267e6	0.0830221
$834$	0	0
$835$	2.30385e7	1.14350
$836$	2.22534e7	1.10124
$837$	0	0
$838$	1.36101e7	0.669502
$839$	−5.77691e6	−0.283329	−0.141664	−	0.989915i	$-0.545245\pi$
$839$	−5.77691e6	−0.283329	−0.141664	+	0.989915i	$0.545245\pi$
$840$	0	0
$841$	3.12173e6	0.152197
$842$	−1.13535e7	−0.551886
$843$	0	0
$844$	−1.06215e7	−0.513253
$845$	1.48317e6	0.0714576
$846$	0	0
$847$	4.63567e7	2.22026
$848$	1.77174e6	0.0846080
$849$	0	0
$850$	−3.00253e6	−0.142541
$851$	3.03625e6	0.143719
$852$	0	0
$853$	−8.29724e6	−0.390446	−0.195223	−	0.980759i	$-0.562543\pi$
$853$	−8.29724e6	−0.390446	−0.195223	+	0.980759i	$0.562543\pi$
$854$	−8.60628e6	−0.403804
$855$	0	0
$856$	3.80571e6	0.177522
$857$	−3.30122e7	−1.53540	−0.767701	−	0.640808i	$-0.778599\pi$
$857$	−3.30122e7	−1.53540	−0.767701	+	0.640808i	$0.778599\pi$
$858$	0	0
$859$	3.20593e7	1.48242	0.741209	−	0.671274i	$-0.234253\pi$
$859$	3.20593e7	1.48242	0.741209	+	0.671274i	$0.234253\pi$
$860$	−1.51356e7	−0.697838
$861$	0	0
$862$	−1.37415e7	−0.629891
$863$	2.38586e7	1.09048	0.545241	−	0.838280i	$-0.316438\pi$
$863$	2.38586e7	1.09048	0.545241	+	0.838280i	$0.316438\pi$
$864$	0	0
$865$	−1.02570e6	−0.0466101
$866$	−2.15446e7	−0.976212
$867$	0	0
$868$	4.66581e6	0.210198
$869$	−745185.	−0.0334746
$870$	0	0
$871$	−1.03716e7	−0.463234
$872$	9.22126e6	0.410676
$873$	0	0
$874$	2.71394e7	1.20177
$875$	−2.32368e7	−1.02602
$876$	0	0
$877$	1.03330e7	0.453656	0.226828	−	0.973935i	$-0.427164\pi$
$877$	1.03330e7	0.453656	0.226828	+	0.973935i	$0.427164\pi$
$878$	1.10776e7	0.484964
$879$	0	0
$880$	−9.67058e6	−0.420965
$881$	−1.29653e7	−0.562785	−0.281393	−	0.959593i	$-0.590796\pi$
$881$	−1.29653e7	−0.562785	−0.281393	+	0.959593i	$0.590796\pi$
$882$	0	0
$883$	−7.73789e6	−0.333980	−0.166990	−	0.985959i	$-0.553405\pi$
$883$	−7.73789e6	−0.333980	−0.166990	+	0.985959i	$0.553405\pi$
$884$	−4.73902e6	−0.203966
$885$	0	0
$886$	−1.17068e6	−0.0501019
$887$	−2.42738e7	−1.03593	−0.517963	−	0.855403i	$-0.673310\pi$
$887$	−2.42738e7	−1.03593	−0.517963	+	0.855403i	$0.673310\pi$
$888$	0	0
$889$	1.83785e6	0.0779931
$890$	5.07121e6	0.214603
$891$	0	0
$892$	1.22915e7	0.517242
$893$	4.87383e7	2.04523
$894$	0	0
$895$	−2.44420e7	−1.01995
$896$	−2.06323e6	−0.0858575
$897$	0	0
$898$	−1.59983e7	−0.662038
$899$	−1.12574e7	−0.464556
$900$	0	0
$901$	−1.21295e7	−0.497772
$902$	4.63818e7	1.89816
$903$	0	0
$904$	−1.04876e7	−0.426829
$905$	1.31738e7	0.534674
$906$	0	0
$907$	−8.52405e6	−0.344055	−0.172027	−	0.985092i	$-0.555032\pi$
$907$	−8.52405e6	−0.344055	−0.172027	+	0.985092i	$0.555032\pi$
$908$	6.23957e6	0.251154
$909$	0	0
$910$	−4.42071e6	−0.176965
$911$	2.72544e7	1.08803	0.544014	−	0.839076i	$-0.316904\pi$
$911$	2.72544e7	1.08803	0.544014	+	0.839076i	$0.316904\pi$
$912$	0	0
$913$	8.56862e7	3.40200
$914$	−1.56331e7	−0.618985
$915$	0	0
$916$	−1.49070e6	−0.0587018
$917$	1.14132e7	0.448214
$918$	0	0
$919$	−4.69088e7	−1.83217	−0.916085	−	0.400984i	$-0.868668\pi$
$919$	−4.69088e7	−1.83217	−0.916085	+	0.400984i	$0.868668\pi$
$920$	−1.17939e7	−0.459396
$921$	0	0
$922$	−2.70653e7	−1.04854
$923$	5.87374e6	0.226940
$924$	0	0
$925$	−366457.	−0.0140821
$926$	2.04509e6	0.0783764
$927$	0	0
$928$	4.97804e6	0.189753
$929$	1.49816e7	0.569533	0.284767	−	0.958597i	$-0.408084\pi$
$929$	1.49816e7	0.569533	0.284767	+	0.958597i	$0.408084\pi$
$930$	0	0
$931$	1.81386e6	0.0685851
$932$	−726972.	−0.0274143
$933$	0	0
$934$	3.34460e7	1.25452
$935$	6.62055e7	2.47665
$936$	0	0
$937$	−1.62498e7	−0.604642	−0.302321	−	0.953206i	$-0.597761\pi$
$937$	−1.62498e7	−0.604642	−0.302321	+	0.953206i	$0.597761\pi$
$938$	3.09134e7	1.14720
$939$	0	0
$940$	−2.11800e7	−0.781821
$941$	3.25833e7	1.19956	0.599778	−	0.800166i	$-0.295255\pi$
$941$	3.25833e7	1.19956	0.599778	+	0.800166i	$0.295255\pi$
$942$	0	0
$943$	5.65656e7	2.07144
$944$	7.22590e6	0.263914
$945$	0	0
$946$	−5.30055e7	−1.92572
$947$	−4.60715e6	−0.166939	−0.0834694	−	0.996510i	$-0.526600\pi$
$947$	−4.60715e6	−0.166939	−0.0834694	+	0.996510i	$0.526600\pi$
$948$	0	0
$949$	−8.68886e6	−0.313182
$950$	−3.27556e6	−0.117754
$951$	0	0
$952$	1.41251e7	0.505124
$953$	4.87957e7	1.74040	0.870200	−	0.492698i	$-0.163989\pi$
$953$	4.87957e7	1.74040	0.870200	+	0.492698i	$0.163989\pi$
$954$	0	0
$955$	−3.59338e7	−1.27496
$956$	−1.02493e7	−0.362703
$957$	0	0
$958$	3.63530e7	1.27975
$959$	2.95450e7	1.03738
$960$	0	0
$961$	−2.32668e7	−0.812695
$962$	−578394.	−0.0201505
$963$	0	0
$964$	−7.51638e6	−0.260505
$965$	−9.75960e6	−0.337376
$966$	0	0
$967$	−7.18367e6	−0.247047	−0.123524	−	0.992342i	$-0.539420\pi$
$967$	−7.18367e6	−0.247047	−0.123524	+	0.992342i	$0.539420\pi$
$968$	−2.35594e7	−0.808119
$969$	0	0
$970$	−6.72866e6	−0.229614
$971$	−5.24904e7	−1.78662	−0.893308	−	0.449444i	$-0.851622\pi$
$971$	−5.24904e7	−1.78662	−0.893308	+	0.449444i	$0.851622\pi$
$972$	0	0
$973$	2.35814e7	0.798523
$974$	992159.	0.0335107
$975$	0	0
$976$	4.37388e6	0.146975
$977$	1.22428e6	0.0410340	0.0205170	−	0.999790i	$-0.493469\pi$
$977$	1.22428e6	0.0410340	0.0205170	+	0.999790i	$0.493469\pi$
$978$	0	0
$979$	1.77595e7	0.592208
$980$	−788244.	−0.0262178
$981$	0	0
$982$	−1.10666e7	−0.366216
$983$	−3.28509e7	−1.08433	−0.542167	−	0.840271i	$-0.682396\pi$
$983$	−3.28509e7	−1.08433	−0.542167	+	0.840271i	$0.682396\pi$
$984$	0	0
$985$	2.03675e7	0.668877
$986$	−3.40800e7	−1.11637
$987$	0	0
$988$	−5.16995e6	−0.168498
$989$	−6.46436e7	−2.10152
$990$	0	0
$991$	−1.02297e6	−0.0330885	−0.0165443	−	0.999863i	$-0.505266\pi$
$991$	−1.02297e6	−0.0330885	−0.0165443	+	0.999863i	$0.505266\pi$
$992$	−2.37126e6	−0.0765068
$993$	0	0
$994$	−1.75072e7	−0.562018
$995$	−7.45835e6	−0.238828
$996$	0	0
$997$	−6.44797e6	−0.205440	−0.102720	−	0.994710i	$-0.532755\pi$
$997$	−6.44797e6	−0.205440	−0.102720	+	0.994710i	$0.532755\pi$
$998$	1.40243e7	0.445711
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	234.6.a.i.1.2		2
3.2	odd	2		78.6.a.h.1.1	✓	2
12.11	even	2		624.6.a.j.1.1		2
39.38	odd	2		1014.6.a.i.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
78.6.a.h.1.1	✓	2	3.2	odd	2
234.6.a.i.1.2		2	1.1	even	1	trivial
624.6.a.j.1.1		2	12.11	even	2
1014.6.a.i.1.2		2	39.38	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:	\( N \)	\(=\)	\( 234 = 2 \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	234.a (trivial)

\(f(q)\)	\(=\)	\(q-4.00000 q^{2} +16.0000 q^{4} +51.9298 q^{5} +125.930 q^{7} -64.0000 q^{8} -207.719 q^{10} -727.438 q^{11} +169.000 q^{13} -503.719 q^{14} +256.000 q^{16} -1752.60 q^{17} -1911.96 q^{19} +830.877 q^{20} +2909.75 q^{22} +3548.63 q^{23} -428.298 q^{25} -676.000 q^{26} +2014.88 q^{28} -4861.37 q^{29} +2315.68 q^{31} -1024.00 q^{32} +7010.38 q^{34} +6539.51 q^{35} +855.613 q^{37} +7647.85 q^{38} -3323.51 q^{40} +15940.1 q^{41} -18216.5 q^{43} -11639.0 q^{44} -14194.5 q^{46} -25491.2 q^{47} -948.690 q^{49} +1713.19 q^{50} +2704.00 q^{52} +6920.88 q^{53} -37775.7 q^{55} -8059.51 q^{56} +19445.5 q^{58} +28226.2 q^{59} +17085.5 q^{61} -9262.73 q^{62} +4096.00 q^{64} +8776.13 q^{65} -61370.4 q^{67} -28041.5 q^{68} -26158.0 q^{70} +34755.8 q^{71} -51413.4 q^{73} -3422.45 q^{74} -30591.4 q^{76} -91606.1 q^{77} +1024.40 q^{79} +13294.0 q^{80} -63760.5 q^{82} -117792. q^{83} -91011.9 q^{85} +72865.9 q^{86} +46556.0 q^{88} -24413.8 q^{89} +21282.1 q^{91} +56778.1 q^{92} +101965. q^{94} -99287.9 q^{95} +32393.1 q^{97} +3794.76 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{2} + 32 q^{4} - 10 q^{5} + 138 q^{7} - 128 q^{8} + 40 q^{10} - 544 q^{11} + 338 q^{13} - 552 q^{14} + 512 q^{16} - 1228 q^{17} - 522 q^{19} - 160 q^{20} + 2176 q^{22} + 1632 q^{23} + 282 q^{25}+ \cdots + 70440 q^{98}+O(q^{100}) \) 2 * q - 8 * q^2 + 32 * q^4 - 10 * q^5 + 138 * q^7 - 128 * q^8 + 40 * q^10 - 544 * q^11 + 338 * q^13 - 552 * q^14 + 512 * q^16 - 1228 * q^17 - 522 * q^19 - 160 * q^20 + 2176 * q^22 + 1632 * q^23 + 282 * q^25 - 1352 * q^26 + 2208 * q^28 - 2208 * q^29 + 874 * q^31 - 2048 * q^32 + 4912 * q^34 + 5792 * q^35 - 2160 * q^37 + 2088 * q^38 + 640 * q^40 + 12638 * q^41 - 17760 * q^43 - 8704 * q^44 - 6528 * q^46 - 4300 * q^47 - 17610 * q^49 - 1128 * q^50 + 5408 * q^52 + 12020 * q^53 - 49136 * q^55 - 8832 * q^56 + 8832 * q^58 + 27532 * q^59 - 38016 * q^61 - 3496 * q^62 + 8192 * q^64 - 1690 * q^65 - 90974 * q^67 - 19648 * q^68 - 23168 * q^70 + 4384 * q^71 - 95312 * q^73 + 8640 * q^74 - 8352 * q^76 - 89392 * q^77 - 62168 * q^79 - 2560 * q^80 - 50552 * q^82 - 137892 * q^83 - 123500 * q^85 + 71040 * q^86 + 34816 * q^88 - 142534 * q^89 + 23322 * q^91 + 26112 * q^92 + 17200 * q^94 - 185368 * q^95 + 4896 * q^97 + 70440 * q^98

Introduction

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